Heike Wersching

Katrin Guske

Sandra Hasenkamp

Claudia Hagedorn

Samira Schiwek

Stefanie Jansen

Veronica Witte

Jürgen Wellmann

Hubertus Lohmann

Kerstin Duning

Prof. Dr Agnes Flöel
email: agnes.floeel@charite.de

Impact of common KIBRA allele on human cognitive functions

Keywords: Aging, memory, executive functions, vascular risk factors, gender, ecogenetic context

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Introduction

Cognitive impairment is of substantial socioeconomic concern in aging populations [START_REF] Ferri | Global prevalence of dementia: a Delphi consensus study[END_REF][START_REF] Wegman | Aging and globalization[END_REF]). Several risk factors for the development of dementia and cognitive impairment have been described, most importantly common vascular and metabolic diseases such as hypertension, diabetes, and atherosclerosis [START_REF] Arvanitakis | Diabetes mellitus and risk of Alzheimer disease and decline in cognitive function[END_REF]; [START_REF] Biessels | Risk of dementia in diabetes mellitus: a systematic review[END_REF]; [START_REF] Elias | Obesity, diabetes and cognitive deficit: The Framingham Heart Study[END_REF]; [START_REF] Knecht | High-normal blood pressure is associated with poor cognitive performance[END_REF]. There is also evidence that lifestyle factors, such as physical activity and dietary habits, exert influence on cognitive functions (Laurin et [START_REF] Zhang | Cognitive flexibility is associated with KIBRA variant and modulated by recent tobacco use[END_REF], reported an interaction between KIBRA rs17070145 genotype and smoking habits and Preuschhof et al. [START_REF] Preuschhof | KIBRA and CLSTN2 polymorphisms exert interactive effects on human episodic memory[END_REF]) showed an interaction between KIBRA and CLSTN2 in the modulation of episodic memory performance. Taken together, the differential impact of the KIBRA rs17070145 polymorphism on cognitive function as well as its clinical significance and interaction with environmental and other genetic factors is not completely understood.

The aim of the present study was to comprehensively test for potential associations of the common KIBRA rs17070450 genotype with cognitive functions in a medically well-characterized cohort of community-dwelling elderly individuals. We focused on cognitive composite scores, reflecting clinically meaningful cognitive domains, and on interactions of rs17070450 with established risk factors for cognitive decline (arterial hypertension, hypercholesterolemia, diabetes and lifestyle habits [START_REF] Daffner | Promoting Successful Cognitive Aging: A Comprehensive Review[END_REF]. We also analyzed gender-specific effects, as gender has been shown to modify the relationship between cognitive decline and environmental as well as genetic traits [START_REF] Kuznetsova | Left ventricular mass in relation to genetic variation in angiotensin II receptors, renin system genes, and sodium excretion[END_REF]Raz et al. 2009b). For the present study 545 individuals were included. Main criteria were age above 50 years and completion of the neuropsychological and laboratory assessments as well as the lifestyle questionnaire. Individuals with a history or imaging evidence of stroke, or other severe neurological conditions were excluded. We further excluded one participant with complex psychiatric co-morbidities. Individuals included in the present study did not differ significantly in terms of age, gender and education from subjects who were excluded from this analysis.

Methods

Subjects and design

Participants

Risk factor assessment

Assessment of cardiovascular risk factors was based on self-reported physician diagnoses of existing or past illnesses, including history of hypertension, history of diabetes and history of hypercholesterolemia. Depression was defined by history, intake of antidepressants or scores higher than 17 on the Beck Depression Inventory. Education was assessed as categorical variable (five vs. seven vs. nine years of secondary school vs. tertiary education according to the German educational system [START_REF] Wersching | Serum C-reactive protein is linked to cerebral microstructural integrity and cognitive function[END_REF]).

Lifestyle index

To create a lifestyle score as described in previous studies [START_REF] Floel | Lifestyle and memory in the elderly[END_REF][START_REF] Kurth | Healthy lifestyle and the risk of stroke in women[END_REF]), we considered self-reported lifestyle information from a detailed questionnaire incorporating information about body mass index (BMI), dietary habits, exercise, smoking, and alcohol consumption. The overall 'lifestyle score' was calculated by assigning scores of 1-5 to each individual variable category, for which a higher point value indicates a healthier behaviour. The healthiest behaviour was defined as a BMI of 18.5-22 kg/m 2 , never smoking, intense physical activity (>13,000 kcal/week), moderate alcohol consumption (4-10 drinks per week), and a dietary pattern rich in fruits, vegetables, wholegrain products and unsaturated fatty acids. For details on this score, please see Floel et al. [START_REF] Floel | Lifestyle and memory in the elderly[END_REF]).

Neuropsychological assessment

The neuropsychological assessment was conducted by trained clinical psychologists supervised by two senior neuropsychologists. A comprehensive neuropsychological test battery was administered to assess the full range of cognitive functions (Table 1). A detailed description of the tests is found in Lezak et al. [START_REF] Lezak | Neuropsychological assessment[END_REF]. A principal component analysis (PCA) was carried out to reduce the single tests to superordinate composite scores using varimax rotation and including coefficients with absolute values above 0.4. We used composite scores to decrease error associated with the analysis of multiple related cognitive outcomes and to extract theoretically relevant cognitive domains. We validated the resulting composite scores with regard to known cognitive domains [START_REF] Lezak | Neuropsychological assessment[END_REF]) and labelled them as 'verbal memory', 'word fluency', 'working memory', and 'executive functions/ psychomotor speed' (Table 1). Using PCA the data were z-transformed resulting in a mean of zero and an SD of 1.00 for each cognitive domain. This allows for a direct comparison of the cognitive domains in terms of SD units. We performed a power analysis which showed that the power to detect a 0.1 mean difference in the z-scores across genotypes is 21%, to detect a mean difference of 0.2 it is 0.64% and for mean difference of 0.3 and higher the power is above 90%.

Genotyping of KIBRA rs17070145 polymorphism

Genomic DNA was extracted from white blood cells using QIAamp DNA Blood Kit (Qiagen, Hilden, Germany). From the published sequence of the KIBRA gene [START_REF] Kremerskothen | Characterization of KIBRA, a novel WW domain-containing protein[END_REF]), a 198 bp fragment was chosen which contained the polymorphic site (rs17070145, GenBank accession number NM_015238) using two primers (forward: atcctcttgaggcttcactgg, reverse: actttcaacacaatgaacaagg). The amplification was performed using the following PCR program: 95°C for 15 min to denature, followed by 94°C for 1 min, 68°C for 1 min, and 72°C for 1 min for 12 cycles, followed by 94°C for 1 min, 56°C for 1 min, 72°C for 1 min for 25 cycles and 72°C for 10 min. PCR products were digested by addition of 1 U of MseI restriction enzyme for 12 hours at 37°C. The C allele lacks a MseI site present in the T allele, so that only in the presence of the T allele, the PCR product (198 bp) was cut into two fragments of 128 bp and 70 bp in length, visualized on ethidium bromide-stained 1.5% agarose gels.

Data Analysis

In line with previous studies, KIBRA rs17070145 CT and TT genotypes were combined and tested against KIBRA rs17070145 CC genotype (=reference) [START_REF] Papassotiropoulos | Common Kibra alleles are associated with human memory performance[END_REF][START_REF] Schaper | KIBRA gene variants are associated with episodic memory in healthy elderly[END_REF]). Student´s T-tests, Fisher´s exact tests and chi-Square tests were used to univariately test for differences in age, gender, education, and risk factors between CT/TT and CC genotypes.

After calculation of cognitive composite scores via principal component analysis, effects of the KIBRA rs17070145 polymorphism on the cognitive domains were tested with multivariate analyses of variance using the general linear model procedure in SAS 9.2. The basic model addressing main effects included the following covariates: Age, gender, education, history of arterial hypertension, intake of antihypertensive medication, history of diabetes, lifestyle index, history of depression, intake of antidepressive medication and neuropsychologists involved. A second model addressing interactions between covariates further included second-order interaction terms among the genotype and the single risk factors including gender (arterial hypertension x rs17070145 genotype, diabetes x rs17070145 genotype, lifestyle index x rs17070145 genotype, depression x rs17070145 genotype, gender x rs17070145 genotype).

Based on these results, further analyses were performed stratified by gender and hypertension.

Statistical analyses were performed using SAS 9.2. All probability values are 2-tailed, and we considered p< 0.05 as statistically significant. We used Bonferroni correction to adjust for multiple testing.

Results

Study population

Of the 545 participants included in this study, 235 (43.12%) were carriers of the CC polymorphism of KIBRA rs17070145 in intron 9, 234 (42.94%) subjects were CT carriers and 76 (13.94%) were homozygous TT carriers. The allelic distribution (35% T allele frequency) was similar to previous studies [START_REF] Almeida | KIBRA genetic polymorphism influences episodic memory in later life, but does not increase the risk of mild cognitive impairment[END_REF][START_REF] Papassotiropoulos | Common Kibra alleles are associated with human memory performance[END_REF][START_REF] Schaper | KIBRA gene variants are associated with episodic memory in healthy elderly[END_REF]) and proved to be in Hardy-Weinberg-equilibrium (chi-square 2.05, 1df, p=0.15). The KIBRA rs17070145 polymorphism was not associated with age, gender, education or any of the specified cerebrovascular risk factors or the lifestyle index (Table 2).

Principal component analysis of the neuropsychological test scores rendered four meaningful cognitive domains: verbal memory, word fluency, executive functions/ psychomotor speed, and working memory (see Table 1 for details).

KIBRA carrier status and cognition

Unadjusted main effects of the KIBRA rs17070145 polymorphism on the cognitive domains are shown in Figure 1. The regression analyses adjusting for the full set of covariates revealed no significant associations of any of the cognitive composite scores with KIBRA rs17070145 genotype showing the following regression estimates: β=-0.118 for executive functions/ psychomotor speed, β=0.037 for verbal memory, β=0.011 for working memory, and β=0.114 for verbal fluency; all p-values > 0.1). Separate regression models for each single memory test of the composite score also revealed no significant association between KIBRA rs17070145 and any of the memory measures with or without adjustment for executive function (supp. Tables 1 and2).

Interactions of KIBRA carrier status and risk factors/ lifestyle

For executive functions/ psychomotor speed there was an interaction of KIBRA genotypes with hypertension status (p=0.049) and gender (p=0.022): In subjects with arterial hypertension, Tallele carriers performed significantly worse in executive functions than non-T-allele carriers (β=-0.365, p=0.0077 and 0.031 after Bonferroni correction), while this was not the case in subjects without arterial hypertension (β=0.031, p=0.756; Fig. 2). Stratification by gender showed that the association of SNP rs17070145 and executive functions was statistically significant in women only (β=-0.273, p=0.012 and 0.048 after Bonferroni correction, men: β=-0.041, p=0.568; Fig. 3). Again female T-allele carriers showed worse performance in executive tasks than female non-T-allele carriers. Stratification in four groups for both gender and hypertension status showed a strong association of SNP rs17070145 with executive function in hypertensive women only (β=-0.500, p=0.0072 and 0.029 after Bonferroni correction, Fig. 4).

There was also a significant interaction of KIBRA genotypes with gender for working memory (p=0.0075), and a trend for interaction of KIBRA genotypes with gender for verbal memory (p=0.107, not shown). The analyses revealed no significant gene-risk factor interaction for the cognitive domain of word fluency.

Discussion

The present study revealed three major findings: (1) We did not find any main effects of the Conflicting results in previous studies may be due to the fact that several single test scores, instead of cognitive domains, were employed, thus increasing the chance of significant correlations in both the positive and the negative direction. On the other hand, inconsistencies might be due to differences in the underlying study population, particularly if sample sizes are very small [START_REF] Schaper | KIBRA gene variants are associated with episodic memory in healthy elderly[END_REF]). Even more important, the impact of environmental factors and traditional vascular risk factors (such as arterial hypertension) has been largely neglected so far.

Besides being an important cardiovascular risk factor, hypertension itself is being influenced by genetic (31%-68%) and environmental factors [START_REF] Ehret | Genome-wide association studies: contribution of genomics to understanding blood pressure and essential hypertension[END_REF]). Genome-wide association studies identified more than a dozen loci that are associated with blood pressure traits in large cohorts.

