Duarte Leite

Pedro Campos
email: pcampos@fep.up.pt

Computational Results of Membership in R&D Cooperation Networks: To Be or Not To Be in a Research Joint Venture

Keywords: R&D, cooperation, networks, spillover, simulation, RJV

In this study, we analyze firms' membership in R&D (Research and Development) cooperation networks trough simulation methods. Our main research hypothesis is that the membership in cooperation networks is related to the degree of the knowledge spillover. The approach has two scenarios: cost symmetry and cost asymmetry. We first develop an analytical model with three stages: firstly, firms decide whether to participate in a cooperative research network; secondly they simultaneously choose the level of R&D output, and finally they choose the level of output. Then we proceed with computational simulations to verify our hypothesis. From our results, we were able to conclude that cooperation leads to an improvement on RJV firms' position in the market as they produce more than others with the same production conditions. Additionally, cooperating firms have to spend fewer resources on research, which turns the network a tremendous success on the productive efficiency level.

Introduction

In recent years, cooperation networks are one of the most appealing topics to study, gathering researchers from different scientific fields, such as biology, economy, management and computer science.

It is generally recognized that R&D activities have some public good features, as firms cannot fully appropriate the returns of their R&D investments, due to the existence of R&D spillovers. As a result, R&D expenditures are usually less than socially optimal. For this reason, R&D cooperation frequently emerges, so as to internalize spillovers.

Cooperation in R&D is usually identified with research collaboration and it is often investigated in the context of two-stage oligopoly models in which firms make their R&D decisions in a first pre-competitive stage and their quantity/price setting in a second stage. The most influential article on R&D cooperation is due to [START_REF] Aspremont | Cooperative and Noncooperative R&D in Duopoly with spillovers[END_REF], who assumed that there are spillovers in R&D output. Another prominent work is [START_REF] Kamien | Research Joint Ventures and R&D Cartels[END_REF], which proposed spillovers in R&D expenditures and allowed for different R&D organization models that may involve R&D expenditures cartelization and/or full information sharing. Since these starting articles, a lot of scientific models emerged around the topic of R&D cooperation, providing numerous extensions to those original models (e.g. [START_REF] Suzumura | Cooperative and Noncooperative R&D in an Oligopoly with Spillovers[END_REF] and [START_REF] Poyago-Theotoky | Equilibrium and Optimal Size of a Research Joint Venture in an Oligopoly with Spillovers[END_REF] on oligopolistic markets; [START_REF] Vonortas | Inter-firm cooperation with imperfectly appropriable research[END_REF] and [START_REF] Amir | One-Way Spillovers, Endogenous Innovator/Imitator Roles, and Research Joint Ventures[END_REF] on R&D spillovers; [START_REF] Cohen | Innovation and Learning: The Two Faces of R&D[END_REF] and [START_REF] Kamien | Meet me halfway: research joint ventures and absorptive capacity[END_REF] on absorptive capacity; [START_REF] Atallah | Research Joint Ventures Cartelization with Asymmetric R&D Spillovers[END_REF] and [START_REF] Atallah | Research Joint Ventures with asymmetric spillovers and symmetric contributions[END_REF] on spillovers asymmetry; [START_REF] Cellini | Dynamic R&D with Spillovers: Competition vs cooperation[END_REF] on welfare). In this study we analyze the membership and profitability of cooperation networks. Our main research hypothesis is that the membership in a cooperation network is associated with the degree of R&D spillovers.

This approach tries to extend existing literature on R&D cooperation networks by focusing on an oligopolistic scenario with asymmetric production costs and a previous membership stage. The work has been conducted in two parts, each of them divided in two steps. In the first part, we assume that firms are symmetric in what respects the marginal cost of production, R&D costs and the spillover outside the network. In the second part, we consider that firms may have different marginal costs of production. As for the steps, we propose developing (i) an analytical model, followed by (ii) a computational approach.

The main conclusions reached driven us to the fact that firms prefer R&D cooperation. Principally when there is cost asymmetry, we find that a network, for some level of spillover, would arise, containing some of the companies in the market. These companies benefit from higher profits and from R&D from other companies which reduced their production costs. The number of firms keeps stable as the firms inside the RJV did not allow new companies to enter. This may allow us to infer that cooperation networks play a great role on firms' interaction and benefit some of them by giving a way to improve efficiency of companies that joined the RJV through a positive impact on internal productive

Part I: R&D Cooperation under Symmetric Marginal Costs

By extending the model of [START_REF] Aspremont | Cooperative and Noncooperative R&D in Duopoly with spillovers[END_REF] to an oligopolistic scenario with partial-industry agreements between cost-symmetric firms, we develop a three-stage game where firms decide about entering in the R&D cooperation network, then about its R&D expenditures and afterwards they compete in the output market.

There are n firms that produce a homogeneous output, whose inverse demand function is given by: . bQ a P -=

where the parameter a captures the market size and b is a constant. Q is the total output given by the expression:

∑ = = n i i q Q 1 (a, b>0 and Q ≤ a/b) . ( 2
)
As it is typical in R&D cooperation models (e.g. [START_REF] Aspremont | Cooperative and Noncooperative R&D in Duopoly with spillovers[END_REF]), we assume that R&D output is cost reducing through an additive formulation, that is:

.

∑ ≠ - - = n i j j i i i x x c β α (3)
where c i is firm i's marginal cost of production, i α accounts for stand-alone marginal costs (0 < i α < a), x i measures firm i's R&D output and β is the spillover (0 ≤ β ≤ 1).

Additionally, and as in [START_REF] Aspremont | Cooperative and Noncooperative R&D in Duopoly with spillovers[END_REF], we consider a specific functional form for the R&D cost function:

C(x i ) = 0,5γx i ² . ( 4
)
where γ > 0 represents the R&D cost. The profit of firm i is then given by:

) ( ) ( i i i i x C q c P - - = π = . ) ( 5 , 0 ) ( 2 1 i i n i j j i i n j j x q x x q b a γ β α - + + - - ∑ ∑ ≠ = (5)
where q i is firm i's output. The game is organized in three stages, as follows:

(i) In the first stage (membership stage), firms decide whether to participate (or not) in a cooperative research network. For simplicity, we will assume that within the cooperative network, the degree of information sharing is set at its maximum level (β=1), a structure known as the Research Joint Venture (RJV) [START_REF] Kamien | Research Joint Ventures and R&D Cartels[END_REF]. Additionally, we will consider that insiders can obstruct the entry of an additional firm if it reduces their profits, while an outside firm will join the RJV only if it increases its profits. The conditions are defined according to those usually adopted in the literature (e.g. [START_REF] Poyago-Theotoky | Equilibrium and Optimal Size of a Research Joint Venture in an Oligopoly with Spillovers[END_REF]; [START_REF] Atallah | Information sharing and the stability of cooperation in research joint ventures[END_REF]): the conditions are the following, for a RJV of size m:

) 1 ( ) ( - ≥ m m m i m i π π . ( 6
) ) 1 ( ) ( - ≥ -m m m n i m i π π . ( 7
) ) 1 ( ) ( + ≥ m m m i m i π π or ) 1 ( ) ( + ≥ - m m m i m n i π π or both . (8)
where ߨ (t) represents the profit of an insider and ߨ ି (t) the profit of an outsider when the RJV is of size t. ߨ (m) represents the average profit of firms in network of size m and ߨ (m-1) represents the average profit of firms in network of size m, not taking into account one specific firm.

(ii) In the second stage (development stage), firms simultaneously choose the level of R&D output, independently or under cooperation. If firms cooperate, then they will coordinate R&D output in order to maximize their joint profits.

(iii) At last, in the production stage, firms simultaneously choose the level of output through Cournot competition.

Due to the complexity of the analytical solution for the equilibrium R&D output, we use numerical simulations to model the game described before. We were able to arrive to some interesting conclusions regarding the behaviour of companies in terms of network formation. Simulation results are presented in Table 1. Firstly, we may observe that all firms outside the RJV want to make part of the network and, simultaneously, insiders do not obstruct the entry of those additional firms. There is stability when all firms are inside the network and therefore there is no entrance or exit of firms to and from the RJV. Whatever is the number of companies in the market and for every spillover levels, companies will sooner or later enter in the RJV and there will be a huge cooperation network between all companies in the market, after interaction starts.

From our computational exercise, we first observe that there is a direct relationship between the R&D spillover outside the network (β) and firms' profit (Figure 1). In fact, we may conclude that an increase of the R&D spillover outside the RJV will make firms to benefit from other firms' knowledge, and, therefore, to increase its profits. And this is true for both RJV and non-RJV firms, while in the first case, profits are higher due to a maximum spillover among cooperating firms. A correlation coefficient of 0,997 between the spillover and firms' profit was found to be statistically significant. Additionally, when we focus on the R&D investment, we observe that it is higher for non-cooperating firms when compared with cooperating firms, except for a high degree of information sharing among non-cooperating firms (Figure 2). At the same time, we observe that for non cooperating firms, there is an inverse relationship

Profits

Beta

RJV Firms

Non-RJV Firms between the level of knowledge spillover and the investment in R&D. This result is rather intuitive: higher degree of information sharing means lower appropriateness of R&D efforts and, therefore, lowers R&D investments. We can also see that whenever we repeat the simulation for different number of firms in the market (n), we observe that as n increases, the R&D output decreases. This fact can be explained by the inverse relationship between the R&D output and the spillover. Therefore, as the number of firms in the market increases the R&D output decreases due to the fact that the spillover is greater for larger number of firms in the market and the need for R&D output is inferior.

Part II: R&D Cooperation under Asymmetric Marginal Costs

The results given in the previous simulation were very interesting and gave some insights on the behaviour of firms. However, it seems worthwhile to consider a more appealing framework where there is cost asymmetry between firms, giving literature a better perspective through simulation.

So, now we considered both equations 1, 2 and 3, that have been introduced in part I, while equation (3') captures the existence of diverse marginal costs of production between firms:

.

