Frederic Tewes
email: ftewes@univ-poitiers.fr

Tania F Bahamondez-Canas

Daniel Moraga-Espinoza

Hugh D C Smyth

Alan B Watts

In vivo Efficacy of a Dry Powder Formulation of Ciprofloxacin-Copper Complex in a Chronic Lung Infection Model of Bioluminescent Pseudomonas aeruginosa

A significant limitation of locally delivered treatments for chronic pulmonary infections is often the short residence time within the airways. Ciprofloxacin (CIP), for example, undergoes rapid absorption from the airway lumen. Previously, we demonstrated that the complexation of CIP with copper (CIP-Cu) reduces its apparent epithelial permeability and pulmonary absorption rate without affecting antimicrobial activity against Pseudomonas aeruginosa grown planktonically or as biofilms. This study aimed to evaluate the in vivo efficacy of CIP-Cu, prepared as a dry powder, in a chronic lung infection model. The powders were prepared by jet milling (CIP-HCl) and by spray drying (CIP-Cu). A bioluminescent strain of P. aeruginosa (PAO1::p16Slux) was used to prepare bacteria-loaded agar beads that were inoculated intratracheally to rats. The dynamics of the infection were monitored using luminometry. The bacteria/beads ratio was optimized to allow the highest luminescence signal and animal survival for 8 days. The efficacy of the treatment was evaluated by luminometry in addition to the endpoint (Day 8) where colony counting was performed after lung harvesting. Luminescent P. aeruginosa entrapped in agar beads were useful to monitor the spatial development of the chronic lung infection in rats. The rats were treated with the dry powders in a nose-only inhalation exposure system (NOIES). CIP-Cu and CIP-HCl powders showed similar aerodynamic properties and comparable CIP lung deposition. However, treatment with CIP-Cu significantly (p<0.01) reduced by 4-log the number of CFU of P. aeruginosa per lung in the chronic infection model, whereas CIP-HCl effect was not different from the untreated control group.

1.

INTRODUCTION

Effective therapy using pulmonary delivery of antibiotics can be limited by the short drug residence time at the site of infection, but a novel complexed form of ciprofloxacin may help overcome this obstacle while maintaining antimicrobial activity against resistant biofilm bacterial infections [START_REF] Tewes | Efficacy of Ciprofloxacin and Its Copper Complex against Pseudomonas aeruginosa Biofilms[END_REF][START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF]. Pulmonary delivery of antibiotics in solution form has demonstrated clinical benefit in the treatment of chronic bacterial lung infections, as it can allow for significantly higher drug concentrations at the site of infection with reduced systemic levels compared to i.v. administration [START_REF] Zhou | Inhaled formulations and pulmonary drug delivery systems for respiratory infections[END_REF]. This is mainly true for antibiotics having very low lungblood barrier permeability, such as the currently used tobramycin, colistin methanesulfonate and aztreonam lysine. However, this approach is less suitable for molecules that readily diffuse through the lung epithelial barrier and rapidly equilibrate with blood concentrations, such as in the case of ciprofloxacin (CIP) [START_REF] Stass | Inhalation of a dry powder ciprofloxacin formulation in healthy subjects: a phase I study[END_REF][START_REF] Stass | Tolerability and pharmacokinetic properties of ciprofloxacin dry powder for inhalation in patients with cystic fibrosis: a Phase I, randomized, dose-escalation study[END_REF][START_REF] Gontijo | Biopharmaceutical characterization of nebulized antimicrobial agents in rats: 1. Ciprofloxacin, moxifloxacin, and grepafloxacin[END_REF][START_REF] Endermann | Treatment of bacterial diseases of the respiratory organs[END_REF]. Thus, for antimicrobial molecules having relatively high permeability across the lung-blood barrier, advanced formulations can be developed to retain the drug in the airways by slowing down the drug flux across this barrier. These advanced formulations generally act either by reducing and sustaining the drug release rate from formulation [START_REF] Gaspar | Pulmonary pharmacokinetics of levofloxacin in rats after aerosolization of immediate-release chitosan or sustained-release PLGA microspheres[END_REF][START_REF] Serisier | Inhaled, dual release liposomal ciprofloxacin in noncystic fibrosis bronchiectasis (ORBIT-2): a randomised, double-blind, placebo-controlled trial[END_REF][START_REF] Nurbaeti | Sustained-release microparticle dry powders of chloramphenicol palmitate or thiamphenicol palmitate prodrugs for lung delivery as aerosols[END_REF][START_REF] Cipolla | Development of liposomal ciprofloxacin to treat lung infections[END_REF][START_REF] Türeli | Ciprofloxacin-loaded PLGA nanoparticles against cystic fibrosis P. aeruginosa lung infections[END_REF][START_REF] Mcshane | Ciprofloxacin Dry Powder for Inhalation (ciprofloxacin DPI): Technical design and features of an efficient drug-device combination[END_REF], or by reducing the apparent permeability of the drug across the epithelium [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF][START_REF] Tewes | Ciprofloxacin-Loaded Inorganic-Organic Composite Microparticles To Treat Bacterial Lung Infection[END_REF][START_REF] Griffith | Aerosol fluoroquinolone formulations for improved pharmacokinetics[END_REF][START_REF] Lamy | Use of to improve aerodynamic properties of ciprofloxacin-loaded maltose microparticles for inhalation[END_REF][START_REF] Brillault | Control of the Lung Residence Time of Highly Permeable Molecules after Nebulization: Example of the Fluoroquinolones[END_REF]. In order to increase CIP lung residency time after its aerosolization, we recently developed a novel positively charged complex made of CIP and copper [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF][START_REF] Tewes | Ciprofloxacin-Loaded Inorganic-Organic Composite Microparticles To Treat Bacterial Lung Infection[END_REF][START_REF] Brillault | In vitro biopharmaceutical evaluation of ciprofloxacin/metal cation complexes for pulmonary administration[END_REF]. This CIPcopper complex (CIP-Cu) allowed a 90% decrease in the apparent permeability of CIP across a Calu-3 cell line model of pulmonary epithelium [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF][START_REF] Brillault | In vitro biopharmaceutical evaluation of ciprofloxacin/metal cation complexes for pulmonary administration[END_REF]. This decrease in permeability translated in vivo as a decrease in the lung-blood absorption rate of CIP and a sustained high CIP concentration in the lung epithelial lining fluid (ELF) after intratracheal delivery to rats [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF].

