Tiphaine Le Roy

Marta Llopis

Patricia Lepage

Aurélia Bruneau

Sylvie Rabot

Claudia Bevilacqua

Patrice Martin

Catherine Philippe

Francine Walker

André Bado

Gabriel Perlemuter

Anne-Marie Cassard-Doulcier

Dr Philippe Gérard
email: philippe.gerard@jouy.inra.fr

Intestinal microbiota determines development of non-alcoholic fatty liver disease in mice

Objective Non-alcoholic fatty liver disease (NAFLD) is prevalent among obese people and is considered the hepatic manifestation of metabolic syndrome. However, not all obese individuals develop NAFLD. Our objective was to demonstrate the role of the gut microbiota in NAFLD development using transplantation experiments in mice. Design Two donor C57BL/6J mice were selected on the basis of their responses to a high-fat diet (HFD). Although both mice displayed similar body weight gain, one mouse, called the 'responder', developed hyperglycaemia and had a high plasma concentration of pro-inflammatory cytokines. The other, called a 'nonresponder', was normoglycaemic and had a lower level of systemic inflammation. Germ-free mice were colonised with intestinal microbiota from either the responder or the non-responder and then fed the same HFD. Results Mice that received microbiota from different donors developed comparable obesity on the HFD. The responder-receiver (RR) group developed fasting hyperglycaemia and insulinaemia, whereas the non-responder-receiver (NRR) group remained normoglycaemic. In contrast to NRR mice, RR mice developed hepatic macrovesicular steatosis, which was confirmed by a higher liver concentration of triglycerides and increased expression of genes involved in de-novo lipogenesis. Pyrosequencing of the 16S ribosomal RNA genes revealed that RR and NRR mice had distinct gut microbiota including differences at the phylum, genera and species levels. Conclusions Differences in microbiota composition can determine response to a HFD in mice. These results further demonstrate that the gut microbiota contributes to the development of NAFLD independently of obesity.

INTRODUCTION

Non-alcoholic fatty liver disease (NAFLD) is considered the hepatic manifestation of metabolic syndrome and is commonly associated with insulin resistance. [START_REF] Bugianesi | Insulin resistance in nonalcoholic fatty liver disease[END_REF] NAFLD affects 20-30% of western countries' population and more than 80% of obese people. It refers to a spectrum of liver damage ranging from simple steatosis to non-alcoholic steatohepatitis, advanced fibrosis, cirrhosis or even hepatocellular carcinoma. An increasing body of literature has recently been generated identifying gut microbiota as a new environmental factor contributing to obesity and NAFLD. 2 3 First, Bäckhed and colleagues 4 5 showed that germ-free (GF) C57BL/6J mice gained less weight than conventional mice when given a sugar and lipids-rich diet despite greater food consumption. Moreover, Rabot et al [START_REF] Rabot | Germ-free C57BL/6J mice are resistant to high-fat-diet-induced insulin resistance and have altered cholesterol metabolism[END_REF] observed that GF mice receiving a high-fat diet (HFD) showed enhanced insulin sensitivity with improved glucose tolerance and reduced insulinaemia. Concurrently, colonisation of GF mice by a gut microbiota from conventional mice produced an increase in body fat content. [START_REF] Backhed | The gut microbiota as an environmental factor that regulates fat storage[END_REF] More recently, an association between the human gut

Significance of this study

What is already known on this subject?

▸ We and others have shown that GF mice are resistant to diet-induced obesity, insulin resistance and steatosis. ▸ The administration of a HFD to conventional mice leads to heterogeneous responses including variable levels of weight gain, glycaemia or steatosis development. ▸ Each mouse and human harbours a different gut microbiota.

What are the new findings?

▸ Gut microbiota markedly impacts the lipid metabolism in the liver, independently of obesity. ▸ The propensity to develop NAFLD features including hyperglycaemia or steatosis is transmissible by means of gut microbiota transplantation. ▸ Bacterial species associated with the NAFLD-resistant and NAFLD-prone phenotypes have been identified.

How might it impact on clinical practice in the foreseeable future?

▸ These findings suggest that manipulation of the gut microbiota may be a new strategy to prevent or treat NAFLD and associated metabolic disorders including type 2 diabetes or metabolic syndrome. In addition, gut microbiota profiling could help predict the susceptibility to develop metabolic disorders.

microbiota and the development of fatty liver due to choline deficiency has been identified. [START_REF] Spencer | Association between composition of the human gastrointestinal microbiome and development of fatty liver with choline deficiency[END_REF] Moreover, it was revealed that changes in the gut microbiota associated with inflammasome defects regulates the progression of NAFLD. [START_REF] Henao-Mejia | Inflammasome-mediated dysbiosis regulates progression of NAFLD and obesity[END_REF] HFD feeding is widely used in rodents to study the onset and progression of obesity and associated metabolic disorders. However, the high-fat-induced phenotype varies distinctly, even within a group of animals with the same genetic background, 9 10 and it has recently been demonstrated that disctinct gut microbiota profiles are associated with different metabolic phenotypes. [START_REF] Serino | Metabolic adaptation to a high-fat diet is associated with a change in the gut microbiota[END_REF] Gut microbiota of humans and mice are more than 95% made up of three phyla: Firmicutes, Bacteroidetes and Actinobacteria. In contrast to the poor diversity at the phylum level, the species level displays a high diversity, with an average human microbiota estimated at 200 prevalent bacterial species and up to 1000 less common species. [START_REF] Ley | Worlds within worlds: evolution of the vertebrate gut microbiota[END_REF] Importantly, in humans but also in mice, the bacterial species profiles are unique for each individual. [START_REF] Zoetendal | Temperature gradient gel electrophoresis analysis of 16S rRNA from human fecal samples reveals stable and host-specific communities of active bacteria[END_REF] We therefore hypothesised that the gut microbiota modulates the effect of a HFD challenge and that the variability of the composition of this microbiota could explain the diversity of responses to HFD.

In the present work, we developed a strategy based on gut microbiota transfer to establish whether we can transmit the different propensity to develop NAFLD in response to HFD by means of gut microbiota transplant.

METHODS

Animal experimentation Stage 1

Eight-week-old conventional male C57BL/6J mice ( Janvier, Le Genest St Isle, France) were fed for 16 weeks a freely available sterilised HFD containing 40% butter and 2.5% soy oil (60% energy from fat, 22.7% from carbohydrates and 17.2% from proteins; SAFE, Augy, France). Body weight was monitored weekly.

Stage 2

GF male C57BL/6J mice were reared from GF breeding pairs at ANAXEM, the GF animal facilities of Micalis ( Jouy-en-Josas, France). Two groups of 8-week-old GF mice were colonised with the gut microbiota of two conventional mice from stage 1 according to the following procedure. The caecal content of donor mice was collected immediately after euthanasia and diluted in liquid casein yeast medium (1 : 100 w/vol) in an anaerobic chamber. Colonisation was achieved by oral-gastric gavage with 250 ml of the diluted caecal content. GF mice were switched from chow to HFD 1 week before the colonisation and were then freely fed the HFD for an additional 16 weeks after colonisation.

All mice were anaesthetised with isoflurane. Procedures were carried out in accordance with the European guidelines for the care and use of laboratory animals and with permission 78-60 of the French veterinary services.

Measurement of plasma parameters and short chain fatty acid concentrations

After 16 weeks of diet, blood taken from the retro-orbital sinus was collected into chilled heparin-coated tubes after 6 h of fasting. Blood glucose was measured using an Accu-Check glucometer (Roche Diagnostics). The remaining blood was then centrifuged at 10 000 g for 10 min. Plasma was aliquoted and frozen at -80°C until analysis.

Alanine and aspartate aminotransferases and lipids were determined using an Olympus AU400 robot. Plasma insulin, leptin, resistin, monocyte chemotactic protein 1, tumour necrosis factor α (TNFα) and interleukin (IL)-6 concentrations were assayed using a Luminex 100 IS system (Luminex Corporation) with a Milliplex MAP mouse serum adipokine panel kit (Millipore). Insulin resistance was estimated by homeostasis model assessment (HOMA index): fasting serum glucose (mg/ dl)×insulin (mU/l)/405. Short-chain fatty acids (SCFA) and branched-chain fatty acids (BCFA) were assayed in caecal samples from receiver mice as previously described. [START_REF] Lan | Survival and metabolic activity of selected strains of Propionibacterium freudenreichii in the gastrointestinal tract of human microbiota-associated rats[END_REF]

Measurement of liver triglycerides

Portions of frozen liver from receiver mice were homogenised in chloroform-methanol (2 : 1) in order to extract total lipids according to the methodology of Folch et al. [START_REF] Folch | A simple method for the isolation and purification of total lipides from animal tissues[END_REF] The organic extract was dried and reconstituted in isopropanol. The triglyceride content was measured with a triglycerides determination kit (Sigma-Aldrich, Saint-Louis, Missouri, USA) according to the manufacturer's instructions and expressed in mmol of triglycerides per milligram of liver.

Liver histology

Thin slices of formaldehyde-fixed, paraffin-embedded liver tissue were stained with haematoxylin and scored for severity of the steatosis and inflammation by an experienced pathologist (FW) blinded to the experiments. The steatosis score was assessed according to the percentage of concerned hepatocytes multiplied by the following grade relying on the size of the fat droplets: grade 1: microvesicular pattern; 2: mixed microvesicular superior to macrovesicular pattern; 3: mixed macrovesicular superior to microvesicular pattern; and 4: macrovesicular ones only.

Gene expression analysis by quantitative PCR

Livers of the receiver mice were disrupted in RNA-PLUS solution (QBiogene). Total RNA was extracted using a modified Chomczynski's procedure. [START_REF] Chomczynski | Single-step method of RNA isolation by acid guanidinium thiocyanate-phenol-chloroform extraction[END_REF] RNA concentration and purity were determined using the Nanodrop ND-1000 spectrophotometer (Thermo Fisher Scientific) at a wavelength of 260/280 nm. RNA integrity was determined with the Agilent bioanalyzer 2100 system with the RNA 6000 Nano LabChip kit. Samples with an RNA integrity number inferior to 8 on a scale ranging from 0 to 10 were eliminated.