∑ ≠ - - = n i j j i i i x x c β α (3')
where α i accounts for stand-alone marginal costs (0 <α i < a), x i measures firm i's R&D output and the R&D spillover has the same properties, β ∈ [0, 1]. As written before, it will be assumed that there are diminishing returns to R&D expenditures, that is, C′( x i ) > 0 and C′′( x i ) > 0.

The profit function has changed so that it reflected the differences on marginal costs of production between firms:

Beta

R&D Investment

RJV firms

Non-RJV firms ) ( ) (

i i i i x C q c P - - = π = . ) ( 5 , 0 ) ( 2 1 i i n i j j i i n j j x q x x q b a γ β α - + + - - ∑ ∑ ≠ = (5')
The same three-stage game used in Part I is proposed: firms decide about cooperation membership, R&D and then about the output. However, and in order to find different structures and the responses of network formation to these types of frameworks, we assume three different decision criteria on the members' decision of letting non-members enter or not the network. Each one of these three options is used to find out how network formation can show different features by changing this condition. Following the literature (e.g. [START_REF] Poyago-Theotoky | Equilibrium and Optimal Size of a Research Joint Venture in an Oligopoly with Spillovers[END_REF]; [START_REF] Atallah | Information sharing and the stability of cooperation in research joint ventures[END_REF]), we add these new three criteria:

(1) The average profit inside the network increases with the entrance of a nonmember firm; (2) The profit of the firm that will enter is higher than the network's average profit;

(3) The profit of the firm with highest income (a kind of "leader" firm) remains the same or increases with the entrance of a new member. From the simulation considered in Part II we could observe several aspects that differ from the previous simulation in Part I. As regarding firms' cost asymmetry, it is possible to say that, in general, firms with low marginal production costs have higher profits, produce more output and are those who do more research, which means that they have higher R&D output. We then test some combinations of firms, with different marginal costs of production, in order to evaluate if there are some changes in the firms within the network. Not in all combinations but in most of them, network stability occurs for low levels of spillover. This means that when the spillover reaches a lower level, firms prefer to join the R&D network in order to benefit from a total share of research output and then have less production costs and so higher profits, contrasting with same background firms. The network advantages make possible firms with same marginal costs of production have different profits, as the ones inside the RJV reduce more their production costs due to R&D output spread between them. That is why their profits are normally superior.

A relevant fact is that some networks are formed by a mix of firms with better and worse production skills. Therefore, with some exceptions, cooperation takes place not only between firms with high or low efficiency, but they are formed by middle marginal cost companies or joint "extreme marginal cost" firms where the ones with the lowest production efficiency are the most benefited by cost reduction. On the profits issue, we can state that they tend to diminish with the fall of the spillover since the marginal production costs are not so reduced by efficiency achievements. Only if the firm belongs to the RJV then its profit increases in the first moment when the company joins the network. From the example in Table 2, where we show one of the simulation results, we see that firms that enter the RJV increase greatly their profits, while the others that have the same level of profits maintain that level and watch them decreasing when spillover increases. Mostly important, we also observe that firms with different levels of marginal costs (reflected in the different levels of profits), join together to form a strong and beneficial network. As we can see, from the three groups of firms here, we have in the network, for a spillover of 0.27, one firm from the first group and two from the third group. These results are confirmed by empirical literature. [START_REF] Cassiman | R&D Cooperation ans Spillovers: Some Empirical Evidence from Belgium[END_REF] study the effects of incoming spillovers and appropriability on having cooperation. Using data from Belgium they found that incoming spillovers have a positive and significant effect on the probability of firms cooperating. They also find that the higher appropriability, the higher the probability of network cooperation. However, they show that it may depend on the kind of partners firms deal with. In addition, some non-spillover determinants of cooperation are also examined, and they found that larger firms are more likely to cooperate, which is partially confirmed in our study as the results show that firms with lower marginal costs usually engage easily in cooperation.

Statistical tests were used in order to corroborate the relationships and the effects of the variables in the experiments. One-way ANOVA has been computed, considering the effect of several values of the marginal cost (here used as a factor/qualitative variable), over the profit of the firms. The overall null hypothesis has been rejected, meaning that different marginal costs produce different levels of profits in the firms. Post-hoc multiple comparisons tests (Tuckey HSD) have been performed and we were able to conclude that lower marginal costs are associated with higher profits.

We have also compared the profits between networked and non networked firms. The result of the Mann-Whitney test is that statistically significant differences exist between those kinds of firms: firms in networks have higher profits than those not in networks.

Concerning R&D output, we can notice that it has a propensity to increase as a result of a diminishing spillover (Table 3). Nevertheless, if some firms start to cooperate and their R&D output decreases, as a result of the full knowledge sharing benefit that the network agreement provides. As they will share all the research they make, they will not spend so many resources on R&D and then their private R&D output decreases. Continuing using the previous example, we see that R&D output decreases with the entrance in the network but it still increases if the spillover decreases outside the network. This happens because firms will have to invest more so that they can maintain themselves producing in the market, and also because their investment will almost not flow to other firms outside the network. Since the spillover is lower, which decreases free riding behavior, firms outside the network need to increase their R&D output. Finally, regarding the output or production level (q), we see that it depends on the type of firms existing in the network, more precisely, the asymmetry between firms' marginal cost of production. In fact, when there are firms with different levels of efficiency, those who are more productive normally tend to increase their output, while the less efficient ones see their output being reduced. However, this situation changes if a network arises: in this case, cooperating firms produce more than in a non-cooperation scenario, mainly if compared with other firms with the same parameter profile.

Concerning the interactions between network formation and stability, we know that companies inside the network have, in this algorithm, the last word on letting or not outside firms to enter in it. Three different entering criteria were defined that turned different numerical results mostly significant on network stability, as it was explained (Figure 3). Considering the above mentioned criteria, the second one, where outsiders enter the network if their initial profit is higher than the average profit, originates the highest frequency of networks. It contrasts with all other criteria, where the last criterion is the one with worse results in what respects network formation.

Generally it is possible to visualize which criterion is more flexible, making the entrance easier to new companies inside the RJV. As we can see in Fig. 3, the spillover necessary to build a network is higher for criterion 2, followed by criterion 1 and then 3, the least advantageous criterion. Considering the third criterion, only in four experiments a network was formed. On the other hand, in the experiments with the second criterion there is always a network formed. Therefore, as explained above, circumstances necessary to generate a RJV are less tough in the second criterion than for the other possible criteria.

Final Remarks

In this study, we analyzed the membership in R&D cooperation networks and its impact on some economic indicators. Our main research hypothesis was that the membership in cooperation networks is related to the degree of the knowledge spillover. We first developed an analytical model where we considered that production costs were symmetric between all companies in the market. We then used numerical simulations to verify our hypothesis. The conclusions obtained were that the profit of firms in the network is higher than the corresponding profit outside the network. We also observed that as the number of firms in the market increases the R&D output decreases due to the fact that the spillover is greater for larger number of firms in the market and the need for R&D output is inferior. However, all firms entered in the network independently of the environment they faced in the market. So, in order to get close to the reality, a new approach was attempted by introducing cost asymmetry. By doing this we could find that a network, for some level of spillover, would arise but without all companies entering it. Only some would be able to join together and benefit from R&D cooperation and then benefit from higher profits and from R&D of other companies which reduced their production costs. The number of firms kept stable as the firms inside the RJV did not allow new companies to enter, maintaining cooperation benefits just for them.

Nevertheless, networks also depended on how companies manage the RJV. The formation and maintenance of a network depended on what were the entrance and exit decisions defined by firms. As seen before, there were types of decisions that leaded to easier network formation and others did not, depending on what were the minimum requirements for companies to belong to the network. By comparing the two approaches we reached the conclusion that the second was more close to reality. In both cases the number of firms in the network kept stable but on the first model companies entered into the network regarding any kind of situation while in the second model there was entrance only for some levels of spillover and in only some circumstances making the simulation more alike to reality.

But, whatever the approach was, the gains to companies that joined the network were higher. And its impact on profits was showed vital to their performance on the model.

Fig. 1 .

 1 Fig. 1. Profits evolution with the spillover between non-cooperating firms (beta).

Fig. 2 .

 2 Fig. 2. R&D output evolution with the spillover between non-cooperating firms.

Fig. 3 .

 3 Fig. 3. Spillover value needed to form a network for each criterion.

Table 1 .

 1 Results of the simulation for n= 5 firms and for different values of Beta (β).

	Marginal cost=50 and R&D cost (γ)=50
	Beta (β)	-	0.2	0.4	0.6	0.81	0.91
	RJV Profits	48.90	49.58	50.09	50.42	50.58	50.59
	Non-RJV profits	24.75	24.92	25.06	25.17	25.25	25.28
	RJV R&D output	0.05	0.11	0.11	0.11	0.11	0.11
	Non-RJV R&D output	0.14	0.11	0.09	0.07	0.05	0.04
	Marginal cost=50 and R&D cost (γ)=90
	Beta (β)	-	0.2	0.4	0.6	0.81	0.91
	RJV Profits	49.34	49.75	50.05	50.25	50.35	50.35
	Non-RJV profits	24.85	24.95	25.03	25.10	25.15	25.17
	RJV R&D output	0.03	0.07	0.07	0.07	0.07	0.07
	Non-RJV R&D output	0.08	0.07	0.06	0.04	0.03	0.02

Table 2 :

 2 Firm's profits determined on the simulation experiments made.

	Firms

Table 3 :

 3 Firm's R&D output determined on the simulation experiments made.

	Firms


María-José Verdecho
email: mverdecho@cigip.upv.es

Juan-José Alfaro-Saiz

Raúl Rodríguez-Rodríguez

A Review of Factors Influencing Collaborative Relationships

Keywords: Collaboration, collaborative networks, conceptual issues

Collaboration is a term commonly used to refer to a type of interorganizational relationship. However, in real business assessments, many collaborative relationships fail due to the lack of understanding of the factors influencing collaboration sustainability. For this reason, enterprises, prior to engage to a collaborative relationship, need to understand further which the main factors influencing collaboration relationships are, how they are structured and how they interact so that decision makers that desire to engage in a collaborative relationship/network focus not only on improving performance indicators but also on the factors that influence the results of those performance indicators. The purpose of this paper is to present a critical literature review of factors influencing collaborative relationships in order to perform a comparative study of the works for identifying main strengths and gaps for future research.