Additionally, the in vitro antibacterial activities of CIP and CIP-Cu evaluated against P. aeruginosa grown in planktonic cultures [START_REF] Tewes | Ciprofloxacin-Loaded Inorganic-Organic Composite Microparticles To Treat Bacterial Lung Infection[END_REF][START_REF] Brillault | In vitro biopharmaceutical evaluation of ciprofloxacin/metal cation complexes for pulmonary administration[END_REF] or as biofilms [START_REF] Tewes | Efficacy of Ciprofloxacin and Its Copper Complex against Pseudomonas aeruginosa Biofilms[END_REF] were also found to be equivalent.

The aim of this study was, therefore, to evaluate the in-vivo efficacy of CIP-Cu complexloaded particles after pulmonary inhalation in model rats that developed chronic pulmonary infection by instillation of Pseudomonas aeruginosa-loaded agar beads [START_REF] Kukavica-Ibrulj | Animal models of chronic lung infection with Pseudomonas aeruginosa: useful tools for cystic fibrosis studies[END_REF][START_REF] Torres | In vitro evaluation of Pseudomonas aeruginosa chronic lung infection models: Are agar and calcium-alginate beads interchangeable?[END_REF]. This approach has been shown to model the chronic infection and inflammation that affect cystic fibrosis patients [START_REF] Kukavica-Ibrulj | Animal models of chronic lung infection with Pseudomonas aeruginosa: useful tools for cystic fibrosis studies[END_REF][START_REF] Bragonzi | Nonmucoid Pseudomonas aeruginosa expresses alginate in the lungs of patients with cystic fibrosis and in a mouse model[END_REF][START_REF] Growcott | Characterisation of a refined rat model of respiratory infection with Pseudomonas aeruginosa and the effect of ciprofloxacin[END_REF]. In order to assess temporal effects of the treatments on the infection, we also evaluated time resolved luminescence of a bioluminescent P. aeruginosa, PAO1 strain to non-invasively monitor the chronic lung infections in rats.