However, the effect sizes of the variants are small and explain only about 1% of the phenotypic variability to date [START_REF] Ehret | Genome-wide association studies: contribution of genomics to understanding blood pressure and essential hypertension[END_REF]. Importantly, in line with our findings, the BDNF Val66Met polymorphism seemed to influence psychomotor speed particularly in women and in subjects with arterial hypertension (Raz et al. 2009b) Our finding of a gender difference in the relationship between KIBRA T-allele carrier status and cognition further argues for a genetically determined modification of the association between hormonal status and cognitive performance. An interaction of estrogen and APOE carrier status on cognitive decline was recently found by Yaffe and colleagues [START_REF] Yaffe | Estrogen use, APOE, and cognitive decline: evidence of gene-environment interaction[END_REF]. KIBRA itself has been shown to play a role in the activation of the estrogen receptor-α in mammalian cells [START_REF] Rayala | Essential role of KIBRA in co-activator function of dynein light chain 1 in mammalian cells[END_REF]. The impact of gender on the associations between KIBRA and cognition can also be seen in light of a differential impact of vascular risk factors on cognitive function: In our study women on average had less vascular risk factors than men, possibly leaving more impact for genetic factors to influence cognitive functions (Raz et al. 2009a).

Mechanistically, KIBRA encodes a postsynaptic scaffold protein, which is predominately expressed in the kidney and the brain ("KIdney and BRAin", [START_REF] Kremerskothen | Characterization of KIBRA, a novel WW domain-containing protein[END_REF]).

KIBRA RNA is expressed in the whole brain with a peak in memory-related structures such as the hippocampus and the temporal lobe [START_REF] Papassotiropoulos | Common Kibra alleles are associated with human memory performance[END_REF]). As a component of the neuronal cytoskeleton, KIBRA might play a role in brain development connecting cytoskeletal and signalling molecules [START_REF] Johannsen | Temporal-spatial expression and novel biochemical properties of the memoryrelated protein KIBRA[END_REF][START_REF] Yoshihama | KIBRA Co-localizes with protein kinase Mzeta (PKMzeta) in the mouse hippocampus[END_REF]. KIBRA is also a substrate for protein kinase Czeta (PKCzeta), which is crucially involved in synaptic plasticity and memory formation [START_REF] Buther | KIBRA is a novel substrate for protein kinase Czeta[END_REF]).

Interestingly, executive functions were only significantly affected in individuals that suffered from hypertension. Prefrontal lobe functioning is dependent to a large degree on subcortical white matter integrity underlying prefrontal cortex. In both healthy individuals (Floel et However, all of these tests crucially depend on an intact function of the prefrontal cortex (therefore, the term "frontal-executive functions" is often used).

Even though the present study included a large and well characterized population in terms of vascular risk factors and cognitive profile, some limitations need to be discussed: First, given the complexity of gene-gene and gene-environment interactions, the statistical model including main effects and second-order interaction terms might not be complex enough to capture the full set of associations. However, to include yet more factors, an even larger cohort will be necessary.

Next, a larger number of common genetic polymorphisms in possibly learning-relevant genes (e. g., BDNF, COMT, dopamine receptor D2, or APOE) could have been included, but similarly, this would have resulted in an excessive number of factors to be included in the model. Third, we analyzed only one polymorphism of the KIBRA gene (SNP rs17070145). Although this approach is suitable for replication of and comparison with previous studies which mainly analyzed the SNP rs17070145, there might be several other SNPs of the KIBRA gene that may impact cognitive function, so that a tagging SNP approach could be helpful for a future study design.

Moreover, the KIBRA rs17070145 polymorphism is found in an intron, and is supposed to be linked to causal polymorphisms, which are yet unknown. Further investigations are needed to determine the causative polymorphism in the KIBRA gene.

Finally the lack of main effects of SNP rs17070145 on cognitive measures in our study might be a problem of study power. We calculated a power of only ~20% to detect a mean difference of 0.1 in the z-scores, which is the about the average difference we observed in the four cognitive domains across genotypes. Hence the power to detect main effects is very low in our cohort.

However, some of the previous studies were based on a smaller study population and yet reported significant findings [START_REF] Nacmias | KIBRA gene variants are associated with episodic memory performance in subjective memory complaints[END_REF][START_REF] Schaper | KIBRA gene variants are associated with episodic memory in healthy elderly[END_REF]). These studies differed from our cohort with respect to the study population in terms of age and premorbidity, respectively. The largest study to date reporting an association between KIBRA and memory function is published by Bates and colleagues [START_REF] Bates | Association of KIBRA and memory[END_REF]). Based on more than 2000 participants the power to detect a significant difference in memory measures across genotypes was very high. However, Bates et al. did not apply tests of executive function, therefore, an even larger effect of rs17070145 on executive function cannot be excluded. Thus, the characteristics of the study participants, the neuropsychological tests applied and possible modifying effects should be carefully evaluated when analyzing and interpreting effects of the KIBRA gene on cognition.

In conclusion, the effect of KIBRA on cognitive functions is more complex than previously thought, depending on the cognitive domain that is being assessed. Furthermore, the effect is modulated not only by gender, but also seems to be modified by arterial hypertension, which itself is being influenced by genetic and environmental factors, and which importantly contributes to atherosclerosis and cognitive decline. Thus, a comprehensive assessment of these factors should be conducted in studies on association of common genetic polymorphisms on cognition, at least in elderly populations. A better understanding of the complex interactions between genetic profile and cardiovascular risk factors will help to develop preventive or therapeutic strategies for memory improvement, ideally tailored to each individual's genetic profile [START_REF] Mayor | Fitting the drug to the patient[END_REF].

Disclosure/Conflict of Interest

The project is supported by several public funding sources, which have no involvement in study design, in the collection, analysis, and interpretation of data, in the writing of the report or in the decision to submit the paper for publication. Heike Wersching, Katrin Guske, Sandra Hasenkamp, Claudia Hagedorn and Mira Schiwek report no disclosure. Stefanie Jansen received support from "Interdisziplinäres Zentrum für klinische Forschung" (IZKF, Floe3/004/08). Note, that except for male non-hypertensives, T-allele carriers performed worse in tests of executive function. This difference is particularly evident in women with arterial hypertension. Table 1:

Titles and legends to Figures

Tables

  al. 2001; Ruscheweyh et al. 2009; Witte et al. 2009). Besides these potentially "modifiable" conditions, genetic factors have been linked to cognitive performance (McClearn et al. 1997; Savitz et al. 2006). Common polymorphisms in several genes have been identified to influence cognitive function, including catechol-Omethyltransferase (Harris et al. 2005; Savitz et al. 2006), dopamine receptor D2 (Duan et al. 2003), brain-derived neurotrophic factor (Chen et al. 2006; Sklar et al. 2002), and apolipoprotein E (Bertram et al. 2007). Rather than directly accounting for cognitive abilities, genetic factors might interact with external modifiers (e.g. cardiovascular risk factors) in determining cognitive functions. Such interactions have already been demonstrated, e.g. for APO-E-carrier status and cholesterol in the development of dementia of the Alzheimer type (AD) (Chandra and Pandav 1998). A single nucleotide polymorphism (SNP) in intron 9 of the KIBRA gene (common C →T substitution, rs17070145, GenBank accession number NM_015238) has recently been associated with episodic memory function and cognitive flexibility (Almeida et al. 2008; Bates et al. 2009; Nacmias et al. 2008; Papassotiropoulos et al. 2006; Schaper et al. 2008; Zhang et al. 2009; Armin Schneider et al. 2010), and even with the risk of developing AD (Burgess et al. 2010; Corneveaux et al. 2008) The KIBRA protein is discussed to be involved in brain development and memory formation as a postsynaptic scaffold protein connecting cytoskeletal and signalling molecules (Johannsen et al. 2008; Yoshihama et al. 2009). However, results for episodic memory have been inconsistent across studies, ranging from the originally reported beneficial effects of the T-allele on memory functions (Almeida et al. 2008; Papassotiropoulos et al. 2006; Schaper et al. 2008; Yasuda et al. 2010; Vassos et al. 2010) to null findings (Need et al. 2008) and to even poorer memory in T-allele carriers (Nacmias et al. 2008). Considering other cognitive functions, two reports looked at an association of executive functions with the rs17070145 polymorphism: Zhang et al. found an inverse association of T-allele carrier status, revealing worse performance in the Wisconsin Card Sorting Test in T-allele carriers (Zhang et al. 2009). Schaper and colleagues, however, found no association of KIBRA with tasks of executive function (Schaper et al. 2008). With exception of the study by Vassos and colleagues (Vassos et al. 2010) who used a memory composite score previous findings were based on single neuropsychological tests (Almeida et al. 2008; Papassotiropoulos et al. 2006; Schaper et al. 2008), rather than composite scores of related cognitive tests. Moreover, only few studies so far have assessed gene-gene or gene-environment interactions, which are of particular relevance in a clinical context. One study by Zhang et al.

  were recruited from the population-based SEARCH-Health study (Systematic evaluation and alteration of risk factors for cognitive health), which aims to examine the impact of modifiable risk factors on cognitive aging (Knecht et al. 2008). The study was approved by the local ethics committee of the Medical Association Westfalen-Lippe. Community-dwelling individuals from 40 to 85 years of age were randomly selected based only on dates of birth from the register of residents from the city of Münster, Germany. They were invited to participate in the study by letter and recruited after giving informed consent. All participants received a structured clinical face-to-face interview, a physical examination by a trained study physician, a non-fasting blood sampling, a comprehensive neuropsychological assessment and cerebral magnetic resonance imaging (MRI) at 3.0 Tesla. The time period of baseline recruitment was 53 month (January 2004 to June 2008). The entire SEARCH baseline cohort comprised 729 individuals.

  KIBRA rs17070145 genotype on cognitive functions. Neither the composite scores nor the single memory tests were significantly associated with T-allele status. Hence we could not replicate recent findings of the association of rs17070145 and immediate and delayed memory performance. (2) For the first time we showed an interaction of the rs17070145 polymorphism (common C→T substitution within the ninth intron) of the KIBRA gene with arterial hypertension: In subjects with arterial hypertension T-allele carriers performed worse in executive functions compared to CC carriers. In subjects without a diagnosis of hypertension there was no effect of rs17070145 genotype on executive functions (3) We found gender to modify the effects of rs17070145 on performance in executive functions/ psychomotor speed showing a negative effect of the T-allele only in females whereas in men there was no difference in executive function across genotypes. When dividing the cohort into four strata by gender and hypertension status we observed an additive effect showing the largest genotype-related difference in executive function for women with arterial hypertension. Moreover, we found a significant interaction of gender and rs17070145 genotype for the cognitive domain of working memory, resulting from a different effect of gender on working memory by rs17070145 genotype. This means, in the group of T-allele carriers women performed significantly worse than men. Hence we see a modulation of gender-specific effects by KIBRA genotype rather than vice versa. A comprehensive evaluation of cognitive function was conducted in this study. A subsequent principal component analysis then reduced the single test scores to four main factors, i.e. the four clusters of tests that best explain the variance of the data, thereby reflecting physiologically meaningful phenotypes. Previous studies used only a restricted number of neuropsychological assessments, and found associations for only subtests of the respective domains tested (Almeida et al. 2008; Papassotiropoulos et al. 2006; Schaper et al. 2008; Zhang et al. 2009). However, even when analyzing the single components of the AVLT, we were not able to replicate the original findings (Papassotiropoulos et al. 2006; Schaper et al. 2008) on episodic memory functions. However, since these first reports, a number of studies have reported no association, or even an opposite association, of T-carrier status for memory performance (Nacmias et al. 2008; Need et al. 2008; Zhang et al. 2009), while others were able to corroborate the finding of a better memory performance in T-allele carriers (Almeida et al. 2008; Bates et al. 2009; Ehret 2010). Similarly, there are conflicting reports on the impact of the rs17070145 polymorphism on executive functions: Zhang et al. found an inverse association of T-allele carriers status, revealing worse performance in the Wisconsin Card Sorting Test in T-allele carriers (Zhang et al. 2009), whereas Schaper and colleagues found no association of KIBRA with the Trail-Making Test and a lexical fluency task (Schaper et al. 2008).