Introduction

In the current complex business environment, enterprises adopt different strategies and business models to remain competitive in the marketplace. In order to be efficient, enterprises focus on their own core competencies and rely on other enterprises for developing activities that are related to non-core competences. In fact, enterprises engage in diverse types of relationships that differ in their scope and broadness ranging from transactional type (punctual relationship) to collaborative relationship (long-term relationship). Enterprises must know which type of relationship they need to engage as not all types of relationships are suitable for all the business purposes. For example, in [START_REF] Camarinha-Matos | Collaborative networked organizations -Concepts and practice in manufacturing enterprises[END_REF], it is presented a classification of four types of relationships among organizations depending on the degree of interaction: networking, coordinated networking, cooperation and collaboration. From a global perspective, collaboration relationships are the ones that involve the greatest degree of interaction and, for that reason, are the most complex. Collaboration involves two or more independent enterprises working together to align their processes with the goal of creating value to end customers and stakeholders with greater success than acting alone [START_REF] Simatupang | Applying the theory of constraints to supply chain collaboration[END_REF]. However, collaborative relationships may fail if they are not properly managed [START_REF] Sabath | The Unfulfilled Promise of Supply Chain Collaboration[END_REF][START_REF] Kampstra | Realities of supply chain collaboration[END_REF]. The Supply Chain Management Review and Computer Science Corporation [START_REF]The second annual global survey of supply chain progress[END_REF] performed a survey and observes that collaboration is cited as the single most important topic; but how to achieve it is not well understood. The survey showed that 44 percent of the organizations in the sample have functions specifically for supplier and customer collaboration. However, only about 35 percent of the collaboration initiatives turned out to be even moderately successful. As [START_REF] Kampstra | Realities of supply chain collaboration[END_REF] suggest, perhaps, it is because not all participants in every supply chain have embedded collaborative values. In [START_REF] Busi | Collaborative performance management: present gaps and future research[END_REF], it is indicated that the lack of understanding of collaboration structure and dynamics is considered the main cause of failure of collaborative initiatives. Then, it seems important that enterprises that desire to engage into a collaborative relationship understand the complexity of collaboration, which are the main factors influencing collaboration, how they are structured and how they interact so that so that decision makers that desire to engage in a collaborative relationship/network focus not only on improving performance indicators but also on the factors that influence the results of those performance indicators.

The aim of this paper is twofold. The first purpose is to provide a literature review of works that analyze factors influencing collaborative relationships (section 2). The second purpose is to perform a comparative study of the works reviewed for identifying main strengths and gaps for future research (section 3 and 4).

Literature Review

In this section, it is presented a review of works that deal with inter-organizational and collaborative relationships in order to analyze the factors that influence interorganizational relationships (in general) and collaborative relationships (in particular). It has to be noted that inter-organizational relationships embed all types of relationships, as collaborative relationships are the most complex type involving the greatest degree of interaction. From the literature review, a typology composed of three groups has been defined (see Fig. 1). The first group consists of works that identify factors influencing interorganizational relationships (type 1). The second group present works that propose conceptual frameworks of collaboration (type 2), and finally, the third group is composed by works that present empirical models of collaboration (type 3). The typology is then structured from a lower to a higher degree of interaction among factors (type 1 only describe some conceptual interaction among factors, type 2 structures the interaction among factors under a conceptual framework and type 3 structures the interaction among factors under a framework and quantifies their relationships empirically).

Works that identify Factors influencing Inter-organizational Relationships

There are numerous works in literature that deal with inter-organizational relationships. It is out of the scope of this paper to review all these works, but a selection has been performed with the aim of identifying the main factors influencing inter-organizational relationships. Within this group of the typology, we have considered two main subgroups of works. One subgroup consists of works that present classifications of inter-organizational environments according to the level of maturity reached in different aspects of their relationships, i.e. they present supply chain evolutionary models (from lower to higher level of collaboration). Within this subgroup, in [START_REF] Sabath | The Unfulfilled Promise of Supply Chain Collaboration[END_REF], it is presented a supply chain classification depending on two main aspects: strategic value of the relationship and technology used to support it. From a process perspective, in [START_REF] Lockamy | The development of a supply chain management process maturity model using the concepts of business process orientation[END_REF], it is developed a model to classify supply chains based on the maturity of their processes. Each level is characterized according to different factors such as alignment of processes, organizational structure, cooperation, process performance and trust. In [START_REF] Lejeune | On characterizing the 4 C's in supply chain management[END_REF], a typology for supply chain characterization related to Fiske [START_REF] Fiske | Relativity within Moose ("Mossi") culture: four incommensurable models for social relationships[END_REF] social relationships theory and the interdependence concept is exposed. In [START_REF] Danese | Collaboration forms, information and communication technologies, and coordination mechanisms in CPFR[END_REF], it is presented a classification of supply chains according to the presence of three factors in the CPFR (Collaborative Planning, Forecasting and Replenishment) process: collaboration form, ICT and coordination mechanisms. In [START_REF] Camarinha-Matos | Collaborative networked organizations -Concepts and practice in manufacturing enterprises[END_REF], organization relationships are classified into four groups depending on four types of factors: communication and information shared; complementary, compatible and/or joint objectives; activity alignment and joint responsibility. In addition, the authors highlight the need of measuring collaborative performance and importance of relationships attributes such as trust.

The second subgroup comprises works that identify the main factors influencing partnerships. In [START_REF] Birnbirg | Control in interfirm co-operative relationships[END_REF] it is considered five factors of inter-organizational relationships: degree of absolute and relative commitment, symmetry of rewards, extent of uncertainty, degree of mutual trust, and length of the relationship. [START_REF] Boddy | Implementing collaboration between organizations: an empirical study of supply chain partnering[END_REF] identifies seven factors for partnering contexts: business processes, people, trust, technology, structure, financial resources and culture. [START_REF] Handfield | power, dependence, and economics: can SCM research borrow paradigms[END_REF] analyzes eight factors from different disciplines of theory (marketing, economy, strategy and management) that may influence the evolution on supply chain management. The eight factors are: trust, power, dependency, economy, collaboration, assets, risk and communication.

From the review, we can group the key factors influencing inter-organizational relationships into: strategic value of the relationship and goal congruence; information shared; ICT and Information Systems (IS) role; joint decision-making, activities and problem solving; process approach and management levels; relationship attributes such as trust, commitment, interdependence and coordination; equity or symmetry of investments, risks and rewards; and performance measurement.

Conceptual Frameworks of Collaboration

In this work, a conceptual framework that characterizes collaboration is defined as 'the set of elements that organized and interrelated allow the representation of the collaboration relationships within inter-organizational environments in a generic manner'. [START_REF] Wilson | An integrated model of buyer-seller relationships[END_REF] proposes a conceptual framework that associates variables of relationships (commitment, cooperation, interdependence/power, technology, trust, mutual goals, etc.) to different stages in relationship process (partner selection, defining purpose, setting relationship boundaries, creating relationship value and relationship maintenance). It exposes the interaction among some of the variables describing how they may evolve when an event happens during the different stages. Nevertheless, the work could be further extended by including collaborative processes and performance measurement.

In [START_REF] Bowersox | How to Master cross-enterprise Collaboration[END_REF], it is developed a conceptual framework for cross-enterprise collaboration that identifies processes, competencies and capabilities. Key process performance objectives for each process are established. In this sense, the inclusion of performance objectives for the leadership process sustained by the 'relationships' competence is noticeable. However, the relationships competence does not consider important relational attributes e.g. trust.

[16] presents a conceptual framework structured in three levels: strategic, collaborative and cultural elements. The framework considers inter-organizational process alignment through the organizations, performance indicators for the whole SC, the connection between SC performance indicators and the cultural elements, and identifies technology as strategic factor. However, the connection among some of the elements is not explicit. For instance, there is no direct connection between performance indicators and joint decision-making.

Other works reviewed are [START_REF] Min | Supply chain collaboration: what's happening?[END_REF], [START_REF] Simatupang | An integrative framework for supply chain collaboration[END_REF], [START_REF] Burgess | A proposed integrated framework for analysing supply chains[END_REF], [START_REF] La Forme | A Framework to analyse Collaborative Performance[END_REF] and [START_REF] Giannakis | Performance Measurement of Supplier Relationships[END_REF]. From the review, we can conclude that although the reviewed conceptual frameworks present some aspects that can be extended, they are a foundation for understanding which are the main factors and interrelations influencing collaborative relationships. Therefore, these works facilitate to analyse further the connection of these collaborative factors although they do not use quantitative techniques that allow measuring the degree of influence among the different factors what is reviewed on the next type of works.

Empirical Models of Collaboration

The works presenting empirical models have gained attention in the literature in the last years as they test relationships among variables (factors) based on real data. [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF] develop a model of commitment-trust theory suggesting that trust and commitment are mediating variables between other factors such as cooperation, conflicts, communication, opportunism and benefits. [START_REF] Mohr | Characteristics of partnership success: Partnership attributes, communication behavior, and conflict resolution techniques[END_REF] identify from an empirical study the characteristics that contribute to successful partnerships: relationship attributes (coordination, commitment, and trust), communication behaviour (participation, and communication quality) and joint problem solving techniques.