Total RNA (2 mg per reaction) was reverse transcribed into complimentary DNA using a high-capacity cDNA reverse transcription kit (Applied Biosystems) according to the manufacturer's instructions. Thereafter, half of the product of each reverse transcription reaction was added to an equal volume of TaqMan universal PCR master mix and then loaded on a TaqMan low density array card (Applied Biosystems). The TaqMan low density array card was centrifuged twice for 1 min at 330g before being sealed. PCR amplification was performed using an Applied Biosystems Prism 7900HT sequence detection system. Data were analysed using SDS V.2.2 software. Statistical analysis was performed using DataAssist v2.0 (Applied Biosystems). The relative gene expressions were normalised to three reference genes: 18S, gapdh and ubc2, chosen on the basis of results obtained from TaqMan mouse endogenous control arrays (Applied biosystems). The non-responder-receiver (NRR) group was used as a reference for the relative expression of genes.

16S rRNA sequencing

Microbiota compositon was thoroughly analysed using 454 pyrosequencing targeting the 16S ribosomal DNA V3-V4 region (V3fwd: 5 0 TACGGRAGGCAGCAG 3 0 , V4rev: 5 0 GGACTACCAGGGTATCTAAT 3 0 ). Forty DNA samples, corresponding to the two caecal samples from donor mice used for inoculation, and faecal samples from receiver mice at two sampling time, were pyrosequenced at Genoscreen (Lille, France) using GS-FLX-Titanium technology. Sequences were trimmed for adaptors and PCR primer removal and binned for a minimal sequence length of 300 bases, a maximum homopolymers length of 6, a minimal base quality threshold set at 27 and a maximum of 15% tolerated low quality bases and N on the overall sequence length. The resulting sequences were assigned to different taxonomic levels (from phylum to genus) using the RDP database (release 10, update 26). [START_REF] Cole | The Ribosomal Database Project: improved alignments and new tools for rRNA analysis[END_REF] Using QIIME, [START_REF] Caporaso | QIIME allows analysis of high-throughput community sequencing data[END_REF] sequences were further clustered in at 97% of identity in operational taxonomic unit (OTU) using cdhit. [START_REF] Li | Cd-hit: a fast program for clustering and comparing large sets of protein or nucleotide sequences[END_REF] OTU were assigned to the closest taxonomic neighbours and relative bacterial species using SEQMATCH and up-to-date 16S rRNA gene RDP database. Estimates of phylotype richness were calculated according to the bias-corrected Chao1 estimator. All statistical analyses were performed using the R program and ade4 package (http://pbil.univ-lyon1.fr/ADE-4/). Principal component analyses (PCA) with the two receiver mice groups at different time points as instrumental variables (interclass PCA) were computed and statistically assessed by a Monte Carlo rank test to observe their net effect on the scattering of the microbiota of different mice. Interclass PCA allows highlighting combinations of variables that maximise variations observed between qualitative variables (eg, specific responder-receiver (RR)/NRR groups). The Wilcoxon test was applied to assess statistical significance in bacterial composition between the different samples.

Statistical analyses

Results are represented as mean±SEM, or median (IQR) for non-parametric data. Statistical analysis was performed by Student's t or Mann-Whitney-Wilcoxon tests, respectively (StatView, SAS Institute Inc, Cary, USA); p<0.05 was considered statistically significant.

RESULTS

Selection of the responder and non-responder donor mice

Conventional mice of the strain C57BL/6J were freely fed a HFD for 16 weeks. HFD treatment led to varying body weights ranging from 33.6 to 47.0 g, with an average final body weight of 39.0±3.9 g (table 1). Despite a majority of the mice developing metabolic disorders, several mice developed high levels of glycaemia, systemic inflammation and steatosis together and were considered as 'responders'. Conversely, several mice did not develop any metabolic disorders and were considered 'nonresponders'. Therefore, we selected from this cohort one responder and one non-responder mouse to verify if we could transmit to GF mice the responder or non-responder phenotypes using gut microbiota transplants. To ensure reliable recovery of the initial gut microbiota in GF mice, transfer of microbiota from donor to receiver mice had to be performed using fresh caecal samples. As a consequence, selection of the donor mice were performed on the basis of parameters available the day of conventional mice euthanasia, including body weight, fat pad and liver weights, food intake, HOMA index and plasma concentrations of pro-inflammatory cytokines (table 1). We intentionally selected two donor mice with similar body weights, fat pad masses and food intake to make sure that the differences in metabolic parameters were not the consequence of a different degree of obesity.

The two groups of receiver mice developed comparable obesity but different metabolic statuses

Two groups of GF male C57BL/6J mice were colonised with the inocula prepared from the two selected donor mice (conventionalisation). The two conventionalised groups were then named NRR and RR, and were subsequently maintained in isolators. After 16-week HFD feeding, both groups showed similar body weight gains (13.2±3.2 g and 14.6±2.6 g for NRR and RR, respectively, p=0.21) and final body weights (figure 1A). Daily food consumption was 2.52±0.92 g and 2.62±0.91 g for NRR and RR mice, respectively, indicating no differences in food intake. Accordingly, epididymal fat pad weights were not found to be different between these two groups (see supplementary table S1, available online only). Conversely, they displayed significantly different fasting glycaemia levels (104.4±26.6 mg/dl for NRR vs 134±34.1 mg/dl for RR) (figure 1B). Concurrently, fasting insulinaemia was lower in NRR mice than in RR mice (665.2±358.9 g/ml vs 1072.4±469.4 pg/ml) (figure 1C). The HOMA index (figure 1D) was consistently 2.4-fold greater in the RR group than in the NRR group, suggesting that the two groups developed different levels of insulin resistance. Leptinaemia (figure 1E) was 75% higher in RR than in NRR mice, whereas the level of resistin was not significantly different in the two groups (data not shown).

The aspartate aminotransferase plasma concentration was three times higher in RR mice than in NRR mice (433.0±201.2 vs 135.9±73.7 UI/l, p=0.0058), whereas the difference did not achieve statistical significance (68.5±56.8 vs 38.1±34.4 UI/l, p=0.0786) for alanine aminotransferase concentrations. Fasting plasma concentrations of triglycerides, cholesterol and highdensity lipoproteins were similar in the two groups (see supplementary table S2, available online only).

Concentrations and proportions of SCFA (acetate, butyrate, propionate, valerate, caproate) were found to be similar in the caecums of NRR and RR groups (see supplementary table S3, available online only). Consistently, the total SCFA concentration was not different in the two groups. Conversely, concentrations and proportions of isobutyrate and isovalerate were significantly higher in the caecum of RR mice (see supplementary table S4, available online only). These BCFA are compounds known to result from the bacterial fermentation of valine and leucine.

RR mice accumulated more triglycerides in the liver than NRR mice

Liver histological analysis showed that the NRR group developed slight to mild steatosis (figure 2A), whereas the RR group developed marked mixed or macrovesicular steatosis (figure 2B).

The steatosis score was found to be higher in RR than in NRR mice (3.00±0.87 vs 1.50±0.61) (figure 2C). Consistent with the morphological changes in the lipid deposition, the liver triacylglycerol concentration was 30% higher in RR mice (figure 2D). No inflammatory infiltrate was observed and no differences in liver weights were found between the two groups (see supplementary table S1, available online only).

RR mice displayed a steatosis-prone hepatic metabolism in contrast to NRR mice

We analysed the hepatic expression of genes involved in lipid uptake, lipogenesis, fatty acid catabolism and very low-density lipoprotein export. The relative expressions of the transcription factors sterol regulatory binding protein (SREBP) 1c, liver X receptor and carbohydrate response element binding protein (ChREBP) are shown figure 3A. These nuclear factors are known to promote de-novo lipogenesis. [START_REF] Musso | Recent insights into hepatic lipid metabolism in non-alcoholic fatty liver disease (NAFLD)[END_REF] The expression of SREBP1c and ChREBP was 1.97 and 2.02 greater in the RR group than in the NRR group, whereas no difference was found in the expression of liver X receptor between the two groups. The relative expressions of three lipogenic enzymes (acetyl-CoA carboxylase 1, stearoyl-CoA desaturase 1 and fatty acid synthase) are shown figure 3B. Stearoyl-CoA desaturase 1 and fatty acid synthase were not differently expressed in the two groups of mice. Conversely, acetyl-CoA carboxylase 1 appeared to be upregulated in RR mice (fold change 1.97). CD36, which imports a large variety of lipids and lipoproteins, was more highly expressed in RR than in NRR mice (fold change 2.32). On the contrary, fatty acid transport protein 5, which essentially transports long-chain fatty acids, was slightly but significantly downregulated (fold change 0.80). No differences were found between the two groups of mice in the messenger RNA levels of carnitine palmytotransferase 1a, a transport protein that regulates mitochondrial β-oxidation, and membrane transport protein, a protein exerting a central regulatory role in lipoprotein assembly and secretion (figure 3C).

No major differences in systemic and hepatic inflammation were detected between the two groups of receiver mice Systemic inflammation was evaluated by assaying the plasma concentration of pro-inflammatory cytokines. No significant differences in the plasma concentrations of these cytokines were found between the two groups of receiver mice (see supplementary table S2, available online only). We then focused specifically on liver inflammation by assaying the hepatic expression of cytokines, markers of liver macrophages and Toll-like receptors (TLR). The relative expression of TNFα, IL-1β, IL-6, IL-10 and transforming growth factor β were found to be similar in the two groups of mice (see supplementary table S5, available online only). Likewise, expressions of CD68 (a marker of macrophages), and TLR (TLR-2, TLR-4, TLR-5 and TLR-9), which play a central role in the innate immune system by recognising bacterial components leading to activation of immune response, appeared to be equivalent in NRR and RR mice (figure 4A,B). Altogether these results indicate that systemic and hepatic inflammation was similar in RR and NRR mice.