MATERIALS AND METHODS

Materials

The bioluminescent PAO1::p16Slux, a PAO1 strain (gift from Professor Shawn Lewenza from the university of Calgary, Canada) tagged by chromosomal integration of p16Slux construction [START_REF] Riedel | Construction of p16Slux, a novel vector for improved bioluminescent labeling of gram-negative bacteria[END_REF], was used to prepare bacteria-loaded agar beads and monitor the chronic lung

Preparation and characterization of the dry powder formulations

Preparation of ciprofloxacin-copper loaded microparticles

Microparticles composed of amorphous calcium carbonate and CIP-Cu complex (Spray dried CIP-Cu) were prepared by spray-drying as previously described [START_REF] Tewes | Ciprofloxacin-Loaded Inorganic-Organic Composite Microparticles To Treat Bacterial Lung Infection[END_REF]. Briefly, two aqueous solutions were prepared. One solution was made of CIP, calcium hydroxide, sodium hyaluronate, formic acid, and copper hydroxide. The other one was made of ammonium carbonate. These solutions were prepared separately and mixed during the microparticle preparation process using a Y-tube connected to the feeding tube of the B-290 mini spray dryer (Büchi, Labortechnik AG, Switzerland). The spray dryer was operated in open mode with 27 mL/min peristaltic pump rate, 15 L/min spraying air flow rate, 630 L/h drying air flow rate, and 120°C inlet temperature. Spray dried CIP-Cu powder was stored at 4°C in desiccated condition.

Preparation of jet milled ciprofloxacin powder

Microparticles made of ciprofloxacin hydrochloride (Micronized CIP-HCl) were micronized by a jet mill Model 00 Jet-O-Mizer™ (Fluid Energy, Telford, PA, USA). For some drugs, this method can produce an easily aerosolizable powder without the use of excipients [START_REF] Brunaugh | Formulation techniques for high dose dry powders[END_REF].

Micronized CIP-HCl powder was stored at room temperature in desiccated condition.

Scanning electron microscopy

Samples mounted on standard aluminum SEM stubs were sputter coated with 12 nm platinum/palladium (Pt/Pd) in an argon atmosphere using a sputter coater 208HR (Cressington Scientific Instruments Ltd., Watford, UK). The particles were imaged using a Zeiss Supra 40VP SEM (Carl Zeiss Microscopy GmbH, Jena, Germany).

Particle size distribution analysis

Particle size distributions (PSD) were determined by laser diffraction using the Sympatec HELOS-R and a dry powder disperser (RODOS -Sympatec GmbH, Clausthal-Zellerfeld, Germany). RODOS was loaded with approximately 3 mg of powder and used at different dispersion pressures in a range of 0.5-4.0 Bar. PSD was calculated using the Fraunhofer approximation (n=3).

Ciprofloxacin assay

CIP was assayed by measuring the CIP fluorescence using a plate reader (Infinite M200, Tecan Systems Inc., CA, USA). CIP standards (concentration ranging from 3.75 to 0.03 µg/mL), controls (3, 0.5 and 0.06 µg/mL) and samples were half-diluted in 0.5 M sodium acetate buffer pH 4.5, containing 0.05M of EDTA to complex the metal cation that could interact with CIP and modify its spectroscopic properties. The maximum excitation wavelength of CIP was 302 nm and its maximum fluorescence emission wavelength was 456 nm.

Preparation and characterization of P. aeruginosa-loaded agar beads

Preparation of the P. aeruginosa-loaded beads

Sterile and bacteria-loaded agar beads were prepared using an adapted version of the method described by Growcott et al. [START_REF] Growcott | Characterisation of a refined rat model of respiratory infection with Pseudomonas aeruginosa and the effect of ciprofloxacin[END_REF]. Fresh suspension of bioluminescent P. aeruginosa (strain PAO1::p16Slux) prepared in BHI broth was cultured to exponential growth phase and then adjusted to an OD 600 of 0.3. Two milliliters of this suspension were washed and concentrated in 1 mL of PBS by centrifugation. The bacteria were embedded into agar beads by mixing 1 mL of this concentrated suspension with 9 mL of molten 2% w/v LB agar (Fisher Scientific, NJ, USA). The mixture was then emulsified into 100 mL of warmed (48 °C) paraffin oil (Sigma Aldrich, USA) containing 0.01% v/v of sorbitan monooleate (SPAN ® 80, Sigma Aldrich, USA) based on the report of Growcott et al. [START_REF] Growcott | Characterisation of a refined rat model of respiratory infection with Pseudomonas aeruginosa and the effect of ciprofloxacin[END_REF]. The emulsion was cooled, the beads centrifuged at 10,000 g for 10 min and then 4-times washed with PBS pH 7.4. Ten milliliters of the bead suspension in PBS were stored up to 2 weeks at 4°C. For sterile agar beads, the volume of bacterial suspension was replaced with PBS.