The works [START_REF] Zineldin | An examination of the main factors affecting trust/commitment in supplier-dealer relationships: an empirical study of the Swedish wood industry[END_REF], [START_REF] Coote | An investigation into commitment in non-Western industrial marketing relationship[END_REF], [START_REF] Wu | The influencing factors of commitment and business integration on supply chain management[END_REF] and [START_REF] Kwon | Trust, commitment and relationships in supply chain management: path analysis[END_REF] expose models with trust as a key factor to increase commitment in relationships as well as a mediator between commitment and the other collaboration factors. The authors suggest that commitment improves with a higher degree of adaptation, shared values, communication, satisfaction and cooperation. [START_REF] Geyskens | The effects of trust and interdependence on relationship commitment: A trans-Atlantic study[END_REF] presents a model with interdependence, commitment and trust factors. This work details that there is two types of commitment, calculated and affective, and analyze to what extent they are influenced by interdependence and trust in the supply chain. [START_REF] Wu | The influencing factors of commitment and business integration on supply chain management[END_REF] also distinguish types of commitment: affective, continuity and normative. The work aims to study the impact of affective and continuity commitment in supply chain integration and increases coordination among partners. On the other side, normative commitment allows establishing objectives and shared values. Other works, [START_REF] Pimentel Claro | Coordinating collaborative joint efforts with suppliers: the effects of trust, transaction specific investment and information network in the Dutch flower industry[END_REF] and [START_REF] Cheng | Trust and knowledge sharing in green supply chains[END_REF] do not include in their works the commitment factor but they consider trust an essential factor of collaboration. [START_REF] Pimentel Claro | Coordinating collaborative joint efforts with suppliers: the effects of trust, transaction specific investment and information network in the Dutch flower industry[END_REF] analyze the impact of three factors: trust, ICT, and information network on the joint efforts (planning and conflict resolution). In [START_REF] Cheng | Trust and knowledge sharing in green supply chains[END_REF] trust is a mediator factor between other factors such as participation, communication, learning capability, and ability to share knowledge to finally obtain inter-organizational knowledge share. Other works reviewed include [START_REF] Matopoulos | A conceptual framework for supply chain collaboration: empirical evidence from the agri-food industry[END_REF] and [START_REF] Handfield | The role of trust and relationship structure in improving supply chain responsiveness[END_REF].

After analyzing the works it is observed that there is not a consensus reached among the authors regarding what are the main factors of collaborative relationships and which are the influences among them. In fact, there are some factors, e.g. trust that is modeled in two different manners (mediator variable and dependent variable). It has to be noted that the most modeled factors are trust, commitment and communication. Each work establishes different objectives for their models but the most commonly used dependent variable are joint action, trust and commitment. This means that this type of models mostly try to analyze the effects of the other collaborative factors on these three factors in order to achieve sustainable relationships.

Comparative Study

Table 1 presents a comparative study of the works reviewed. For that purpose, eight most commonly cited factors have been considered so that different perspectives of enterprises are covered: strategic, process, culture, communication infrastructure and performance measurement (management). This is because the works that integrate different enterprise perspectives constitute a sound basis for understanding collaboration relationship dynamics. The reviewed factors are: common strategy (S), collaborative processes (P), collaborative culture (in general) (CC), trust (T), commitment (C), information shared (IS), ICT and performance measurement (PM). The criteria established to compare the works ranges from absence of the factor (blank) to high degree of consideration (++).

Conclusions and research implications

From the table, the main conclusion obtained is that there is not a consensus regarding what are the main factors of collaborative relationships and which are the influences among them. First, the different groups present different insight into the factors. For example, type 2 and 3 provide more insight than type 1. However, type 2 and 3 differ on the factors treated. Type 2 deepens more on strategy, processes, information shared and ICT while type 3 deepens more on collaborative culture, trust, commitment and information shared. It has to be noted that other factors such as performance measurement are only treated in low detail in these works. Thus, one future research line is to analyse how performance measurement frameworks developed for collaborative enterprises can be connected to the conceptual frameworks and empirical models reviewed in this work to conceptualize further and measure factors influencing collaborative relationships. Second, the interrelations among collaborative elements identified by the different works of type 2 and 3 of the typology differ and only cover some of the factors. Even in the type 3, there are some factors, e.g. trust that is modeled in two different manners (mediator variable and dependent variable). Further research is needed to model all the factors influencing collaborative relationships, establishing the interactions among them and quantifying under the same model all the existing influences. It may seem that complex models do not capture the essence of what is analysed. However, simple models do not capture the dynamics of complex phenomena. Further research should encounter a balance of both aspects for the different situations.

Third, some interactions among factors have been analysed in more detail e.g. trust and commitment, while other interesting interactions have been less treated e.g. strategy and trust or strategy and performance measurement. Further research should consider these issues into their scope. Finally, it is important to note that it will be interesting to analyse combinations of all types of factors as all of them influence each other in a collaborative relationship.

Fig. 1 .

 1 Fig. 1. Relevant factors influencing collaborative relationships: a typology.

Table 1 .

 1 Comparative study of the works reviewed.

	Tip.	References	S	P	CC	T	C	IS	ICT	PM
		Birnbirg (1998)			+	++ ++		+	+
		Boddy et al. (2000)	+	+	+				+	+
		Sabath and Fontanella (2002)	++	+	+			+	++
		Handfield and Bechtel (2004)	+		+	++		++		+
	1	Lockamy and McCormack (2004)	+	++	+	+		+		+
		Lejeune and Yakova (2005)	+		+	++		++
		Danese (2006)	+		+			+	++
		Camarinha-Matos et al. (2009)	++	+	+	+	+	+	+	+
		Wilson (1995)	+		+	+	+	+	+	+
		Bowersox et al. (2003)	++ ++	+		+	++	++	+
		Barratt (2004)	+	++	+	+	+	+	+	+
		Min et al. (2005)	++ ++	+			++	+	+
	2	Simatupang and Sridharan (2005)	++ ++	+	+	+	++	++	+
		Burgess and Singh (2006)	++ ++	+	+		++	++	+
		Giannakis (2007)	++ ++	+	++ ++	+	+
		Gruat La Forme et al. (2007)	+	++	+			++		+
		Morgan and Hunt (1994)	+		++	++ ++ ++		+
		Mohr and Spekman (1994)			++	++ ++ ++		+
		Geyskens et al. (1996)			+	++ ++
		Zineldin and Jonsson (2000)	+		++	++ ++ ++		+
		Coote et al. (2003),			++	++ ++ ++
	3	Wu et al. (2004)	+	++	++	++ ++ ++
		Handfield and Bechtel (2002)	+		+	++		++		+
		Kwon and Suh (2005)			+	++ ++		++
		Pimentel et al. (2006)			+	++		++
		Matopoulos et al. (2007)	+	++	+	+		++	++
		Cheng et al. (2008)	+		+	++		++	+


domain: math stat

text:
Abdus Saboor
email: saboorhangu@gmail.com

Ibrahim Elbatal
email: iielbatal@imamu.edu.sa

Muhammad Nauman Khan

Gauss M Cordeiro
email: gausscordeiro@gmail.com

Rodrigo R Pescim
email: rrpescim@gmail.com

The Beta Exponentiated Nadarajah-Haghighi Distribution: Theory and Applications

Keywords: Beta distribution, Moment generating function, Maximum likelihood estimation, Monte Carlo simulation, Return level, Wind data. 2000 MSC: 60E05, 62E15, 62E20

We introduce and study the beta exponentiated Nadarajah-Haghighi model, which has increasing, decreasing, upside-down bathtub and bathtub shaped hazard functions. Some of its mathematical properties are determined including a power series for the quantile function. The model parameters are estimated by the method of maximum likelihood. The new distribution and some other competitive models are fitted to a meteorological data set. We prove empirically that it has a superior performance compared with other distributions based on some goodness-of-fit statistics. As a result, some parameters such as return level and mean deviation about the return level are estimated under the best model. We define a new parametric model based on the introduced distribution.

Introduction

The exponential distribution was the first lifetime model for which statistical methods were extensively developed in the life testing literature. In many applied sciences such as medicine, engineering and finance, among others, modeling and analyzing lifetime data are crucial. Several lifetime distributions have been used

to model these types of data, including the exponential, Weibull, gamma and Rayleigh distributions and their generalizations (see, e.g., [START_REF] Gupta | Generalized exponential distributions[END_REF][START_REF] Nadarajah | The beta exponential distribution[END_REF]). Each distribution has its own characteristics due specifically to the shapes of the hazard rate function (hrf), which can be monotonically decreasing or increasing, bathtub and unimodal.

Nadarajah and Haghighi [START_REF] Nadarajah | An extension of the exponential distribution[END_REF] introduced an extension of the exponential distribution as an alternative to the gamma, Weibull and exponentiated exponential (EE) distributions. The cumulative distribution function (cdf) of the Nadarajah-Haghighi (NH) distribution is given by

G(x) = 1 -e 1-(1+λx) α , x > 0, (1)
where λ > 0 and α > 0 are scale and shape parameters, respectively. The corresponding probability density function (pdf) and hrf are given by

g(x) = α λ (1 + λ x) α-1 e 1-(1+λx) α (2)
and h(x) = α λ (1 + λx) α-1 .

Equation [START_REF] Nadarajah | The beta exponential distribution[END_REF] has two parameters just like the gamma, Weibull and EE distributions. Note also that the NH model has closed-form survival and hrfs just like the Weibull and EE distributions. For α = 1, it reduces to the exponential distribution. For general properties about the NH model, the reader is referred to [START_REF] Nadarajah | An extension of the exponential distribution[END_REF]. The NH distribution has its mode at zero and allows for increasing, decreasing and constant hrfs.

Lemonte [START_REF] Lemonte | A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function[END_REF] proposed a three-parameter generalization of the NH distribution called the exponentiated NH (ENH) model, whose cdf is given by

G(x, α, λ, θ) = { 1 -e 1-(1+λx) α } θ , (3)
where the parameters α > 0 and θ > 0 control the shapes of the distribution and λ > 0 is the scale parameter.