Gut microbiota differs between RR and NRR mice

Caecal samples used for inoculation as well as faecal samples from receiver mice after 3 (T3) and 16 (T16) weeks of HFD were analysed by pyrosequencing. A total of 188 058 sequences was obtained and after trimming, 120 241 sequences were further analysed (approximately 3000 sequences/sample). To evaluate similarity among the samples, interclass PCA was performed based on their microbial composition. The main genera between caecal samples from donor mice and faecal samples from corresponding receiver mice were conserved but showed different proportions (see supplementary figure S1, available online only). These differences may relate to sampling location or specific selection within receivers. At a genus level, the microbiota from NRR and RR mice significantly clustered separately at T3 (Monte Carlo test; p=0.001) but adopted a closer configuration at T16 (figure 5A). Barnesiella and Roseburia genera were represented more in RR mice at T3 and T16, whereas Allobaculum was higher in NRR mice (figure 5B). Moreover, HFD treatment increased Barnesiella and Allobaculum and decreased Lactobacilli in the two groups (figure 5B). The microbial phylotype richness assessed by the Chao1 estimator was very similar between the two groups whatever the sampling time (figure 5C). At the phylum level, RR mice harboured significantly increased numbers of sequences belonging to Firmicutes at T3 ( p=0.0041; figure 5D). Finally, two main bacterial species (Lachnospiraceae bacterium 609 and a relative of Barnesiella intestinihominis), corresponding to more than 10% of sequences, were found to be significantly overrepresented in RR mice at T3 ( p=2.16e-5 and p=0.00025, respectively) and to a lesser extent at T16. Conversely, a significantly increased number of sequences related to Bacteroides vulgatus was found in NRR mice ( p=0.00041; figure 5E).

DISCUSSION

HFD feeding is extensively used in rodents for developing obesity, steatosis and insulin resistance. In fact, these models are considered more relevant to human metabolic diseases than models of gene inactivation. [START_REF] Fearnside | Phylometabonomic patterns of adaptation to high fat diet feeding in inbred mice[END_REF] Nevertheless, wide variations in the susceptibility to develop HFD-associated features have been observed, 9 10 and the factors that make some rodents susceptible and others resistant to metabolic disorders induced by HFD are still indefinite. Using GF animal models, we and others have previously demonstrated that mice devoid of gut microbiota are resistant to diet-induced obesity, steatosis and insulin resistance. 5 6 Like humans, each conventional mouse harbours a unique gut microbiota, and it was recently shown that diabetessensitive and diabetes-resistant mice are associated with a specific gut microbiota. [START_REF] Serino | Metabolic adaptation to a high-fat diet is associated with a change in the gut microbiota[END_REF] In the present study, we first submitted mice of a common biological lineage to the same HFD and laboratory environment, and obtained, as expected, heterogeneous responses including variable weight gain, steatosis, HOMA index and systemic inflammation levels. Then, we inoculated two groups of GF mice with the caecal microbiota of two mice whose responses to HFD were discrepant. The outcomes of HFD feeding were found to be different in these two groups of mice associated with distinct microbiota. Gut microbiota dysbiosis has recently been associated with liver diseases. 7 8 Here, we further demonstrated for the first time the causal role of the gut microbiota in the susceptibility to develop NAFLD features in response to a HFD challenge. Using pyrosequencing, we identified phyla and genera differently represented in the gut microbiota of the two receiver groups. Moreover, two bacterial species were found to be dominant in RR mice only. Interestingly, Barnesiella intestinihominis belongs to the family Porphyromonadaceae, which was increased in inflammasomedeficient mice associated with exacerbated hepatic steatosis and inflammation. [START_REF] Henao-Mejia | Inflammasome-mediated dysbiosis regulates progression of NAFLD and obesity[END_REF] On the other hand, Bacteroides vulgatus, previously found to be decreased in the faecal microbiota of patients with diabetes, [START_REF] Wu | Molecular characterisation of the faecal microbiota in patients with type II diabetes[END_REF] was found to be overrepresented in NRR mice, suggesting that this bacterial species may exert protective effects against the development of metabolic disorders. The concept that insulin resistance is a common process to all stages of NAFLD is now accepted. We have previously demonstrated that GF mice are resistant to diet-induced insulin resistance. [START_REF] Rabot | Germ-free C57BL/6J mice are resistant to high-fat-diet-induced insulin resistance and have altered cholesterol metabolism[END_REF] In the present study, we further revealed that the insulin resistance index is a transmissible feature by means of gut microbiota transplantation. Both glucose and insulin were found to be higher in RR mice, indicating that hyperplycaemia is likely to result from insulin resistance rather than insulin secretion deficit. We also observed that NRR and RR mice displayed similar body and epididymal fat pad weights. Despite not measuring mesenteric and subcutaneous fats, our results suggest that insulin resistance develops from separate mechanisms to those responsible for obesity. This supports a recent study revealing that the gut microbiota impacts adipose macrophage accumulation and systemic glucose metabolism by independent mechanisms. [START_REF] Caesar | Gut-derived lipopolysaccharide augments adipose macrophage accumulation but is not essential for impaired glucose or insulin tolerance in mice[END_REF] This further confirms a previous study demonstrating that the modulation of gut microbiota using antibiotics in a diet-induced obesity mouse model showed a significant improvement of fasting glycaemia and insulin resistance independently of food intake or adiposity. [START_REF] Membrez | Gut microbiota modulation with norfloxacin and ampicillin enhances glucose tolerance in mice[END_REF] In addition, improved insulin sensitivity was found to correlate with reduced hepatic lipogenesis and steatosis in these antibiotic-treated mice, [START_REF] Membrez | Gut microbiota modulation with norfloxacin and ampicillin enhances glucose tolerance in mice[END_REF] implying that the influence of the gut microbiota on host glucose metabolism and on liver functions may result from dependent or identical mechanisms.

It has been proposed that the ability of GF mice to resist diet-induced insulin resistance could be due partly to their low plasma TNFα concentrations. [START_REF] Rabot | Germ-free C57BL/6J mice are resistant to high-fat-diet-induced insulin resistance and have altered cholesterol metabolism[END_REF] Our results, showing different levels of HOMA index but similar TNFα plasma concentrations and hepatic expression levels, indicate rather that the impact of our two different transplanted microbiota is independent of TNFα. It corroborates recent results showing similar TNFα and IL-6 expression in the livers of diabetic sensitive and resistant mice, despite different microbiota profiles. [START_REF] Serino | Metabolic adaptation to a high-fat diet is associated with a change in the gut microbiota[END_REF] Overall, all parameters related to inflammation were similar in our two receiver groups and were then unaffected by the transplantation of different gut microbiota. This indicates that in our conditions the gut microbiota affects host metabolism by mechanisms that might be independent of the immune system. This finding opens new perspectives in the field indicating that gut communities are able to drive host metabolism by cross-talk mechanisms still unknown that seem to be independent of a systemic pro-inflammatory state.

Dysregulation of hepatic metabolism has been implicated in the development of NAFLD. Nevertheless, HFD, which causes increased influx of fatty acids without primary impairment of very low-density lipoprotein export, induced merely steatosis, but not liver inflammation. [START_REF] Shindo | Involvement of microsomal triglyceride transfer protein in nonalcoholic steatohepatitis in novel spontaneous mouse model[END_REF] Previous studies have demonstrated that GF mice display increased fatty acid oxidation and decreased lipogenesis. [START_REF] Backhed | The gut microbiota as an environmental factor that regulates fat storage[END_REF][START_REF] Backhed | Mechanisms underlying the resistance to diet-induced obesity in germ-free mice[END_REF][START_REF] Rabot | Germ-free C57BL/6J mice are resistant to high-fat-diet-induced insulin resistance and have altered cholesterol metabolism[END_REF] Our results from gene expression in the liver indicate that the increased steatosis observed in the livers of RR mice results mainly from increased lipid uptake and de-novo lipogenesis, suggesting increased energy availability. Indeed, a higher energy harvest from the diet resulting in elevated SCFA production by the gut microbiota has been demonstrated in ob/ob mice. [START_REF] Turnbaugh | An obesity-associated gut microbiome with increased capacity for energy harvest[END_REF] Nevertheless, the role of SCFA is controversial as they are known to stimulate de-novo synthesis of triglycerides in the liver but were also proposed in the prevention of type 2 diabetes. [START_REF] Roelofsen | The interaction of short-chain fatty acids with adipose tissue: relevance for prevention of type 2 diabetes[END_REF] In our study, only the concentrations of BCFA were higher in the caecums of RR mice indicating more pronounced protein fermentation, which is commonly associated with a negative health effect. [START_REF] Macfarlane | Bacteria, colonic fermentation, and gastrointestinal health[END_REF] Moreover, our results corroborate a recent study indicating a higher faecal concentration of isobutyrate and isovalerate but not SCFA in obese compared to normal weight individuals. [START_REF] Tiihonen | Effect of overweight on gastrointestinal microbiology and immunology: correlation with blood biomarkers[END_REF] Finally, the increased de-novo lipogenesis observed in RR mice may be due to the increased levels of blood glucose and insulin. Indeed, these levels are known to activate the transcription factors ChREBP and SREBP, which control the transcription of lipogenic enzyme genes. Therefore, we may postulate that the influence of the gut microbiota on glucose homeostasis may constitute a possible new mechanism explaining the impact of this microbiota on steatosis and NAFLD. This study, showing the feasibility of transferring host metabolic phenotype by means of the gut microbiota, further underlines the power of gnotobiotic models to decipher the involvement of the gut microbiota in health and disease. It has recently been demonstrated that human gut microbiota can be transferred to GF mice with remarkable preservation of structure and diversity, [START_REF] Turnbaugh | The effect of diet on the human gut microbiome: a metagenomic analysis in humanized gnotobiotic mice[END_REF] and previous studies established that specific enzymatic activities of the human gut microbiota can be transmitted to GF rodents on colonisation with a faecal sample. 31 32 This opens possibilities to transfer gut microbiota from humans with pathologies to GF mice to create humanised gnotobiotic mice and conduct preclinical studies.