The number of bacteria per bead was determined from a CFU count performed with 100 µL of the bead suspension diluted in 5 mL of PBS and homogenized for 1 min (Polytron PT2100, Kinematica AG, Switzerland). Ten-fold serial dilutions of this suspension were spread on LB agar plates (Hardy Diagbostics, CA, USA) for colony counts. Bead concentrations were determined using a hemocytometer that allowed calculating the number of bacteria per beads.

Size distribution of blank beads was determined by laser diffraction using the Sympatec HELOS-R equipped with a Cuvette module (Sympatec GmbH, Clausthal-Zellerfeld, Germany).

A reference measurement was performed on the cell filled with 50 mL of PBS. After the reference measurement, an aliquot of sterile beads suspension was added to the cell to achieve optical concentrations between 10-20 % (n=3).

Time-luminescence intensity kinetics of bacteria-loaded agar beads

In vitro behavior of the luminescent P. aeruginosa in the agar beads was evaluated by following the luminescence produced by the bacteria within the beads versus time in the presence of various CIP or CIP-Cu complex concentrations. One hundred and eighty microliters of agar bead suspension at 1.5 x 10 5 beads/mL and containing 25 bacteria/bead were added into the wells of a black 96-well plate with flat clear bottom (Corning Inc. NC, US). Wells were supplemented with 20 μL of 10-times concentrated two-fold dilution of CIP or CIP-Cu complex in BHI. Final CIP concentrations ranged from 64 to 0.062 µg/mL. Then, the plates were covered using an air-permeable membrane and incubated at 37 °C in the plate reader (SpectraMax M3, Molecular Devices LLC, CA, USA). Luminescence intensity emitted from the bottom of the wells was measured during 1 second per well every 20 min. Plates were shaken before each measurement (n=3).

Development of the chronic lung infection model

Animal model

The protocol for the animal study was approved by the Institutional Animal Care and Use Committee (IACUC) at the University of Texas at Austin, Austin, TX. Sprague-Dawley rats weighting between 225 and 275 g were purchased from Charles River Laboratories (Wilmington, MA). Animals were housed one per cage, subjected to 12 h/12 h light and darkness cycles with access to food and water ad libitum.

Inoculation of rats with bioluminescent P. aeruginosa-loaded agar beads

On day 1, anesthetized (Ketamine/xylazine (80/6 mg/Kg) i. p.) male Sprague-Dawley rats were inoculated with various dose of P. aeruginosa (0 to 5 x 10 7 CFU) embedded into agar beads. Intratracheal (i.t.) inoculation was performed by instilling 100 µL of agar bead suspension loaded in a syringe connected to a dry powder insufflator cannula (Penn-Century Inc, Philadelphia, USA). This cannula has a blunt-tip designed to pass between the vocal cords without cutting blood vessels or damaging the epithelium. The vocal cords were visualized using a special laryngoscope suitable for rodent (Penn-Century Inc, Philadelphia, USA). Animals were kept on a warming pad until full recovery from anesthesia. Body weight and luminescence emitted from the rats anesthetized with isoflurane were measured daily for 8 days using an in vivo imaging system IVIS ® Spectrum (PerkinElmer, Houston, TX, USA) with an exposure time of 3 min per rat.

In vivo evaluation of ciprofloxacin-copper inhalation treatment

Pulmonary dosing and regimen

A nose-only inhalation exposure system (NOIES) capable of dosing 24 rats at a time was used (CH Technologies (USA) Inc., Westwood, NJ, USA). Prior to dosing, the animals were individually acclimated for 30 min per day for four days in restraint tubes. Animals were dosed on day 4 and day 6 post infections (Figure 1). Powders for inhalation were loaded into a feed cylinder connected to a rotating brush aerosol generator (RGB 1000). At an airflow rate of 0.5 L/min, animals treated with spray dried CIP-Cu were dosed for 35 min and animals treated with micronized CIP-HCl were dosed for 20 min, so that equivalent doses of CIP (1.65 mg/animal) were given. The purpose of the 0.5 L/min flow was to transport the aerosolized particles to the animal at a flow slightly higher than the minute ventilation of a rat. Following dosing, animals were placed back into their respective cages. These times of exposures were calculated to provide the same Total inhaled dose (TIDmg) by rats and determined according to Eq. ( 1) as described by Alexander et al. [START_REF] Alexander | Association of Inhalation Toxicologists (AIT) Working Party Recommendation for Standard Delivered Dose Calculation and Expression in Non-Clinical Aerosol Inhalation Toxicology Studies with Pharmaceuticals[END_REF].

