Consumption of financial services: developing a conceptual framework

Keywords: Financial services, Financial consumption, Banks, Service relationships

Consumption of financial services is central to understanding economic lives of contemporary consumers. An abundance of empirical contributions on the ways in which people manage their economic capital and interact with their financial service providers have been spread across different fields. This paper contributes to the rich body of literature dealing with different aspects of consumer behaviour in the financial context by systematizing the accumulated knowledge and proposing a conceptual framework that encompasses relevant aspects of consumption of financial services. In this framework the financial consumption process is presented as a multifaceted construct that incorporates selection and purchase of financial products and services, relationships between consumers and their financial service providers as well as the broader sociocultural aspects of financial consumption beyond the immediate situation of purchase. Finally, the paper develops a research agenda for future studies, providing suggestions for future research.

Introduction

Lives of modern consumers have become closely connected with the financial world.

Over the last two decades researchers from different fields have shown an increased interest in the ways consumers manage their economic capital and interact with their financial service providers. The large interdisciplinary body of literature on this topic consists of multiplicity of contributions from the fields of bank marketing, service marketing, economics, economic psychology, economic sociology and consumer research.

The majority of research is empirical in nature and embraces a wide variety of topics: selection of the financial institution [START_REF] Mckechnie | Consumer Buying Behaviour in Financial Services: An overview[END_REF][START_REF] Lee | How consumers choose a financial institution: decision-making criteria and heuristics[END_REF][START_REF] Söderberg | Relationships between advisor characteristics and consumer perceptions[END_REF] and related issues of customer involvement and customer segmentation [START_REF] Kamakura | Applying latent trait analysis in the evaluation of prospects for cross-selling of financial services[END_REF][START_REF] Harrison | Mapping Customer Segments for Personal Financial Services[END_REF]Foxall & Pallister 1998;[START_REF] Beckett | An exposition of consumer behaviour in the financial services industry[END_REF][START_REF] Aldlaigan | Consumer involvement in financial services : an empirical test of two measures[END_REF], financial consumption patterns and practices [START_REF] Bernthal | Credit Cards as Lifestyle Facilitators[END_REF][START_REF] Henry | Social class, market situation, and consumers' metaphors of (dis) empowerment[END_REF]Cook et al. 2009;[START_REF] Peñaloza | Living US Capitalism: The Normalization of Credit/Debt[END_REF], innovation adoption [START_REF] Johns | The impact of internet banking on business-customer relationships (are you being self-served?)[END_REF][START_REF] Grabner-Kräuter | Consumer acceptance of internet banking: the influence of internet trust[END_REF][START_REF] Chiou | The antecedents of online financial service adoption: the impact of physical banking services on Internet banking acceptance[END_REF], personal financial strategies [START_REF] Lunt | Psychological, social and economic determinants of saving: comparing recurrent and total savings[END_REF][START_REF] Browning | Household Saving : Micro Theories and Micro Facts[END_REF][START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF][START_REF] Henry | Social class, market situation, and consumers' metaphors of (dis) empowerment[END_REF][START_REF] Patel | Debt and disadvantage: the experience of unmanageable debt and financial difficulty in England and Wales[END_REF]) and determinants of customer loyalty to the financial institution [START_REF] Çalık | Consumer satisfaction and loyalty derived from the perceived quality of individual banking services: A field study in Eskişehir from Turkey[END_REF][START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF][START_REF] Cooil | A Longitudinal Analysis of Customer Satisfaction and Share of Wallet: Investigating the Moderating Effect of Customer Characteristics[END_REF][START_REF] Keisidou | Customer satisfaction, loyalty and financial performance: A holistic approach of the Greek banking sector[END_REF].

However, despite the abundance of empirical studies, research contributions are spread across different fields, and there have scarcely been any attempts to systematize the accumulated body of knowledge or to develop a conceptual framework that would provide an overall view of different aspects of consumption of financial services. That is, a framework that would view the consumption process in the financial context not only as the selection and purchase of financial products and services, but as a multifaceted construct that also encompasses relationships between consumers and their financial service providers as well as the broader sociocultural aspects of financial consumption beyond the immediate situation of purchase [START_REF] Mccracken | Culture and Consumption: A Theoretical Account of the Structure and Movement of the Cultural Meaning of Consumer Goods[END_REF]. This paper contributes to the rich body of literature dealing with different aspects of consumer behaviour in the financial context, by examining the accumulated knowledge on these three aspects of financial service consumption. It adopts the sociocultural approach that regards consumption as "the selection, purchase and use of goods and services" [START_REF] Campbell | The Romantic Ethic and the Spirit of Modern Consumersm Kindle edi[END_REF], proposing a conceptual framework that incorporates relevant contributions from the field. In this paper I revisit the last two decades of literature, systematizing the research streams across three dimensions: 1) different stages of the financial consumption process, 2) relationships between consumers and their financial service providers, and 3) sociocultural aspects of financial consumption. The first section explores literature on the financial consumption process. It encompasses three subsections: the pre-purchase stage of financial services consumption or the bank selection, the different aspects of purchase situation and consumer decision making, and finally the post-purchase stage and the different ways consumers use financial products and services as well as consumers' personal financial management strategies.

The second section provides an overview of extant knowledge on customer-bank relationships.

The third section deals with sociocultural aspects of financial services consumption. Finally the contributions from these three streams of literature are summarized into a three-dimensional conceptual framework, which incorporates relevant literature on different aspects of financial service consumption. In the end of the paper I develop a research agenda for future studies, providing suggestions for future research.

Stages of financial consumption process

Pre-purchase consumption: bank selection

Bank selection is an important step in the financial consumption process, as financial consumption evolves over time, involving customers and service providers in a number of continuous transactions [START_REF] Mckechnie | Consumer Buying Behaviour in Financial Services: An overview[END_REF]. As the financial industry tends to have high switching costs, once consumers have chosen their bank, they tend to stay with it over prolonged periods of time, gradually increasing the amount of transactions and building their personal assets [START_REF] Levesque | Determinants of customer satisfaction in retail banking[END_REF][START_REF] Bell | Customer Relationship Dynamics: Service Quality and Customer Loyalty in the Context of Varying Levels of Customer Expertise and Switching Costs[END_REF]. A number of researchers explored the factors influencing the bank selection process over the years. However, despite multiple attempts to find a universal solution, researchers couldn't single out one dominant influence on customers' bank choice, concluding that customers tend to consider multiple factors when choosing a financial institution [START_REF] Lee | How consumers choose a financial institution: decision-making criteria and heuristics[END_REF]. In her overview of consumer buying behaviour in financial services [START_REF] Mckechnie | Consumer Buying Behaviour in Financial Services: An overview[END_REF] has listed some of the common choice criteria: customer confidence, trust and loyalty, dependability and size of the institution, location, convenience and ease of transactions, professionalism of bank personnel and availability of loans. [START_REF] Lee | How consumers choose a financial institution: decision-making criteria and heuristics[END_REF] examined the influence of convenience, low fees, minimum balances and ranges of services, stating that while all these factors are important, customers consider several of them at the same time when making a choice.

Purchase of financial products and services

Purchase situations

A lot of empirical studies over the years have been dedicated to exploration of behaviour of financial services consumers in purchase situations. In recent experimental study of Swedish customers [START_REF] Söderberg | Relationships between advisor characteristics and consumer perceptions[END_REF] found out that certain advisor characteristics, such as gender and mood, can influence customer perceptions and willingness to follow their advice. Thus customers were more willing to follow advice of a smiling female advisor, despite perceiving it to be associated with greater degree of risk. However, customers of both genders perceived male advisors as more credible and subsequently their advice to have a lower degree of risk. A number of studies suggested that purchase decisions can also be affected by ethnic background of customers, criticizing the lack of cultural awareness of financial institutions that operate colour-blind marketing strategies [START_REF] Burton | Ethnicity and consumer financial behaviour: a case study of British Asians in the pensions market[END_REF][START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF]. However, exploration of purchase situations per se has not been as much in the focus as the attempts of customer profiling and segmentation and exploration of customer involvement.

Customer profiling

The majority of studies of consumer behaviour in purchase situations have been concerned with consumer profiling based on a variety of dimensions, as managerially-oriented researchers tried to determine the most profitable customer segments. For instance, [START_REF] Harrison | Mapping Customer Segments for Personal Financial Services[END_REF] combined dimensions of financial maturity [START_REF] Kamakura | Applying latent trait analysis in the evaluation of prospects for cross-selling of financial services[END_REF] and perceived level of financial knowledge in order to segment bank customers, which resulted in four segments, that he labelled financially confused (low maturity, low knowledge), apathetic minimalists (low knowledge, high maturity), cautious investors (good knowledge, moderate maturity) and capital accumulators (high knowledge, high maturity). He subsequently found that customers with the highest and lowest levels of perceived knowledge are the least profitable, as the customers who don't know anything about financial services usually have low interest in acquiring new products, while those with extremely high levels of perceived financial knowledge are highly sophisticated and often prefer to use external financial experts, thus not generating fees for the bank. [START_REF] Beckett | An exposition of consumer behaviour in the financial services industry[END_REF] conducted a focus group study in the UK and developed a typology of financial services customers based on the dimensions of consumer confidence and involvement, identifying four ideal types of repeat passive, rational active, no purchase and relational dependent customers. The follow-up study by [START_REF] Howcroft | Customer involvement and interaction in retail banking: an examination of risk and confidence in the purchase of financial products[END_REF] combined focus groups with survey questionnaires and divided customers into six clusters using dimensions of involvement risk and confidence: action inert, repeat passive, emerging on line, emerging repeat-passive, rational active and emerging profit potential customers. In general, the findings indicated that customers did not possess high levels of interest in the financial products and services and did not perceive many differences between financial services providers. Other attempts of consumer profiling used dimensions of trust, commitment and involvement [START_REF] Tsarenko | Examining customer privacy concerns in dealings with financial institutions[END_REF][START_REF] Sunikka | Empirical investigation into the multifaceted trust in the wealth management context[END_REF].

Customer involvement

Customer involvement in financial services has been studied extensively in relation to buying behaviours of bank clients. Often defined as a "person's perceived relevance of the object based on inherent needs, values and interests" (Zaichkowsky 1985, p.342), it has attracted considerable attention of researchers, who had tried to measure it or explore its nature in different contexts. In the financial services marketing, a certain amount of studies has been dedicated to exploration of the consumer purchase behaviour in the financial context using a number of measurement instruments. The main measurement scales, applied to this context, have been the Personal Involvement Inventory (PII) [START_REF] Zaichkowsky | Measuring the Involvement Construct[END_REF], purchase-decision involvement scale (PIS) [START_REF] Mittal | A Causal Model of Consumer Involvement[END_REF]) and consumer involvement profile (CIP) [START_REF] Laurent | Measuring Consumer Involvement Profiles[END_REF].