In conclusion, our results demonstrate that gut microbiota influences whole body glucose homeostasis and liver lipid metabolism, indicating that the gut microbiota constitutes an environmental factor driving the progression of NAFLD. They support the idea that targeting the gut microbiota could be a new prevention or therapeutic approach for improving type 2 diabetes and NAFLD. Finally, faecal transplants in humans could be envisaged in the future to treat these pathologies, as already applied for Clostridium difficile infections. [START_REF] Rohlke | Fecal flora reconstitution for recurrent Clostridium difficile infection: results and methodology[END_REF]

Figure 1

 1 Figure 1 Metabolic responses to high-fat diet of non-responder-receiver (NRR) and responder-receiver (RR) mice. (A) Body weight curves; (B) fasting glycaemia; (C) fasting insulinemia; (D) homeostasis model assessment-insulin resistance Index; (E) fasting leptinaemia. All mice were fasted for 6 h before blood sampling. Data are mean±SEM, n=18 for NRR and n=22 for RR mice. **p<0.01; ***p<0.001 (Student's t test).

Figure 2

 2 Figure 2 Representative H&E stains of paraformaldehyde-fixed liver sections prepared from non-responder-receiver (NRR) (A) and responder-receiver (RR) mice (B). The steatosis score corresponds to the percentage of hepatocytes presenting with steatosis multiplied by the intensity of the steatosis of the concerned hepatocytes (C). (D) Hepatic triglyceride concentrations (mmol of triglycerides per milligram of liver). Data are shown as mean±SEM, n=18 for NRR and n=22 for RR mice. *p<0.05; **p<0.01 (Student's t test).

Figure 3

 3 Figure 3 Relative hepatic gene expression analyses of transcription factors (A), lipogenic enzymes (B) and lipids transporters (C). Results were normalised on the non-responder-receiver (NRR) group. Data are shown as mean±SEM, n=14 for NRR and n=16 for responder-receiver (RR) mice. *p<0.05; ***p<0.001. ACC, acetyl-CoA carboxylase; ChREBP, carbohydrate response element binding protein; CPT1a, carnitine palmytotransferase 1a; FAS, fatty acid synthase; FATP5, fatty acid transport protein 5; LXR, liver X receptor; MTP, membrane transport protein; SCD1, stearoyl-CoA desaturase 1; SREBP, sterol regulatory binding protein.

Figure 4

 4 Figure 4 Relative hepatic gene expression analyses CD68 (A), and Toll-like receptors (TLR) 2, 4, 5 and 9 (B). Results were normalised on the non-responder-receiver (NRR) group. Data are shown as mean±SEM, n=14 for NRR and n=16 for responder-receiver (RR) mice.

Figure 5

 5 Figure 5 Microbiota composition and diversity in non-responder-receiver (NRR) and responder-receiver (RR) mice (A) Interclass principal component analyses with phenotype as environmental factor was performed based on the genera abundance. Mice (light green circles, NRR_T3; green circles, NRR_T16, light red, RR_T3 and red, RR_T16) were clustered (ellipses) and the centre of gravity computed for each class. The p value of the link between phenotype and genera abundance was assessed using a Monte Carlo test (999 replicates) p=0.001. (B) Main genus composition as a percentage of total assigned sequences (sequences unassigned to any genus represent 46% of total sequences in average, SD 13.94%). (C) Estimates of bacterial richness as assessed by the Chao1 estimate (median number of OTU NRR_T3: 1052; NRR_T16: 1230; RR_T3: 1080; RR_T16: 1146). (D) Bacterial phyla distribution as a percentage of total sequences. (E) Differentially represented bacterial species (or relatives of bacterial isolates). All presented results are statistically significant ( p<0.05) as assessed by the Wilcoxon test after applying Bonferroni correction.

Table 1

 1 Metabolic status of conventional and donor mice after 16 weeks of HFD

		Body		Average	Liver weight	Epididymal fat	Fasting	Fasting
		weight	Final body	food intake	(% of body	pads (% of	glycaemia	insulinaemia		MCP-1	TNF-α
		gain (g)	weight (g)	(g/day)	weight)	body weight)	(mg/dl)	(pg/ml)	HOMA-IR	(pg/ml)	(pg/ml)
	Conventional HFD	15.3±3.3 39±3.9	4.8±0.3	4.65±0.21	5.02±0.17	164±39	1944±820	19.46±9.58 36.2±47.1 5.4±4.4
	mean
	Responder donor	17.6	40.2	4.7	3.77	5.03	191	3673	42.94	199	13.1
	Non-responder	15.7	39.4	5.1	4.38	4.49	90	2054	11.31	4	3.5
	donor

Data obtained from a cohort of 24 conventional mice, data are mean±SEM. HFD, high-fat diet; HOMA-IR, homeostasis model assessment of insulin resistance; MCP-1, monocyte chemotactic protein 1.

group.bmj.com on February 18, 2014 -Published by gut.bmj.com Downloaded from

Acknowledgements The authors would like to thank Martine Bensaada, Anne Foussier, Bénédicte Langelier, Camille Mayeur and Laura Boschat for technical assistance, Alain Joffre for breeding and husbandry of GF mice, Donald White for the English correction.

Contributors TLR: acquisition of data, analysis and interpretation of data; ML: acquisition of data; ABr: technical support; SR, CB, PM: material support; PL, FW, ABa, GP, AMCD: analysis and interpretation of data; PG: study concept and design, analysis and interpretation of data. TLR and PG: drafted manuscript, which all authors have commented on.

Funding This work was supported by the Science Committee Syndifrais/CNIEL and a grant was obtained from CNIEL (Centre National Interprofessionnel de l'Economie Laitière), grant number S2234.

Competing interests None.

Provenance and peer review Not commissioned; externally peer reviewed.

Updated information and services can be found at:


Olivier Lopez
email: lopez@ensai.fr.

On the estimation of the joint distribution in regression models with censored responses

Keywords: distribution function, right censored regression, law of large numbers, asymptotic normality, kernel estimators. MSC 2000 : 62N01, 62G30, 60F05

In a regression model with univariate censored responses, a new estimator of the joint distribution function of the covariates and response is proposed, under the assumption that the response and the censoring variable are independent conditionally to the covariates. This estimator is an extension of the multivariate empirical function used in the uncensored case. Furthermore, under some simple additional identiability assumption, this estimator is not sensible to the "curse of dimensionality", so that it allows to infer on models with multidimensional covariates. Integrals with respect to this empirical measure are considered. Consistency and asymptotic normality of these integrals over a class of functions is obtained, by deriving asymptotic i.i.d. representations. Several applications of the new estimator are proposed.

Introduction

Under random censoring, estimation of the distribution of a single variable Y is traditionally carried out by using the Kaplan-Meier estimator (see [START_REF] Kaplan | Nonparametric estimator from incomplete observations[END_REF]). A vast scope of approaches has been developed to study the theoretical behavior of this estimate, and of Kaplan-Meier integrals (K M -integrals in the following). See e.g. Gill (8), Stute and Wang [START_REF] Stute | The strong law under random censorship[END_REF], Stute (20), Akritas [START_REF] Akritas | The Central Limit Theorem with Censored Data[END_REF]. A crucial identiability assumption to obtain convergence is the independence of Y and C, the censoring variable. In presence of (uncensored) covariates X, it seems natural to extend Kaplan-Meier's approach, but now to estimate a multivariate distribution function, that is F (x, y) = P(X ≤ x, Y ≤ y). However, traditional approaches to this kind of problem typically face two major kind of drawbacks, that is either they do not allow to handle multivariate X, or they rely on strong identiability assumptions which restrain the eld of applications. The aim of this paper is to propose a new approach which circumvents these two important limitations, by generalizing the notion of the (multivariate) empirical distribution function.

Indeed, a crucial point in censored regression is to extend the identiability assumption on the independence of Y and C (used in the univariate case) to the presence of covariates. In the spirit of KM -estimator, one may impose that Y and C are independent conditionally to X, which seems to be the slightest identiability assumption.

Under this assumption, Beran [START_REF] Beran | Nonparametric regression with randomly censored survival data[END_REF] provided an estimate of the conditional distribution function F (y | x) = P (Y ≤ y | X = x). In this approach, kernel smoothing is introduced into Kaplan-Meier's approach to account for the information on the interest variable contained in the covariates. Dabrowska (3), (4) studied asymptotics of Beran estimate. Van Keilegom and [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF] proposed, with some additional assumptions on the regression model, a modication Beran's approach and derived asymptotic properties of their estimate in the case X ∈ R. A major diculty in studying this kind of estimate stands in the non-i.i.d. sums that may be involved in. Therefore, several asymptotic i.i.d. representations of the estimated conditional distribution function have been proposed, all in the case where x is univariate, see e.g. Van Keilegom and Akritas [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], Van Keilegom and Veraverberke [START_REF] Van Keilegom | Estimation and Bootstrap with censored data in xed design nonparametric regression[END_REF]. In particular, Du and Akritas [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF] proposed an uniform i.i.d. representation that holds uniformly in y and x.

When it comes to the multivariate distribution function F (x, y), Stute (21) proposed an extension of KM -estimator, and furnished asymptotic representation of integrals with respect to this estimator that turned out to have interesting practical applications for regression purpose in some situations, see also Stute (22), Gonzalez-Manteiga, Sanchez-Sellero and Van Keilegom [START_REF] Sánchez Sellero | Uniform representation of product-limit integrals with applications[END_REF], Delecroix, Lopez and Patilea [START_REF] Delecroix | Nonlinear Censored Regression Using Synthetic Data[END_REF], Lopez and Patilea (14).