  While previous studies consistently corrected for age, gender, and education, they largely missed adjustment for vascular risk factors and lifestyle choices. Since the latter factors are known to influence development of atherosclerotic disease and dementia, they should be taken into account when examining gene-cognition associations, particularly in an elderly population in which a large proportion shows one or more of these risk factors (Wolf-Maier et al. 2003). Gene-risk factor interactions regarding cognitive performance have been found for other genes and vascular risk factors: APOE carrier status modified the negative effects of atherosclerosis and diabetes on cognition (Dore et al. 2009; Haan et al. 1999). Brain Derived Neurotrophic Factor (BDNF), which is involved in brain development and memory formation (Savitz et al. 2006) modified the effects of vascular risk factors on cognitive skills (Raz et al. 2008).

  al. 2009; Gold et al. 2007) and patient populations (Guerini et al. 2004; Delano-Wood et al. 2008) integrity of white matter fiber tracts connecting the prefrontal cortex to other cortical, as well as subcortical structures has been shown to correlate with success of learning and memory formation. Thus, prefrontal functions are particularly vulnerable to cerebral vascular damage (Prins et al. 2005). Our findings of worse effects for T-allele carrier status with regard to executive functions, as well as effect modification by the presence of arterial hypertension (resulting in even worse performance for executive functions in hypertensive T-allele carriers) may thus be due to a differential modification of prefrontal functions by vascular risk factors. An alternative explanation for the differential effects of T-allele carrier status on different cognitive domains has been proposed by Zhang and colleagues (Zhang et al. 2009): The authors postulated a differential effect of KIBRA carrier status on long-lasting memory (hippocampus-dependent) on the one hand, and cognitive flexibility, short-term memory and working memory (prefrontal-cortex-dependent) on the other hand. However, note that the classification of cognitive tests varies across studies. In our study, using a principal component analysis first to identify superordinate composite scores for cognitive domains, tests of cognitive flexibility fell within the category of executive functions/psychomotor speed. In the study by Zhang et al. the Wisconsin Card Sorting tests, another test of cognitive flexibility was designated as a test of working memory. Moreover, 5 min delayed free recall in the study by Papassotiropoulos et al. has been termed 'short-term memory' as well (Papassotiropoulos et al. 2006). Further studies will have to pay close attention to the definition of clinically significant cognitive domains from multiple neuropsychological tests first before progressing to correlation analyses. The use of composite scores per se needs to be handled with caution when trying to identify molecular pathways related to human cognition. Many SNP effects have shown to be highly specific (Egan et al. 2003) and therefore might be distorted when constructing artificial composite scores. We therefore used a principal component analysis to reveal the internal structure of the present data (Tabachnik and Fedell 2011). The composite scores extracted this way comprise tests that share common variance, i.e., they reflect underlying neuropsychological functions. For the cognitive domain of verbal memory the composite score is highly specific, reflecting hippocampal-dependent episodic memory performance. The domain "executive function" is of a more widespread character. Consisting of the subtests TMT-A, TMT-B, Stroop, and DSST, it measures multiple related skills such as cognitive flexibility and speed, attention, and visuomotor coordination (Lezak 2004). Hence, physiologically, several brain regions are involved.

Figure 1 :Figure 2 :

 12 Figure 1: Cognitive performance by cognitive domain and KIBRA rs17070145 genotype.

Figure 3 :

 3 Figure 3: Composite score of executive functions/ psychomotor speed by rs17070145 genotype,

Figure 4 :

 4 Figure 4: Composite score of executive functions/ psychomotor speed by rs17070145 genotype,

  Neuropsychometric test batterySeventeen neuropsychological subtests were grouped into clinically significant cognitive domains using principal component analysis.

Table 2 :

 2 Sample characteristics

	Characteristic	CT/TT genotypes	CC genotype	p-value 1
		N= 310	N= 235
	Age, years, mean ± SD	63.2 ± 7.2	64.0 ± 6.7	0.235
	Female, %	56.5	54.9	0.728
	Education
	-five years of secondary school, %	30.0	35.7
	-seven years of secondary school, %	22.9	20.9
	-nine years of secondary school, %	11.0	8.1
	-tertiary education, %	36.1	35.3	0.431
	Arterial hypertension, %	35.2	34.9	1.000
	Diabetes, %	5.2	3.4	0.401
	Lifestyle index, mean ± SD	13.75 ± 3.00	14.10 ± 2.96	0.182

p-value denoting significance between groups

Acknowledgements

This work has been supported by a grant from the "Interdisziplinäres Zentrum für klinische Forschung" (IZKF, Floe3/004/08), the DFG to Agnes Flöel (Fl 379-4/2, FL 379-8/1; and DFG-Exc 257), the BMBF to Agnes Flöel (FKZ 0315673A and 01EO0801), the BMBF-Competence Network Mednet Atrial Fibrillation, the European Commission (MRTN), and the Neuromedical Foundation Muenster. Stefan Knecht has received speaker's honoraria from Janssen-Cilag and Boehringer Ingelheim and travel support from Bayer Health Care. Eva Brand is supported by a Heisenberg professorship from the DFG (Br 1589/8-2). Joachim Kremerskothen and Kerstin Duning were supported by IMF Münster 2009-2011 and DFG Förderung Pavenstädt/Kremerskothen PA 483/14-2.

Veronica Witte, Jürgen Wellmann and Hubertus Lohmann report no disclosure. Kerstin Duning and Joachim Kremerskothen were supported by DFG grant PA 483/14-2 and received support by IMF Münster 2009-2011. Stefan Knecht has received speaker's honoraria from Janssen-Cilag and Boehringer Ingelheim and travel support from Bayer Health Care. Eva Brand is supported by a Heisenberg professorship from the DFG (Br 1589/8-1 and BR 1589/8-2). Agnes Flöel is supported by a grant from the "Interdisziplinäres Zentrum für klinische Forschung" (IZKF, Floe3/004/08), the DFG (FL 379-4/2, FL 379-8/1; and DFG-Exc 257), the BMBF (FKZ 0315673A and 01EO0801), the BMBF-Competence Network Mednet Atrial Fibrillation, the European Commission (MRTN), and the Neuromedical Foundation Muenster.

Supplementary Table 1:

Regression analyses on the effect of rs17070145 on the AVLT subtests (main analysis, CC is reference group, basic model)


Jean-Pierre Stockis

Joseph Tadjuidje-Kamgaing

Jürgen Franke
email: franke@mathematik.uni-kl.de

A note on the identifiability of the conditional expectation for the mixtures of neural networks

Keywords: Mixture Models, Neural networks, Identifiability

We consider a generalized mixture of nonlinear AR models, a hidden Markov model for which the autoregressive functions are single layer feedforward neural networks. The non trivial problem of identifiability, which is usually postulated for hidden Markov models, is addressed here.

A c c e p t e d m a n u s c r i p t 1 Introduction

Neural networks provide a flexible tool for learning an unknown mapping from input space to output space in the presence of noise. It has been proven that neural networks have nice asymptotic properties under some general assumptions (see, e.g, White (1984) or [START_REF] Franke | Estimating Market risk with neural networks[END_REF] and some of the references therein). We assume in this paper that our time series data obey the following hidden Markov model

X t = K k=1 S tk (m k (X t-1 , • • • , X t-p ) + t ) (1)
with The process {X t } passes through different dynamics depending on {Q t }. This type of mixture of nonparametric autoregressive models plays an important role in practice, for example in econometrics (see, e.g. [START_REF] Franke | Statistics of Financial Markets[END_REF], [START_REF] Gourieroux | ARCH Model and Financial Application[END_REF]) where it is often not realistic to assume that the process is driven by only one trend function throughout time.

S tk = 1 for Q t = k, 0 otherwise. ( 2
)
{Q t } is a hidden Markov chain with values in {1, • • • , K}, K ∈ N,
Now, a natural way of estimating this type of mixture models is to replace the unknown m k (x) in equation ( 1) by

m k (x) = ν 0k + H h=1 ν hk ψ(< α hk , x > +β hk ), k = 1, • • • , K (3)
where m k (x) is a one layer feedforward neural network with H hidden neurons, <, > is the classical scalar product on R p and ψ the activation functions, taken here, for sake of simplicity to be the logistic function. Further, we remind that the network parameters ν hk , β hk ∈ R and α hk ∈ R p .

Compared to a single neural network this framework has the advantage of mimicking well the structure of the data and also it allows for the detection of the change points in the dynamics of the models.

Considering the asymptotic behavior of the parameter estimates, it is well known that an important theoretical problem with neural networks is the unidentifiability of the parameters (for a definition see, e.g. [START_REF] Hwang | Prediction Intervals for Artificial Neural networks[END_REF]. If identifiability is not satisfied, asymptotic properties like consistency and asymptotic normality of the parameter estimates are not guaranteed anymore and resampling techniques like bootstrap will not work properly (see, [START_REF] Franke | Bootstrapping Neural Networks[END_REF]). In this paper we are providing sufficient conditions for identifiability (up to the family of transformations generated by equation (5) in [START_REF] Hwang | Prediction Intervals for Artificial Neural networks[END_REF]); in the remainder of this paper for conciseness we shall simply write identifiability.

A c c e p t e d m a n u s c r i p t 3 Conclusion

In conclusion, we can notice that the only condition on our driving hidden Markov chain {Q t } for ensuring the identifiability of the parameters of our mixture model is a weak condition, namely the invertibility of the transition probability matrix. Furthermore, this result can obviously be extended to a large variety of parametric models with switching regimes driven by a hidden Markov process.

Under those rather weak conditions, we can identify the parameter from the conditional expectation (5). For practical use, we have to be able to estimate (5) and, as an auxiliary step, S t for an increasing set of time points t. If the state space of the hidden process {Q t } is infinite, that is possible. For a hidden Markov chain with infinite state space, we have to impose some additional assumptions which guarantee that not all states are transient.

The identifiability of the parameters of our model is an important step towards the proof of asymptotic results for the estimators, e.g. their consistency. Asymptotic inference and practical results based on our model are beyond the scope of this paper and will be analyzed in further research.

  Pötscher, B.M.,Prucha, I.R. (1997), Dynamic Nonlinear Econometric Models:Asymptotic Theory (Springer, Germany).White, H. (1984), Asymptotic Theory for Econometricians (Academic Press, New York).

  and with transition probability matrix A = (a ij ) 1≤i,j≤K . m k , k = 1, • • • , K, are unknown real valued functions. The random errors t are i.i.d. with mean zero and support R.

Acknowledgements

The authors would like to thank the referees for their helpful comments.

The work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the priority research program 1114 Mathematical Methods of Time Series and Digital Image Analysis, the center of excellence Dependable Adaptive Systems and Mathematical Modelling funded by the state of Rhineland-Palatinate as well as the "Graduiertenkolleg Mathematik und Praxis" and by the Fraunhofer ITWM.

A c c e p t e d m a n u s c r i p t

To achieve this, we will mainly generalize the works done for simple neural networks by Hwang and Ding (1997, p. 749) and Farinas et al. (2004).

Results

Let us define S t = (S t1 , • • • , S tK ) and assume

(4)

Now we are focusing on the identifiability of

While it is clear that the expectation needs to be conditioned on the X t-1 , • • • , X t-p , at first sight it may be surprising to condition also on S t-1 . Notice first that without S t-1 , terms of the form P(S t | S t-1 , X t-1 , X t-2 , • • • ) would appear in the conditional expectation, terms which are in general not tractable in practice. Moreover, we condition on S t-1 and not on S t and although we do not know S t-1 either, it is possible to get good estimates for it given X t-1 , X t-2 , • • • Finally, from a theoretical point of view, we can consider S t-1 as an exogenous variable as in Farinas et al (2004).

Let us now focus on the conditional expectation,

Now, we recall that A = (a ij ) 1≤i,j≤K is the transition probability matrix of the hidden process {Q t } with

If we consider for example S

Similarly, if S t-1 is the jth unit vector,

A c c e p t e d m a n u s c r i p t

one can hence write

where <, > is the classical scalar product on R K .

To achieve the announced results we shall need the following assumptions

A. 1 ∀k, m k is not redundant ( i.e. there exists no other network with fewer hidden neurons that represents exactly the same relationship function).

A. 2 The transition probability matrix A of the hidden process {Q t } is invertible.

Proposition 1 Consider that assumptions A.1, A.2 and A.3 hold. Then, the identifiability of the parameters of our model is achieved.

Proof: In the following proof, we will focus on the case m : R -→ R. The multidimensional case can easily be reduced to the one dimensional case in a similar way as in the paper by [START_REF] Hwang | Prediction Intervals for Artificial Neural networks[END_REF].