In the financial services marketing all three instruments of measuring consumer involvement have been explored in relation to their applicability to the financial services consumption context. Foxall and Pallister (1998) compared PII and PIS scales and concluded that in the financial services the rational dimensions of consumer involvement overrun the emotional dimensions, as consumers didn't become emotionally involved with the financial services brand, with the notable exception of mortgage buyers. [START_REF] Aldlaigan | Consumer involvement in financial services : an empirical test of two measures[END_REF] further applied the PII and CIP instruments to the financial services, and confirmed the dominance of rational involvement in the financial services context, providing evidence that while risk importance and interest are the highest rated dimensions of involvement in the financial services, the sign value, pleasure and interest (the PII dimension reflecting personal interests, motivations and values of the customer) are insignificant and less relevant for this context. By empirically testing these two measures on the sample of 120 students, they were able to classify financial services according to their levels of involvement -thus mortgages, investments and savings were deemed high involvement services, while checking accounts, personal loans and cash machine can be considered medium involvement. Interestingly enough, no services in the financial context have been discovered to be low involvement.

Post-purchase: personal financial strategies and financial consumption patterns

Consumption of financial products and services was studied across multiple streams of research. In bank marketing literature a significant amount of attention was dedicated to customer adoption of innovative products and services, while in consumer research, economics, economic psychology and economic sociology the main focus has been on usage of financial products in relation to broader issues of personal financial strategies particularly personal financial planning and debt.

Adoption of innovative products and services

In the field of bank marketing, a blooming body of literature adoption of internet and mobile banking has emerged over the last fifteen years. Most of the studies have been dealing with consumer attitudes and the factors influencing the adoption decision. Relationships with the bank -especially consumers' trust and commitment to the financial institution have been found to have a significant impact on the technology perceptions and adoption decisions [START_REF] Johns | The impact of internet banking on business-customer relationships (are you being self-served?)[END_REF][START_REF] Grabner-Kräuter | Consumer acceptance of internet banking: the influence of internet trust[END_REF][START_REF] Chiou | The antecedents of online financial service adoption: the impact of physical banking services on Internet banking acceptance[END_REF] 2012), a study conducted in the UK attributes adoption decisions to customer-specific factors saying that highly innovative customers value usefulness of the service while customers with high levels of discomfort and insecurity value ease of use [START_REF] Yousafzai | Understanding customer-specific factors underpinning internet banking adoption[END_REF], and the study conducted in Saudi Arabia just states that mobile banking services are primarily used by young males [START_REF] Sohail | Attitudes towards mobile banking: are there any differences between users and non-users? Behaviour & Information Technology[END_REF]. A deeper qualitative exploration of these consumption patterns could give us more insights into the contextual influences behind this variety.

Savings, investments and financial planning

Early studies in the financial services literature were based on the life cycle model of saving [START_REF] Ando | The" life cycle" hypothesis of saving: Aggregate implications and tests[END_REF] and viewed consumption of financial services in terms of hierarchy of investment objectives along consumers' life cycle. According to this model, consumers plan their consumption and savings evenly along their lifetime, keeping their spending levels approximately the same in every period in order to maintain stable lifestyles and planning their consumption and savings behaviour in long-term perspective. [START_REF] Kamakura | Applying latent trait analysis in the evaluation of prospects for cross-selling of financial services[END_REF] developed a hierarchical model of consumption of financial services based on consumer financial maturity and the difficulty of acquisition of the services. According to the hierarchical model consumers tend to pattern their investments along the pyramid of financial independence. In the beginning they are concerned with acquiring foundational products (such as checking accounts and personal loans) as well as risk management (insurance, pensions) and creating emergency cash reserves. As their household income grows, investor's objectives shift to acquiring products that can facilitate their financial growth to offset inflation (e.g. stocks, mutual funds, cash management accounts), and finally the products that deal with risk and tax protection (offshore accounts in tax shelters, municipal bonds, real estate) are positioned on the top of the pyramid. The financial maturity of the investor grows as they acquire products on higher levels. The latent maturity is positively related to the household income, household net worth and the occupation of the investor and negatively related to the rented household (an indicator of a lack of financial resources for investment).

It has been repeatedly argued that savings increase with the growth of income reaching its peak before retirement, that highest saving rates occur among consumers with higher levels of income, wealth and education as well as among the homeowners and married couples without children [START_REF] Lunt | Psychological, social and economic determinants of saving: comparing recurrent and total savings[END_REF][START_REF] Browning | Household Saving : Micro Theories and Micro Facts[END_REF] and that loss of income can result in debt problems [START_REF] Patel | Debt and disadvantage: the experience of unmanageable debt and financial difficulty in England and Wales[END_REF].

In recent years some researchers have challenged the universal applicability of the life cycle model. Rather than viewing financial maturity as a linear function of lifetime's income fluctuations, consumer researchers highlighted the importance of sociocultural context in explaining financial decisions. While the life-cycle model can be viewed as an ideal type approximation [START_REF] Weber | The Methodology of the Social Sciences[END_REF] of consumer behaviour, it assumes even levels of consumption across lifetime and does not account for a number of sociocultural influences that may result in other varieties of saving and consumption levels. For instance, the religious orientation of consumers may affect their financial decisions -a study of professional South Asian families in Canada by [START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF] showed that Indian families prioritize saving and investment goals over consumption goals. As parents' primary purpose was to save money to provide for their children's education, they were able to manage their consumption levels in order to achieve their investment goals. The authors attributed this pattern to the particularity of the Hindu worldview, which prioritizes wealth acquisition over consumption along individual's lifetime. Social class is another factor that needs to be taken into consideration - [START_REF] Henry | Social class, market situation, and consumers' metaphors of (dis) empowerment[END_REF] argued that self-perceptions of consumers belonging to different social classes can result in distinctively different financial consumption patterns. According to Henry, young professionals' empowered self-perceptions encourage them to adopt elaborate budget planning strategies in pursuit of future opportunities, while working class consumers who feel disempowered and see future as a threat manage their budgets with the primary goal of maintaining stability and avoiding future risks.

Credit cards, mortgages and debt

In recent decades debt has become a continuous, lived experience of consumption boosted by the democratization of revolving consumer credit and the introduction of credit cards (Marron 2007, p. 120). The adoption of the practice of sub-prime lending by the financial institutions broadened possibilities of consumer involvement in the financial system [START_REF] Langley | Sub-prime mortgage lending: a cultural economy[END_REF]. Consumer credit has come to play an increasingly central role in the practices of consumer subjects, wiping away the need to earn and/or save money before a purchase can be made [START_REF] Langley | Uncertain subjects of Anglo-American financialization[END_REF]).

Debt has long been a sensitive issue in financial consumption research. Predatory lending policies of financial institutions that boost consumer debt have been repeatedly mentioned among the primary reasons of major financial crises [START_REF] Langley | Sub-prime mortgage lending: a cultural economy[END_REF][START_REF] Schularick | Credit Booms Gone Bust : Monetary Policy, Leverage Cycles, and Financial Crises, 1870-2008[END_REF], and a potential solutions for encouraging responsible consumer behaviour and predicting potential risks of over-indebtedness have been discussed, for instance consumer education [START_REF] Baumann | Getting Cinderella to the ball: putting education at the heart of financial education[END_REF], increasing financial literacy [START_REF] Gathergood | Self-control, financial literacy and consumer over-indebtedness[END_REF] or evaluating consumer meaning patterns [START_REF] Peñaloza | Living US Capitalism: The Normalization of Credit/Debt[END_REF].

It has been argued that socially excluded groups are particularly vulnerable to overindebtedness [START_REF] Patel | Debt and disadvantage: the experience of unmanageable debt and financial difficulty in England and Wales[END_REF]) and some authors (e.g. [START_REF] Williams | Empowerment of Whom and for What? Financial Literacy Education and the New Regulation of Consumer Financial Services[END_REF]) condemn the practice of blaming irresponsible consumers for financial crises, arguing that the praise of consumer education, financial literacy and empowerment shifts the focus from financial institutions and puts consumers at the centre of a problem. However, a number of studies showed that increasing levels of financial literacy indeed help in lowering levels of consumer debt [START_REF] Huhmann | A model of consumer financial numeracy[END_REF][START_REF] Gathergood | Self-control, financial literacy and consumer over-indebtedness[END_REF][START_REF] West | Financial literacy education and behaviour unhinged: combating bias and poor product design[END_REF][START_REF] Sevim | The effects of financial literacy on the borrowing behaviour of Turkish financial consumers[END_REF]Disney & Gathergood 2013;[START_REF] Jappelli | Investment in financial literacy and saving decisions[END_REF].

Recently, sociocultural-oriented consumer research studies started ascribing more agency to consumers, transcending their perception as impotent reactors who can be manipulated into desirable or undesirable consumption patterns by financial institutions and governments. Cook, Smith and Searle (2009) argued against the dominant metaphor of "duped debtor" in the financial consumption literature and showed how consumers can be actively engaged in the financialization of domestic space through exploration of flexible options of their mortgages. Bernthal, Crocket and Rose (2005) demonstrated how credit card practices can be either used to facilitate overlapping lifestyle dimensions -achieving lifestyles, where consumers pursue their lifestyle objectives by means of their credit card consumption, or coping lifestyles (debtor's prison) where consumers accumulate increased levels of debt. In order to assert control over the levels of their debt consumers employ a variety of credit card practices, for example credit card shuffling, debt conversion or requesting additional credit. Moreover, while dealing with debt, consumers learn from their direct experience [START_REF] Peñaloza | Living US Capitalism: The Normalization of Credit/Debt[END_REF], being able to distinguish good debt (i.e. the debt that can potentially generate returns in future) from bad debt (transient expenses).

Relationships between consumers and financial institutions

In the extremely competitive financial services industry consumption revolves around regular transactions between customers and service providers over extended period of time [START_REF] Mckechnie | Consumer Buying Behaviour in Financial Services: An overview[END_REF]. Therefore, long-term profitable relationships with customers become a key driver of success. The majority of research on this matter assumes linear trajectory of consumerbank relationship development that begins with customer first opening a student savings account, then taking a next step opening a credit account and finally arriving to procure home loans and complex financial instruments once they have stable income and improve their financial situation [START_REF] Bell | Customer Relationship Dynamics: Service Quality and Customer Loyalty in the Context of Varying Levels of Customer Expertise and Switching Costs[END_REF]. It is argued that as the time passes, the longer customers hold their accounts with the same bank and the higher are their personal assets, they become less price-sensitive and prioritize long-term relationships over service fees in evaluation of their service experiences [START_REF] Levesque | Determinants of customer satisfaction in retail banking[END_REF][START_REF] Lee | How consumers choose a financial institution: decision-making criteria and heuristics[END_REF].