Eq. ( 1)

where C is the measured particle concentration in air (mg/L), RMV is the species-specific respiratory minute volume or the volume of air inhaled in one minute per animal (L/min), D is the duration of exposure (min), and P is the particle drug potency (mass of CIP base per mass of particles -0.4 for CIP-Cu and 0.86 for CIP•HCl). The RMV for rats was calculated according to the Eq. ( 2) [START_REF] Alexander | Association of Inhalation Toxicologists (AIT) Working Party Recommendation for Standard Delivered Dose Calculation and Expression in Non-Clinical Aerosol Inhalation Toxicology Studies with Pharmaceuticals[END_REF].

Eq. ( 2)

where BW (kg) is the body weight of the rats. The aerosol concentration in air C, was calculated based on the average mass of samples deposited on filters (n = 3) taken from the NOIES inhalation chamber throughout the exposure using a vacuum flow of 0.5 L/min. To determine the portion of the TID that was deposited in the lungs, the percent deposition in relationship to the aerodynamic PSD was estimated based on rodent dosimetry findings reported by Kuehl et al. [START_REF] Kuehl | Regional particle size dependent deposition of inhaled aerosols in rats and mice[END_REF]. Thus, a Mercer-style mini cascade impactor (CH Technologies (USA) Inc., Westwood, NJ, USA) was used to determine the aerodynamic PSD of both aerosolized powders present within the nose-only exposure chamber. This impactor is a seven-stage cascade impactor that has been manufactured to be plugged directly to the nose-only exposure system, in exactly the same position as the animal restraint tubes used to connect the animal to the inhalation chamber [START_REF] Kuehl | Regional particle size dependent deposition of inhaled aerosols in rats and mice[END_REF][START_REF] Carvalho | Characterization and pharmacokinetic analysis of crystalline versus amorphous rapamycin dry powder via pulmonary administration in rats[END_REF]. The particles were collected at a rate of 0.5 L/min on this seven impactor stages having the following cut-off diameters at this specific airflow rate (5.14, 3.04, 1.89, 1.16, 0.77, 0.48, 0.35 µm). The MMADs were determined by differential weight analysis of the stages before and after impaction using a microbalance (METTLER TOLEDO, Columbus, OH, USA) and plotting the inverse of the standard normal cumulative mass distribution less than stated size cut-off against the natural logarithm of the cut-off diameter of the respective stages. The MMAD of the particles distribution was defined as the particle size at which the line crosses the 50% mark.

Lung harvest for microbiological assessment

Eight days after infection, the animals in each group were euthanized by CO 2 inhalation, right after measuring the luminescence. Animals were cleaned with 70% v/v ethanol solution before lungs collection. Lungs were homogenized in PBS and serial dilutions of these homogenates were plated onto LB agar plates to measure the number of CFU per lung. (n = 4 for the untreated group; n = 4 for the CIP-HCl group, and n = 7 for the CIP-Cu group)

Statistical analysis

Statistical analyses were performed using GraphPad Prism 6.0 (GraphPad Software Inc.). All data were expressed as means ± SD and analysed with the Dunn's Kruskal-Wallis multiple comparisons test. p < 0.05 was considered statistical significance.

RESULTS

Preparation and characterization of the dry powder formulations

The SEM micrographs of spray dried CIP-Cu and micronised CIP-HCl powders showed different morphologies (Figure 2A). Spray dried CIP-Cu particles were spherical hollow brittle particles while jet milled CIP-HCl particles were solid and non-spherical. The RODOS dry powder disperser was used to compare the dispersibility of the powders at different pressures (Figure 2B). The parameters D 10 , D 50 and D 90 , represent the 10%, 50% and 90% point in the cumulative undersize PSD, respectively. The increase in the dispersion pressure decreased mainly the D 90 and D 50 of the powders. This behavior was more noticeable for the spray dried CIP-Cu powder than for the jet milled CIP-HCl. However, both powders had similar PSD when they were dispersed with a pressure of 3 bar or above.

Characterization of the P. aeruginosa-loaded agar beads

The geometric size distribution (weighed in volume) of the sterile agar beads was monodisperse with a median value of 88 ± 3 µm as indicated by the D 50 (Figure 3). Prior to animals inoculation, we evaluated the in vitro susceptibility of P. aeruginosa embedded in the agar beads to CIP-HCl and the CIP-Cu complex by recording bioluminescence curves over time in the presence of various equivalent concentrations of CIP base (Figure 4).