The ENH density function is given by

g(x, α, λ, θ) = αλθ(1 + λx) α-1 e 1-(1+λx) α { 1 -e 1-(1+λx) α } θ-1 . (4)
The beta generalized family pioneered by Eugene et al. [START_REF] Eugene | Beta-normal distribution and its applications[END_REF] includes nearly all of well-known models as special or limiting cases such as those exponentiated distributions. It is a rich class of generalized distributions, which allows for greater flexibility of its tails and can be widely applied in many areas such as engineering, biology and medicine, among others. One major benefit of this family is its ability of fitting skewed data that cannot be properly fitted by existing distributions.

Several models have been investigated in the beta family in the last ten years. In fact, this class has been studied in the literature for some special baselines, among them, we cite the beta normal (BN) [START_REF] Eugene | Beta-normal distribution and its applications[END_REF], beta exponential (BE) [START_REF] Nadarajah | The beta exponential distribution[END_REF], beta generalized half-normal (BGHN) [START_REF] Pescim | The beta generalized half-normal distribution[END_REF], beta Weibull geometric (BWG) [START_REF] Cordeiro | The beta-Weibull geometric distribution[END_REF], beta generalized exponential (BGE) [START_REF] Souza | The beta generalized exponential distribution[END_REF], beta generalized Weibull (BGW) [START_REF] Singla | The Beta Generalized Weibull distribution:Properties and applications[END_REF], beta gamma (BG) [START_REF] Kong | On the Properties of Beta-Gamma Distribution[END_REF] and beta exponentiated Weibull (BEW) [START_REF] Cordeiro | The beta exponentiated Weibull distribution[END_REF] distributions.

The cdf of the beta-G class is given by

F(x) = 1 B(a, b) ∫ G(x) 0 w (a-1) (1 -w) b-1 dw = B(G(x); a, b) B(a, b) , (5)
where G(x) denotes the baseline cdf, a > 0 and b > 0 are two additional parameters,

B(a, b) = ∫ 1 0 t a-1 (1-t) b-1 dt is the beta function and B(z; a, b) = ∫ z 0 t a-1 (1 -t) b-1
dt is the incomplete beta function. The role of these parameters is to control skewness and vary tail weights of the generated model.

The pdf corresponding to (5) is given by

f (x) = 1 B(a, b) G(x) a-1 {1 -G(x)} b-1 g(x), (6)
where g(x) = d dx G(x). The density f (x) will be most tractable when G(x) and g(x) have simple forms. The rest of the paper is organized as follows. In Section 2, we define the beta exponentiated Nadarajah-Haghighi (BENH) distribution by using the NH model as baseline in the beta family. We show the flexibility of its hrf and present some special models. In Section 3, we study some of its structural properties, which include ordinary and conditional moments, mean deviations and Bonferroni and Lorenz curves. We derive a useful power series for the quantile function (qf) in Section 4. A Monte Carlo simulation study is performed in section 5. In Section 6, we obtain the maximum likelihood estimates (MLEs) of the unknown parameters.

In Section 9.1, we provide an application from meteorology data to prove empirically that the new model can yield better fits than some other known lifetime models. Finally, Section 10 offers some conclusions.

The BENH distribution

In this section, we introduce the five-parameter BENH distribution by taking G(x) in Equation ( 3) to be the cdf of the ENH distribution. Let φ = (λ, θ, α, a, b) be the parameters of the proposed model, where λ > 0, α > 0, θ > 0, a > 0 and b > 0. By using (3) in Equation ( 5), the cdf of the BENH distribution reduces to

F(x; φ) = B ( { 1 -e 1-(1+λx) α } θ ; a, b ) B(a, b) . (7)
Henceforth, if X is a random variable with cdf (7), we use the notation X ∼ BENH a,b (λ, θ, α). The pdf of X is given by

f (x; φ) = αθλ B(a, b) (1 + λx) α-1 e 1-(1+λx) α { 1 -e 1-(1+λx) α } θ a-1 [ 1 - { 1 -e 1-(1+λx) α } θ ] b-1 . (8)
Clearly, the sequential double construction beta class→exponentiated class→NH model yields the BEHN density function [START_REF] Cordeiro | The beta-Weibull geometric distribution[END_REF]. The derivations of some properties of the BEHN model can be facilitated by this construction. We eliminate sometimes the dependence of the parameters and write

F(x) = F(x; φ), f (x) = f (x; φ), etc.
The hazard rate function of X follow from equations ( 7) and [START_REF] Cordeiro | The beta-Weibull geometric distribution[END_REF] as

h(x) = f (x)/[1 -F(x)].

Shapes of the density and hazard functions

The BENH distribution given by ( 8) is much more flexible than the ENH distribution since it allows for greater flexibility of the tails. In fact, it can approach different distributions when its parameters change.

Special models of the BENH distribution

The BENH a,b (λ, θ, α) model contains as special cases some well-known distributions. The BENH 1,1 (λ , θ , α) model is the ENH [START_REF] Lemonte | A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function[END_REF], BENH a,b (λ, θ, 1) is the beta generalized exponential (BGE) distribution [START_REF] Souza | The beta generalized exponential distribution[END_REF],

BENH a,b (λ, 1, 1) is the beta exponential (BE) distribution pioneered by [START_REF] Nadarajah | The beta exponential distribution[END_REF], BENH 1,1 (λ, 1, α) is identical to the NH model [START_REF] Nadarajah | An extension of the exponential distribution[END_REF] and BENH 1,1 (λ, θ, 1) coincides with the generalized exponential (GE) distribution [START_REF] Gupta | Discriminating between the Weibull and the GE distributions[END_REF].

Simulation

The BEHN distribution is easily simulated from [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] as follows: if v is a generated beta variate with shape parameters a and b, then

x = λ -1 { [ 1 -log ( 1 -v 1/θ )] 1/α -1 } (9)
is a generated variate having density (8).

Linear representation

Equations ( 7) and ( 8) are straightforward to be evaluated using any software with algebraic facilities. In this section, we present a linear representation for the BENH density in terms of EHN densities.

If | z |< 1 and b > 0 is real non-integer, the power series holds

(1 -z) b-1 = ∞ ∑ j=0 (-1) j Γ(b) Γ(b -j) j! z j , (10)
where Γ(•) is the gamma function. By using [START_REF] Souza | The beta generalized exponential distribution[END_REF], the pdf (8) can be rewritten as

f (x; φ) = αλθ B(a, b) ∞ ∑ j=0 (-1) j Γ(b) Γ(b -j) j! e 1-(1+λ x) α (1 + λx) α-1 { 1 -e 1-(1+λx) α } θ (a+j)-1
and then

f (x; φ) = ∞ ∑ j=0 v j f ENH (x; α, λ, (a + j)θ), ( 11
)
where f ENH (x; α, λ, (a + j)θ) denotes the ENH density with parameters α, λ and (a + j)θ and the coefficient v j (for j ≥ 0) is given by

v j = (-1) j Γ(b) (a + j) j!Γ(b -j)B(a, b) .
Equation ( 11) is the main result of this section. It reveals that the BENH density is an infinite mixture of ENH densities with parameters α, λ and (a + j)θ (for j ≥ 0) (See [START_REF] Lemonte | A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function[END_REF]).

Mathematical properties

In this section, we study some structural properties of the BENH distribution, specifically moments, cumulants, mean deviations and Bonferroni and Lorenz curves. Established algebraic expansions to determine some structural properties of this distribution can be more efficient than computing those directly by numerical integration of the density function [START_REF] Cordeiro | The beta-Weibull geometric distribution[END_REF], which can be prone to rounding off errors among others. The formulae derived throughout the paper can be easily handled in softwares such as Mathematica and Maple.

Theorem 1. The rth ordinary moment of X ∼ BENH a,b (λ, θ, α) is given by

µ ′ r = E(X r ) = θ λ r ∞ ∑ j,k=0 r ∑ i=0 (-1) r+k-i e k+1 v j (k + 1) i α +1 ( (a + j)θ -1 k ) ( r i ) Γ ( i α + 1, k + 1 ) . ( 12
)
Proof. We can write

µ ′ r = E(X r ) = ∫ ∞ 0 x r f (x; φ)dx = ∞ ∑ j,k=0 (-1) k e k+1 v j ( (a + j)θ -1 k ) ∫ ∞ 0 x r (1 + λx) α-1 e -(k+1)(1+λx) α dx. ( 13
)
It follows, since r > 0 is an integer, that

∫ ∞ 0 x r (1 + λx) α-1 e -(k+1)(1+λx) α dx = (-1) r-i αλ r+1 (k + 1) i α +1 r ∑ i=0 ( r i ) Γ ( i α + 1, k + 1 ) , where Γ(a, x) = ∫ ∞
x w a-1 e -w dw denotes the complementary incomplete gamma function. Then, we have

µ ′ r = θ λ r ∞ ∑ j,k=0 r ∑ i=0 (-1) r+k-i e k+1 v j (k + 1) i α +1 ( (a + j)θ -1 k ) ( r i ) Γ ( i α + 1, k + 1 ) , (14)
which completes the proof.

For lifetime models, it is also of interest to find the conditional moments and the mean residual lifetime function.

Remark 1. The conditional moments of X, say E(X s | X > t) = ∫ ∞ t x s f (x; φ)dx, are given by E(X s | X > t) = αλθ ∞ ∑ j,k=0 (-1) k e k+1 v j ( (a + j)θ -1 k ) ∫ ∞ t x s (1 + λx) α-1 e -(k+1)(1+λx) α dx = λ -s θ ∞ ∑ j,k=0 s ∑ i=0 (-1) s+k-i e k+1 v j (k + 1) i α +1 ( (a + j)θ -1 k ) ( s i ) Γ ( i α + 1, (k + 1)(1 + λt) α
) .