In the financial services industry building strong relationships can be quite a challenge, as often customers tend not to perceive significant differences between financial services providers [START_REF] Howcroft | Customer involvement and interaction in retail banking: an examination of risk and confidence in the purchase of financial products[END_REF]) and there is a blurred distinction between the brand and the company [START_REF] Aggarwal | The Effects of Brand Relationship Norms on Consumer Attitudes and Behavior[END_REF]Farquhar 2011). While de Chernatony (1999) claimed that brand culture might be important for financial services, empirical studies repeatedly indicated that consumers primarily rely on convenience-and price-related factors over brand image or affective components while establishing (Foxall & Pallister 1998;[START_REF] Lee | How consumers choose a financial institution: decision-making criteria and heuristics[END_REF][START_REF] Petruzzellis | Branding relationships in financial services: Paradigm shift in Mediterranean countries[END_REF], maintaining [START_REF] Bell | Customer Relationship Dynamics: Service Quality and Customer Loyalty in the Context of Varying Levels of Customer Expertise and Switching Costs[END_REF][START_REF] Çalık | Consumer satisfaction and loyalty derived from the perceived quality of individual banking services: A field study in Eskişehir from Turkey[END_REF][START_REF] Roig | Social value in retail banking[END_REF], or deciding to terminate the relationship [START_REF] Panther | Consumer responses to dissatisfaction with financial service providers: An exploration of why some stay while others switch[END_REF][START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF]) with their financial service provider.

While the word "relationships" is widely used in the literature, in fact, the body of knowledge on consumer-bank relationships is currently dominated by survey studies that explore antecedents of customer loyalty. That is, the main focus of consumer-bank relationships research is not on relationships as an ongoing interactive process, but rather on relationship outcomes.

Antecedents of customer loyalty in financial services

Customer loyalty is believed to be key element of maintaining successful relationships in the financial services industry. While there exists a multitude of loyalty definitions and researchers keep on coming up with new ones regularly, it been often defined in terms of customer's intent to stay with the same financial services organization (Eisingerich & Bell 2006), or a commitment to make repeat purchases with the same service provider [START_REF] Oliver | Whence Consumer Loyalty[END_REF]).

Beerli, Martin and Quintana (2004) suggested a distinction between 1) loyalty based on inertia, where customers stay with the same service provider out of habit and would not hesitate to switch to another one provided there is a convenient reason for it, and 2) true brand loyalty, which can be described as a conscious repeat purchasing behaviour accompanied with positive attitude and high degree of commitment to the brand. Other distinctions distinguish behavioural loyalty from attitudinal loyalty, as well as cognitive and affective loyalty aspects (for detailed review see [START_REF] Lewis | The antecedents of consumer loyalty in retail banking[END_REF].

Research has shown that in financial services industry customers tend to lack in emotional attachment to their banks, primarily associating loyalty with the value they get from their bank and being willing to switch to other service providers in case they find better rates and conditions [START_REF] Lewis | The antecedents of consumer loyalty in retail banking[END_REF]. Another common occurrence in the financial services industry is polygamous customer loyalty, when customers simultaneously maintain relationships with different financial service providers, as empirical evidence suggests that many prefer to use the services of different banks at the same time [START_REF] Çalık | Consumer satisfaction and loyalty derived from the perceived quality of individual banking services: A field study in Eskişehir from Turkey[END_REF][START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF][START_REF] Cooil | A Longitudinal Analysis of Customer Satisfaction and Share of Wallet: Investigating the Moderating Effect of Customer Characteristics[END_REF] It has been argued that customer satisfaction is one of the most important antecedents of customer loyalty in the financial services [START_REF] Nguyen | The mediating role of corporate image on customers' retention decisions: an investigation in financial services[END_REF][START_REF] Beerli | A model of customer loyalty in the retail banking market[END_REF]. A number of studies has tried to identify the determinants of customer satisfaction, citing competitive interest rates, core performance, service quality, convenience, fair service, trust, employee behaviour and consumer demographics among key satisfaction antecedents [START_REF] Levesque | Determinants of customer satisfaction in retail banking[END_REF][START_REF] Cooil | A Longitudinal Analysis of Customer Satisfaction and Share of Wallet: Investigating the Moderating Effect of Customer Characteristics[END_REF][START_REF] Chen | The impact of financial services quality and fairness on customer satisfaction[END_REF][START_REF] Kaura | Antecedents of customer satisfaction: a study of Indian public and private sector banks[END_REF][START_REF] Keisidou | Customer satisfaction, loyalty and financial performance: A holistic approach of the Greek banking sector[END_REF][START_REF] Seiler | The influence of socio-demographic variables on customer satisfaction and loyalty in the private banking industry[END_REF]. Switching costs and convenience-related factors are the second most mentioned determinant of customer loyalty, as indicated in the literature. In fact, sometimes convenience trumps satisfaction as the loyalty determinant, when dissatisfied customers prefer to stay with the bank in order to avoid the hassle of switching service providers [START_REF] Panther | Consumer responses to dissatisfaction with financial service providers: An exploration of why some stay while others switch[END_REF].

Functional benefits of service such as service quality, time required to process applications, money savings, simplicity and convenience, as well as emotional benefits (feelings and emotions generated by the products, social bonds with service providers and social approval) [START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF][START_REF] Roig | Social value in retail banking[END_REF]) can also influence the degree of customer loyalty. It is generally believed that, while functional benefits dominate as loyalty determinants in the financial services, emotional benefits should be also taken into account while explaining customer's decision to make a repeat purchase versus switching the service provider.

The length of relationship with the service provider is another suggested loyalty antecedent. It has been claimed, that the customers who stayed with the same bank for the longer periods of time are less likely to switch [START_REF] Panther | Consumer responses to dissatisfaction with financial service providers: An exploration of why some stay while others switch[END_REF][START_REF] Çalık | Consumer satisfaction and loyalty derived from the perceived quality of individual banking services: A field study in Eskişehir from Turkey[END_REF], which has been attributed to the fact that their share of wallet in the given bank is higher, and it would be too inconvenient to change the service provider at this point [START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF]). According to Eisingerich and Bell (2006, p.90), customer education, defined as "financial adviser's willingness to explain financial concepts and the pros and cons of recommended investment opportunities to their clients", can also have a positive influence on increasing consumer loyalty by encouraging customer participation, defined as "client's willingness to make constructive suggestions to the firm on how to improve its service offerings". Other possible antecedents of customer loyalty, quoted in the literature, include corporate social responsibility and price fairness [START_REF] Matute-Vallejo | The Influence of Corporate Social Responsibility and Price Fairness on Customer Behaviour: Evidence from the Financial Sector[END_REF]) and customer demographic factors [START_REF] Çalık | Consumer satisfaction and loyalty derived from the perceived quality of individual banking services: A field study in Eskişehir from Turkey[END_REF][START_REF] Seiler | The influence of socio-demographic variables on customer satisfaction and loyalty in the private banking industry[END_REF]. It has been argued, that customer loyalty can be measured by the share of wallet [START_REF] Cooil | A Longitudinal Analysis of Customer Satisfaction and Share of Wallet: Investigating the Moderating Effect of Customer Characteristics[END_REF][START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF], defined by a share of purchases in the category [START_REF] Jones | Why Satisfied Customers Defect[END_REF].

A small number of studies attempted to take a broader view of relationships, examining different forms of interactions between customers and their banks, using in-depth research methods [START_REF] O'loughlin | Emerging perspectives on customer relationships, interactions and loyalty in Irish retail financial services[END_REF]Elsharnouby & Parsons 2010). Their findings mostly support the conclusion that loyalty to financial service providers is fundamentally conveniencebased and that customers show a notable lack of involvement in the financial services as a whole, while at times they might show positive attitudes to particular staff members. It is important to note though, that findings can differ tremendously between studies conducted in different cultural contexts. The study of retail banking customers in Ireland showed that customer experience with the bank was mostly transactional and functionally-oriented, as customers treated their financial consumption as everyday necessity or means to the end, perceiving little differences between different suppliers and not willing to build any relationships with their service providers beyond merely opportunistic ones [START_REF] O'loughlin | Emerging perspectives on customer relationships, interactions and loyalty in Irish retail financial services[END_REF]). In the context of Egyptian financial services, however, researchers have found that some customers can show affective commitment to bank personnel to a certain degree (Elsharnouby & Parsons 2010).

Sociocultural aspects of financial consumption

It has been argued that our lives have become closely connected with the financial world [START_REF] Cobb | Offshore financial services and the Internet: creating confidence in the use of cyberspace?[END_REF][START_REF] Langley | Uncertain subjects of Anglo-American financialization[END_REF]. The wish to control future uncertainties by taking responsibility for building personal wealth engaged a large number of individuals in a number of investment activities and attempts to improve the financial literacy of themselves and their families. This trend towards integrating the world of finance in the everyday lives of consumers came to be known as the financialization of daily life [START_REF] Martin | Financialization of daily life[END_REF].

Financial institutions have played an important facilitating role in the growth of consumer society. The financial globalization, the rise of the offshore trade and the internationalization of the stock trade has spread the influence of financialization worldwide [START_REF] Aitken | The democratic method of obtaining capital-culture, governmentality and ethics of mass investment[END_REF][START_REF] Cobb | Offshore financial services and the Internet: creating confidence in the use of cyberspace?[END_REF]. The participation of individuals and households in stock markets has been rapidly growing in many countries [START_REF] Aspara | Aesthetics of stock investments[END_REF]. The redefinition of family home as an object of speculation and credit, has further involved the domestic life of the households in the financial activities [START_REF] Martin | Financialization of daily life[END_REF]Cook et al. 2009). Some consumers are even taking a step forward from individual consumption, taking up the role of co-producers of local community banks (Cutcher 2010).