Moreover, in this approach, the covariates do not need to be one-dimensional. Nevertheless, consistency of Stute's estimator relies on assumptions that may be unrealistic in some situations, especially when C and X are not independent. On the other hand, under the more appealing assumption that Y and C are independent conditionally to X, Lo and Singh [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF] and Van Keilegom and Akritas [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF] used an empirical integral of Beran estimator. Van Keilegom and Akritas (25) also provided some alternative estimate in their so-called scale-location model. To our best knowledge, i.i.d. representations of integrals with respect to these estimated distributions have not been provided yet. Moreover, it is particularly disappointing to see that, in the uncensored case, the empirical distribution function of (X , Y ) can not be seen as a particular case of these approaches. On the contrary, KM -estimator is a generalization of the (univariate) empirical distribution function. As a large amount of statistical tools are seen to be integrals with respect to the empirical distribution function, it is still of interest to produce some procedure that would generalize this simple and classical way to proceed to the censored framework. In fact, an important preoccupation in the study of censored regression is to extend procedures existing in the uncensored case. For this reason, it is of real interest to use the most natural extension of the uncensored case's concepts.

In this paper, the new estimator is an extension of the notion of the multivariate empirical distribution function, and it can also be seen as a generalization of univariate Kaplan-Meier estimator. Using the results of Dabrowska (4) and Du and Akritas (6), we provide some i.i.d. representation of integrals with respect to this estimator, uniformly over a class of functions. Furthermore, we propose a reasonable modication of the identiability assumption that may allow us to consider multivariate covariates. The paper is organized as follows. In section 2, we present the model and motivate the introduction of our new estimator of F (x, y). In section 3, we present the asymptotic properties of integrals with respect to this estimate. Section 4 is devoted to some applications of these results, and section 5 to some simulation study, while section 6 gives the proof of some technical results.

Femp (x, y) = 1 n n i=1 1 X i ≤x,Y i ≤y , (2.1)
where 1 A denotes the indicator function of the set A. If we are interested in estimating E[φ(X, Y )] = φ(x, y)dF (x, y) for some measurable function φ, we can proceed by using

φ(x, y)d Femp (x, y) = 1 n n i=1 φ (X i , Y i ) .
Studying the behavior of these integrals is then more general than simply studying the distribution function (2.1). Asymptotic results on these empirical integrals may be derived by applying the classical strong law of large numbers and the central limit theorem. In a censored regression model, the situation is dierent since the variable Y is not directly available. Indeed, instead of Y , one observes

T = Y ∧ C, δ = 1 Y ≤C .
Observations consist of a n-sample (X i , T i , δ i ) 1≤i≤n . In this framework, the empirical distribution function can not be computed, since it depends on unobserved quantities Y i . In absence of covariates X, the univariate distribution function P(Y ≤ y) can be F km (y) = 1 -

T i ≤y 1 - 1 n j=1 1 T j ≥T i δ i .
Asymptotics of F km and of integrals with respect to F km can be found in Stute and Wang [START_REF] Stute | The strong law under random censorship[END_REF] and Stute [START_REF] Stute | The central limit theorem under random censorship[END_REF]. Conditions for convergence are essentially of two kinds : moment conditions (which can be interpreted as assumptions on the strength of the censoring in the tail of the distributions, see condition (1.6) in Stute [START_REF] Stute | The central limit theorem under random censorship[END_REF], and an identiability condition that are only needed to ensure that F km converges to the proper function. This identiability condition, in the univariate case, reduces to Y and C are independent.

(2.2)

In a regression framework, an important question is to extend condition (2.2) to the presence of covariates. A rst way to proceed would be to assume that (X , Y ) and C are independent.

(2.3) However, assumption (2.3) is too restrictive, since, in several frameworks, the censoring variable may depend on X. Stute (21) proposed to replace this assumption by assumption (2.2) and

P (Y ≤ C | X, Y ) = P (Y ≤ C | Y ) . (2.4)
Under these assumption (2.2) and (2.4), Stute (21) studied the asymptotics of the following estimate, that is

F S (x, y) = n i=1 W in 1 X i ≤x,T i ≤y ,
where W in denotes the jump of F km at the i-th observation. Observing that

W in = 1 n δ i 1 -G km (T i -)
,

where G km (t) denotes the Kaplan-Meier estimate of G(t) = P (C ≤ t) (see Satten and Datta (17)), this estimator may be rewritten as

F S (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -G km (T i -)
.

(2.5)

From this writing, one may observe two interesting facts. First, this estimate is a generalization of the empirical distribution function used in the uncensored case. Indeed, in absence of censoring, 1 -G km (t) ≡ 1 for t < ∞, and δ = 1 a.s. Second, F S can be seen as an approximation of the empirical function Y and C are independent conditionally to X.

FS (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -G (T i -
(

Inspired by the empirical distribution function, we are searching for an estimate which puts mass only at the uncensored observations, that is of the form

1 n n i=1 δ i W (X i , T i )1 X i ≤x,T i ≤y , (2.7)
where W (X i , T i ) is some weight which has to be chosen in order to compensate the bias due to censoring (F S is an estimator of the type (2.7), however, under (2.6), it is biased).

An ideal way to proceed would be to use weights such as, for any function φ,

E [δ i W (X i , T i )φ(X i , T i )] = φ(x, y)dF (x, y) ,
so that integrals with respect to the measure dened by (2.7) converge to the proper limit by the law of large numbers. In this case, (2.7) would appear to be a sum of i.i.d. quantities converging to F (x, y) from the strong law of large numbers. Under (2.6), observe that,

for any function W , E[δ i W (X i , T i )φ(X i , T i )] = E[{1 -G(Y i -|X i )}W (X i , Y i )φ(X i , Y i )], (2.8)
where G(y | x) denotes P(C ≤ y | X = x). Hence, a natural choice of W would be

W (X i , T i ) = 1 1 -G (T i -| X i ) . F (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -G (T i -| X i )
.

(2.9)

Unfortunately, G(y | x) is unknown. However, it can be estimated using Beran's estimator (1981). This estimator is dened, in the case d = 1, by Ĝ(y | x) = 1 -

T i ≤y 1 - w in (x) n j=1 w in (x)1 T j ≥T i 1-δ i , (2.10)
where, introducing a kernel function K,

w in (x) = K X i -x h n j=1 K X j -x h
.

The estimator of F that we propose is then

F (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -Ĝ (T i -| X i ) . (2.11)
This type of approach is quite natural in censored regression, see e.g. van der Laan and Robins [START_REF] Van Der Laan | Unied Methods for Censored Longitudinal Data and Causality[END_REF] or Koul, Susarla, Van Ryzin [START_REF] Koul | Regression analysis with randomly right censored data[END_REF]. From this denition, we see that this estimate generalizes the empirical distribution function for the same reasons (2.5) does.

Now if we consider a function φ(x, y), we can estimate φ(x, y)dF (x, y) by

φ(x, y)d F (x, y) = 1 n n i=1 δ i φ (X i , T i ) 1 -Ĝ (T i -| X i ) .
(2.12)

This estimator is more dicult to study than (2.9), since, as it is the case for Kaplan-Meier integrals, the sums in (2.11) and (2.12) are not i.i.d. In fact, each term depends on the whole sample since Ĝ is computed from the whole sample. In section 3, we will show

that φ(x, y)d F (x, y) = φ(x, y)d F (x, y) + S n (φ) .
From (2.8), the classical strong law of large numbers and the central limit theorem, the rst integral will converge to φ(x, y)dF (x, y) at rate n -1/2 , while S n (φ), under suitable conditions, can be written as an i.i.d sum which only contributes to the asymptotic variance, but does not modify the limit.

2.2

Comparison with other approaches Under (2.6), most of the eorts have been concentrated in estimating F (y

| x) = P(Y ≤ y | X = x).
Y = m (X) + σ (X) ε, (2.13)
for some location function m, some scale function σ, and ε independent from X.

When it comes to the estimation of the estimation of F (x, y), the only approach that has been used consists of considering

x -∞ F (y | u)d F (u) , (2.14)
where F (x) denotes the empirical distribution function of X. 9) and [START_REF] Heuchenne | Nonlinear censored regression[END_REF].

On the other hand, the estimator proposed in (2.12) is not of the type (2.14). It still relies on Beran's estimator, so that its asymptotical behavior will only be carried out for d = 1. However, in section 2.3, we propose a modication of this estimator to handle the case d > 1, by slightly strengthening the condition (2.6).

2.3

The case d > 1

In (2.
P (Y ≤ C | X, Y ) = P (Y ≤ C | g(X), Y ) .
(2.16)

In particular, condition (2.16

) will hold if L(C | X, Y ) = L(C | g(X), Y ), that is if C
depends only on g(X) and Y . As an important example, denote X = (X (1) , ..., X (d) ). In some practical situations, one may suspect the censoring variable to depend only on X (k) for some k known.

Another interesting advantage of this model is that it may permit us to consider discrete covariates. If we refer to the approach of Van Keilegom and Akritas [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], we can only consider continuous covariates. Here, we will only have to assume that g(X) has a density (but not necessary all component of X). Under this new set of identiability conditions, we propose to use

F (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -G (T i -| g(X i ))
,

(2.17)

F (x, y) = 1 n n i=1 δ i 1 X i ≤x,T i ≤y 1 -Ĝ (T i -| g(X i )
) .

(2.18)

Note that using the set of condition (2.15)-(2.16) does not permit to prevent the estimators of type (2.14) from the curse of dimensionality. In fact, using estimators (2.14), we still need to estimate F (y | x), no matter the identiability conditions.

Asymptotics

To simplify the notation, let z = g(x) and Z i = g(X i ). We provide asymptotic i.i.d.

representations of integrals of the type (2.12) which hold uniformly over a class of functions F.

Assumptions

We list here some assumptions that are needed to ensure consistency and asymptotic normality of our estimate. We will use the following notations to refer to some (sub-)distribution functions,

H (t) = P (T ≤ t) , H (t | z) = P (T ≤ t | Z = z) , H 0 (t | z) = P (T ≤ t, δ = 0 | Z = z) , H 1 (t | z) = P (T ≤ t, δ = 1 | Z = z) .
Assumptions on the model.

Assumption 1

The variable Z = g(X) belongs to a compact subset X ⊂ R. The distribution function of Z has three bounded derivatives on the interior of X . Furthermore, the

density f Z (z) satises inf z∈X f Z (z) > 0. Assumption 2 Let τ H,z = inf{t | H(t | z) < 1}.
There exists some real number τ < τ H,z for all z ∈ X .