The mean residual lifetime function

µ(t) = E(X | X > t) -t is given by µ(t) = θ λ ∞ ∑ j,k=0 1 ∑ i=0 (-1) 1+k-i e k+1 v j ( (a + j)θ -1 k ) ( 1 i ) Γ ( i α + 1, (k + 1)(1 + λt) α ) -t.
Generally, there has been a great interest in obtaining the first incomplete moment of a distribution. Based on this quantity, we can obtain, for example, the mean deviations that provide important information about characteristics of a population. Indeed, the amount of dispersion in a population may be measured to some extent by all the deviations from the mean and median. One way to measure the amount of scatter in a population is the totality of deviations about the mean and the median. The mean deviations of X about the mean µ ′ 1 = E(X) and about the median M can be expressed as

δ 1 = 2µ ′ 1 F(µ ′ 1 ) -2m 1 (µ ′ 1 )
and δ 2 = µ -2m 1 (M), respectively, where F(µ ′ 1 ) is easily evaluated from [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF], the median M follows from [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] as the solution of F(M; φ) = 0.5 and

m 1 (z) = ∫ z 0 x f (x)dx. We can write m 1 (z) = α λ θ ∞ ∑ j,k=0 (-1) k e k+1 v j ( (a + j)θ -1 k ) ∫ z 0 x (1 + λx) α-1 e -(k+1)(1+λx) α dx = θ λ ∞ ∑ j,k=0 1 ∑ i=0 (-1) 1+k-i e k+1 v j ( (a + j)θ -1 k ) ( 1 i
)

Γ ( i α + 1, k + 1 ) { 1 -(1 + λz) α } . ( 15
)

Quantile function

In this section, we derive a power series for the qf of the BENH distribution obtained by inverting [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] as

Q(u) = F -1 (u)
. First, we can expand (9) using Mathematica as

x = ∞ ∑ i=1 s i v i/θ , (16)
where

s 1 = 1/(λα), s 2 = 1/(2λα 2 ), s 3 = (α 2 + 1)/(6λα 3 ), s 4 = (α 3 + 4α 2 + 1)/(24λα 4 ), etc.
An expansion for the inverse of the incomplete beta function B(v; a, b) = u can be found in Wolfram website 1 as

v = B -1 (u; a, b) = ∞ ∑ j=1 q j y j , (17)
where y = y(u) = [aB(a, b)u] 1/a and q 1 = 1, q 2 = (b -1)/(a + 1), etc.

Combining ( 16) and ( 17), the BENH qf can be expressed as

x = Q(u) = ∞ ∑ i=1 s i         ∞ ∑ j=1 t j u j/a         i/θ , (18)
where t j = q j a j B(a, b) j and the q ′ j s are given above. By expanding ( ∑ ∞ j=1 t j u j/a

) i/θ in Taylor series, we can rewrite [START_REF] Huster | Modelling paired survival data with covariates[END_REF] as

        ∞ ∑ j=1 t j u j/a         i/θ = ∞ ∑ k=0 f k (i θ -1 )         ∞ ∑ j=1 t j u j/a         k , (19)
where We now use an equation of Gradshteyn and Ryzhik [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] for a power series raised to a positive integer k

f k (p) = ∑ ∞ m=k (-1) m-k ( m k ) (p) m /m! (for k ≥ 0)
        ∞ ∑ j=0 a j z j         k = ∞ ∑ j=0 c k, j z j , (20)
where the coefficients c k, j (for k, j = 1, 2, . . .) are obtained from the recurrence equation

c k, j = (k a 0 ) -1 k ∑ m=1 [m(k + 1) -j] a m c k, j-m (21)
and, for k ≥ 0, c k,0 = a k 0 . The coefficient c k, j can be determined from c k,0 , . . . , c k, j-1 and then from the quantities a 0 , . . . , a k listed above.

We can write using (20) and ( 21)

        u 1/a ∞ ∑ j=0 p j u j/a         k = u k/a ∞ ∑ j=0 d k, j u j/a
where p j = t j+1 for j ≥ 0, and, for k ≥ 0, d k,0 = p k 0 and, for j ≥ 1,

d k, j = (k p 0 ) -1 ∑ k m=1 [m(k + 1) -j] p m d k, j-m .
Inserting the last equation in (19), we obtain

        ∞ ∑ j=1 t j u j/a         i/θ = ∞ ∑ k, j=0 f k (i θ -1 ) d k, j u ( j+k)/a
and then Q(u) in ( 16) can be expressed as

Q(u) = ∞ ∑ k, j=0
e k, j u ( j+k)/a , where e k, j = ∑ ∞ i=1 s i f k (i θ -1 ) d k, j (for j, k ≥ 0).

Finally, we can write

Q(u) = ∞ ∑ r=0 h r u r/a , (22)
where h r = ∑ ∞ k, j=0 k+ j=r e k, j .

Equation ( 22) reveals that the qf of the BEHN distribution is just a power series. Then, several mathematical quantities of X can be expressed in terms of integrals over (0, 1). In practical terms, the upper limit ∞ in the above sums can be substituted by a number like twenty or thirty. In fact, if W(•) be any integrable function in the real line, we can write

∫ ∞ 0 W(x) f (x)dx = ∫ 1 0 W        ∞ ∑ r=0 h r u r/a        du. ( 23
)
Equations ( 22) and ( 23) are the main results of this section since we can obtain from them numerically various BENH structural quantities. They can be determined by using the integral on the right-hand side for special W(•) functions, which can be simpler than if they are evaluated from the left-hand integral.

A simulation study

We perform a Monte Carlo simulation study to assess the finite sample behavior of the MLEs of λ, θ, α, a and b. The results were obtained from 2,000 Monte Carlo replications and the simulations were carried out using the R statistical software. In each replication, a random sample of size n is drawn from the BENH a,b (λ, θ, α)

distribution and the parameters were estimated by the maximum likelihood method. The random variable X was generated using the inversion method. We considered two setups with the following values for the parameters 1 and2, respectively.

In both setups, we note that the biases and RMSEs of the MLEs of λ, θ, α, a and b decay toward zero when the sample size increases, as expected. There is a small sample bias in the estimation of the model parameters.

Future research should be conducted to obtain bias corrections for these estimators.

Estimation and inference

Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The MLEs enjoy desirable properties and can be used when constructing confidence intervals for the parameters and also in test statistics. Given the observed values x 1 , . . . , x n , the MLEs of the model parameters of the BENH distribution are determined by maximization the log-likelihood function given by

ℓ(φ) = ℓ(x i , α, λ, θ, a, b) = n [ log α + log θ + log λ -log {B(a, b)} ] + (α -1) n ∑ i=1 log(1 + λx i ) n ∑ i=1 {1 -(1 + λx i ) α } + (θa -1) n ∑ i=1 log { 1 -e 1-(1+λx i ) α } + (b -1) n ∑ i=1 log [ 1 - { 1 -e 1-(1+λx i ) α } θ ] . (24)
The above log-likelihood can be maximized numerically using by using the R (optim function), SAS (PROC NLMIXED), Ox program (sub-routine MaxBFGS), Nmaximize command in Mathematica, among others.

To check the goodness of fit of all statistical models, some goodness-of-fit statistics shall be used are computed using computational package Mathematica. The MLEs are evaluated using Nmaximize as well as the goodness-of-fit statistics including the Anderson-Darling (A * ), Cramér-von Mises (W * ), Liao-Shimokawa (L-S) and Kolmogrov-Simnorov (K-S) statistics (with their p-values). These statistics are used as goodness-offit measures and to evaluate how closely a specific density fits the corresponding histogram of a given data set.

Usually, the distribution with smallest statistic values will yield a better fit. These statistics are given by

A * = ( 2.25 n 2 + 0.75 n + 1
)        -n - 1 n n ∑ i=1 (2i -1) log (z i (1 -z n-i+1 ))        , L -S = 1 √ n n ∑ i=1 Max [ i n -z i , z i -i-1 n ] √ z i (1 -z i ) , W * = ( 0.5 n + 1 )        n ∑ i=1 ( z i - 2i -1 2n
) 2 + 1 12n        , K -S = Max [ i n -z i , z i - i -1 n ] ,
where z i = cdf(y i ) and the y ′ i s are the ordered observations.

The log-beta exponentiated Nadarajah-Haghighi distribution

Let X be a random variable having the BENH density function [START_REF] Cordeiro | The beta-Weibull geometric distribution[END_REF]. The random variable Y = log(X) defines the log-beta exponentiated Nadarajah-Haghighi (LBENH) distribution. The density function of Y reparameterized in terms of λ = e -µ can be expressed as

f (y) = α θ B(a, b) e y-µ ( 1 + e y-µ ) α-1 e 1-(1+e y-µ ) α { 1 -e 1-(1+ e y-µ ) α } θ a-1 [ 1 - { 1 -e 1-(1+ e y-µ ) α } θ ] b-1 , ( 25
)
where y, µ ∈ R, θ > 0, α > 0, a > 0 and b > 0.

If Y is a random variable having density function (25), we can write Y ∼ LBENH(µ, θ, α, a, b). Thus, if The survival function corresponding to (25) is a,b) .

X ∼ BENH(λ, θ, α, a, b) then Y = log(X) ∼ LBENH(µ, θ, α
S (y) = 1 - B [ { 1 -e 1-(1+ e y-µ ) α } θ ; a, b ] B(
We define the standardized random variable Z = (Y -µ) having density function given by

f (z) = α θ B(a, b) e z (1 + e z ) α-1 e 1-(1+e z ) α { 1 -e 1-(1+ e z ) α } θ a-1 [ 1 - { 1 -e 1-(1+ e z ) α } θ ] b-1 , z ∈ R (26)

The LBENH regression model for censored data

A parametric regression model that provides a good fit to lifetime data tends to yield more precise estimates of the quantities of interest. Next, we construct a linear regression model for the response variable y i and the explanatory variable vector v T i = (v i1 , . . . , v ip ) based on the LBENH distribution given by

y i = v T i β + z i , i = 1, . . . , n. (27)
Here, the random error z i is assumed to have the density function (26) with unknown parameters µ i ∈ R, α > 0, θ > 0, a > 0 and b > 0. The parameter µ i = v T i β is the location of y i . The vector µ = (µ 1 , . . . , µ n ) T of location parameters is represented by a linear model µ = Vβ, where V = (v 1 , . . . , v n ) T is a known model matrix. The LBENH model (27) opens new possibilities for fitting many different types of data.