In modern world consumption cannot be studied outside of its relationship with culture [START_REF] Mccracken | Culture and Consumption: A Theoretical Account of the Structure and Movement of the Cultural Meaning of Consumer Goods[END_REF], and different, sometimes contradicting results of financial service consumption studies coming from different cultural contexts seem to illustrate this idea very well. Researchers have long argued that macro-structural accounts of financial and economic research need to be analysed at the level of everyday practices of the daily life and culture [START_REF] Aitken | The democratic method of obtaining capital-culture, governmentality and ethics of mass investment[END_REF][START_REF] Bernthal | Credit Cards as Lifestyle Facilitators[END_REF]. However, the field of financial consumption studies has yet to transcend its almost exclusive focus on individual factors that determine financial consumption and open up to acknowledging broader sociocultural dimensions. As studies of cognitive and affective aspects of financial consumption currently dominate the field, its cultural aspects have been significantly under researched, with notable exceptions of several works [START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF][START_REF] Bernthal | Credit Cards as Lifestyle Facilitators[END_REF][START_REF] Henry | Social class, market situation, and consumers' metaphors of (dis) empowerment[END_REF]Cook et al. 2009;[START_REF] Peñaloza | Living US Capitalism: The Normalization of Credit/Debt[END_REF].

A number of contributions point us to the importance of embracing the sociocultural perspective and the need of further studies in this direction. For instance financial consumption decisions and choice of banks can be affected not only by income levels, but also by consumer ethnicity [START_REF] Burton | Ethnicity and consumer financial behaviour: a case study of British Asians in the pensions market[END_REF], religious preferences [START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF][START_REF] Sayani | Determinants of bank selection in the United Arab Emirates[END_REF] and social class [START_REF] Henry | Social class, market situation, and consumers' metaphors of (dis) empowerment[END_REF], and there are even differences in buying behaviours between customers of national and local banks in the US [START_REF] Kaynak | American consumers' attitudes towards commercial banks: A comparison of local and national bank customers by use of geodemographic segmentation[END_REF]. The relationships between customers and banks can also vary greatly in different cultural contexts [START_REF] O'loughlin | Emerging perspectives on customer relationships, interactions and loyalty in Irish retail financial services[END_REF]Elsharnouby & Parsons 2010). As the banks are slowly shifting towards providing tailored options to their customers [START_REF] Shanmuganthaan | Ethnic banking in the USA[END_REF]), sociocultural context of financial consumption can no longer be ignored. In the next section the reviewed literature is summarized into a multidimensional conceptual framework of financial consumption that includes its cultural aspects as well as different stages of consumption process and consumer relationships with their banks.

Discussion and conclusion: conceptual framework and suggestions for future research

This work revisited the last two decades of literature on financial consumption, presenting the author's conceptualization of consumption of financial services as a multidimensional phenomenon that encompasses customer relationships with their service providers, the different stages of consumption process pre-, during-and post-purchase, and that is inherently connected with and occurs within the broader sociocultural context. The visual representation of the suggested conceptual framework is shown in Figure 1.

This framework is based on the review of literature covering a range of subjects related to different aspects of financial consumption in an attempt to capture the bigger picture of financial consumption process in its different manifestations. While some of the subjects, like customer loyalty or customer decision making, have been extensively covered in the literature, other subjects, particularly related to sociocultural aspects of financial services consumption, have been significantly under researched. Future research could address this imbalance by addressing a number of questions that post financial consumption within sociocultural context. Some suggestions for future research are provided below.

Figure 1. Conceptual framework of financial consumption

Embracing globalization and the emergence of new consumer groups

As distant localities become linked with each other, intensifying social relations [START_REF] Giddens | The Consequences of Modernity[END_REF], financial services industry is striving to keep up with challenges of globalization. Banks are trying to actively participate in the globalization processes by interacting with consumers with the help of internet-based technologies and complex financial instruments (Wilson et al. 2010;Beaverstock et al. 2011). However, existing research on financial services consumption does not adequately grasp the globalization tendency regarding consumer financial practices and relationships with their financial service providers, as most of the studies adopt a sedentarist perspective, assuming that consumers remain grounded in a local context along the course of their lifetime. In the globalizing world where the flow of people and information is changing consumer landscape [START_REF] Sheller | The new mobilities paradigm[END_REF], more and more consumers are travelling for personal or business purposes and have simultaneous relationships with different banks [START_REF] Lam | Customer tradeoffs between key determinants of SME banking loyalty[END_REF], it is important to take consumer relocation into account when developing a theory of financial consumption.

Moreover, new consumer groups emerge as the global flows of culture and labour result in increasing number of transnational migrants and professionals (Appadurai 1988;[START_REF] Bauman | Liquid Modernity[END_REF][START_REF] Sheller | The new mobilities paradigm[END_REF][START_REF] Bardhi | Liquid Relationship to Possessions[END_REF]. Do they exhibit the same financial consumption and service relationships patterns as local consumers, and will the marketing strategies tailored for particular national cultures work on them, or are they carriers of universal culture [START_REF] Leung | Culture and International business: Recent Advances and Their Implications for Future Research[END_REF]) that financial consumption researchers have yet to explore?

Acculturation to financial services in different countries

Another interesting research opportunity is exploration of processes of consumer acculturation to the financial services contexts. The concept of consumer acculturation, reflecting the processes of movement and adaptation of individuals to a new consumer cultural environment has been first introduced by [START_REF] Peñaloza | Atraversando Fronteras/Border Crossings: A Critical Ethnographic Exploration of the Consumer Acculturation of Mexican Immigrants[END_REF] in her study of Mexican immigrants in the U.S. This process occurred in three stages: a movement to a new country was followed by processes of translation while adapting to the new country's language, currency and social relations, and finally adaptation to the new environment. The following consumer acculturation outcomes included assimilation, maintenance, resistance and segregation. The assimilation occurred when informants adopted new products and services, characteristic to the new culture, however at the same time they also managed to maintain their own cultural aspects. The pressure to change sometimes caused the informants to resist the pulls of competing cultures and, finally, some of the informants were segregated from the dominant culture by means of geographical location. Differences in acculturation outcomes can be used as explanations for different consumption patterns. One interesting research direction for acculturation informed financial consumption study is a study of consumption habits of different consumer groups, for instance, financial consumption of globally mobile consumers -professionals, free movers and ethnic immigrants in different countries and the issues of culture-specific consumption patterns, consumer adaptation to new service environments. Of particular interest are the ways in which these consumer groups acculturate to financial services in different countries and the trust issues that arise between the immigrants and the local service providers [START_REF] Peñaloza | Atraversando Fronteras/Border Crossings: A Critical Ethnographic Exploration of the Consumer Acculturation of Mexican Immigrants[END_REF][START_REF] Jain | Money matters: An exploratory study of the socio-cultural context of consumption, saving, and investment patterns[END_REF][START_REF] Peñaloza | The Marketer Acculturation : The Changer and the Changed[END_REF][START_REF] Chai | The influence of acculturation on consumer relational bonding in banking relationships[END_REF][START_REF] Chai | How consumer acculturation influences interpersonal trust[END_REF][START_REF] Bardhi | Liquid Relationship to Possessions[END_REF]Figueiredo & Uncles 2014).

Towards a broader view of relationships

In the relationships domain, a potential opportunity for future research can lay in going beyond the exploration of consumer attitudes towards acquiring a broader view of relationship as a whole. More in-depth qualitative inquiries guided by the focus on consumer practices can provide a viable alternative to the exclusive focus on loyalty aspects of relationships in order to understand how service relationships unfold, evolve and disintegrate. Taking consumer mobility into consideration can also provide us with interesting areas of exploration. How do consumers maintain relationships with their financial services providers in many different countries? And can we still assume the linear development of relationships between consumers and their banks, when a consumer acquires the first checking account in their student years and then stays with the same bank, acquiring more and more services as their income grows? And what happens when consumers build new service relationships in a new country or several new countries, as is the case with transnational professionals? How do they navigate cultural norms of financial consumption while dealing with banks in different countries simultaneously? In the era of instantaneous and short-term relationships [START_REF] Bauman | Liquid Modernity[END_REF][START_REF] Urry | Sociology beyond Societies: Mobilities for the twenty-first century[END_REF], can we still rely on old models of customer loyalty in financial institutions? Or are there new varieties of customer loyalty that emerge in the mobility context? Are mobile consumers responsive to the same satisfaction determinants as local consumers?

Service encounters

In the context of study of consumption of financial services we need to study new ways of services encounters. A stream of research has examined factors influencing the adoption of new technologies in financial services among local consumers, however, now it's time to take a step further and see what is going on after the adoption process, particularly how these technologies are being used on the everyday level. How do consumers use internet and mobile banking in order to consume services, make financial transaction or navigate relationships with their financial services providers? Are new ways of virtual consumption emerging as an alternative to heaviness and solidity of brick-and-mortar service experiences?

From attitudes to practices

The attitudinal focus of the majority of current studies in the financial services literature sets limits to our knowledge of the cultural influences on processes of consumption. We know how consumers think or feel about their banks but our knowledge of what is going on after the purchase event is currently limited. Ìn order to gain broader understanding of contextual factors and social processes that may affect this variety of empirical contributions we need to abandon the exclusive focus on cognitive and affective aspects of consumption in order to understand what consumers actually do with their finances. More focus on financial consumption practices by individuals can provide us with understanding of what happens after the financial service have been acquired and how the globalized financial reality is lived by individuals on an everyday level [START_REF] Aitken | The democratic method of obtaining capital-culture, governmentality and ethics of mass investment[END_REF]. It has been argued that consumer expertise in dealing with financial services can affect their satisfaction levels [START_REF] Jamal | Customer satisfaction and retail banking: an assessment of some of the key antecedents of customer satisfaction in retail banking[END_REF][START_REF] Bell | Customer Relationship Dynamics: Service Quality and Customer Loyalty in the Context of Varying Levels of Customer Expertise and Switching Costs[END_REF]). An interesting opportunity for future research is to explore how development of consumer financial expertise is related to the demands of the context. Will different consumption contexts require a certain set of consumption practices and related skills, created as an answer to the demands of the environment? And how are financial practices of different consumer groups consumers manifested in their consumption patterns?

The field of financial consumption studies needs a broader understanding of the consumption process that spans beyond the immediacy of purchase situation and issues of customer loyalty. Up until recently it has been mostly dominated by study of individual factors that affect financial consumption choices. The proposed conceptual framework views financial consumption as a complex multidimensional phenomenon that incorporates different stages of the consumption process, relationships between customers and their service providers as well as the broader sociocultural context, calling for attention to sociocultural aspects of financial consumption and providing suggestions for future research. It calls for attention to the bigger picture -not only what and how consumers buy and whether they make repeat purchases or not, but how they use the purchased financial products and services, how they build and maintain relationships with their financial service providers, and finally what are the broader sociocultural influences on the financial service consumption.