Assumption 2 has to be connected with the bad performances of Beran's estimator in the tail of the distribution. This assumption is present in Du and Akritas [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF]. In Van their scale-location regression model.

The important situation that we have in mind in which Assumption 2 holds, is when, for all x, the support of the conditional law

L(Y | Z = z) is [a(z), τ H ], for some τ H ≤ ∞
and a(z) ≥ -∞, where the upper bound τ H does not depend on z and can be nite or not (for example, this condition is fullled when Y is gaussian conditionally to Z = g(X)).

In this case, τ can be chosen arbitrary close to τ H .

Assumptions on the regularity of the (sub-)distribution functions.

We will assume that the variable Z = g(X) is continuous, but the responses may not be. For any function J(t | z) we will denote by J c (t | z) the continuous part, and

J d (t | z) = J(t | z) -J c (t | z).
|H c (t 1 | z) -H c (t 2 | z)| ≤ |L 1 (t 1 ) -L 1 (t 2 )| , ∂H c ∂z (t 1 | z) - ∂H c ∂z (t 2 | z) ≤ |L 2 (t 1 ) -L 2 (t 2 )| , ∂H 0c ∂z (t 1 | z) - ∂H 0c ∂z (t 2 | z) ≤ |L 3 (t 1 ) -L 3 (t 2 )| ,
the last two assumptions implying the same kind for ∂H 1c /∂z.

s i = sup z |F (d i -| z) -F (d i | z)| , s i = sup z ∂F ∂z (d i -| z) - ∂F ∂z (d i | z) , r i = sup z |G(d i -| z) -G(d i | z)| , r i = sup z ∂G ∂z (d i -| z) - ∂G ∂z (d i | z) .
Then

d i ≤τ s i + s i + r i + r i < ∞.
Assumptions on the kernel.

Assumption 8 The kernel K is a symmetric probability density function with compact support, and K has bounded second derivative.

Assumption 9 The bandwidth h satises (log log n)n -1 h -2 = O(1), and nh 5 (log n) -1 =

O(1).

Assumptions on the family of functions. To achieve uniform consistency over a class of functions, it is necessary to make assumptions on the class of functions F.

Assumption 10 The class F is P-Glivenko-Cantelli (cf. Van der Vaart and Wellner, 1996, page 81) and has an integrable enveloppe Φ satisfying Φ(t) = 0 for t ≥ τ, for some τ as dened in Assumption 1.

For asymptotic normality, we will need more restrictions on the class F. Dene N (ε, F, L 2 ) denote the covering number (cf. Van der Vaart and Wellner (24) page 83) of the class F relatively to the L 2 -norm.

Assumption 11 N (ε, F, L 2 ) ≤ Aε -V for some A and V > 0, and F has a square integrable envelope Φ, satisfying Φ(x, t) = 0 for t ≥ τ, for some τ as dened in Assumption 1.

Particular case of classes satisfying Assumption 11 are V C-subgraph classes of functions (see Van der Vaart and Wellner ( 24)). We also need some dierentiability conditions on functions φ.

• The envelope Φ is square integrable.

• Let F Z (x, y) = P(X ≤ x, Y ≤ y | Z), and for any function φ, dene φz (s) =

X ×R 1 s≤y φ(x, y)dF z (x, y).

Let X δ be the set of all points at a distance at least δ > 0 from the complementary of X . Assume that there is a nite number K(F) such as, for all φ ∈ F,

φ(X, Y ) = K(F ) i=1 φ i (X, Y )1 g(X)∈I i ,
where I i ⊂ X δ , and φi,z is twice continuously dierentiable with respect to z, with

sup s≤τ,z |∂ z φi,z (s)| + |∂ 2 z φi,z (s)| ≤ M < ∞, for some constant M not depending on φ i . • Φ is bounded on X δ ×] -∞; τ ],
and has bounded partial derivatives with respect to z.

The reason for introducing the set X δ is to prevent us from some boundary eects which happen while obtaining uniform convergence rate for kernel estimates, see the proof of our Theorem 3.3 below. Note that it is possible to replace X δ by a set growing with n, that is X δ(h) . For the sake of simplicity, we do not consider this situation, and prefer to focus on a xed δ. Consequently, if we consider the case g(x) = x, to estimate the distribution function F (x 0 , y 0 ), we should consider the function φ(x, y) = 1 x≤x 0 ,y≤y 0 . This function does not satisfy Assumption 12, but we can still consider 1 x≤x 0 ,y≤y 0 1 x∈X δ . This will lead to an asymptotically biased estimate, but this bias can be taken arbitrary small, as in the approach of Van Keilegom and Akritas (25).

Consistency

Theorem 3.1 Under Assumptions 1, 2, 8, 10, and with h → 0, and nh → ∞, sup φ∈F φ (x, y) d F (x, y) -φ (x, y) dF (x, y) → a.s. 0.

Proof. Write, from the denition (2.12) of I(φ),

I(φ) = 1 n n i=1 δ i φ (X i , T i ) 1 -G (T i -| Z i ) + 1 n n i=1 δ i φ(X i , T i )[ Ĝ(T i -|Z i ) -G(T i -|Z i )] [1 -G (T i -|Z i )][1 -Ĝ (T i -|Z i )] (3.1)
= I 0n + I 1n .

From the strong law of large numbers, the rst term converges almost surely to the expectation φ (x, y) dF (x, y) (uniformly over F since F is P -Glivenko Cantelli), while, for the second,

|I 1n | ≤ O P (1) × sup t≤τ,x∈χ Ĝ(t -|z) -G (t -|z) × 1 n n i=1 δ i |Φ(X i , T i )| [1 -G(T i -|Z i )] 2 .
The empirical sum converges almost surely, while the supremum tends to zero almost surely from Corollary 2.1 of Dabrowska (4).

Asymptotic normality

Theorem 3.1 is not sucient when it comes to proving asymptotic normality of integrals of type (2.12). As in the case of Kaplan

ξ z (T i , δ i ; t) = [1 -G(T i -|z)](1 -δ i )1 T i ≤t [1 -G(T i |z)][1 -H(T i -|z)] - t -∞ 1 T i ≥s [1 -G(s -|z)]dH 0 (s|z) [1 -G(s|z)][1 -H(s -|z)] 2 = ψ 1,z (T i , δ i )1 T i ≤t - t -∞ ψ 2,x (T i , s)dH 0 (s|z),
the authors showed that, under Assumptions 1 to 9 and for t ≤ τ, Ĝ(t|z) -G(t|z)

1 -G(t|z) = 1 n n i=1 w in (z)ξ z (T i , δ i ; t) + R G n (z, t), with sup z∈X δ ,t≤τ |R G n (z, t)| = O a.s. ((log n) 3/4 n -3/4 h -3/4
). Actually, the authors provide an uniform rate over the whole set X , but this is due to the fact that they overlook the Taylor expansion problem near the boundaries of X . See formulas (A.13) and (A.14) in Du and Akritas [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF], where their O(h 3 n ) should be O(h 2 n ). See also the proof of our Theorem 3.3.

The representation of Du and Akritas can be rewritten, analogously to the expansion of Kaplan-Meier's estimate, Ĝ(t|z) -G(t|z)

1 -G(t|z) = t -∞ dM n,z (y) [1 -G(y|z)][1 -F (y -|z)] + R G n (z, t), (3.2)
where

M n,z (t) = n i=1 w ni (z) (1 -δ i )1 T i ≤t - t -∞ 1 T i ≥y dG(y|z) 1 -G(y -|z) .
Observe that, contrary to the i.i.d. representation of Kaplan-Meier estimate, M n,z is not a martingale with respect to the natural ltration

H t = σ({X i 1 T i ≤t , T i 1 T i ≤t , δ i 1 T i ≤t , i = 1, ..., n}), since it is biased. In fact, we have E[ξ Z i (T i , δ i ; t)|X i ] = 0, (3.3)
but E[ξ z (T i , δ i ; t)] = 0. However, from (3.2) and (3.3), it seems natural to dene

M i (t) = (1 -δ i )1 T i ≤t - t -∞ 1 T i ≥y dG(y|Z i ) 1 -G(y -|Z i ) ,
which is a martingale which will naturally appear in the development of the integral of type (2.12).

Theorem 3.2 Under Assumptions 1 to 12,

φ (x, y) d( F -F ) (x, y) = 1 n n i=1 φZ i (s)dM i (s) [1 -F (s-| Z i )][1 -G(s | Z i )] +R n (φ) , with sup φ∈F |R n (φ)| = O P ((log n) 3/4 n -3/4 h -3/4 ) + O(h 2 )
, and φ dened in Assumption 12.

In particular, we see that, choosing h such as nh 4 → 0 and such as (log n) 3/4 nh 3 → ∞, the remainder term is o P (n -1/2 ). Also note that, if we do not wish to restrain ourselves to the set X δ dened in Assumption 12, we should add an O(h) in the remainder term.

φ(x, y)d( F -F )(x, y) = φ(x, y)[ Ĝ(y -|z) -G(y -|z)] [1 -G(y -|z)] d F (x, y) + φ(x, y)[ Ĝ(y -|z) -G(y -|z)] 2 [1 -Ĝ(y -|z)][1 -G(y -|z)] d F (x, y) = I 1 (φ) + I 2 (φ). For I 2 (φ), observe |I 2 (φ)| ≤ C × sup z∈X δ ,y≤τ Ĝ(y-| z) -G(y-| z) 1 -Ĝ(y-| z) 2 Φ(x, y)d F (x, y).
From Proposition 4.3 in Van Keilegom and Akritas [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], deduce that

sup φ∈F |I 2 (φ)| = O P (n -1 h -1 [log(h -1 )] 1/2 ).
Applying the representation (3.2) from Du and Akritas (6),

I 1 (φ) = 1 n i,j δ i φ(X i , T i )w nj (Z i )ξ Z j (Z i , T i -) [1 -G(T i -|Z i )] + 1 n n i=1 δ i φ(X i , T i )R G n (Z i , T i -) [1 -G(T i -|Z i )] = y- -∞ φ(x, y)dM n,z (y ) [1 -F (y |z)][1 -G(y -|z)] d F (x, y)
+R (1) n (φ).