Overall, CIP-HCl and CIP-Cu induced similar effect on the bacteria proliferation profiles. For concentrations above 0.5 µg/mL and up to 32 µg/mL, both treatments induced a continuous decrease in bacterial luminescence. Interestingly, for concentration between 0.063 and 0.25 µg/mL (Figure 4, top), the initial decrease in luminescence was followed by an increase up to 15 -20 h.

Development of the chronic lung infection model with bioluminescent bacteria

To develop the chronic lung infection model, we evaluated different bacteria/bead ratios for i.t. inoculation of the animals. The luminescence emitted by P. aeruginosa from the thorax of the inoculated rats was recorded daily and the average radiance values (photon/sec/cm 2 /sr) were plotted versus time after inoculation (Figure 5A). Just after inoculation (day 1), a large part of the signal was located at the carina as indicated by the luminescence images obtained with IVIS (Figure 5B), and no significant differences in luminescence were observed between the inoculums. On day 2, the signal sharply increased for the three inoculums tested in proportion to CFU numbers of the inoculum. In all cases, the increase in signal on day 2 was associated with a In the highest inoculum (5 x 10 7 CFU; 500 CFU/bead), the signal increased 100-times compared to day 1, however, none of the animals survived until day 3. For the 2 lower inoculums (1.5 x 10 7 and 6 x 10 6 CFU/animal), after a sharp luminescence increase on day 2, the signals decreased on day 3 to values close to these obtained on day 1 and then were stable from day 3 to day 8. The inoculum with 6 x 10 6 CFU (20 CFU/bead) gave the best survival rate associated to the highest and stable signal for 3 days. On day 8, the lungs of these rats had visible nodules (yellow arrows, Figure 5C) that correlated with values of luminescence and the number of CFU recovered from the lungs after plating. This inoculum was selected to evaluate the in vivo efficacy of the CIP powders after pulmonary inhalation.

In vivo evaluation of ciprofloxacin-copper inhalation treatment

Powders were administered to rats by the inhalation route using the NOIES that allowed dosing multiple animals in a time (Figure 6A). In order to compare the efficacy of the CIP-HCl and CIP-Cu powders in vivo, the same equivalent dose of CIP, with a similar lung deposition, had to be delivered to rats. Therefore, we evaluated the aerodynamic properties of the powders present in impactor and the dosimetry results obtained on rats reported by Kuehl et al [START_REF] Kuehl | Regional particle size dependent deposition of inhaled aerosols in rats and mice[END_REF], the estimated lung depositions were calculated and similar values were obtained between the two powders (Figure 6C). The in vivo efficacy of both formulations was measured by luminometry and by the number of CFU in the lung (CFU/lung) (Figure 7). Infected animals were dosed with the formulations at day 4 and 6 after infection. Both treatments induced a decrease in luminescence compared to the untreated group, however, due to the variability and small groups, these differences were not statistically significant (Figure 7A). On the other hand, a significant decrease of 4 logs in the number of CFU/lung was obtained with the CIP-Cu compared to the non-treated group (p<0.01; Figure 7B). In the same condition, CIP-HCl effect was not different from the control group.

Mercer-Style mini-cascade impactor

Estimated lung deposition (mg) 1.59 0.11 1.74 0.14

Figure 7:

In vivo efficacy in a P. aeruginosa chronic lung infection model. Rats were infected with 6 x 10 6 CFU/animal, then treated on days 4 and 6 with CIP-Cu or CIP-HCl, and the lungs were harvested on day 8. CIP dose was 1.65 mg/animal. A) Changes in the luminescence (photons per s -1 cm -2 steradian -1 ). B) Box and whisker plot of P. aeruginosa surviving colony forming units (CFU) per lung on day 8. Whiskers represent the minimal and maximal values (n = 4 for the untreated group; n = 4 for the CIP-HCl group, and n = 7 for the CIP-Cu group).

DISCUSSION

Pulmonary delivery of ciprofloxacin-copper complex significantly reduced the P. aeruginosa lung infection in the in vivo chronic lung infection model.