  . The other factors that affect technology acceptance are consumer innovativeness (Aldás-Manzano et al. 2009), information and guidance provided by the bank (Laukkanen & Kiviniemi 2010), service quality (Yap et al. 2010), perceived usefulness, social norms and social risk (Riquelme & Rios 2010), tradition and dispositional resistance to change (Chemingui & Iallouna 2013; Mzoughi & M'Sallem 2013), technology readiness and demographic characteristics (Yousafzai & Yani-de-Soriano 2012). Most of the studies have been conducted in local contexts using survey research methodology and do not account for sociocultural factors, which sometimes leads to confusing and contradictory results. For instance, a study conducted in Singapore place brick-and-mortar relationships as the central influence on the consumers' adoption decision (e.g. Chiou & Shen

  ; Keisidou 2013; O'Loughlin & Szmigin 2006; Elsharnouby & Parsons 2010).

 Marketing,16(5),


Christophe Chesneau
email: christophe.chesneau@gmail.com

Proposal of a modified Clayton copula: Theory, properties and examples

Keywords: Clayton copula, Archimedean copula, correlation measures, Gaussian distribution 2010 AMS Subject Classification: 60E05, 62E15, 62H99

The Clayton copula is a mathematical tool used in copula theory to model dependence between random variables. It is a notable member of the Archimedean copula family and is mainly known for its ability to capture tail dependence. In this article, we present a new modified variant of the Clayton copula that aims at improving its flexibility. The proposed modification scheme perturbs its Archimedean nature by integrating a bivariate product of logarithmic functions and an additional tuning parameter. The elaborated copula benefits from a more nuanced representation of the copula density, and the negative dependence can be reached in a regular manner. We investigate its properties, including limit results revealing some connection with the Gumbel-Barnett copula, important related functions, modifications, and extensions, various lower and upper bounds, diverse tail dependences, and the correlation properties through the medial correlation and the Kendall tau. As an example of probability application, a new modified bivariate Gaussian distribution is presented via equations and graphics.

Finally, two special copula cases are discussed, including a simple one with a single parameter that is designed to be a practical alternative to the Clayton copula. The overall finding contributes to advancing the theoretical foundations of copula-based modeling techniques.

Introduction 1.Context

In the field of statistical modeling, the copula concept has emerged as a powerful tool for characterizing dependence structures between random variables. It was introduced by Sklar in 1959 (see [START_REF] Sklar | Fonctions de repartition à n dimensions et leurs marges[END_REF]), and has gained importance over time in various fields such as finance, insurance, hydrology, genetics, neuroscience, environmental science, telecommunications, and reliability engineering. The basis of copula theory lies in the Sklar theorem, which states that any multivariate distribution can be decomposed into its marginal distributions and a copula function. This separation allows researchers and practitioners to independently analyze particular dependence behaviors, facilitating a more nuanced understanding of complex data sets. The established copulas are numerous, as evidenced by the books of [START_REF] Joe | Multivariate Models and Dependence Concepts[END_REF], [START_REF] Nelsen | An Introduction to Copulas[END_REF] and [START_REF] Sempi | Principles of Copula Theory[END_REF]. From the fundamental Gaussian copula, often used in financial modeling due to its simplicity, to the more complex copulas of the Archimedean family, like the Ali-Mikhail-Haq, Clayton, Gumbel, and Frank copulas, each differs in their features for capturing dependence structures and tail dependence characteristics. They provide a versatile toolbox for modeling different degrees of association, from linear to nonlinear and asymmetric relationships. We may refer to [START_REF] Nelsen | An Introduction to Copulas[END_REF] and, for a modern and global study, to [START_REF] Kularatne | On the use of Archimedean copulas for insurance modelling[END_REF]. Recent copula creation works include the diverse extended FGM copula families (see [START_REF] Bairamov | Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions[END_REF], [START_REF] Amini | Aspects of dependence in generalized Farlie-Gumbel-Morgenstern distributions[END_REF], and [START_REF] Domma | A copula-based approach to account for dependence in stress-strength models[END_REF])) and the variablepower copula families (see [START_REF] Chesneau | Theoretical validation of new two-dimensional one-variable-power copulas[END_REF] and [START_REF] Chesneau | Some new developments on variable-power copulas[END_REF]). Additionally, the introduction of vine copulas further expands this diversity by allowing the combination of simpler copulas to represent complex dependence structures in high-dimensional data (see [START_REF] Kurowicka | Dependence Modeling: Vine Copula Handbook[END_REF]).

On the Clayton copula

To comprehend the underlying motivation behind this article, a retrospective examination of the Clayton copula is imperative. To begin, it was created by Clayton in 1978 (see [START_REF] Clayton | A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence[END_REF]) and, as mentioned previously, it is one of the most famous Archimedean copulas (see again [START_REF] Joe | Multivariate Models and Dependence Concepts[END_REF], [START_REF] Nelsen | An Introduction to Copulas[END_REF] and [START_REF] Sempi | Principles of Copula Theory[END_REF]). By considering a parameter a > 0, the standard version of the Clayton copula is derived from the following strict generator function:

φ † (t) = t -a -1, t > 0,
and can be expressed as

C † (u, v) = φ -1 † [φ † (u) + φ † (v)] = u -a + v -a -1 -1/a , (u, v) ∈ [0, 1] 2 .
The coefficients of lower and upper dependence are λ l = 2 -1/a and λ u = 0, respectively, implying that the Clayton copula is lower tail dependent but not upper tail dependent. Furthermore, the Clayton copula converges to the independence copula when a → 0 + , and to the minimum copula C • (u, v) = min(u, v) when a → +∞. The Kendall tau has the unit range [0, 1]. Hence, it can only take into account the positive dependence. On the other hand, a well-known modified Clayton copula allowing the negative dependence can be presented. By considering a ∈ [-1, +∞)/{0}, it is derived from the following (non-strict) generator function:

φ ‡ (t) = t -a -1 a , t > 0,
and the related pseudo-inverse function φ

[-1] ‡ (t), i.e., φ [-1] ‡ (t) = φ -1 ‡ (t) = (at + 1) -1/a for t ∈ [0, φ ‡ (0)], and φ [-1] ‡ (t) = 0 for t > φ ‡ (0) (with φ ‡ (0) = -1/a for a ∈ [-1, 0)
), and can be expressed as

C ‡ (u, v) = φ [-1] ‡ [φ ‡ (u) + φ ‡ (v)] = max u -a + v -a -1 , 0 -1/a = (u -a + v -a -1) -1/a , (u, v) ∈ [0, 1] 2 such that u -a + v -a > 1, 0, elsewhere.
When a → -1 + , it tends to the maximum copula C (u, v) = max(u + v -1, 0). The negative dependence is achieved for a ∈ [-1, 0). In this case, this modified Clayton copula is nonregular; its support depends on a and corresponds to a restricted region, breaking the symmetry about the diagonal from the corner points (0, 1) to (1, 0). Furthermore, it is also proved that it cannot properly capture higher negative dependence (see [START_REF] Joe | Multivariate Models and Dependence Concepts[END_REF], [START_REF] Nelsen | An Introduction to Copulas[END_REF] and [START_REF] Sempi | Principles of Copula Theory[END_REF]).

The Clayton copula, in its standard or maximum modified form, has been at the heart of several important applied studies, including [START_REF] Darabi | Application of Clayton copula in portfolio optimization and its comparison with Markowitz mean-variance analysis[END_REF], [START_REF] El-Sherpieny | Bivariate generalized rayleigh distribution based on Clayton Copula[END_REF], [START_REF] Novianti | Application of Clayton copula to identify dependency structure of COVID-19 outbreak and average temperature in Jakarta Indonesia[END_REF], [START_REF] Schneider | Clayton copula for survival data with dependent censoring: An application to a tuberculosis treatment adherence data[END_REF] and [START_REF] Michimae | Likelihood inference for copula models based on left-truncated and competing risks data from field studies[END_REF]. It has also served as the main example to test an artificial intelligence system in [START_REF] Górecki | Pair programming with ChatGPT for sampling and estimation of copulas[END_REF], mainly because of its simple expression and all the deep computational knowledge around it.

In order to provide suitable alternatives with the same mathematical ingredients, several articles have focused on extensions of the Clayton copula by modifying its strict generator function. In particular, in [START_REF] Cooray | Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family[END_REF], the following weighted Clayton strict generator function is proposed:

φ (t) = t -a -1 a (1 -bt a ), t > 0,
under the following parameter conditions: b ∈ [0, 1], a ≥ 0, or b < 0 and b(a -1) + a + 1 ≥ 0. The associated Archimedean copula has a sophisticated expression (see [START_REF] Cooray | Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family[END_REF]Remark 2.2]), but it has the advantages of being regular and reaching negative dependence thanks to the combined action of the parameters a and b. It is shown in [START_REF] Cooray | Strictly Archimedean copulas with complete association for multivariate dependence based on the Clayton family[END_REF]Subsection 3.4] that the corresponding Kendall tau has the optimal range [-1, 1]. In the same vein, a more simple approach was studied in [START_REF] Chesneau | Parametric extensions of some referenced two-dimensional strict Archimedean copulas[END_REF]. It consists of considering the following parameter-extended Clayton strict generator function:

φ (t) = (at -b -1) c -(a -1) c , t > 0,
under the following parameter conditions: b > 0, c > 0 and a ≥ max[1, (b + 1)/(bc + 1)]. The corresponding Archimedean copula is expressed as

C (u, v) = φ -1 [φ (u) + φ (v)] = a 1/b [(au -b -1) c + (av -b -1) c -(a -1) c ] 1/c + 1 -1/b , (u, v) ∈ [0, 1] 2 .
This copula is regular, and it can reach the negative dependence thanks to the combined action of the parameters a, b, and c. For some parameter-value tests, it is shown in [3, Tables 1 and2] that the corresponding Kendall tau has the range [-0.31, 0.75], but a more large numerical analysis suggests the range [-0.31, 1]. The two above copulas thus provide interesting alternatives to the Clayton copula because of their regularity and possible negative dependence. However, this is achieved with the direct modification of the generator function and the manipulation of several parameters.