Now decompose I 1 (φ) into I 1 (φ) = y- -∞ φ(x, y)dM n,z (y ) [1 -F (y |z)][1 -G(y -|z)] dF (x, y) + y- -∞ φ(x, y)dM n,z (y ) [1 -G(y -|x)][1 -F (y |z)] d( F -F )(x, y) +R (1) n (φ) = I 0 (φ) + R (2) n (φ) + R (1) n (φ).
From the rate of convergence of R G n , and the fact that |φ| ≤ Φ, we obtain the rate sup φ∈F |R (1)

n (φ)| = O P ((log n) 3/4 n -3/4 h -3/4 ). In Lemma 6.1, we show that R (2) n (φ) = O P ((log n)n -1 h -1 ) + O P (h 2 )
, so that only I 0 (φ) remains to be studied. Applying Fubini's theorem, rewrite

I 0 (φ) = 1 n n j=1
w nj (z)ξ z (T j , δ j ; y-)φ(x, y)dF (x, y)

= 1 nh n j=1 K Z j -z h ξ z (T j , δ j ; y-)φ(x, y)dF (x, y) f Z (z)    + 1 nh n j=1 K Z j -z h ξ z (T j , δ j ; y-)φ(x, y)dF (x, y) [f Z (z) -fZ (z)] -1 [f Z (z)] 2 + 1 nh n j=1 K Z j -z h ξ z (T j , δ j ; y-)φ(x, y)dF (x, y) [f Z (z) -fZ (z)] -2 [f Z (z)] 2 [ fZ (z)] .    = I 00 (φ) + R (3) n (φ). (3.4)
We show in Lemma 6.2 that sup φ∈F |R

n (φ)| = O P (n -1 h -1 log n + h 2 ) (3)
. By a change of variable, I 00 (φ) can be rewritten as

1 n n j=1 K (u) ξ Z j +hu (T j , δ j ; y-)φ(x, y)dF (x, y | Z j + hu)du = 1 n n j=1 K (u) ψ 1,Z j +hu (T j , δ j ) φZ j +hu (T j )du -K (u) ψ 2,Z j +hu (T j , s) φZ j +hu (s)dH 0 (s | Z j + hu)du. (3.5)
We now use Assumption 12. By linearity, we only have to consider φ(x, y) = φ 1 (x, y)1 x∈I 1 , satisfying Assumption 12. Under Assumption 3, the function ψ i have two bounded derivatives with respect to z. To use a Taylor expansion in (3.5), we must check that Z j and Z j + hu are interior points of X . This is the reason why we introduced the set X δ , and why it should appear in Du and Akritas [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF] to control the bias of their estimate of the conditional distribution function (for example, observe in Du and Akritas [START_REF] Du | I.i.d representations of the conditional Kaplan-Meier process for arbitrary distributions[END_REF], equations (A.13) and (A.14) that the rate is not obtained uniformly in x 0 ∈ X , because x 0 + hu is not an interior point of X for all x 0 ∈ X ). Now consider some j 0 such as Z j 0 ∈ I 1 ⊂ X δ . X j 0 is an interior point of X . Furthermore, since u takes values only in a compact interval (K has a compact support), Z j 0 + hu is almost surely an interior point of X for n large enough (only depending on δ). From a

Taylor expansion and Fubini's Theorem, the two integrals appearing in (3.5) corresponding to the index j 0 can be rewritten as φ1,Z j 0 (s)dM j 0 (s)

[1 -F (s-| Z j 0 )][1 -G(s | Z j 0 )] + O(h 2 ),
where we used uK(u)du = 0, u 2 K(u)du < ∞, and where the O(h 2 )-rate depends only on δ.

Now we have to consider the index j such as :

1. X j + hu ∈ I 1 and X j / ∈ I 1 ,

2. X j ∈ I 1 and X j + hu / ∈ I 1 .

To simplify the discussion, we will assume that I 1 = [a; b]. The contribution of these terms to (3.5) is

1 n n j=1 1 Z j ∈I 1 ,Z j +hu / ∈I 1 φ1,Z j (s)dM j (s) [1 -F (s-| Z j )][1 -G(s | Z j )] + 1 n n j=1 1 Z j / ∈I 1 ,Z j +hu∈I 1 ] φ1,Z j (s)dM j (s) [1 -F (s-| Z j )][1 -G(s | Z j )] + R (4) n (φ 1 ),
where we can bound

|R 4 n (φ 1 )| ≤ M × 1 n n j=1 K(u)[1 Z j ∈I 1 ,Z j +hu / ∈I 1 + 1 Z j / ∈I 1 ,Z j +hu∈I 1 ]du,
where M is a positive constant, and where we used that |φ 1 | ≤ Φ, with Φ bounded on

X δ ×] -∞; τ ].
The expectation of the right hand can be bounded by

M × K(u)[F Z (a + h) -F Z (a -h) + F Z (b + h) -F Z (b -h)]du,
where F Z is the cumulative distribution function of Z. Using Assumption 1, a Taylor expansion shows that this term is O(h 2 ).

Applications

Regression analysis

To simplify, assume that d = 1. Consider the following regression model,

E[Y | X, Y ≤ τ ] = f (θ 0 , X),
where f is a known function and θ 0 ∈ Θ ⊂ R k an unknown parameter, and τ is as in Assumption 1. Once again, introducing τ is a classical way to proceed for mean-regression under (2.6). See e.g. Heuchenne and Van Keilegom [START_REF] Heuchenne | Nonlinear censored regression[END_REF]. If we assume that θ 0 is the unique minimizer of

M (θ) = E {Y -f (θ, X)} 2 1 Y ≤τ,X∈X δ ,
we can estimate θ 0 by θ = arg min

θ∈Θ x∈X δ ,y≤τ [y -f (θ, x)] 2 d F (x, y).
As a consequence of Theorem 3.1 and Theorem 3.3, the following proposition furnishes the asymptotics for θ.

Proposition 4.1 If F = {x → f (θ, x), θ ∈ Θ} is P -Glivenko-Cantelli, we have θ → θ 0 a.s. (4.1)
Furthermore, let ∇ θ (resp. ∇ 2 θ ) denotes the vectors of partial derivatives with respect to θ (resp. the Hessian matrix) and assume that F = {x → ∇ θ f (θ, x), θ ∈ Θ} and F = {x → ∇ 2 θ f (θ, x), θ ∈ Θ} are P -Glivenko-Cantelli with an integrable enveloppe. We have, if nh 4 → 0 and under Assumptions 1 to 12 for φ(x, y) = ∇ θ f (θ 0 , x)[y -f (θ 0 , x)],

√ n( θ -θ 0 ) ⇒ N (0, Ω -1 V Ω -1 ), (4.2)
with

Ω = E [∇ θ f (θ 0 , X)∇ θ f (θ 0 , X) ] , V = V ar φ(x, y)d F (x, y) + 1 n n i=1 [1 -G(s|X i )] -1 φX i (s)dM i (s) [1 -F (s -|X i )] . Proof. Let M n (θ) = x∈X δ ,y≤τ [y -f (θ, x)] 2 d F (x, y).
θ -θ 0 = ∇ 2 θ M n (θ 1n ) -1 ∇ θ M n (θ 0 ),
for some θ 1n between θ 0 and θ. Apply Theorem 3.1 to see that we have ∇ 2 θ M n (θ 1n ) -1 → Ω -1 a.s., and Theorem 3.3 to obtain that ∇ θ M n (θ 0 ) ⇒ N (0, V ).

Density estimation

In this section, we consider a random variable Y with Lebesgue density f that we wish to estimate. Estimation of the density of Y received a lot interest in the case where Y and C are independent. See e.g. Mielniczuk (15). This assumption may not hold in several practical situations. In such cases the estimator of Mielniczuk ( 15) is biased. An alternative is to consider that we are under (2.6) or (2.15)- (2.16), where X represent some auxiliary variables which are observed. In this framework, our estimate F will permit us to estimate the density f , for example through the use of kernel smoothing. Let K be a compact support function, h 1 some positive parameter tending to zero, and dene

fδ (y) = h -1 1 X δ ×R K y -y h 1 d F (x, y ). (4.3)
Observe that, since K has compact support, if we choose h 1 small enough, the integral

in (4.3) is only on X δ ×] -∞; τ ] for some τ < τ H . Let Kh 1 ,y = K((y -.)h -1 1 ). As an immediate corollary of Theorem 3.3, deduce that fδ (y) = h -1 1 X δ ×R Kh 1 ,y (s)d F (x, y ) + 1 nh 1 n i=1 1 X i ∈X δ KX i .-y h 1 dM i (s) [1 -F (s-| X i )][1 -G(s | X i )] +R n (y), (4.4)
with

sup y≤τ |R n (y)| = O P (h -1 1 (log n) 3/4 n -3/4 h -3/4 ) + O P (h 2 h -1 1 ).

Simulation study

We present in this section a comparison between the behavior of our new estimator F , and the estimator of Van Keilegom and Akritas ((25)). As already mentioned, their estimator was only proposed for the case d = 1, and we would like to emphasize that it is does behaves as well in the multivariate case. For d = 1 and d = 3, we consider the following regression models,

• X = (X (1) , ..., X (d) ) ∼ U[0, 1] ⊗d . • Y = d -1 d i=1 (-1) i X (i) + ε.
• ε ∼ N (m, 1) independent from X, m was taken 1.7, corresponding to 45% of censoring approximatively.

• C|X ∼ E(e β 0 X /5), independent from ε, where β 0 = (1/d, ..., 1/d).