The conditions of Theorem 2.3(a) of [START_REF] Hwang | Prediction Intervals for Artificial Neural networks[END_REF] and thus the identifiability of the parameters are satisfied if

To show this implication, it suffices to show that

Now consider assumption A.2 and

and hence ν 0k = ν hk = 0, ∀ h, k (see Lemma 2.7 of [START_REF] Hwang | Prediction Intervals for Artificial Neural networks[END_REF]).


Baisuo Jin

Guangming Pan

Baiqi Miao

Central limit theorem of random quadratics forms involving random matrices

Keywords: Central limit theorem, Large dimensional matrix, Quadratic

Introduction

Let random variables {v ij , i, j = 1, • • • } be independent and identically distributed (i.i.d.) with Ev 11 = 0 and Ev 2 11 = 1. Define

s i = 1 √ N (v i,1 , • • • , v i,N ) T , i = 1, • • • , K, S 1 = (s 2 , s 3 , • • • , s K ), S = (s 1 , s 2 , • • • , s K ). Let λ 1 ≤ λ 2 ≤ • • • ≤ λ N be the eigenvalues of N K S 1 S T 1

A c c e p t e d m a n u s c r i p t

and F N be the empirical spectral distribution of N K S 1 S T 1 , i.e.,

F N (x) = N i=1 I(λ i ≤ x) N .
It is well known that if Ev 2 11 < ∞ and lim

N →∞ K N = y > 0, then F N (x)
a.s.

-→ F y (x),

where F y (x) denotes the limiting spectral distribution. There are many literatures about the convergence of F N (x) (see [START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices, a review[END_REF] and the references given there).

Lately, the theory of large dimensional random matrices has been a popular tool in the research of the asymptotic problem of code division multiple-access (CDMA) communication systems. For example, one may refer to the work of [START_REF] Tse | Linear multiuser receivers :effective interference, effective bandwidth and capacity[END_REF] and [START_REF] Tse | Linear multiusers in random enviroments[END_REF]. The application in the communication also puts forward challenges to the large dimensional matrix theory. For instance, to analyze the large system performance of the signal-to-inference (SIR) of some kind of receiver, the random quadratic forms involving random matrix, s t 1 (S 1 S T 1 ) i s 1 and s t 1 (SS T ) i s 1 , were introduced by Trichard, et al. (2002). However, only the convergence in probability for a special random sequence was given in their work. As we know, the central limit theorem is more powerful than the convergence in probability. Even in wireless communications, central limit theorem can also be used to characterize the fluctuation of the performance of SIR around the asymptotic value. The main object of this paper is to prove that s t 1 (SS T ) i s 1 is asymptotic normal under the assumption Ev 4 11 < ∞. Here we would like to emphasize that the random vector s 1 is not independent of the random matrix SS T .

Main results

For convenience, some notations are firstly introduced. Let

α (N ) i = s T 1 (S 1 S T 1 ) i s 1 , β (N ) i = s T 1 (SS T ) i s 1 , Xi = y i (1+ √ y) 2 (1- √ y) 2 t i dW 0 (F y (t)) y N = K N , φ i (y N ) = y i N x i dF y N (x), X i = Xi + ξ √ 2 y i x i dF y (x)

A c c e p t e d m a n u s c r i p t

where F y (t) is the limiting spectral distribution, W 0 (t) is a Brownian bridge independent of ξ, which is N(0, Ev 4 11 -1). Here F y N (x) is obtained from F y (x) by replacing y with y N and other analogues below such as φ i (y N ) have similar meanings. Moreover, i ≥ 1 is required in the definition of X i . Our main results are as below.

Theorem 2.1 Assume that {v ij } are independent and identically distributed (i.i.d.) for all i and j with Ev 11 = 0, Ev 2 11 = 1 and Ev 4 11 < ∞. In addition, suppose lim

N →∞ K N = y > 0. Then for i ≥ 1 N 2 (β (N ) i -β i (y N )) D → Y i
where

Y i = X i + φ 0 (y)Y i-1 + β i-1 (y)X 0 + φ 1 (y)Y i-2 + β i-2 (y)X 1 + • • • + φ i-1 (y)Y 0 + β 0 (y)X i-1
and

β i (y N ) = φ i (y N ) + φ 0 (y N )β i-1 (y N ) + φ 1 (y N )β i-2 (y N ) + • • • + φ i-2 (y N )β 1 (y N ) + φ i-1 (y N )β 0 (y N ), with Y 0 = X 0 = ξ √ 2 , φ 0 (y N ) = φ 0 (y) = β 0 (y N ) = β 0 (y) = 1
and the corresponding term being zero if some subscript is negative.

Further, we have a multidimensional version of central limit theorem.

Theorem 2.2 If the conditions of Theorem 2.1 are satisfied, then

{ N 2 (β (N ) i -β i (y N ))} ∞ i=1 D -→ {Y i } ∞ i=1 ,
where D denotes weak convergence on R ∞ .

A c c e p t e d m a n u s c r i p t 3 The proof of theorem

To simplify the notations, the constant c may denote different values at different occasions in what follows. Before proceeding, we first need some lemmas.

Lemma 3.1 Let g N (s 1 ) be a function of the random vector s 1 and ξ N be a random variable. Suppose that for

u ∈ C(F ), v ∈ C(G), lim N →∞ P (ξ N ≤ u|s 1 ) = F (u), a.s., lim N →∞ P (g N (s 1 ) ≤ v) = G(v)
where F (u) and G(v) both are the non-random distribution functions, C(F ) and C(G) denote the sets of the continuity points of F (u) and G(v), respectively. Then

lim N →∞ P (ξ N ≤ u, g N (s 1 ) ≤ v) = F (u)G(v).

Proof.

It is observed that

P (ξ N ≤ u, g N (s 1 ) ≤ v) = I(g N (s 1 ) ≤ v)P (ξ N ≤ u|s 1 )dP.
It follows, with the help of the Fatou lemma, that lim sup

N →∞ |P (ξ N ≤ u, g N (s 1 ) ≤ v) -F (u)P (g N (s 1 ) ≤ v)| ≤ lim sup N →∞ | I(g N (s 1 ) ≤ v)[P (ξ N ≤ u|s 1 ) -F (u)]dP | ≤ lim sup N →∞ |[P (ξ N ≤ u|s 1 ) -F (u)]|dP = 0.
On the other hand

I(g N (s 1 ) ≤ v)F (u)dP → F (u)G(v).
Thus the proof is complete.

Based on Theorem 4.1 of Silverstein (1990), we next establish a lemma concerning multidimensional central limit theorem.

A c c e p t e d m a n u s c r i p t

Lemma 3.2 Under the assumptions of Theorem 2.1, the following holds:

N 2 α (N ) i -φ i (y N ), • • • , α (N ) 1 -φ 1 (y N ), α (N ) 0 -1, (s T 1 s 1 ) 2 -1 D -→ (X i , • • • , X 1 , η 1 , η 2 )
where

η 1 = ξ √ 2 , η 2 =
√ 2ξ and D denotes weak convergence.

Proof. For any real numbers t -1 , t 0 , t 1 , • • • , t i and any positive integer i, write

N 2 t i α (N ) i + • • • +t 1 α (N ) 1 + t 0 α (N ) 0 + t -1 (s T 1 s 1 ) 2 -t i φ i (y N ) -• • • -t 1 φ 1 (y N ) -t 0 -t -1 I (1) N + I (2) N
where

I (1) N = N 2 [t i ( α (N ) i ||s 1 || 2 - 1 N tr(S 1 S T 1 ) i )||s 1 || 2 + t i 1 N tr(S 1 S T 1 ) i )(||s 1 || 2 -1) + • • • + t 1 ( α (N ) 1 ||s 1 || 2 - 1 N tr(S 1 S T 1 ))||s 1 || 2 + t 1 1 N tr(S 1 S T 1 )(||s 1 || 2 -1)
+t 0 (α

(N ) 0 -1) + t -1 ((s T 1 s 1 ) 2 -1)
] and

I (2) N = N 2 ( 1 N t i tr(S 1 S T 1 ) i +• • •+ 1 N t 1 tr(S 1 S T 1 )-t i φ i (y N )-• • •-t 1 φ 1 (y N )),
where • denotes the Euclidean norm. We first consider the term I

(1)

N . Define I (3) N = √ N(s T 1 s 1 -1), f (u) = t i √ 2 y i x i dF y (x) + t i-1 √ 2 y i-1 x i-1 dF y (x) + • • • + t 1 √ 2 y xdF y (x) + t 0 √ 2 u + t -1 √ 2 u 2 ,
and

ξ N = N 2 [t i ( α (N ) i ||s 1 || 2 - 1 N tr(S 1 S T 1 ) i )||s 1 || 2 +• • •+t 1 ( α (N ) 1 ||s 1 || 2 - 1 N tr(S 1 S T 1 ))||s 1 || 2 ]

A c c e p t e d m a n u s c r i p t

By the classical central limit theorem,

I (3) N D -→ ξ,
Here ξ is defined in the second part. This, together with Theorem 2.5.2 of Chen (1999), implies that

√ N(f (||s 1 || 2 ) -f (1)) D -→ aξ (3.1)
where

a = t i √ 2 y i x i dF y (x) + t i-1 √ 2 y i-1 x i-1 dF y (x) + • • • + t 1 √ 2 y xdF y (x) + t 0 √ 2 + √ 2t -1 .
Appealing to the strong law of large numbers gives lim

N →∞ ||s 1 || 2 = 1, lim N →∞ N i=1 ( 1 √ N v 1i ||s 1 || ) 4 = 0 a.s..

It follows that lim

N →∞ P (ξ N ≤ u|s 1 ) = P (t i Xi + t i-1 Xi-1 + • • • + t 1 X1 ≤ u) a.s., (3.2)
where we make use of Theorem 4.1 of Silverstein (1990), Theorem 4.1 of Billingsley (1968) and the fact that s 1 is independent of S 1 S T 1 . Combining (3.1), (3.2) with Lemma 3.1, we obtain lim

N →∞ P (ξ N ≤ u, √ N(f (||s 1 || 2 ) -f (1)) ≤ v) = P (t i Xi + t i-1 Xi-1 + • • • + t 1 X1 ≤ u, aξ ≤ v)
Hence, using Theorem 4.1, Corollary 1 to Theorem 5.1 of Billingsley (1968) and the fact that lim

N →∞ 1 N tr(S 1 S T 1 ) i = y i x i dF y (x) a.s. ( 3.3)
( One may see [START_REF] Jonsson | Some limit theorems for the eigenvalues of a sample covariance matrix[END_REF] or Yin (1986)), we have

I (1) N D → t i X i + t i-1 X i-1 + • • • + t 1 X 1 + t 0 η 1 + t -1 η 2 .

A c c e p t e d m a n u s c r i p t

Finally, it suffices to show that

I (2) N i.p.
-→ 0.

However this is immediately given by Theorem 1.1 of Bai and Silverstein ( 2004). Thus we are done according to Crammer-Wold method.

From now on, we begin with the proof of Theorem 2.1. Recall the definitions of β (N ) i , we know that the present difficulty is that s 1 is not independent of SS T . To overcome it, we mainly employ the induction.

Proof of Theorem 2.1. First, one can easily derive the following equality:

β (N ) i = α (N ) i + α (N ) 0 β (N ) i-1 + α (N ) 1 β (N ) i-2 + • • • + α (N ) i-2 β (N ) 1 + α (N ) i-1 β (N ) 0 , (3.4) where α (N ) i = β (N ) i = 0, if i < 0 and α (N ) 0 = β (N ) 0 = s T 1 s 1 .
We proceed by induction on i. By Lemma 3.2, for i = 1, we have

N 2 (β (N ) 1 -β 1 (y N )) = N 2 (α (N ) 1 -φ 1 (y N )+(s T 1 s 1 ) 2 -1) D → Y 1 X 1 + √ 2ξ.
Here, we should mention a key fact that the asymptotic distribution of the random variable β and (s T 1 s 1 ) 2 . For i ≤ m, we need make the following two inductive hypothesis : a): Theorem 2.1 is true for i ≤ m , b): For i ≤ m, the asymptotic distribution of the random variable β (N ) i can be derived by the asymptotic distribution of the linear combination of the random variables {α (N ) j , 0 ≤ j ≤ i} and (s T 1 s 1 ) 2 . In other words, there exist constant number t 0 , t 1 • • • , t i , t -1 such that N 2 (β

(N ) i -β i (y N )) - N 2 t i α (N ) i + • • • + t 0 α (N ) 0 + t -1 (s T 1 s 1 ) 2 -t i φ i (y N ) -• • • -t 0 -t -1 i.p.
-→ 0 (3.

  the asymptotic distribution of the linear combination of the random variables α (N ) 1

5 ) 1 ) 9 )

 519 2m → y 2m x 2m dF y (x), as N → ∞. Based on (3.6)-(3.8), via Theorem 4.1 of Billingsley (By the above argument, one see easily that inductive hypothesis b) is also true for i = m + 1. Hence the proof is complete. Proof of Theorem 2.2: This Theorem follows easily upon Cramer-Wold method and the argument of Theorem 2.1. Details are omitted.