Contributions

In this article, we develop an original approach to extend the standard Clayton copula beyond the modification of the corresponding strict generator function. We aim to determine a simple bivariate function ψ(u, v) such that

C(u, v) = u -a + v -a + ψ(u, v) -1 -1/a , (u, v) ∈ [0, 1] 2 , (1)
is a valid copula that is regular, covers both negative and positive dependence, and depends on a reasonable number of parameters. It can also be expressed as

C(u, v) = φ -1 † [φ † (u) + φ † (v) + ψ(u, v)] or C(u, v) = [C † (u, v)] -a + ψ(u, v) -1/a .
In this way, we intentionally perturb the Archimedean structure inherent in the standard Clayton copula while retaining its fundamental mathematical components in an effort to broaden its modeling capabilities. Our approach considers ψ(u, v) as a product of logarithmic functions modulated by a single tuning parameter, strategically designed to establish a connection with the Gumbel-Barnett copula, a well-known copula exhibiting negative dependence (refer to [START_REF] Nelsen | An Introduction to Copulas[END_REF]). We first establish the conditions on the parameters that make the new copula valid. Subsequently, we conduct a comprehensive examination of its properties, encompassing limit results, associated functions, random number generation, natural extensions, exhaustive bounds, tail dependences, correlation measures elucidating both negative and positive dependences, and bivariate distribution generation with an example related to the bivariate Gaussian distribution. We supplement the analysis with pertinent figures and numerical tables. Additionally, we showcase two practical versions of the copula. In light of the widespread applications of the Clayton copula, we anticipate a similar perspective on our proposed modification. The applied aspect is, however, left to future work.

Article organization

The rest of the article is as follows: Section 2 describes the proposed Clayton copula. Section 3 is devoted to its main properties. Some special cases are discussed in Section 4. A conclusion is provided in Section 5.

Modified Clayton copula

Let us present the copula concept in the standard bivariate absolutely continuous setting, as recalled below (see [START_REF] Nelsen | An Introduction to Copulas[END_REF]).

Definition 1 Under the bivariate absolutely continuous setting, the following definition of a copula is adopted: A copula is a function defined on [0, 1] 2 , say A(u, v), (u, v) ∈ [0, 1] 2 , differentiable on (0, 1) and satisfying the two assumptions below.

(a) A(u, 1) = u, A(1, v) = v, A(u, 0) = A(0, u) = 0, (b) ∂ 2 ∂u∂v A(u, v) ≥ 0.
Throughout the article, the bivariate absolutely continuous setting will be taken into consideration. Mathematically speaking, proving assumption (b) can be more difficult than

proving assumption (a); it may require extensive mathematical developments (tedious differentiations, factorizings, inequalities, etc.).

In light of Definition 1 and the copula construction elucidated in Equation ( 1) with a judicious function ψ(u, v), the proposition below introduces our modified Clayton copula.

Proposition 2.1 Let us consider the following bivariate function:

C(u, v) = u -a + v -a + b log(u) log(v) -1 -1/a , (u, v) ∈ [0, 1] 2 . ( 2
)
Then it is a copula for a > 0 and a(a + 1) ≥ b ≥ 0.

Proof. The proof is based on the verification of (a) and (b) in Definition 1. First, let us focus on (a). For any u ∈ [0, 1], since log(1) = 0, we immediately have

C(u, 1) = u -a + 1 -a + b log(u) log(1) -1 -1/a = (u -a ) -1/a = u.
By using a similar approach, for any v ∈ [0, 1], we also obtain

C(1, v) = v.
On the other hand, for any u ∈ [0, 1], since a > 0, we have lim v→0 + v a log(v) = 0. This implies that

C(u, 0) = lim v→0 + u -a + v -a + b log(u) log(v) -1 -1/a = lim v→0 + v u -a v a + 1 + bv a log(u) log(v) -v a -1/a = lim v→0 + v × (0 + 1 + 0 -0) -1/a = 0.
With a similar rationale, for any v ∈ [0, 1], we obtain C(0, v) = 0. As a result of the above developments, (a) is fulfilled.

Let us now consider (b). For any (u, v) ∈ (0, 1) 2 , through the use of several differentiation rules and appropriate factorizings, we establish that

∂ 2 ∂u∂v C(u, v) = 1 a 2 [bu a v a log(u) log(v) + u a (1 -v a ) + v a ] 2 × u a-1 v a-1 u -a + v -a + b log(u) log(v) -1 -1/a [aJ(u, v) + K(u, v)] ,
where

J(u, v) = a 2 + bu a (v a -1) -(a + 1)bu a log(v) -bv a + a and K(u, v) = -bv a log(u) a 2 -bu a log(v) + a .
Therefore, given a > 0, proving J(u, v) ≥ 0 and K(u, v) ≥ 0 are sufficient to establish

∂ 2 C(u, v)/(∂u∂v) ≥ 0, i.e., (b).
Since a > 0 and b ≥ 0, we have -bv a log(u) ≥ 0, a 2 ≥ 0 and -bu a log(v) ≥ 0, which imply that K(u, v) ≥ 0. On the other hand, under the assumptions a > 0 and a(a

+ 1) ≥ b ≥ 0, since b(1 -u a )(1 -v a ) ≥ 0 and -(a + 1)bu a log(v) ≥ 0, we have J(u, v) = a 2 + bu a v a -bu a -bv a + a -(a + 1)bu a log(v) = a(a + 1) -b + b(1 -u a )(1 -v a ) -(a + 1)bu a log(v) ≥ a(a + 1) -b ≥ 0.
Hence, (b) is satisfied, ending the proof.

Let us call the copula in Equation ( 2) the logarithmic-modified Clayton (LMC) copula. It corresponds to the copula described in Equation ( 1) with

ψ(u, v) = b log(u) log(v).
Clearly, by taking b = 0, it corresponds to the standard Clayton copula and, for any b ≥ 0, the following copula ordering holds: For any (u, v)

∈ [0, 1] 2 , since b log(u) log(v) ≥ 0, we have C(u, v) ≤ C † (u, v).
This inequality implies that certain correlation measures associated with the LMC copula are smaller than those of the Clayton copula, among other considerations. Further exploration of this assertion will be conducted in greater detail in Subsection 3.5. While the conditions a > 0 and a(a + 1) ≥ b ≥ 0 may seem stringent due to the interdependence of a and b, practical alternatives exist, such as b = ac with c ∈ [0, 1], or b = a(a + 1). These specific configurations will be discussed in Section 4.

In order to have a representative idea of the LMC copula, Figure 1 displays it for some values of a and b satisfying the required assumptions. The package plotly of the R software was used in this regard (see [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF] and [START_REF] Sievert | Interactive Web-Based Data Visualization with R, plotly, and shiny[END_REF]). This illustrates the validity of the copula on the one hand and its versatility on the other.

(i) (ii) (iii) (iv)
In the rest of the study, we will investigate the main properties of the LMC copula, and discuss how it contributes to the developments on modified Clayton copulas.

Properties

In this section, a focus is put on limit results, related functions, number generation, copula bounds, tail dependence, certain correlation measures, and a new bivariate Gaussian distribution, all derived from the LMC copula.

Limit results

Some copula limits on the LMC copula are now examined. The next proposition investigates the limit of the LMC copula when a → 0 + and an additional assumption on b.

Proposition 3.1 Let (u, v) ∈ [0, 1]
2 and C(u, v) be the LMC copula with a > 0 and a(a + 1) ≥ b ≥ 0. Suppose that lim a→0 + b = 0 and lim a→0 + b/a = c with c ∈ [0, 1], then we have

lim a→0 + C(u, v) = C (u, v), where C (u, v) = uv exp [-c log(u) log(v)]
is the Gumbel-Barnett copula with parameter c (see [START_REF] Nelsen | An Introduction to Copulas[END_REF]). When c = 0, it is reduced to the independence copula.

Proof. By the basic properties of a copula, the result is immediate for the corner points (0, 0), (0, 1), (1, 0) and (1, 1), so let us consider (u, v) ∈ (0, 1) 2 . Owing to the equivalence exp(t) ∼ 1 + t when t → 0, we get

lim a→0 + C(u, v) = lim a→0 + u -a + v -a + b log(u) log(v) -1 -1/a = lim a→0 + exp - 1 a log {exp[-a log(u)] + exp[-a log(v)] + b log(u) log(v) -1} = lim a→0 + exp - 1 a log [1 -a log(u) -a log(v) + b log(u) log(v)] .
Using the limit assumptions lim a→0 + b = 0, it is clear that lim a→0 + a log(u) + a log(v)b log(u) log(v) = 0. This, combined with the equivalence log(1 + t) ∼ t when t → 0, gives

lim a→0 + C(u, v) = lim a→0 + exp log(u) + log(v) - b a log(u) log(v) = uv lim a→0 + exp - b a log(u) log(v) .
The assumption lim a→0 + b/a = c with c ∈ [0, 1] implies that

lim a→0 + C(u, v) = uv exp [-c log(u) log(v)] = C (u, v
), which ends the proof.

These limit results are sufficient to justify the study of the LMC copula. Indeed, it means that, for a and b/a closed to 0, the LMC copula can have the behavior of the Gumbel-Barnett copula, and it is known that the Gumbel-Barnett copula can reach the negative dependence (with a Kendall tau of the range [-0.361, 0]). Therefore, under some values of the parameters, it is expected that the LMC copula attains both negative and positive dependences. This comment will be illustrated numerically in Subsection 3.5.

The proposition below shows that the limit property of the standard Clayton copula for a → +∞ still holds for the LMC copula. Proposition 3.2 Let (u, v) ∈ [0, 1] 2 and C(u, v) be the LMC copula with a > 0 and a(a + 1) ≥ b ≥ 0. Then we have

lim a→+∞ C(u, v) = C • (u, v). We recall that C • (u, v) = min(u, v).
Proof. By the basic properties of a copula, the result is immediate for the corner points (0, 0), (0, 1), (1, 0) and (1, 1), so let us consider (u, v) ∈ (0, 1) 2 . We have

lim a→+∞ C(u, v) = lim a→0 + u -a + v -a + b log(u) log(v) -1 -1/a = lim a→+∞ exp - 1 a log {exp[-a log(u)] + exp[-a log(v)] + b log(u) log(v) -1} .
For any a(a + 1) ≥ b ≥ 0, even if b depends on a (such as b = a(a + 1) at the maximum order with respect to a), for a → +∞, since u -a = exp[-a log(u)] → +∞ and v -a = exp[-a log(v)] → +∞ with an exponential rate, we have

u -a + v -a + b log(u) log(v) -1 ∼ u -a + v -a ∼ [min(u, v)] -a = [C • (u, v)] -a .
Therefore, we have

lim a→+∞ C(u, v) = lim a→+∞ exp - 1 a log [C • (u, v)] -a = C • (u, v).
The proof is achieved.

Hence, the important min-copula limit property of the Clayton copula is conserved for the LMC copula.

These comprehensive results are advantages for the LMC copula, among other properties to be examined in the next sections.

Related functions

Several important functions derived from the LMC copula are now described.