Consider a sample (y 1 , v 1 ), . . . , (y n , v n ) of n independent observations, where the random response is defined by y i = min{log(x i ), log(c i )}. We assume non-informative censoring such that the observed lifetimes and censoring times are independent. Let F and C be the sets of individuals for which y i is the log-lifetime and log-censoring, respectively. The log-likelihood function for the vector of parameters η = (β T , θ, α, a, b) T from model ( 27) is given by l

(η) = ∑ i∈F log[ f (y i )] + ∑ i∈C log[S (y i )],
where f (y i ) is the density function (25) and S (y i ) is the survival function (26) of Y i . The log-likelihood function for η reduces to

l(η) = r { 1 + log ( α θ σ B(a, b) )} + ∑ i∈F y i -v T i β + (α -1) ∑ i∈F log ( 1 + e y i -v T i β ) - ∑ i∈F ( 1 + e y i -v T i β ) + (θ a -1) ∑ i∈F log        1 -e 1- ( 1+ e y i -v T i β ) α        + (b -1) ∑ i∈F log          1 -        1 -e 1- ( 1+ e y i -v T i β ) α        θ          + ∑ i∈C log                          1 - B                 1 -e 1- ( 1+ e y i -v T i β ) α        θ ; a, b          B(a, b)                          , ( 28
)
where r is the number of uncensored observations (failures).

The MLE η of η can be obtained by maximizing the log-likelihood function (28). We use the statistical R software to compute the estimates. From the fitted model ( 27), the survival function for y i can be estimated by

S (y i ; βT , α, θ, â, b) = 1 - B                     1 -e 1- ( 1+ e y i -v T i β ) α          θ ; â, b           B(â, b) . (29)
Under general regularity conditions, the asymptotic distribution of ( η -η) can be approximated by the multivariate normal N p+4 (0, J(θ) -1 ), where J(θ) is the observed information matrix. Then, the inference on the parameter vector η can be based on this approximation distribution for η. The likelihood ratio (LR) statistic can be used to discriminate between the LBENH and LENH regression models since they are nested models.

Applications

In this section, we provide two applications to show the usefulness of the BENH distribution and the new LBENH regression model.

Meteorological application

We fit some well-known statistical distributions and the BENH distribution to a meteorology data set and determine the best fitted distribution among the fitted models. We also provide some meteorological parameters for these data under the best model.

Model fitting

We consider the following models as competitive for the proposed distribution:

• The Gumbel model with density function

f (x) = e -e -x-µ σ -x-µ σ σ , x, µ ∈ R, σ > 0 ,
• The gamma Weibull (GW) [START_REF] Provost | The gamma-Weibull distribution[END_REF] model with density function

f (x) = k λ -k-ξ x ξ+k-1 e -λ -k x k Γ(1 + ξ/k) , x > 0 , ξ + k, λ > 0 ,
• The BE [START_REF] Nadarajah | The beta exponential distribution[END_REF] model with density function

f (x) = λ e -(bλx) ( 1 -e -λx ) a-1 B(a, b) , x > 0, a, b, λ > 0 ,
• The ENH [START_REF] Lemonte | A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function[END_REF] model with density function

f (x) = αλθ(1 + λx) α-1 e 1-(1+λx) α { 1 -e 1-(1+λx) α } θ-1 , x > 0, α, θ, λ > 0,
• The BG [START_REF] Kong | On the Properties of Beta-Gamma Distribution[END_REF] model with density function

f (x) = x ρ-1 e -x/λ λ ρ B(α, β)Γ(ρ) { γ(ρ, x/λ) Γ(ρ) } α-1 { 1 - γ(ρ, x/λ) Γ(ρ) } β-1 , x > 0, α, β, ρ, λ > 0,
• The BEW [START_REF] Cordeiro | The beta exponentiated Weibull distribution[END_REF] model with density function

f (x) = α cλ c x c-1 e -(λ x) c ( 1 -e -(λ x) c ) α a-1 ( 1 - ( 1 -e -(λ x) c ) α ) b-1 B(a, b) , x > 0, a, b, c, λ, β > 0.
The application refers to the "yearly maxima wind" data, which represents the yearly maxima wind speed (miles per hour) registered at a certain location over a period of 50 years as reported in Table 1.1 by [START_REF] Castillo | Extreme Value and Related Models with Applications in Engineering and Science[END_REF].

These data may help in the design of structures and buildings.

The maximum likelihood method is used to obtain the parameter estimates and their standard errors (computed by inverting the observed information matrix) given in Table 3 for the current data. Further, all mentioned goodness-of-fit statistics are determined for each fitted distribution and listed in Table 4.

Based on the figures in Table 4, we note that the BENH distribution has the smallest statistics. Therefore, we can conclude that this distribution gives the best fit to these data among the other distributions. Also, this fact is confirmed by observing the plots of the estimated cdfs of these distributions and the empirical cdf given in the Appendix. In fact, the estimated BENH cdf and the empirical cdf are quite close. Further, the LR statistic can be used to compare a distribution having extra parameters with some of its sub-models. Then, we apply the LR statistic for comparing the BENH and ENH distributions. The LR statistic w for testing the following hypotheses: H 0 : a = b = 1 (ENH) versus H 1 : a b 1 (BENH) yields the value w = 12.3273 (p-value=0.0021). Thus, the BENH model provides a better fit to the yearly maxima wind data when compared with the ENH model.

The 95% and 99% confidence intervals for these parameters are listed in Table 5.

Meteorological parameters

We prove empirically that the BENH distribution is the best model for the yearly maxima wind data. So, it is of interest to obtain the following meteorological parameters for these data.

Return level

A return period is an estimate of the likelihood of an event, like a flood or wind speed to occur and it is the inverse of the probability that the event will be exceeded in any one year (or the expected waiting time or the mean number of years that will be taken for an exceeding event to occur). The probability, return period and return level (high quantile of the distribution) of the event can be estimated by P(

x T ) = 1 -F(x T ), T = 1/P(x T ) and x T = F -1 ( 1 -1 T )
, respectively, where F -1 (•) is the qf and P(x T ) is called the exceedance probability (Nadarajah and Choi [START_REF] Nadarajah | Maximum daily rainfall in South Korea[END_REF]).

The return level x T under the BENH distribution is obtained by solving

B ( 1 -e -(λ x T +β x k T ) ;α, ρ ) = B(α, ρ) ( 1 - 1 T
) .

where x T > 0 and T ≥ 1. Table 6 provides the estimates of the return level x T for the yearly maxima wind speed data in Table 3 by replacing the model parameters by their MLEs. The mean deviation about the return level is the mean of the distances of each value from their return level.

Analogously to the mean deviation about the mean given in Section 4, the mean deviation about return level is given by

η = ∫ ∞ 0 |x -x T | f (x)dx = 2x T F(x T ) -x T + µ -2m(x T ),
where m(x T ) is given by ( 15) and f (x) is the density of the BENH distribution.

Diabetic Retinopathy Study

We consider a data set analyzed by Huster et al. [START_REF] Huster | Modelling paired survival data with covariates[END_REF]. Patients with diabetic retinopathy in both eyes and 20/100 or better visual acuity for both eyes were eligible for the study. The patients were followed for two con- The explanatory variable x 1 is marginally significant for the model at the significance level of 5%.

In order to assess if the model is appropriate, the empirical survival function and the estimated survival function (29) from the fitted LBENH regression model are plotted in Figure 6. In fact, the LBENH regression model provides a good fit for these data.

Figures

1 and 2 Figure 1 :Figure 2 :Figure 3 :

 2123 Figure 1: Plots of the BENH density function. (a) a = 1.9, b = 3.9, λ = 0.5, θ = 2.3 and α = 2 (dotted line), α = 3 (dashed line), α = 4 (solid line), α = 5 (thick line). (b) a = 2.9, b = 1.9, λ = 1.3, α = 1.3 and θ = 1 (dotted line), θ = 3 (dashed line), θ = 5 (solid line), θ = 7 (thick line). (c) a = 9, b = 1.5, α = 3, θ = 1.5 and λ = 1 (dotted line) λ = 1.3, (dashed line), λ = 1.5 (solid line), λ = 1.7 (thick line).

Figure 4 :

 4 Figure 4: Plots of the BENH hazard function. (h) a = 4, b = 11, α = 0.4, λ = 15, θ = 1.7 and (i) a = 3.2, b = 1.5, α = 1.3, λ = 0.8, θ = 8.

  and (p) m = p(p -1) . . . (pm + 1) is the descending factorial.

  of the model: λ = 1.5, θ = 10.0, α = 2.0, a = 4.0, and b = 2.0 (setup 1) and λ = 3.0, θ = 5.0, α = 1.5, a = 2.5, and b = 3.0 (setup 2). The mean estimates of the five model parameters and the corresponding root mean squared errors (RMSEs) for the sample sizes n = 50, 100 and 200 are listed in Tables

Figure 5 :

 5 Figure 5: Plots of the LBENH density function. (j) µ = 0.5, α = 0.1, a = 0.5, b = 0.9 and θ = 5 (dotted line), θ = 10 (dashed line), θ = 30 (solid line), θ = 70 (thick line). (k) µ = 7, θ = 0.8, α = 3, b = 0.8 and a = 1 (dotted line), a = 1.5 (dashed line), a = 2.3 (solid line), a = 4 (thick line). (l) µ = 0.8, θ = 11, α = 1.2, a = 5 and b = 1.5 (dotted line) b = 5, (dashed line), b = 15 (solid line), b = 55 (thick line).

secutively completed 4

 4 month follow-ups and the endpoint was the occurrence of visual acuity less than 5/200. We choose only the treatment time. A 50% sample of the high-risk patients defined by diabetic retinopathy criteria was taken for the data set (n = 197) and the percentage of censored observations was 72.4%. The variables involved in the study are: ti -failure time for the treatment (in min); censoring indicator (0=censoring, 1=lifetime observed); xi1 -age (0 = patient is an adult diabetic, 1 = patient is a juvenile diabetic). We adopt the model y i = β 0 +β 1 x i1 +z i , where the random variable Y i follows the LBENH distribution (25) for i = 1, . . . , 197. The MLEs of the model parameters are determined using the statistical software R. In order to estimate α, θ, a, b and β, we perform iterative maximization of the logarithm of the likelihood function (28) starting with initial values for α and β taken from the fitted log-NH regression model (with a = b = θ = 1). The MLEs (approximate standard errors and p-values in parentheses) are: α = 0.4536 (0.1887), θ = 10.7832 (0.1247), â = 2.5057 (2.5872), b = 0.0219 (0.0327), β0 = -3.6222 (1.1698) (0.4994) and β1 = 0.6399 (0.3984) (0.0274).