Microparticles loaded with CIP-Cu produced a significant reduction in bacterial lung burden compared to untreated rats (4-log reduction, Figure 7B), while the micronized powder of CIP-HCl did not change it significantly. Yet, with both powders, the total and peripheral doses of CIP delivered into the lungs of rats were identical (Figure 6). Moreover, it has been previously demonstrated that CIP or CIP-Cu solutions have the same efficacy against P. aeruginosa PA01 cultured either planktonically or as biofilms [START_REF] Tewes | Efficacy of Ciprofloxacin and Its Copper Complex against Pseudomonas aeruginosa Biofilms[END_REF][START_REF] Brillault | In vitro biopharmaceutical evaluation of ciprofloxacin/metal cation complexes for pulmonary administration[END_REF]. The better efficacy observed with the CIP-Cu over the CIP-HCl formulation could be attributed to a higher CIP lung residence time obtained with the CIP-Cu microparticles. Indeed, a previous study has shown that intratracheal (i.t.) administration to rats of CIP-Cu-loaded particles provides a mean AUC ratio of the CIP concentration kinetic profiles in ELF on the CIP concentration kinetic profiles in plasma 100fold higher than after administration of a CIP solution [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF]. After CIP-Cu i.t. administration, CIP half-life in rat lung ELF is higher than 2 h, while it is lower than 1 h after a CIP solution nebulization [START_REF] Lamy | New aerosol formulation to control ciprofloxacin pulmonary concentration[END_REF]. At the opposite, previous studies showed that CIP-HCl has poor lung targeting following oral inhalation; the drug is rapidly absorbed into the systemic circulation with a halflife of less than 1 h in the rat lungs, making CIP-HCl suboptimal for pulmonary delivery [START_REF] Stass | Tolerability and pharmacokinetic properties of ciprofloxacin dry powder for inhalation in patients with cystic fibrosis: a Phase I, randomized, dose-escalation study[END_REF][START_REF] Mcshane | Ciprofloxacin Dry Powder for Inhalation (ciprofloxacin DPI): Technical design and features of an efficient drug-device combination[END_REF][START_REF] Weers | Comparison of Phospholipid-Based Particles for Sustained Release of Ciprofloxacin Following Pulmonary Administration to Bronchiectasis Patients[END_REF]. To improve the lung targeting of CIP by inhalation, Bayer AG also aimed at prolonging the residence time of CIP in the lungs. To do this, they have developed a slow-dissolving CIP formulation using the neutral (zwitterionic) solid form of the drug (CIP-betaine) instead of the faster dissolving CIP-HCl [START_REF] Mcshane | Ciprofloxacin Dry Powder for Inhalation (ciprofloxacin DPI): Technical design and features of an efficient drug-device combination[END_REF]. When used as a dry powder, this neutral form of CIP enhances the efficacy of CIP in the treatment of a model of pulmonary P. aeruginosa infection in the rat.

After i.t. administration at a dose of 10 mg/kg, which is 2 times the total dose used in this study, a 4 to 10-log reduction in P. aeruginosa CFU count was observed with the neutral form of CIP versus a 1.1 to 2.8-log reduction for CIP-HCl [START_REF] Mcshane | Ciprofloxacin Dry Powder for Inhalation (ciprofloxacin DPI): Technical design and features of an efficient drug-device combination[END_REF]. In this approach, CIP lung targeting and efficacy was improved only by controlling the dissolution rate of the CIP. Indeed, after dissolution, CIP-betaine and CIP-HCl lead to the presence of the same zwitterionic in solution at the pH of the pulmonary fluids. In the approach presented in this study, CIP lung targeting and efficacy were improved by reducing CIP apparent permeability across the bloodlung barrier [START_REF] Brillault | In vitro biopharmaceutical evaluation of ciprofloxacin/metal cation complexes for pulmonary administration[END_REF].

P. aeruginosa growing as aggregates, or biofilms, is the main responsible of the recurrent lung infections that affect cystic fibrosis patients [START_REF] Barclay | Adaptive resistance to tobramycin in Pseudomonas aeruginosa lung infection in cystic fibrosis[END_REF][START_REF] Bjarnsholt | Pseudomonas aeruginosa biofilms in the respiratory tract of cystic fibrosis patients[END_REF]. Biofilms can be highly resistant to antibiotics compared to free swimming planktonic bacteria [START_REF] Høiby | Pseudomonas aeruginosa biofilms in cystic fibrosis[END_REF]. For example, about ten-to hundred-times more CIP is necessary to inhibit the growth of bacteria as biofilms compared to planktonic bacteria [START_REF] Torres | In vitro evaluation of Pseudomonas aeruginosa chronic lung infection models: Are agar and calcium-alginate beads interchangeable?[END_REF][START_REF] Walters | Contributions of antibiotic penetration, oxygen limitation, and low metabolic activity to tolerance of Pseudomonas aeruginosa biofilms to ciprofloxacin and tobramycin[END_REF][START_REF] Bandara | Magnetic fields suppress Pseudomonas aeruginosa biofilms and enhance ciprofloxacin activity[END_REF]. P. aeruginosa-entrapped in agar beads once in the lungs for several days resembles P. aeruginosa from biofilms observed in chronic infections [START_REF] Bjarnsholt | Pseudomonas aeruginosa biofilms in the respiratory tract of cystic fibrosis patients[END_REF][START_REF] Sønderholm | Pseudomonas aeruginosa Aggregate Formation in an Alginate Bead Model System Exhibits In Vivo-Like Characteristics[END_REF][START_REF] Gnanadhas | Chronic lung infection by Pseudomonas aeruginosa biofilm is cured by L-Methionine in combination with antibiotic therapy[END_REF].