Useful functions

The LMC copula density is a hidden ingredient of the proof of (b) in Proposition 2.1; it is given by

c(u, v) = ∂ 2 ∂u∂v C(u, v) = 1 a 2 [bu a v a log(u) log(v) + u a (1 -v a ) + v a ] 2 × u a-1 v a-1 u -a + v -a + b log(u) log(v) -1 -1/a × a a 2 + bu a (v a -1) -(a + 1)bu a log(v) -bv a + a -bv a log(u) a 2 -bu a log(v) + a , (u, v) ∈ [0, 1] 2 ,
with the limit values at the corner points (0, 0), (0, 1), (1, 0), and (1, 1). In full generality, the copula density is of interest because the more flexible the copula density, the more the related copula model is adaptable to versatile dependence structures. Due to the functional complexity of c(u, v), a graphical study is preferable to an analytical study. Thus, Figure 2 displays it for some values of a and b, satisfying the required assumptions. On the other hand, the two conditional LMC copulas are given by

(i) (ii) (iii) (iv)
C 1 (u, v) = ∂ ∂u C(u, v) = - 1 au b log(v) -au -a u -a + v -a + b log(u) log(v) -1 -1-1/a (3)
and

C 2 (u, v) = ∂ ∂v C(u, v) = C 1 (v, u) = - 1 av b log(u) -av -a u -a + v -a + b log(u) log(v) -1 -1-1/a , (u, v) ∈ [0, 1] 2 . ( 4
)
These functions are useful for simulating a couple of values (u ι , v ι ) from a random vector (U, V ) having the LMC copula as a cumulative distribution function. The standard simulation scheme is as follows:

• Simulate a couple of independent values (u ι , w), each from the uniform distribution over [0, 1].

• Determine numerically the value v ι satisfying the following nonlinear equation: w = C 1 (u ι , v ι ), where C 1 (u, v) is given by Equation (3).

• Consider (u ι , v ι ) as the generated couple of values.

For any positive integer n, we can repeat the process n times to have n couples of values from (U, V ). As an illustration, for four different parameter configurations on a and b, Figure 3 depicts n = 100 such simulated couples of values. The package rootSolve of the R software was used for the second step of the process (see [START_REF] Soetaert | rootSolve: Nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations[END_REF]). From this figure, we see the different kinds of dependence trends, with a more or less clear structure and with some specific clusters of points.

(i) (ii) (iii) (iv)
Simulated couples of values can also be used to test the behavior of diverse parametric estimation procedures by varying the number n; the larger n is, the more the procedures must be proved to be efficient. We leave this practical aspect for future studies.

On the other hand, the conditional copulas are also involved in the definition of correlation measures, such as the Kendall tau, as shown later.

Derived copulas

The LMC copula can be used to derive other new copulas with little effort. Based on some schemes presented in [START_REF] Nelsen | An Introduction to Copulas[END_REF], still under the assumptions a > 0 and a(a + 1) ≥ b ≥ 0, a short list of them is given below.

• We define the x-flipping LMC copula by

C(u, v) = v -C(1 -u, v) = v -(1 -u) -a + v -a + b log(1 -u) log(v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
• We introduce the y-flipping LMC copula by

C(u, v) = u -C(u, 1 -v) = u -u -a + (1 -v) -a + b log(u) log(1 -v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
• We define the survival LMC copula by

C(u, v) = u + v -1 + C(1 -u, 1 -v) = u + v -1 + (1 -u) -a + (1 -v) -a + b log(1 -u) log(1 -v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
All of them offer a new dependence model on a Clayton copula basis.

Another interesting scheme is the one elaborated in [START_REF] Durante | Construction of non-exchangeable bivariate distribution functions[END_REF]. It allows for flexibility and asymmetry of a given copula. In particular, with the use of power functions as in [10, Corollary 4], the following result holds: For any a > 0, a(a + 1) ≥ b ≥ 0, c ∈ [0, 1] and d ∈ [0, 1], we define a new copula by

C (u, v) = u c v d C(u 1-c , v 1-d ) = u c v d u -a(1-c) + v -a(1-d) + b(1 -c)(1 -d) log(u) log(v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
Clearly, for c = d, it is not diagonally symmetric. The parameters c and d inject more versatility into the LMC copula but complicate it on the mathematical plan; there is a risk of the over-parameterization phenomenon from a practical viewpoint.

Other copula-generated schemes are possible; the LMC copula may offer new possibilities for advancing research in this direction.

Copula bounds

As for any copula, the Fréchet-Hoeffding result holds (see [START_REF] Nelsen | An Introduction to Copulas[END_REF]). It ensures that the LMC copula satisfies the following copula inequalities:

C (u, v) ≤ C(u, v) ≤ C • (u, v), i.e., max(u + v -1, 0) ≤ u -a + v -a + b log(u) log(v) -1 -1/a ≤ min(u, v).
In fact, for the LMC copula, these general bounds can be improved under some additional assumptions on a and b. An improved lower bound that has a quadrant dependence interpretation is described below.

Proposition 3.3 Let (u, v) ∈ [0, 1] 2 and C(u, v) be the LMC copula with a > 0 and a 2 ≥ b ≥ 0. Then we have

C(u, v) ≥ uv.
Proof. For a > 0 and b ≤ a 2 , we have

u -a + v -a + b log(u) log(v) -1 ≤ u -a + v -a + a 2 log(u) log(v) -1 = u -a + v -a + log(u -a ) log(v -a ) -1.
Now, by using the inequality log(t) ≤ t -1 for t > 0, since u -a ≥ 1 and v -a ≥ 1, we have 0

≤ log(u -a ) ≤ u -a -1 and 0 ≤ log(v -a ) ≤ v -a -1, which imply that log(u -a ) log(v -a ) ≤ (u -a -1)(v -a -1)
. Therefore, we have

u -a + v -a + b log(u) log(v) -1 ≤ u -a + v -a + (u -a -1)(v -a -1) -1 = u -a v -a .
By raising the two sides at the negative power -1/a, we get

C(u, v) = u -a + v -a + b log(u) log(v) -1 -1/a ≥ (u -a v -a ) -1/a = uv.
The desired limit result is demonstrated.

It follows from Proposition 3.3 that, under the assumptions a > 0 and a 2 ≥ b ≥ 0, the lower bound C (u, v) is improved and that the LMC copula has the positive quadrant dependence property. This was known for the Clayton copula, i.e., b = 0, and now extends to the LMC copula with a 2 ≥ b ≥ 0.

Under a complementary assumption, the next result suggests an improved upper bound for the LMC copula.

Proposition 3.4 Let (u, v) ∈ [0, 1] 2 and C(u, v) be the LMC copula with a > 0 and a 2 ≤ b ≤ a(a + 1). Then we have

C(u, v) ≤ [C • (u, v)] -α + (1 -u a )(1 -v a ) -1/a , upper bound which is cleary smaller than C • (u, v).
Proof. For a > 0 and a 2 ≤ b ≤ a(a + 1), we have

u -a + v -a + b log(u) log(v) -1 ≥ u -a + v -a + a 2 log(u) log(v) -1 = u -a + v -a + log(u -a ) log(v -a ) -1.
Now, by using the inequality log(t) ≥ (t -1)/t for t > 0, since u -a ≥ 1 and v -a ≥ 1, we have log(u -a ) ≥ (u -a -1)/u -a ≥ 0, log(v -a ) ≥ (v -a -1)/v -a ≥ 0, implying that log(u -a ) log(v -a ) ≥ (u -a -1)(v -a -1)/(u -a v -a ). Therefore, we have

u -a + v -a + b log(u) log(v) -1 ≥ u -a + v -a + 1 u -a v -a (u -a -1)(v -a -1) -1 = u -a + v -a -u a -v a + (uv) a ≥ [min(u, v)] -α + 1 -u a -v a + (uv) a = [C • (u, v)] -α + (1 -u a )(1 -v a )
By raising the two sides at the negative power -1/a, we get

C(u, v) = u -a + v -a + b log(u) log(v) -1 -1/a ≤ [C • (u, v)] -α + (1 -u a )(1 -v a ) -1/a .
The stated upper bound is obtained.

The interest in the obtained bounds is mainly theoretical but contributes to the comprehension of the LMC copula.

Tail dependence

Diverse kinds of tail dependence in the LMC copula are now investigated. To begin, the lower left tail dependence parameter is given by

λ 1 = λ l = lim u→0 + C(u, u) u = lim u→0 + {2u -a + b[log(u)] 2 -1} -1/a u = 2 -1/a .
It is the same as the standard Clayton copula; the parameter b plays no role in it. Thus, the LMC copula is lower left tail dependent. In fact, it is the only tail dependence it has. Indeed, the lower right tail dependence parameter is obtained as

λ 2 = lim u→0 + u -C(1 -u, u) u = lim u→0 + u -[(1 -u) -a + u -a + b log(1 -u) log(u) -1] -1/a u = 0, since C(u, v
) is symmetric, the corresponding upper left tail dependence parameter is determined by

λ 3 = lim u→0 + u -C(u, 1 -u) u = λ 2 = 0
and the upper right tail dependence parameter is given by

λ 4 = λ u = lim u→1 - 1 -2u + C(u, u) 1 -u = lim u→1 - 1 -2u + {2u -a + b[log(u)] 2 -1} -1/a 1 -u = 0.
Thus, we have λ 2 = λ 3 = λ 4 = 0, supporting the claim.

Correlation measures

Let us now examine some correlation measures associated with the LMC copula, namely the medial correlation and Kendall tau. We again refer to [START_REF] Nelsen | An Introduction to Copulas[END_REF] for the details on these measures.

Medial correlation

First, the medial correlation of the LMC copula is indicated as follows:

M = 4C 1 2 , 1 2 -1 = 4 2 a+1 + b[log(2)] 2 -1 -1/a -1.
In particular, it is non-positive for a > 0 and

a(a + 1) ≥ b ≥ max 1 [log(2)] 2 1 + 2 2a -2 a+1 , 0 .
These conditions happen when a is small enough, which is consistent with the limit copula result described in Proposition 3.1. We illustrate this claim in Tables 1 and2 by calculating M for selected values of a and b satisfying a > 0 and a(a + 1) ≥ b ≥ 0. In fact, Table 2 focuses on the special case b = a(a + 1) and small values for a.

Table 1: Values of the medial correlation of the LMC copula for some values of a and b satisfying a > 0 and a(a + 1) ≥ b ≥ 0 (when it is not true, the cross symbol × is put). From these tables, based on the selected values of the parameters only, we find that M ∈ [-0.377, 0.812], but more numerical tests with large a extend this interval to [-0.377, 1].

× × × × × × × × a = 0.1 0.046 -0.15 -0.307 × × × × × × × × × × ×
This demonstrates the fact that the LMC copula can reach both negative and positive dependences. This is particularly true for the single-parameter LMC copula defined with b = a(a + 1) and a > 0, as observed in Table 2.