Table 1 :

 1

	n	Parameter	Mean	RMSE
		λ	1.1910	1.2007
		θ	13.3681	7.8019
	50	α	3.6020	2.7661
		a	7.6681	8.0093
		b	3.9407	5.1279
		λ	1.3289	1.0321
		θ	11.6246	5.0460
	100	α	2.8785	1.7866
		a	5.9771	5.3011
		b	3.1445	3.3964
		λ	1.4131	0.7452
		θ	10.7861	3.5364
	200	α	2.4810	1.2138
		a	4.8404	3.0189
		b	2.5862	2.2210

Monte Carlo simulation results for Setup 1: Mean estimates and RMSEs of λ, θ, α, a and b.

Table 2 :

 2

	n	Parameter	Mean	RMSE
		λ	2.7536	2.5369
		θ	8.8775	10.0651
	50	α	2.6233	1.9267
		a	4.6692	4.0526
		b	4.8437	5.1057
		λ	2.9982	2.2718
		θ	7.4226	7.5222
	100	α	2.2102	1.3976
		a	3.9831	3.0587
		b	4.0726	3.5745
		λ	2.9956	2.0220
		θ	6.8338	6.2232
	200	α	2.0991	1.2690
		a	3.7933	2.7461
		b	3.6886	2.8409

Monte Carlo simulation results for Setup 2: Mean estimates and RMSEs of λ, θ, α, a and b.

Table 3 :

 3 MLEs of the parameters (standard errors in parentheses) for yearly maxima wind data

	Distributions	Estimates
	Gumbel(µ, σ)	29.448375	6.728401
		(0.986481)	(0.811886)
	GW(ξ, k, λ)	16.37274	0.636736	0.191498
		(3.81043)	(0.102927)	(0.250299)
	BE( λ , a , b)	0.174056	99.999996	0.667858
		(0.033464)	(63.3027)	(0.214419)
	ENH(α, λ, θ)	1.090938	0.098860	29.99999
		(0.230356)	(0.0479538)	(14.0858)
	BG(α, β, ρ, λ)	82.54150	75.99302	0.285808	595.603
		(89.3679)	(98.5805)	(0.280189)	(873.244)
	BEW(α, λ, a, b, c)	89.96635	0.299499	14.99929	0.365674	0.992785
		(98.776)	(0.358327)	(16.1477)	(0.351981)	(0.57515)
	BENH(λ, θ, α, a, b)	0.440040	41.48587	0.899999	88.218035	0.364810
		(0.24795)	(68.1182)	(0.197686)	(144.411)	(0.194824)

Table 4 :

 4 Goodness-of-fit statistics for yearly maxima wind data

	Distributions	A *	W *	L-S	K-S	p-value
	Gumbel(µ, σ)	1.0083	0.149555	5.73492	0.130391	0.363071
	GW(ξ, k, λ)	1.81995	0.276761	6.2735	0.130799	0.359295
	BE(λ , a , b)	0.995606	0.160182	3.41307	0.12506	0.414754
	ENH(α, λ, θ)	1.32567	0.204332	3.84876	0.126957	0.395883
	BG(α, β, ρ, λ)	1.83913	0.282842	7.01132	0.134266	0.328222
	BEW(α, λ, a, b, c)	0.644849	0.107817	2.89811	0.112035	0.556868
	BENH(λ, θ, α, a, b)	0.561666	0.094480	2.75311	0.107788	0.606717

Table 5 :

 5 Confidence intervals for the model parameters

	CI	λ	β	k	α	ρ
	95%	[0 0.926]	[0 174.9]	[0.512 1.287]	[0 371.2]	[0 0.746]
	99%	[0 1.079]	[0 217.2]	[0.389 1.410]	[0 460.7]	[0 0.867]

Table 6 :

 6 Return level estimates xT for the T-year of the maxima wind speed data

	T	Return level estimate of wind data (mm)
	2	31.4687
	5	40.1367
	10	46.6312
	20	53.1969
	50	61.9929
	100	68.7284
	200	75.5285
	9.1.4. Mean deviation about the return level

http://functions.wolfram.com/06.23.06.0004.01.

Acknowledgement

The research of Abdus Saboor has been supported in part by the Higher Education Commission of Pakistan under NRPU project No. 3104. The research of Gauss M. Cordeiro has been supported by CNPq (Brazil).

Conclusions

We introduce a five-parameter lifetime model called the beta exponentiated Nadarajah-Haghighi (BENH) distribution, which extends several distributions proposed and widely used in the literature. The new distribution is useful to model lifetime data with increasing, decreasing, upside-down bathtub and bathtub shaped hazard functions. It can be much more flexible than the exponentiated Nadarajah-Haghighi, beta exponentiated Weibull, beta gamma, beta exponential, gamma Weibull and Gumbel distributions. We provide some closed-form expressions for the ordinary, incomplete and conditional moments, mean deviations and quantile and generating functions. By using nine goodness-of-fit statistics we prove empirically that the new model can be superior to some distributions generated from other families in terms of model fitting by means of a meteorological application.

Appendix

The symbol ψ (1) (•) denotes the trigamma function, and the polygamma function of order m is denoted by: 7 displays the estimated cdfs for some models and the empirical cdf for yearly maxima wind data.

The BENH distribution


Thomas Maullin-Sapey

Peter Williams

Guillaume Flandin

Thomas E Nichols

Camille Maumet

Viewing FSL results with SPM and vice versa

Introduction

A growing number of efforts are emerging in the neuroimaging community to increase reproducibility of research findings (e.g. [START_REF] Gorgolewski | org: a web-based repository for collecting and sharing unthresholded statistical maps of the human brain[END_REF][START_REF] Poldrack | Toward open sharing of task-based fMRI data: the OpenfMRI project[END_REF] ). In an attempt to facilitate publishing of neuroimaging data and metadata, the NeuroImaging Data Model (NIDM) defines a set of specifications based on semantic web technologies.

The NIDM-Results specification was recently introduced, providing a machine-readable representation of mass univariate statistical results, including extensive metadata and key images summarising the findings [START_REF] Maumet | Sharing brain mapping statistical results with the neuroimaging data model[END_REF] . The main motivation for NIDM-Results was to make publishing of neuroimaging results data and metadata as effortless as possible and hence facilitate compliance with statistical guidelines (e.g. [START_REF]SAMPL guidelines for statistical reporting[END_REF] ), increase reproducibility and enable image-based meta-analyses.

An NIDM-Results pack is a ZIP archive containing a Resource Description Framework (RDF) metadata file, which stores provenance information, alongside several NIfTI images and CSV files. NIDM-Results packs can be generated via NIDM-Results exporters, which are available for Statistical Parametric Mapping (SPM) [START_REF] Penny | Statistical parametric mapping: the analysis of functional brain images: the analysis of functional brain images[END_REF] and the FMRIB Software Library (FSL) [START_REF] Jenkinson | [END_REF] . However few tools exist to read and display NIDM-Results. This is hindering the distribution and acceptance of NIDM within the neuroimaging community.

Here, we introduce two NIDM-Results viewers, one each oriented for users accustomed to SPM and FSL, two of the most widely-used neuroimaging analysis softwares [START_REF] Carp | The secret lives of experiments: methods reporting in the fMRI literature[END_REF] .

Methods

The SPM and FSL viewers were written in programming languages already in use in each of the neuroimaging software packages, namely MATLAB and Python, respectively. Each viewer, provides an HTML display of NIDM-Results packs formatted according to the layout of the respective results page. The Matlab-based viewer reads a json-ld version of the RDF metadata (as presently there is no Matlab SPARQL query engine), while the Python-based viewer uses RDFlib [9] to directly query the metadata.

Both applications were tested on a set of publicly available NIDM-Results packs generated with SPM and FSL ( http://neurovault.org/collections/1692/ ) [9] and designed to cover the main use-cases of fMRI analyses [START_REF] Pauli | Exploring fMRI Results Space: 31 Variants of an fMRI Analysis in AFNI, FSL, and SPM[END_REF] .

Results

The SPM and FSL viewers are publically available at https://github.com/incf-nidash/nidmresults-spmhtml a n d https://github.com/incf-nidash/nidmresults-fslhtml . Fig. 1 presents sample results page for an example FSL and SPM study respectively. HTML versions of these results can be found here: http://bit.ly/2hQHQhz .

Conclusions

We hope that the viewer will facilitate the adoption of the NIDM-Result format for sharing of statistical results in the neuroimaging community as well as interoperability across software packages. For example, it will make it easy to share results with a collaborator using different a different software package. Another important use for NIDM-Results is sharing of findings for meta-analyses, ensuring all vital metadata is retained. This work is part of a growing ecosystem of tools for NIDM and it is our intention to develop more applications to further to increase the practical utility of NIDM-Results. Future work will include development of NIDM-Results packs exporter from other neuroimaging software packages such as AFNI, as well as the creation of a software independent Javascript viewer for NIDM-Results which may be embedded in HTML or run locally. Other projects also include the creation tools for meta-analysis of neuroimaging data.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Examples of SPM HTML display for (A) SPM subject-level analysis, (B) FSL subject-level analysis, (C) SPM group-level analysis and (D) FSL group-level analysis .

Acknowledgments

We acknowledge the work of all INCF task force members as well as the Wellcome Trust for support of CM and TEN.