Acknowledgments

The authors would like to thank an anonymous referee for his constructive comments.

Research supported by Grant 10471135 and 10571001 from the National Natural Science Foundation of China. MSC 2000 subject classifications. Primary 60F05; secondary 62P30.

A c c e p t e d m a n u s c r i p t

Suppose i = m + 1. It follows from (3.4) that

By Lemma 3. On the other hand, by Lemma 2.7 of Bai and Silverstein (1998), we have


Roel Braekers
email: roel.braekers@uhasselt.be

Noël Veraverbeke

A conditional Koziol-Green model under dependent censoring

Keywords: Archimedean copula, dependent censoring, fixed design regression, Koziol-Green model MSC2000 Classification: Primary 62N01, Secondary 62N02, 62G08

 to also accommodate for dependent censoring and in this way introduce a model with two different types of informative censoring. We derive in this model a copula-graphic estimator for the conditional distribution of the lifetime and establish an exponential bound and an almost sure asymptotic representation which serve as starting points for an almost sure consistency and an asymptotic normality result. Afterwards we apply this estimator on a real data set about the survival of Atlantic halibut.

Introduction

At fixed design points 0 ≤ x 1 ≤ . . . ≤ x n ≤ 1, we have nonnegative responses Y 1 , . . . , Y n such as survival times or failure times. These responses are independent random variables and the distribution function of the response Y i at x i will be denoted by F xi (t) = P (Y i ≤ t). In many clinical or industrial trials, the responses Y 1 , . . . , Y n are subject to random right censoring. For each response, there is a censoring variable C i with conditional distribution function G xi (t) = P (C i ≤ t). The observed random variables at design point x i are in fact Z i and δ i (i = 1, . . . , n), with

Z i = min(Y i , C i )
and

δ i = I(Y i ≤ C i ).
At a given fixed design value x ∈ [0, 1], we write F x , G x , H x for the distribution function of respectively the response Y x , the censoring variable C x and the observed variable Z x = min(Y x , C x ) at x. Also we will write δ x = I(Y x ≤ C x ). Note that for the design variables x i , we write

Y i , C i , Z i , F i , . . . instead of Y xi , C xi , Z xi , F xi , . . ..
In order to estimate uniquely the distribution function F x from the observed data, we have to make an assumption about the dependence between the Y i and C i for each i [START_REF] Tsiatis | A nonidentifiability aspect of the problem of competing risks[END_REF]). It is very common in survival analysis to assume independence between these random variables (conditional on the covariate).

However we see that in some practical situations this assumption clearly does not hold. For example in medicine when the event of interest is death due to a given disease and the censoring event is death due to other diseases. In industrial testing, it may occur that some piece of equipment is taken away (is censored) because it shows some sign of future failure. Therefore a dependence model is used in which the dependence structure is given by specifying a copula for the joint distribution of Y x and C x . Assume that the joint survival function of the response Y x and the censoring variable C x at x can be written as

S x (t 1 , t 2 ) = P (Y x > t 1 , C x > t 2 ) = C x ( Fx (t 1 ), Ḡx (t 2 ))
where C x is a known copula function depending in a general way on x and Fx (t) (resp. Ḡx (t)) is the survival function of Y x (resp. C x ) at x. Without covariates x, this idea was introduced by [START_REF] Zheng | Estimates of marginal survival for dependent competing risks based on an assumed copula[END_REF]. However their copula-graphic estimator had no closed form expression. [START_REF] Rivest | A martingale approach to the copula-graphic estimator for the survival function under dependent censoring[END_REF] got around this problem by focusing on the class of Archimedean copulas. As in [START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF], we will extend their ideas to the fixed design regression case. We assume that at a fixed design value x ∈ [0, 1], the joint survival function is given by

S x (t 1 , t 2 ) = ϕ [-1] x (ϕ x ( Fx (t 1 )) + ϕ x ( Ḡx (t 2 ))) (1)
where, for each x, ϕ x : [0, 1] → [0, +∞] is a known continuous, convex, strictly decreasing function with

ϕ x (1) = 0. ϕ [-1] x
is the pseudo-inverse of ϕ x , as defined in [START_REF] Nelsen | An introduction to copulas[END_REF] and given by

ϕ [-1] x (s) = ϕ -1 x (s) 0 ≤ s ≤ ϕ x (0) 0 ϕ x (0) ≤ s ≤ +∞ .
We note from (1) that,

1 -H x (t) = Hx (t) = S x (t, t) = ϕ [-1] x (ϕ x ( Fx (t)) + ϕ x ( Ḡx (t))). (2)
However in the design of some clinical trials, we see another type of informative censoring in which the distribution function of the failure time and the censoring time are related. [START_REF] Koziol | A Cramér-von Mises statistic for randomly censored data[END_REF] considered a sub-model for the Kaplan-Meier estimator in which they assumed that the survival function of the censoring variable is a power of the survival function of the failure time. This sub-model has the advantage that the estimator for the distribution function of the failure time has a simpler form.

Other results, like a test-procedure to check the validity of this assumption [START_REF] Csörgő | On the Koziol-Green model for random censorship[END_REF], [START_REF] Csörgő | Testing for the proportional hazards model of random censorship[END_REF]) were derived. [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF] extended this model for the fixed design regression situation.

In this paper we will further extend this sub-model to the case where the failure time Y x depends on the censoring variable C x . We therefore use the fact that the classical Koziol-Green model is characterized by the conditional independence of Z x and δ x . Translating the latter property into our model (1) leads to the following assumption: for each covariate value x ∈ [0, 1],

Ḡx (t) = ϕ [-1] x (β x ϕ x ( Fx (t))), ∀t > 0 (3)
where β x > 0 is a constant depending only on x.

When we consider both types of informative censoring, we rewrite (2) as

Hx (t) = ϕ [-1] x ϕ x ( Fx (t)) + β x ϕ x ( Fx (t)) = ϕ [-1] x (β x + 1)ϕ x ( Fx (t)) . (4)
This relation will be used to find a conditional distribution estimator F xh for F x where x ∈ ]0, 1[ is a fixed design value. We organize this paper as follows. In Section 2, we define the distribution function estimator F xh and show that it is an extension of the conditional Koziol-Green estimator, as it was studied by [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF]. After specifying some assumptions in Section 3, we give for this estimator an exponential bound with consistency result in Section 4. In Section 5, we derive an almost sure asymptotic representation which we use in Section 6 to find an asymptotic normality result for this estimator. In Section 7 we apply the estimator to a practical situation in which we explore different choices for the generator function ϕ x . In the Appendix we give the proofs of the results in Section 4, 5 and 6.

The conditional Koziol-Green model

We develop in this section a non-parametric estimator for the conditional distribution function F x of the failure time Y x under two different types of informative censoring. As first type, we assume that the failure time Y x depends on the censoring variable C x as given in (1). While for the second type, we assume that the distribution of the censoring variable C x is related to the distribution function of Y x as given by (3).

From (4), we find an estimator for F x since we can rewrite this equation as

Fx (t) = ϕ [-1] x γ x ϕ x ( Hx (t)) (5)
with

γ x = 1 β x + 1 = P (δ x = 1).
The last equation follows from

p x1 = P (δ x = 1) = +∞ 0 - ∂ ∂t 1 S x (t 1 , t 2 ) t=t1=t2 dt = +∞ 0 ϕ ′ x ( Fx (t)) ϕ ′
x ( Hx (t))

dF x (t)

p x0 = P (δ x = 0) = +∞ 0 - ∂ ∂t 2 S x (t 1 , t 2 ) t=t1=t2 dt = β x +∞ 0 ϕ ′ x ( Fx (t)) ϕ ′
x ( Hx (t))

dF x (t) so that β x = px0 px1 .
To find an estimator for F x in this model, we replace in (5), the different quantities H x (t) and γ x by estimators. As in other work with a non-parametric regression problem [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF], Braekers and Veraverbeke ( 2005)), we consider estimators which involve a sequence of smoothing weights {w ni (x, h n )}, depending on a positive bandwidth sequence {h n }, tending to zero, as n → +∞. In our present situation of fixed design points, it is customary to take the Gasser-Müller type weights, given by

w ni (x, h n ) = 1 c n (x, h n ) xi xi-1 1 h n K x -z h n dz, (i = 1, . . . , n) c n (x, h n ) = xn 0 1 h n K x -z h n dz. (6)
Here x 0 = 0 and K is a known probability density function (kernel).

For the conditional distribution function H x (t), we take a Stone type estimator [START_REF] Stone | Consistent nonparametric regression[END_REF]) given by

H xh (t) = n i=1 w ni (x, h n )I(Z i ≤ t).
A similar estimator is taken for the exponent γ x and is given by

γ xh = n i=1 w ni (x, h n )I(δ i = 1).
Hence we find an estimator for the conditional distribution function F x (t) by

Fxh (t) = ϕ [-1] x γ xh ϕ x ( Hxh (t)) .
Note that the estimator Fxh (t) has a simpler structure than the copula-graphic estimator of [START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF] for the more general model under dependent censoring. Furthermore we see that in our estimator the estimator for γ x only depends on the δ i while the estimator for H x (t) only depends on the Z i . This result follows from assumption (3), which is equivalent to the assumption that Z x and δ x are conditionally independent.

If we take ϕ x (t) = -log(t), we see that this estimator equals the estimator of [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF] as we expected.

Regularity conditions

For the design points x 1 , . . . , x n we write ∆ n = min 1≤i≤n (x i -x i-1 ) and ∆n = max

1≤i≤n (x i -x i-1 ). The notations K ∞ = sup u∈I R K(u), K 2 2 = +∞ -∞ K 2 (u)du, µ K 1 = +∞ -∞ uK(u)du, µ K 2 = +∞ -∞
u 2 K(u)du will be used for the kernel K.

We use the following assumptions on the design and on the kernel.

(C1) x n → 1, ∆n = O(n -1 ), ∆n -∆ n = o(n -1 ). (C2) K is a probability density function with finite support [-M, M ] for some M > 0, µ K 1 = 0 and K Lipschitz of order 1.
The assumption (C1) expresses that the chosen design points are asymptotically equidistant points, selected uniformly over the whole interval [0, 1]. This implies that, for c n (x, h n ) defined in (6), c n (x, h n ) = 1 for n sufficiently large. Therefore we may take c n (x, h n ) = 1 in all proofs of the asymptotic results.

If L is any distribution, then T L denotes the right endpoint of its support (T L = inf{t : L(t) = L(+∞)}). We note that T Hx = T Fx = T Gx . To obtain our results, we need some smoothness conditions. For a fixed 0

< T < T Fx , (C3) Ḟx (t) = ∂ ∂x F x (t), Fx (t) = ∂ 2 ∂x 2 F x (t) exist and are continuous in (x, t) ∈ [0, 1] × [0, T ] (C4) βx = ∂ ∂x β x , βx = ∂ 2 ∂x 2 β x exist and are continuous in x ∈ [0, 1]
The generator ϕ x (v) of the Archimedean copula needs to satisfy the following properties.

(C5) ϕ ′ x (v) = ∂ ∂v ϕ x (v) and ϕ ′′ x (v) = ∂ 2 ∂v 2 ϕ x (v)
are Lipschitz in the x-direction with a bounded Lipschitz constant, and

ϕ ′′′ x (v) = ∂ 3 ∂v 3 ϕ x (v) ≤ 0 exists and is continuous in (x, v) ∈ [0, 1]×]0, 1].
These assumptions and the fact that ϕ x is a generator for an Archimedean copula, give that ϕ

′ x (v) is monotone increasing with ϕ ′ x (v) < 0 and ϕ ′′ x (v) is monotone decreasing with ϕ ′′ x (v) ≥ 0.