Kendall's tau

Based on C 1 (u, v) and C 2 (u, v) specified in Equations ( 3) and ( 4), respectively, the Kendall tau of the LMC copula has the following integral expression:

τ = 1 -4 1 0 1 0 C 1 (u, v)C 2 (u, v)dudv = 1 -4 1 0 1 0 1 a 2 uv b log(v) -au -a b log(u) -av -a × u -a + v -a + b log(u) log(v) -1 -2-2/a dudv.
The integrated term is too sophisticated from an analytical viewpoint to expect a nice expression. Nevertheless, based on the behavior of the medial correlation and the limit result in Proposition 3.1, it is plausible that this measure is negative for small values of a.

We support this claim in Tables 3 and4 with a numerical work; we determine τ for selected values of a and b satisfying a > 0 and a(a + 1) ≥ b ≥ 0. Table 4 focuses on the special case b = a(a + 1) and small values for a. For the considered parameter scenarios, we find that τ ∈ [-0.356, 0.778], but more numerical tests with large a extend it to τ ∈ [-0.356, 1]. Thus, the analysis of the Kendall tau also confirms that the LMC copula can reach both negative and positive dependences.

Again, this is particularly true for the single-parameter LMC copula defined with b = a(a+1)

and a > 0, as seen in Table 4.

The LMC Gaussian distribution

The LMC copula can be used to generate a plethora of bivariate distributions. In particular, by considering the LMC copula and the standard Gaussian distribution for the marginal distributions, we define a new bivariate Gaussian distribution by the following cumulative distribution function:

F (x, y) = C[Φ(x), Φ(y)] = [Φ(x)] -a + [Φ(y)] -a + b log[Φ(x)] log[Φ(y)] -1 -1/a , (x, y) ∈ R 2 ,
where Φ(x) denotes the cumulative distribution function of the standard Gaussian distribution, i.e.,

Φ(x) = 1 √ 2π x -∞ exp - u 2 2 du, x ∈ R.
Let us call the corresponding bivariate distribution the LMC Gaussian distribution. Based on the LMC copula density, the corresponding probability density function is obtained as

f (x, y) = ϕ(x)ϕ(y)c[Φ(x), Φ(y)] = ϕ(x)ϕ(y) a 2 {b[Φ(x)] a [Φ(y)] a log(u) log(v) + [Φ(x)] a {1 -[Φ(y)] a } + [Φ(y)] a } 2 × [Φ(x)] a-1 [Φ(y)] a-1 [Φ(x)] -a + [Φ(y)] -a + b log[Φ(x)] log[Φ(y)] -1 -1/a × a a 2 + b[Φ(x)] a {[Φ(y)] a -1} -(a + 1)b[Φ(x)] a log[Φ(y)] -b[Φ(y)] a + a -b[Φ(y)] a log[Φ(y)] a 2 -b[Φ(x)] a log[Φ(y)] + a , (x, y) ∈ R 2 ,
where ϕ(x) denotes the probability distribution function of the standard Gaussian distribution, i.e.,

ϕ(x) = 1 √ 2π exp - x 2 2 , x ∈ R.
In order to see the shape ability of this probability density function, Figure 4 depicts it for some values of a and b satisfying the required assumptions. From this figure, we see how the classical Gaussian bell shape is simultaneously skewed and deformed at its basis, with various weights on the tails. For the configuration (ii), we see a multimodal phenomenon, certainly caused by the large value of a, the reasonable value of b, and the combined presence of power and logarithmic functions. This versatility in shape makes the LMC Gaussian distribution of interest for further investigation in various bivariate modeling scenarios (bivariate noise, regression modeling, bivariate data analysis beyond the standard Gaussian case, etc.).

Special copulas

Two special cases of the LMC copula are now highlighted because of their practical designs.

Two-parameter practical LMC copula

By choosing b = ac with c ∈ [0, 1], which is possible since ac ≤ a ≤ a(a + 1), the LMC copula can be expressed as

C(u, v) = u -a + v -a + ac log(u) log(v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
It is of particular interest because a and c are totally independent, since lim a→0 + b = 0 and lim a→0 + b/a = c with c ∈ [0, 1], owing to Proposition 3.1, the Gumbel-Barnett copula with parameter c is obtained as a limit when a → 0 + , the Clayton copula is recovered with c = 0, and it can reach negative and positive dependences. This last claim is supported by Table 5. Despite a slight restriction on the parameter domain, this variant can be viewed as a practical version of the LMC copula because of the independence of the parameters and the fact that the main properties are conserved. Therefore, it is sufficiently motivated for various applications where the Clayton copula model is not the optimal choice.

Single-parameter LMC copula

A special single-parameter LMC copula has been sketched in several parts of the article. It is defined by choosing b = a(a + 1) with a > 0, and is expressed as

C(u, v) = u -a + v -a + a(a + 1) log(u) log(v) -1 -1/a , (u, v) ∈ [0, 1] 2 .
This version is of interest because, with the tune of only one parameter, since lim a→0 + b = 0 and lim a→0 + b/a = 1, owing to Proposition 3.1, the Gumbel-Barnett copula with parameter 1 is obtained as a limit when a → 0 + , and it can reach the negative and positive dependences as already demonstrated in Tables 2 and4. However, the Clayton copula cannot be recovered.

For this reason, it must be viewed as a real alternative with similar mathematical ingredients.

Conclusion

In conclusion, this article introduces a novel modification to the Clayton copula, deviating from its Archimedean nature to enhance flexibility. The proposed variant is characterized by an additional bivariate product of logarithmic functions and a tuning parameter. It offers a nuanced representation of the corresponding copula density. Regularity and negative dependence can be achieved simultaneously. The study explores the properties of the new copula, connecting it with the Gumbel-Barnett copula, elucidating various bounds and tail dependences, and examining correlation properties. Additionally, the article gives a probability application with a modified bivariate Gaussian distribution. Furthermore, two special copula cases are discussed. The findings contribute to the advancement of theoretical foundations in copula-based modeling techniques. Some possible perspectives on this work are formulated below.

• One can consider other choices for the function ψ(u, v) in Equation (1), perhaps of not constant sign for a higher level of perturbation of the Clayton copula.

• One can investigate a higher-dimensional version of the proposed copula, such as the trivariate variant indicated as C(u, v, w) = u -a + v -a + w -a + b log(u) log(v) log(w) -1 -1/a , (u, v, w) ∈ [0, 1] 3 , with a and b are two parameters in ranges of values to determine (even if the values of b are logically negative to make sense to the power -1/a).

• More generally, beyond the Clayton copula setting, based on a well-established strict generator function φ (t) and a well-chosen perturbation function ψ(u, v), one can explore the modeling horizons opened by copulas of the following form:

C (u, v) = φ -1 [φ (u) + φ (v) + ψ(u, v)] , (u, v) ∈ [0, 1] 2 .
These ideas need further examination, which we leave for future work.

Figure 1 :

 1 Figure 1: Contour plots of the LMC copula for (i) a = 1 and b = 2, (ii) a = 10 and b = 1, (iii) a = 0.1 and b = 0.05, and (iv) a = 1.5 and b = 2.

Figure 2 :

 2 Figure 2: Contour plots of the LMC copula density for (i) a = 1 and b = 2, (ii) a = 10 and b = 1, (iii) a = 0.1 and b = 0.05, and (iv) a = 1.5 and b = 2.

Figure 3 :

 3 Figure 3: Plots of n = 100 simulated couples of values from a random vector (U, V ) having the LMC copula as cumulative distribution function for (i) a = 1 and b = 2, (ii) a = 10 and b = 1, (iii) a = 0.1 and b = 0.05, and (iv) a = 1.5 and b = 2.

Figure 4 :

 4 Figure 4: Contour plots (left column) and shape plots (right column) of the LMC Gaussian probability density for (i) a = 1 and b = 2, (ii) a = 10 and b = 1, (iii) a = 0.1 and b = 0.05, and (iv) a = 1.5 and b = 2.

Remark 4 . 1

 41 The choice b = a 2 in the LMC copula is really different from the choice b = a(a + 1) because it does not permit the negative dependence. This can be deduced from Proposition 3.3, the inequality C(u, v) ≥ uv implying that M ≥ 0 and τ ≥ 0.

Table 2 :

 2 Values of the medial correlation of the LMC copula for some small values of a > 0 and b = a(a + 1).

	a	0.01	0.11	0.21	0.31	0.41	0.51	0.61	0.71	0.81	0.91

Table 3 :

 3 Values of the Kendall tau of the LMC copula for some values of a and b satisfying a > 0 and a(a + 1) ≥ b ≥ 0.

	b	0.0	0.05	0.1	0.2	0.5	0.9	1.1	1.3	1.5	2	2.5	3	4	5
	a = 7	0.778	0.778	0.778	0.778	0.777	0.777 0.776 0.776 0.776 0.775 0.775 0.774 0.773 0.772
	a = 2	0.5	0.498	0.496	0.491	0.478	0.462 0.453 0.445 0.438 0.418	0.4	0.382 0.349 0.317
	a = 1	0.333	0.324	0.314	0.296	0.244	0.183 0.154 0.127 0.101 0.041	×	×	×	×
	a = 0.5	0.2	0.17	0.142	0.09	-0.043	×	×	×	×	×	×	×	×	×
	a = 0.1 0.048 -0.142 -0.286	×	×	×	×	×	×	×	×	×	×	×

Table 4 :

 4 Values of the Kendall tau of the LMC copula for some small values of a > 0 and b = a(a + 1).

	a	0.01	0.11	0.21	0.31	0.41	0.51	0.61	0.71	0.81	0.91
		-0.356 -0.306 -0.259 -0.214 -0.171 -0.131 -0.092 -0.056 -0.021 0.013

Table 5 :

 5 Values of the Kendall tau of the LMC copula for some some values of a and b such that b = ac with a > 0 and c ∈ [0, 1].

	c	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
	a = 0.1 0.048 0.004 -0.036 -0.073 -0.108 -0.142 -0.173 -0.203 -0.232 -0.259 -0.286
	a = 0.4 0.167 0.134	0.103	0.074	0.047	0.020	-0.005 -0.029 -0.052 -0.075 -0.096
	a = 1	0.333 0.314	0.296	0.278	0.261	0.244	0.228	0.213	0.197	0.183	0.168
	a = 6	0.750 0.749	0.748	0.746	0.745	0.744	0.743	0.742	0.741	0.739	0.738

Funding

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Ethical approval

This article does not contain any studies with human participants performed by the author.

Conflicts of interest

The author declares no conflict of interest.