In this framework, we are focusing on estimating a truncated mean, that is E[φ(Y )]

where φ(y) = y1 y≤τ , we took τ = 100. In the one-dimensional case, there is no big dierence between both estimators. When it comes to dimension 3, performances of the estimator based on the technique of Van Keilegom and Akritas shrink. It is obvious for the small values of h, and for h = 0.9, the bias stays approximatively three times bigger than the one obtained using our new estimator, while the variance is still important.

6 Technical lemmas Lemma 6.1 Under Assumptions 1, 2, 6, 9 and 11,

sup φ∈F |R (2) n (φ)| = O P ((log n)n -1 h -1 ) + O P (h 2 ). Proof. Let U i,j (φ) = δ i φ(X i , T i ) [1 -G(T i -|Z i )] K Z i -Z j h fZ (Z i ) -1 ξ Z i (T j , δ j ; T i -) -φ(x, y)K z -Z j h fZ (z) -1 ξ z (T j , δ j ; y-)dF (x, y). Let W j = (X j , Y j , C j ). We can decompose U i,j into U i,j (φ) = 4 k=1 U i,j k (φ), where U i,j 1 (φ) = δ i φ(X i , T i )ξ Z i (T j , δ j ; T i -) f Z (Z i )[1 -G(T i -|Z i )] K Z i -Z j h -E φ(X, Y )ξ Z (T j , δ j ; Y -) f Z (Z) K Z -Z j h |W j , U i,j 2 (φ) = δ i φ(X i , T i )ξ Z i (T j , δ j ; T i -)K Z i -Z j h ( fZ (Z i ) -E[ fZ (Z i )|Z i ]) -1 f Z (Z i ) 2 [1 -G(T i -|Z i )] -E   φ(X, Y )ξ Z (T j , δ j ; Y -)K Z-Z j h ( fZ (Z) -E[ fZ (Z)|Z]) -1 f Z (Z) 2 |W j   , U i,j 3 (φ) = δ i φ(X i , T i )E[ fZ (Z i )|Z i ]ξ Z i (T j , δ j ; T i -) f Z (Z i ) 2 [1 -G(T i -|Z i )] K Z i -Z j h -E φ(X, Y )E[ fZ (Z)|Z]ξ Z (T j , δ j ; Y -) f Z (Z) 2 K Z -Z j h |W j , U i,j 4 (φ) = δ i φ(X i , T i )[ fZ (Z i ) -fZ (Z i )] 2 ξ Z i (T j , δ j ; T i -)K Z i -Z j h f Z (Z i ) 2 fZ (Z i )[1 -G(T i -|Z i )]
φ(x, y)[ fZ (z) -fZ (z)] 2 ξ x (T j , δ j ; y-)K z-Z j h f Z (z) 2 fZ (z) dF (x, y).

Observe that, for any k = 1, ..., 4, U i,i k (φ) = 0. We have, for some constant M , 1 n 2 i,j [f Z (z)] 2 .

|U i,j 4 (φ)| ≤ M n 2 × sup z∈X | fZ (z) -f Z (z)| 2 fZ (z) i,j K Z i -Z j h Φ(X i , T i ) = O P (n -1 log n) + O P (h 2 ),
The rst part can be written as

1 n 2 h 2 j,i K( Z j -z h )ξ z (T j , δ j ; y-){K( Z i -z h ) -E[K( Z -z h )]} × φ(x, y)dF (x, y) [f Z (z)] 2 .
Observe that the terms for i = j are negligible, since

1 n 2 h 2 n i=1 K Z i -z h 2 ξ z (T i , δ i ; y-)φ(x, y)dF (x, y) [f Z 2 ≤ M n 2 h 2 n i=1 K Z i -z h 2 Φ(x, y)dF (x, y) = O P (n -1 h -1 ), 1 n 2 h 2 n i=1 K Z i -z h E K Z-z h
ξ z (T i , δ i ; y-)φ(x, y)dF (x, y)

[f Z (z)] 2 ≤ M n 2 h 2 n i=1 K Z i -z h E K Z -z h Φ(x, y)dF (x, y)
= O P (n -1 h -1 ).

Let N j (z, y) = K Z j -z h ξ z (T j , δ j ; y-) -E K Z j -z h ξ z (T j , δ j ; y-) . We have to consider .

1 n 2 h 2 n j =i N j (z, y) (K( Z i -z h )-E[K( Z-z h )])φ(x,
Take the expectation of the rst parenthesis to see that this expectation is O(n -2 h -2 ),

while the second parenthesis is nite from the square integrability of Φ and Assumptions 1 and 2.

Lemma 6.3 Let w = (x, D, t), and dene the set of functions indexed by φ ∈ F, and h > 0,

G 1 = ψ φ,h : (w 1 , w 2 ) → D 1 φ(x 1 , t 1 )ξ x 1 (t 2 , D 2 ; t 1 -)K x 1 -x 2 h f X (x 1 )[1 -G(t 1 -|x 1 )] 2 .
Under Assumptions 1, 2, 9, and 12, for all ε > 0 and for some V 1 < ∞,

N (ε, G 1 , L 2 ) ≤ A 1 ε -V 1 .
Proof. Let g = D 1 ξ x 1 (t 2 , D 2 ; t 1 -) f X (x 1 )[1 -G(t 1 -|x 1 )] 2 .

The class of functions g×F as a polynomial covering number, from Lemma A.1 in Einmahl and Mason [START_REF] Einmahl | An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators[END_REF] (the function g is bounded). Let G 1 = {K x 1 -x 2 h , h > 0}. This class is uniformly bounded. Apply Lemma A.1 of Einmahl and Mason [START_REF] Einmahl | An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators[END_REF] to conclude on the covering number of G 1 = G 1 × (g × F).

  Dabrowska (3) and (4) studied uniform consistency and asymptotic normality of Beran's estimator. Van Keilegom and Veraverberke[START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], in the case of a xed design, provided an asymptotic i.i.d. representation of F (y | x), that is a representation of F (y | x) as a mean of i.i.d. quantities plus a remainder term which becomes negligible as n grows to innity. Recently, Du and Akritas (6) provided an analogous representation holding uniformly in y and x for a random X. Van Keilegom and Akritas[START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF] proposed an alternative to Beran's estimate under some restrictions on the regression model. In particular, they assumed

Assumption 5 Assumption 6 F 2 z

 562 The jumps of F (. | z) and G(. | z) are the same for all z. Let (d 1 , d 2 , ...) be the atoms of G. (. | z) and G(. | z) have two derivatives with respect to z, with the rst derivatives uniformly bounded (on [-∞; τ ]). The variation of the functions ∂ z F (. | z) and ∂ F (. | z) on [-∞; τ ] is bounded by a constant not depending on z.

Apply Theorem 3 . 1

 31 to obtain sup θ |M n (θ) -M (θ)| = o P (1), and hence (4.1) follows. For(4.2), observe that, from a Taylor expansion,

  For each model, we generate 100 samples of size n = 100 and we estimate E[φ(Y )] using either the estimator of the distribution function dened in ((25)), or our new estimator. We then estimate the absolute value of the bias, that is |E[ φ] -E[φ(Y )]|, and the variance E[ φ2 ] -E[ φ] 2 . We consider dierent values for the smoothing parameter h. In the gure below, model 1 stands for d = 1 while model 3 stands for d = 3.

Figure 1 :

 1 Figure 1: Bias and variance of estimators of E[Y 1 Y ≤100 ] in function of the smoothing parameter h for n = 100 and 45% censoring.

  ξ z (T j , δ j ; y-)φ(x, y){ fZ (z) -E[ fZ (z)]}dF (x, y) [f Z (z)] ξ z (T j , δ j ; y-)φ(x, y)(E[ fZ (z)] -f Z (z))dF (x, y)

  Assumption 3 Functions H and H c (and consequently H d ) have two derivatives with respect to z. Furthermore, these derivatives are uniformly bounded for y < τ .

Assumption 4

For some positive nondecreasing bounded (on [-∞; τ ]) functions L 1 , L 2 , L 3 , we have, for all z,

  y)dF(x,y)

	value is bounded by
	 	1 n 2 h 2	i =j	N j (z, y){K(	Z i -z h	) -E[K(	h Z -z	)]}	2	dF (x, y)	  1/2
							×	f Z (z) 4 Φ(x, y) 2 dF (x, y)	1/2
							f Z (z) 2				,	(6.2)
		1 nh 2	n i=1	E[N 1 (x,y)](K(	Z i -z h )-E[K( Z-z h )])φ(x,y)dF(x,y) f Z (z) 2	.	(6.3)

From a Taylor expansion, and from Assumptions 1, 6, and 9, we have h -1 E [N j (z, y)] = h 2 C(z, y), with C(z, y) bounded for z ∈ X and y ≤ τ. Consequently one readily sees that, uniformly over F, (6.3) is O P (h 2 ). For (6.2), by Cauchy-Schwarz inequality, the absolute

from the uniform convergence rate of fX , see Einmahl and Mason [START_REF] Einmahl | An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators[END_REF].

Since we have E[U i,j 1 (φ)|W j ] = 0, we see that the process dened by

is a degenerate U -process of order 2. From Lemma 6.3, deduce that this U -process is indexed by a class of functions with polynomial covering number. From Corollary 4

in Sherman [START_REF] Sherman | Maximal inequalities for degenerate U-processes with applications to optimization estimators[END_REF],

). Moreover, from a change of variable and a

Taylor expansion,

where, for some constant M and using Assumption 6 and uK(u)du = 0,

The rst term in (6.1) is zero from (3.3). Finally, we have obtained

Using the same arguments, the terms n -1 h -1 i,j U i,j k for k = 2, 3 can be decomposed into a degenerate U -process of order greater than 2 indexed by a polynomial class, plus a "bias" term of order O P (h 2 ) uniformly over F. Hence, for k = 2, 3,

Lemma 6.2 Under Assumptions 1, 2, 6, 9, and 11,

Proof. From (3.4), we see that the second term of R

n (φ) has order O P (n -1 h -1 log n)

from Lemma 4.3 of Van Keilegom and Akritas [START_REF] Van Keilegom | Transfer of tail information in censored regression models[END_REF], and the fact that |φ| ≤ Φ. The rst