Exponential Bound and Strong Consistency

In the first theorem we establish conditions under which F xh (t) is a strongly consistent estimator for F x (t) and we also obtain the rate of this convergence. These results follow from an exponential inequality result. We postpone the proofs of these results to the Appendix.

Theorem 1. Assume (C1) -(C5), h n → 0, T < T Fx , (a) For ε > 0, n sufficiently large and

η 1 ≥ 2( γx ∞ ∆n + γx ∞ µ K 2 h 2 n ), η 2 ≥ max √ 6 K 2 (nh n ) -1/2 , 2( Ḣx ∞ ∆n + Ḧx ∞ µ K 2 h 2 n )
we have

P sup 0≤t≤T |F xh (t) -F x (t)| > ε ≤ 4e -C1η 2 1 nhn + d 0 η 2 nh n e -d1nhnη 2 2 /4
where C 1 , d 0 , d 1 are constants and η 1 ,

η 2 satisfy η 1 = ε 2M1(γx-η1) and η 2 = ε 2M2( Hx(T )-η2) with M 1 (z) = max 0≤y≤1 -ϕx(y) ϕ ′ x (ϕ -1 x (zϕx(y))) and M 2 (z) = max 0≤y≤1 yϕ ′ x (z) ϕ ′ x (ϕ -1 x (zϕx(y))) . (b) If nh 5 n log n = O(1), then, as n → +∞, sup 0≤t≤T |F xh (t) -F x (t)| = O (nh n ) -1/2 (log n) 1/2 a.s.

Almost sure representation

As in the work of [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF] for the conditional Koziol-Green estimator and in [START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF] for the copula-graphic estimator, we derive in this section an almost sure representation for the estimator F xh (t). In this result we rewrite this estimator as a weighted sum plus a remainder term. This representation forms a basic tool in finding further asymptotic results.

For the proof of Theorem 2 we again refer to the Appendix.

Theorem 2. Assume (C1) -(C5), h n → 0, log n nhn → 0,

nh 5 n log n = O(1), T < T Fx . Then, for t < T Fx , F xh (t) -F x (t) = n i=1 w ni (x, h n )g tx (Z i , δ i ) + R n (x, t)
where

g tx (Z i , δ i ) = - ϕ x ( Hx (t)) ϕ ′ x ( Fx (t)) (I(δ i = 1) -γ x ) + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) (I(Z i ≤ t) -H x (t))
and as n → +∞, sup

0≤t≤T |R n (x, t)| = O((nh n ) -1 log n) a.s.

Asymptotic normality

In Theorem 3, we show the asymptotic normality of (nh n ) 1/2 (F xh (t) -F x (t)). Due to the asymptotic representation in Theorem 2, we only have to use the main term to find the bias and variance expressions of this result. The proof of Theorem 3 is given in the Appendix.

Theorem 3. Assume (C1) -(C5), T < T Hx , (a) if nh 5 n → 0 and (nh n ) -1/2 log n → 0, then for t ≤ T , as n → ∞,

(nh n ) 1/2 (F xh (t) -F x (t)) → N (0, s 2 x (t)) (b) If h n = Cn -1/5
for some C > 0, then for t ≤ T , as n → ∞,

(nh n ) 1/2 (F xh (t) -F x (t)) → N (b x (t), s 2 x (t))
where

b tx = 1 2 µ K 2 C 5/2 -ϕ x ( Hx (t)) ϕ ′ x ( Fx (t)) γx + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) Ḧx (t) s 2 x (t) = ||K|| 2 2 ϕ x ( Hx (t)) ϕ ′ x ( Fx (t)) 2 γ x (1 -γ x ) + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) 2 H x (t)(1 -H x (t)) .

Example: Survival of Atlantic Halibut

We apply in this section the conditional Koziol-Green estimator on a practical data set about survival of Atlantic halibut, studied by [START_REF] Neilson | Survival of Atlantic halibut (Hippoglossus hippoglossus) caught by longline and otter trawl gear[END_REF]. An important issue was the survival time of the fish after it was caught and handled as in the commercial fishery. For this purpose they had installed special holding tanks on the research vessel in which they placed the fish. Each fish was followed until it died. However some fish, mainly large fish, were removed after 48 hours to make space for other experimental animals. So the time until death was censored by the time that the animal had spent in the holding tank. Also the fish that were alive at the end of the experiment, were treated as censored observations. The researchers recorded several covariates among which we focus on the fork length of the fish. In previous analyses of the data set, a significant effect of fork length on survival time had been found. In Figure 1 we show a scatter plot of the survival time versus the fork length of each animal, where we use + for uncensored observations and O for censored observations. The main causes of death for the fish were the stress of the new environment and infections caused by sick fishes in the tank. Therefore we believe that the survival time Y x of a fish depends on the time that this fish has spent in the holding tank C x , where the time in the holding tank has a negative influence on the survival time. For these data, we construct a conditional Koziol-Green estimate for different choices of ϕ x at fork lengths 32 cm and 53 cm, representing typical small fishes and typical large fishes. The four choices of the function ϕ x that we will consider here, will lead each time to a different association for the dependence structure between the survival time and the time spent in the holding tank. The first choice is the independent copula (ϕ x (t) = -log(t)). This is the (possibly wrong) choice used in previous analyses of the data. In the other choices of ϕ x we express that the time spent in the holding tank has a negative influence on the survival time. [START_REF] Nelsen | An introduction to copulas[END_REF] formally defined this as discordance. For the second choice of ϕ x we take the Fréchet-Hoeffding lower bound (ϕ x (t) = 1 -t), which is the most extreme discordance that can be considered. In the next choices we allow the generator function ϕ x to depends on the fork length x. Our third choice is the Frank family 1 copula given by

ϕ x (t) = -log e (x-20)t -1 e x-20 -1
and the fourth choice is the Frank family 2 copula given by ϕ x (t) = -log e (60-x)t -1 e 60-x -1 .

The Frank family 1 copula gives a stronger discordant association for larger fishes than for small fishes, while for the Frank family 2 copula, there is a stronger discordant association for small fishes.

The choices of ϕ x in the first group have in common that they assume a large discordant association between survival time and time spent in the holding tank for larger fishes. In the second group, the choices of ϕ x assume practically no discordant association for larger fishes. This influences the estimates for the survival function. With a ϕ x from the first group, the estimated survival function for larger fish is higher than with a ϕ x from the second group (in particular the ϕ x that describes independence). This means that when we ignore stress in the fish caused by the catch, the handling and the living conditions in the holding tank, we underestimate the true survival time for larger fishes. To finish this section, we compare the estimates for the Koziol-Green sub-model with the estimates for the general copula-graphic model of [START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF]. We note that there is not much difference between the plots in the two columns. Therefore we know that the Koziol-Green assumption is satisfied in this data set and the Koziol-Green sub-model gives a better insight into the data.

Appendix: Proofs

Proof of Theorem 1: (a) To establish strong consistency of the estimator F xh (t), we first use the bivariate mean value theorem.

F xh (t) -F x (t) = Fx (t) -Fxh (t) = - ϕ x (ε 2 (t)) ϕ ′ x (ϕ -1 x (ε 1 ϕ x (ε 2 (t)))) (γ xh -γ x ) + ε 1 ϕ ′ x (ε 2 (t)) ϕ ′ x (ϕ -1 x (ε 1 ϕ x (ε 2 (t)))) (H xh (t) -H x (t))
with ε 1 between γ xh and γ x , and ε 2 (t) between Hxh (t) and Hx (t).

Hence for all ε > 0, η 1 > 0, η 2 > 0,

P sup 0≤t≤T |F xh (t) -F x (t)| > ε ≤ P M 1 (ε 1 )|γ xh -γ x | > ε 2 + P sup 0≤t≤T M 2 (ε 2 (t)) sup 0≤t≤T |H xh (t) -H x (t)| > ε 2 ≤ P M 1 (ε 1 )|γ xh -γ x | > ε 2 , |γ xh -γ x | ≤ η 1 + P (|γ xh -γ x | > η 1 ) + P sup 0≤t≤T |H xh (t) -H x (t)| > η 2 + P sup 0≤t≤T M 2 (ε 2 (t)) sup 0≤t≤T |H xh (t) -H x (t)| > ε 2 , sup 0≤t≤T |H xh (t) -H x (t)| ≤ η 2
If we assume that 0 < η 1 < γ x and 0 < η 2 < Hx (T ), by Lemma 1 below, this expression is bounded by

P |γ xh -γ x | > ε 2M 1 (γ x -η 1 ) + P (|γ xh -γ x | > η 1 ) + P sup 0≤t≤T |H xh (t) -H x (t)| > η 2 + P sup 0≤t≤T |H xh (t) -H x (t)| > ε 2M 2 ( Hx (T ) -η 2 )
.

Choosing η 1 and η 2 such that η 1 = ε 2M1(γx-η1) and η 2 = ε 2M2( Hx(T )-η2) , we have that

P sup 0≤t≤T |F xh (t) -F x (t)| > ε ≤ 2P (|γ xh -γ x | > η 1 ) + 2P sup 0≤t≤T |H xh (t) -H x (t)| > η 2 . (7)
For the second probability on the right hand side of (7), we have the bound

P sup 0≤t≤T |H xh (t) -H x (t)| > η 2 ≤ 1 2 d 0 η 2 nh n e -d1nhnη 2 2 /4
where d 0 and d 1 are constants. This result is given by Van Keilegom and [START_REF] Van Keilegom | Estimation and bootstrap with censored data in fixed design nonparametric regression[END_REF] under the condition that

η 2 ≥ max √ 6 K 2 (nh n ) -1/2 , 2 Ḣx ∞ ∆n + 2µ K 2 Ḧ ∞ h 2 n .
From [START_REF] Veraverbeke | Estimation of the conditional distribution in a conditional Koziol-Green model[END_REF], we find a similar result for the first probability of (7),

P (|γ xh -γ x | > η 1 ) ≤ 2e -C1η 2 1 nhn
with C 1 an absolute constant and under the condition that

η 1 ≥ 2( γx ∞ ∆n + γx ∞ µ K 2 h 2 n ).
Combining these results gives the exponential bound for F xh (t).

(b) From Lemma 2 below, we find, for small ε > 0, a new upper bound

P sup 0≤t≤T |F xh (t) -F x (t)| > ε ≤ 4e - C 1 ε 2 nhn 4(M 1 (γx)+1) 2 + d 0 ε 2 nh n e - d 1 nhn 4 ε 2 4(M 2 ( Hx(T ))+1) 2 If we take ε n = C 2 (nh n ) -1/2 (log n) 1/2 ,
we note that ε n is small for large n and by the Borel-Cantelli Lemma we find the result.

Lemma 1. We have, for z ∈ ]0, 1[, that

M 1 (z) = max 0≤y≤1 -ϕ x (y) ϕ ′ x (ϕ -1 x (zϕ x (y)))
and

M 2 (z) = max 0≤y≤1 yϕ ′ x (z) ϕ ′ x (ϕ -1 x (yϕ x (z)
)) are non-increasing functions of z.

Proof: Due to the assumptions on ϕ x (t) and ϕ ′ x (t), we see that, for every y, z ∈ ]0, 1[ -ϕ x (y) ϕ ′

x (ϕ -1 x (zϕ x (y)))

and yϕ ′ x (z) ϕ ′ x (ϕ -1 x (yϕ x (z)))
are well-defined. Furthermore we can show that, for every z ∈ ]0, 1[, 0 ≤ lim y→0 -ϕx(y) ϕ ′

x (ϕ -1

x (zϕx(y))) ≤ 1, lim y→1 -ϕx(y) ϕ ′

x (ϕ -1

x (zϕx(y))) = 0, 0 ≤ lim

y→0 yϕ ′ x (z) ϕ ′ x (ϕ -1 x (yϕx(z))) < +∞, lim y→1 yϕ ′ x (z) ϕ ′ x (ϕ -1
x (yϕx(z))) = 1 such that M 1 (z) and M 2 (z) are well-defined. To prove that these functions are non-increasing, we fix for the moment a value of y ∈ [0, 1]. For z 1 , z 2 ∈ ]0, 1[ with z 1 < z 2 , we find that

-ϕ x (y) ϕ ′ x (ϕ -1 x (z 1 ϕ x (y))) ≥ -ϕ x (y) ϕ ′ x (ϕ -1 x (z 2 ϕ x (y))) and yϕ ′ x (z 1 ) ϕ ′ x (ϕ -1 x (yϕ x (z 1 ))) ≥ yϕ ′ x (z 2 ) ϕ ′ x (ϕ -1 x (yϕ x (z 2 )))
if we rewrite the second inequality as a constraint optimization problem. Hence M 1 (z 2 ) ≤ M 1 (z 1 ) and

M 2 (z 2 ) ≤ M 2 (z 1 ).
Lemma 2. If ε is sufficiently small, we find that

ε 2(M 1 (γ x ) + 1) ≤ min(η 1 , γ x ) ≤ min(η * 1 , γ x ) ε 2(M 2 ( Hx (T )) + 1) ≤ min(η 2 , Hx (T )) ≤ min(η * 2 , Hx (T ))
where

η 1 , η * 1 , η 2 , η * 2 > 0 is, respectively, a root of η 1 = ε 2M1(γx-η1) , M 1 (γ x -η * 1 ) = M 1 (γ x ) + 1, η 2 = ε 2M2( Hx(T )-η2) and M 2 ( Hx (T ) -η *
2 ) = M 2 ( Hx (T )) + 1.

Proof: We only prove the results for η 1 . For η 2 we work similarly.

From Lemma 1, we know that M 1 (γ x -η 1 ) is a non-decreasing function of η 1 on the interval [0, γ x ]. Therefore there either exists a value η * 1 in this interval such that M 1 (γ

x -η * 1 ) = M 1 (γ x ) + 1 or we have that M 1 (γ x -η 1 ) < M 1 (γ x ) + 1, ∀ η 1 ∈ [0, γ x ]. Hence M 1 (γ x -η 1 ) ≤ M 1 (γ x ) + 1, ∀ η 1 ≤ min(η * 1 , γ x ).
Define ε * := 2(M 1 (γ x ) + 1) min(η * 1 , γ x ). We note that for ε

< ε * , ε 2(M 1 (γ x ) + 1) < ε * 2(M 1 (γ x ) + 1) = min(η * 1 , γ x ).
Furthermore we see that if

η * 1 exists, η * 1 M 1 (γ x -η * 1 ) = (M 1 (γ x ) + 1)η * 1 = ε * 2 else γ x M 1 (0) < (M 1 (γ x ) + 1)γ x = ε * 2 . Since the function η 1 M 1 (γ x -η 1
) is an increasing function of η 1 on the interval [0, γ x ], this means that for ε < ε * , the root η 1 of

η 1 = ε 2M 1 (γ x -η 1 ) ⇔ η 1 M 1 (γ x -η 1 ) = ε 2 is smaller than η * 1 . If η 1 M 1 (γ x -η 1 ) < ε 2 , ∀ η 1 ∈ [0, γ x ], we take η 1 = γ x . So min(η 1 , γ x ) ≤ min(η * 1 , γ x ).
If η 1 exists, we have that

ε 2(M 1 (γ x ) + 1) M 1 γ x - ε 2(M 1 (γ x ) + 1) ≤ ε(M 1 (γ x ) + 1) 2(M 1 (γ x ) + 1) = ε 2 = η 1 M 1 (γ x -η 1 ), such that ε 2(M1(γx)+1) ≤ η 1 . If η 1 does not exists, we see that ε 2(M 1 (γ x ) + 1) ≤ ε * 2(M 1 (γ x ) + 1) = γ x . Hence ε 2(M 1 (γ x ) + 1) ≤ min(η 1 , γ x ).
Proof of Theorem 2: To find an asymptotic representation of F xh (t), we use a second order Taylor expansion.

F xh (t) -F x (t) = ϕ -1 x γ x ϕ x ( Hx (t)) -ϕ -1 x γ xh ϕ x ( Hxh (t)) = - 1 ϕ ′ x ( Fx (t)) γ xh ϕ x ( Hxh (t)) -γ x ϕ x ( Hx (t)) + ϕ ′′ x (ϕ -1 x (ε(t))) 2ϕ ′ x (ϕ -1 x (ε(t))) 3 γ xh ϕ x ( Hxh (t)) -γ x ϕ x ( Hx (t))
2 with ε(t) between γ xh ϕ x ( Hxh (t)) and γ x ϕ x ( Hx (t)).

After adding and subtracting some terms, and applying a second order Taylor expansion to the second term, we get

= - 1 ϕ ′ x ( Fx (t)) ϕ x ( Hx (t))(γ xh -γ x ) + γ x (ϕ x ( Hxh (t)) -ϕ x ( Hx (t))) - (ϕ x ( Hxh (t)) -ϕ x ( Hx (t)))(γ xh -γ x ) ϕ ′ x ( Fx (t)) + ϕ ′′ x (ϕ -1 x (ε(t))) 2ϕ ′ x (ϕ -1 x (ε(t))) 3 γ xh ϕ x ( Hxh (t)) -γ x ϕ x ( Hx (t)) 2 = - ϕ x ( Hx (t)) ϕ ′ x ( Fx (t)) (γ xh -γ x ) + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) (H xh (t) -H x (t)) + R n1 (t) + R n2 (t) + R n3 (t) = n i=1 w ni (x, h n )g tx (Z i , δ i ) + R n1 (t) + R n2 (t) + R n3 (t) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT with R n1 (t) = - γxϕ ′′ x (η(t)) 2ϕ ′ x ( Fx(t)) (H xh (t) -H x (t)) 2 , R n2 (t) = -(ϕx( Hxh (t))-ϕx( Hx(t)))(γxh-γx) ϕ ′ x ( Fx(t)) , R n3 (t) = ϕ ′′ x (ϕ -1 x (ε(t))) 2ϕ ′ x (ϕ -1 x (ε(t))) 3 γ xh ϕ x ( Hxh (t)) -γ x ϕ x ( Hx (t))
2 where η(t) lies between Hxh (t) and Hx (t). In the remaining part of this proof we show the rate of convergence for each of the remainder terms. For R n1 (t), we find sup

0<t≤T |R n1 (t)| ≤ ϕ ′′ x (η(T )) sup 0<t≤T (H xh (t) -H x (t)) 2 .
While for R n2 (t), we use a first order Taylor expansion and have

sup 0<t≤T |R n2 (t)| ≤ ϕ ′ x (ν(T )) sup 0<t≤T |H xh (t) -H x (t)|.|γ xh -γ x |
where ν(t) lies between Hxh (t) and Hx (t). After adding and subtracting some terms, and also using a first order Taylor expansion, we find for R n3 (t),

R n3 (t) = ϕ ′′ x (ϕ -1 x (ε(t))) 2ϕ ′
x (ϕ -1 x (ε(t))) 3 ϕ x ( Hxh (t))

)) = n i=1 w 2 ni (x, h n ) ϕ x ( Hx (t)) 2 ϕ ′ x ( Fx (t)) 2 γ xi (1 -γ xi ) + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) 2 H xi (t)(1 -H xi (t)) = K 2 2 nh n ϕ x ( Hx (t)) 2 ϕ ′ x ( Fx (t)) 2 γ x (1 -γ x ) + γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t)) 2 H x (t)(1 -H x (t)) + o((nh n ) -1 )
The asymptotic normality result for the estimator can now be obtained by checking Liapunov's condition for (nh n ) 1/2 n i=1 w ni (x, h n ) (g tx (Z i , δ i ) -E[g tx (Z i , δ i )]). The (b)-part of Theorem 3 deals with the optimal bandwidth h n = Cn -1/5 for some C > 0, which minimizes the mean squared error.

Figure 1 :

 1 Figure 1: Atlantic halibut data set: Survival times (in hours) versus fork length (in cm). Fish died in the holding tanks: +, fish removed from the holding tanks or alive at the end of the study: O.

  2 (γ xh -γ x ) 2 + γ 2 x ϕ ′ x (ξ(t)) 2 (H xh (t) -H x (t)) 2 -2γ x ϕ x ( Hxh (t))ϕ ′ x (ξ(t))(γ xh -γ x )(H xh (t) -H x (t))with ξ(t) between Hxh (t) and Hx (t). Since H x (T ) < 1 and H xh (T ) → H x (T ) a.s. (Lemma A2, Van Keilegom and Veraverbeke (1997)), we may suppose that T < T H xh . Furthermore we have that supProof of Theorem 3: To prove asymptotic normality, we first calculate the bias and variance expressions. Due to Theorem 2 we only need to consider the main term in the asymptotic representation. After a straightforward calculation (see e.g. Aerts, Janssen and Veraverbeke (1994)), we find

					n i=1	w ni (x, h n ) -	ϕ x ( Hx (t)) ϕ ′ x ( Fx (t))	(γ xi -γ x ) +	γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t))	(H xi (t) -H x (t))
	=	1 2	µ K 2 h 2 n -	ϕ x ( Hx (t)) ϕ ′ x ( Fx (t))	γx +	γ x ϕ ′ x ( Hx (t)) ϕ ′ x ( Fx (t))	Ḧx (t) + o(h 2 n ) + O(n -1 )
	n
	w 2 ni (x, h
	i=1

0<t≤T

|H xh (t) -H x (t)| → 0 a.s. and γ xh → γ x a.s.

(Lemma A4, Van Keilegom and Veraverbeke (1997)

, Lemma A1,

[START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF]

). Hence, we have that sup

0<t≤T |R n1 (t)|, sup 0<t≤T |R n2 (t)| and sup 0<t≤T |R n3 (t)| are all O((nh n ) -1 log n) a.s. n i=1 w ni (x, h n )E[g tx (Z i , δ i )] = n )Var(g tx (Z i , δ i

Acknowledgement

Financial support from the IAP research network nr P5/24 of the Belgian Government (Belgian Science Policy) is gratefully acknowledged.

In Figure 2, we show in the right column the Koziol-Green estimates for the conditional survival function Fx (t) at fork lengths of 32 cm and 53 cm for bandwidth 20. In the left column we give the copula-graphic estimates of [START_REF] Braekers | A copula-graphic estimator for the conditional survival function under dependent censoring[END_REF] for this conditional survival function under a general model. By comparing both estimates we can verify whether the Koziol-Green sub-model is satisfied. In each of the four plots, we construct the copula-graphic estimator for the four choices of ϕ x . We use in this data set the Gasser-Müller weights with a biquadratic kernel given by K(z) = (15/16)(1 -z 2 ) 2 I(|z| ≤ 1). As we saw in Figure 1, the covariate fork length of a fish is measured crudely on a scale of whole centimeters so that the observations form vertical lines on this plot. It is therefore possible to treat this covariate as fixed. It is also easy to see that our results for the copula-graphic estimator remain valid in the interval [25,60] for the covariate x, instead of the standard interval [0, 1]. The choice of the bandwidth is selected here for illustration purpose only. We also calculated estimates under other bandwidths but the results did not change. It is possible to set up a bandwidth selection criterium using, for example, the asymptotic mean squared error expression, but this would lead us into a field of research that we do not enter at this moment.

We note in Figure 2 that the different estimates for the conditional survival function lie close together in each of the four plots and lie even almost on top of each other in the plots of the first row. This means that the choice of the generator function ϕ x does not have a great influence on the survival time of small fishes. In the plots at the fork length of 53 cm we see that both the copula-graphic and the Koziol-Green estimates can be divided in two groups. The Fréchet-Hoeffding lower bound copula and the Frank family 1 copula give estimates that lie almost on top of each other but that are clearly different from the estimates of the independent copula and the Frank family 2 copula which form the second group. By this division in two groups, we see that this data set reacts differently to two different situations.


Marcin Kozak
email: m.kozak@omega.sggw.waw.pl

Andrzej Zieliński

Sarjinder

Sarjinder Singh

Stratified two

Keywords: domain-orientated approach, optimum sample allocation, survey cost

published or not. The documents may come

1 Introduction [START_REF] Kozak | On stratified two-stage sampling: Optimum stratification and sample allocation between strata and sampling stages[END_REF] presented basic concepts of stratified two-stage sampling design, in which a population of primary sampling units is subdivided into strata. He provided formulas for optimum sample allocation between strata and sampling stages under two schemes of the design: (i) in which sample size of secondary sampling units (SSUs) from primary sampling units (PSUs) is fixed, and (ii) with self-weighting design in strata. [START_REF] Kozak | Sample allocation between domains and strata[END_REF], on the other hand, presented basic concepts of a problem of sample allocation between domains and strata in case when domains are subdivided into strata. They considered (i) a so-called domain-orientated approach to the sample allocation, in which one requires precise estimation in all the domains, and (ii) sample allocation orientated towards minimizing total survey cost subject to fixed level of precision of estimation in the domains. In this paper, we introduce a hybrid of these two designs, namely stratified two-stage sampling in domains. Such a design can be of practical use when a population is subdivided into domains, each domain comprising some number of strata consisting of PSUs. A simple random sample of SSUs is to be taken without replacement from PSUs. We consider a situation in which sample size of SSUs from PSUs is fixed; in such a case, one obtains a sample of fixed size (and fixed total cost). We give formulas for sample allocation between domains, strata and sampling stages (i) under domain-orientated approach, and (ii) orientated towards minimizing a total survey cost.

2 Estimation for domains under stratified two-stage sampling in the domains: basic ideas Basic concepts of stratified sampling and two-stage sampling, which lay the basis for the design introduced in this section, may be found, e.g., in [START_REF] Särndal | Model Assisted Survey Sampling[END_REF] or [START_REF] Singh | Advanced Sampling Theory with Applications. How Michael "Selected[END_REF]. Consider a population U comprising N elements. 1) is given by [START_REF] Kozak | On stratified two-stage sampling: Optimum stratification and sample allocation between strata and sampling stages[END_REF])

V ( Ŷd ) = H d h=1 M dh m dh (M dh -m dh )S 2 1dh + 1 n dh M dh g=1 N dhg (N dhg -n dh )S 2 2dhg (2)
where

S 2 1dh = (M dh -1) -1 M dh g=1 (Y dhg -Y dh ) 2 , Y dhg = N dhg j=1 y dhgj , Y dh = N -1 dh M dh g=1 N dhg j=1 y dhgj , S 2 2dhg = (N dhg -1) -1 N dhg j=1 (y dhgj -Y dhg ) 2 , Y dhg = N -1 dhg M dhg j=1 y dhgj .
Note again that sample sizes n dh in the variance (2) refer to sample sizes n dhg , which are assumed to be the same for all g = 1, . . . , N dh in a particular section domain d×stratum h. Hence, for the sake of convenience, we write n dh instead of n dhg , keeping in mind that n dh is the sample size of SSUs from every gth PSU of the hth stratum in the dth domain. An ordinary unbiased estimator of the variance (2) is obtained by replacing the population quantities S 2 1dh and S 2 2dhg with their sample estimators; the summation in (2) is to be done by sampled PSUs in each hth stratum from the dth domain.

The coefficient of variation of the estimator Ŷd , say δ( Ŷd ), is

δ( Ŷd ) = V ( Ŷd ) Y d , d = 1, . . . , D
In this paper, we understand optimum conditions of a design as the ones for which some function of δ( Ŷd ) is minimum. Let the overall survey cost C be

C = C 0 + D d=1 H d h=1 m dh c 1dh + n dh c 2dh (3)
where C 0 is the fixed survey cost, c 1dh is the cost of selecting one PSU from the hth stratum of the dth domain, and c 2dh is the cost of obtaining the information on Y value in one SSU from the hth stratum of the dth domain.

3 Optimizing a design under domain-orientated approach

Here we apply a domain-orientated approach to the design presented in previous section. It aims at precise estimation for each domain U d of the population U [START_REF] Kozak | Sample allocation between domains and strata[END_REF]. Let g = (g 1 , . . . , g D ) T be a vector of important weights of the domains. Following [START_REF] Kozak | Sample allocation between domains and strata[END_REF], the optimum design is the one under which the smallest common value ϕ of g -1 d δ( Ŷd ), d = 1, . . . , D, is obtained. Thus, we require coefficient of variation of the estimator Ŷd of the population total in the dth domain to satisfy the condition [START_REF] Kozak | Sample allocation between domains and strata[END_REF]

δ( Ŷd ) = g d ϕ, d = 1, . . . , D (4)
Then, our aim is to find optimum values of n dh and m dh (d = 1, . . . , D, h = 1, . . . , H d ) under fixed overall survey cost (3) equal C (given c 1dh and c 2dh ) so that the condition ( 4) is satisfied and the common value ϕ is minimum. We will optimize the design based on the assumption that the survey variable is the same as the auxiliary variable used to allocate the survey cost. Of course, in practice, it is an untrue situation; instead of the population values, the quantities originating from recent censuses or previous/pilot surveys are used.

Theorem 1. When a population U is subdivided into D domains and stratified two-stage sampling with fixed sample size of secondary sampling units from primary sampling units is to be applied within the domains, under a cost function (3), given survey costs C, C 0 , c 1dh and c 2dh , the smallest common value ϕ of g -1 d δ( Ŷd ), d = 1, . . . , D is obtained when for d = 1, . . . , D, h = 1, . . . , H d ,

n dh = c 1dh c 2dh M dh g=1 N 2 dhg S 2 2dhg M dh S 2 1dh -M dh g=1 N dhg S 2 2dhg m dh = (C -C 0 )v d M dh M dh S 2 1dh -M dh g=1 N dhg S 2 2dhg Y d √ c 1dh D e=1 v e Y -1 e H e i=1 √ M ei Z ei where Z ei = c 1ei M ei S 2 1ei -M ei k=1 N eik S 2 2eik + c 2ei M ei k=1 N 2 eik S 2 2eik and v = (v 1 , . . . , v D )
T is the eigenvector connected with the largest eigenvalue of the matrix

F = (C -C 0 ) -1 AB T -diag(E) , where A = (A 1 , . . . , A D ) T , B = (B 1 , . . . , B D ) T and E = (E 1 , . . . , E D ) T , pro- vided that M dh S 2 1dh - M dh g=1 N dhg S 2 2dhg > 0 (5)
for each d = 1, . . . , D, and h = 1, . . . , H d .

Proof. To prove Theorem 1, a procedure developed by Niemiro and Weso lowski ( 2001) may be used. It was recently applied in sample allocation between domains and strata by [START_REF] Kozak | Sample allocation between domains and strata[END_REF]. Consider the following Lagrange function:

L = ϕ - D d=1 λ d 1 Y 2 d H d h=1 1 m dh u dh + w dh n dh -x dh - 1 Y 2 d H d h=1 M dh S 2 1dh -g 2 d ϕ 2 -α D d=1 H d h=1 m dh (c 1dh + n dh c 2dh ) -(C -C 0 ) (6)
where λ d and α are the Lagrange multipliers,

w dh = M dh M dh g=1 N 2 dhg S 2 2dhg , u dh = M 2 dh S 2 1dh , x dh = M dh M dh g=1 N dhg S 2 2dhg
, and Y d is the population total of Y in the dth domain. Differentiation of (6) with respect to m dh , n dh , λ d , and α and solving the obtained equations yield the results presented in Theorem 1. A detailed proof may be obtained from the authors upon request.

Remark 1. If any of the conditions (5) or any of the following conditions

2 ≤ n dh ≤ N dh ; 2 ≤ m dh ≤ M dh f or d = 1, . . . , D, h = 1, . . . , H d , (7)
is not fulfilled, the values of n dh and m dh from Theorem 1 are not real numbers, so they are not optimum. In such a case, the optimum n dh and m dh are the solution of the following numerical problem:

1 Y 2 d H d h=1 M dh m dh M dh -m dh S 2 1dh + 1 n dh M dh g=1 N dhg N dhg -n dh S 2 2dhg = g 2 d ϕ 2 D d=1 H d h=1 m dh (c 1dh + n dh c 2dh ) = C -C 0 M dh S 2 1dh - M dh g=1 N dhg S 2 2dhg > 0 f or each d = 1, . . . , D, and h = 1, . . . , H d 2 ≤ n dh ≤ N dh ; 2 ≤ m dh ≤ M dh f or d = 1, . . . , D, h = 1, . . . , H d .
4 Optimizing a design subject to constraints connected with domain precisions

Here we consider a question dual to the problem presented in previous section. We aim at minimizing a total survey cost C given in (3) subject to

1 Y 2 d H d h=1 M dh m dh M dh -m dh S 2 1dh + 1 n dh M dh g=1 N dhg N dhg -n dh S 2 2dhg = δ 2 d , d = 1, . . . , D, (8)
where δ d is the fixed value of coefficient of variation of Ŷd . Thus, this time we consider a design in which we look for optimum values of n dh and m dh for which the constraint ( 8) is fulfilled and the cost (3) is minimum.

Theorem 2. When a population U is subdivided into D domains and stratified two-stage sampling with fixed sample size of secondary sampling units from primary sampling units is to be applied within the domains, under a cost function (3), given survey costs C 0 , c 1dh and c 2dh , and under the condition (8) (for δ d being fixed), the minimum total survey cost C is obtained when for d = 1, . . . , D, h = 1, . . . , H d ,

n dh = c 1dh c 2dh M dh g=1 N 2 dhg S 2 2dhg M dh S 2 1dh -M dh g=1 N dhg S 2 2dhg m dh = M dh M dh S 2 1dh -M dh g=1 N dhg S 2 2dhg Y d √ c 1dh • H d i=1 D i δ 2 d + Y -2 d H d h=1 M dh S 2 1dh D i = c 2di M di M di S 2 1di - M di k=1 N dik S 2 2dik + M di M di k=1 N 2 dik S 2 2dik provided that M dh S 2 1dh - M dh g=1 N dhg S 2 2dhg > 0 (

  minimize f (n 1 , m 1 ), . . . , (n D , m D ); ϕ = ϕ, where n d = n d1 , . . . , n dH d T and m d = m d1 , . . . , m dH d T for d = 1, . . . , D subject to:

  9)for each d = 1, . . . , D, and h = 1, . . . , H d .Proof. Consider the following Lagrange function L = C 0 + λ d are the Lagrange multipliers and u dh , w dh , and x dh are the same as defined in previous section. Differentiating of (10) with respect to m dh , n dh , and λ d and solving the obtained equations lead to the results presented in Theorem 1. A detailed proof may be obtained from the authors upon request.Remark 2. If any of the conditions (9) or any of the conditions (7) is not fulfilled, the values of n dh and m dh from Theorem 2 are not real numbers, so they are not optimum. In such a case, the optimum n dh and m dh are the solution of the following numerical problem: minimize f (n 1 , m 1 ), . . . , (n D , m D ); ϕ = C 0 + D d=1 H d h=1 m dh c 1dh +n dh c 2dh ,

  dh ; finally, each stratum U dh is subdivided into M dh separate PSUs U dhg . This division can be presented asThe gth PSU from the hth stratum in the dth domain comprises N dhg SSUs, which are the population elements. Let N d indicate the number of SSUs in the dth domain and N dh indicate the number of SSUs in the hth stratum of the dth domain. Ŷd is the estimator of Y d , Y d being the population total of the variable Y restricted to the dth domain; Ŷdh is the estimator of Y dh , Y dh being the population total of the variable Y restricted to the hth stratum of the dth domain; m dh is the sample size of PSUs from the hth stratum of the dth domain; n dhg is the sample size of SSUs from the gth PSU of the hth stratum of the dth domain; and y dhgi is the Y value in the ith SSU (population element) of the gth PSU from the hth stratum in dth domain. In both cases (i.e., when sampling PSUs from strata and when sampling SSUs from PSUs), simple random sample is to be taken without replacement.Let us consider a two-stage sampling scheme in which we deal with a fixed size of sample of SSUs. In our design, it consists in sampling the same number n dh ( n dhg = n dh for each combination of d = 1, . . . , D and h = 1, . . . , H d ) of SSUs from PSUs in each section domain d × stratum h. Under such a design, the variance of the estimator (

	where
		D		H d	M dh
	U =					U dhg , U d ∩ U d = ∅ f or d, d = 1, . . . , D, d = d ; and
		d=1	h=1	g=1
	U dh ∩ U dh = ∅ f or d = 1, . . . , D, h, h = 1, . . . , H d , h = h ; and
	U dhg ∩ U dhg = ∅ f or d = 1, . . . , D, h = 1, . . . , H d , g, g = 1, . . . , M dh , g = g .
	Let a population parameter investigated be the population total of Y , Y being
	a characteristic studied. For the dth domain its estimator is given by
	Ŷd =	H d h=1	Ŷdh =	H d h=1	M dh m dh	m dh g=1	N dhg n dhg	n dhg i=1	y dhgi	(1)

The population is subdivided into D domains U d (d = 1, . . . , D); each domain is subdivided into H d non-overlapping strata U

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

where