The results of the present study indicate that complexation of CIP with Cu 2+ can effectively lead to enhanced local concentration in the lung necessary to reduce P. aeruginosa biofilm infection.

The in vivo chronic lung infection model with bioluminescent PA01 was highly resistant and allowed for monitoring of the progression of infection in the same animal.

A non-invasive technique was used to monitor the progression of the pulmonary infection caused by a luminescent strain of P. aeruginosa (PAO1::p16Slux) entrapped in agar beads (Figure 5). Riedel et al. demonstrated that the transformation of P. aeruginosa with the p16Slux system does not affect bacterial pathogenesis in rats after intranasal inoculation [START_REF] Riedel | Construction of p16Slux, a novel vector for improved bioluminescent labeling of gram-negative bacteria[END_REF]. They also demonstrated that after acute PAO1::p16Slux infection, luminescence from dissected organs linearly correlated with their bacterial loads, including lungs. Animal inoculation with bacterialaden beads has been extensively used to study novel therapeutic approaches to fight chronic lung infections [START_REF] Meers | Biofilm penetration, triggered release and in vivo activity of inhaled liposomal amikacin in chronic Pseudomonas aeruginosa lung infections[END_REF] as it resembles the lung disease observed in humans with P. aeruginosa chronic pneumonia with formation of pulmonary biofilm [START_REF] Cash | A rat model of chronic respiratory infection with Pseudomonas aeruginosa[END_REF]. The P. aeruginosa-laden beads used in this study had a median diameter D 50 of 88 ± 3 µm. Growcott et al. showed that beads with a D 50 of 155 µm were widely distribute in the lungs of rats, and were found as aggregates in the airways and alveolar spaces after i.t. inoculation [START_REF] Growcott | Characterisation of a refined rat model of respiratory infection with Pseudomonas aeruginosa and the effect of ciprofloxacin[END_REF]. The development of the infection in the whole lungs was confirmed in the present study by luminescence measurements from the thoracic region of the rats (Figure 5B) as well as pulmonary lesions observed after 8 days postinoculation (Figure 5C).

P. aeruginosa-entrapped in agar beads present in the lungs resembles P. aeruginosa from biofilms observed in chronic infections [START_REF] Bjarnsholt | Pseudomonas aeruginosa biofilms in the respiratory tract of cystic fibrosis patients[END_REF][START_REF] Sønderholm | Pseudomonas aeruginosa Aggregate Formation in an Alginate Bead Model System Exhibits In Vivo-Like Characteristics[END_REF][START_REF] Gnanadhas | Chronic lung infection by Pseudomonas aeruginosa biofilm is cured by L-Methionine in combination with antibiotic therapy[END_REF]. Biofilms bacterial population is heterogeneous and often harbor several subpopulations of bacteria that are resistant, persistent and/or tolerant to antimicrobial [START_REF] Lewis | Persister cells[END_REF][START_REF] Flemming | Biofilms: an emergent form of bacterial life[END_REF]. Tolerance and persistence are phenomena of increased bacterial survival in the presence of an antibiotic without an increase in the MIC [START_REF] Balaban | Definitions and guidelines for research on antibiotic persistence[END_REF]. Tolerant and persistent bacteria often are in dormancy. They do not grow, or very slowly, and have reduced metabolism activity when compared with growing cells [START_REF] Balaban | Definitions and guidelines for research on antibiotic persistence[END_REF]. To distinguish between the in-vivo antimicrobial activity of CIP-Cu and CIP-HCl, the number of colonies appearing in 24 hours from lung homogenate was counted (Figure 7.B). This method allows the quantification mainly of the surviving bacterial population able to grow, but is not adapted for bacteria in dormancy. Similarly, luminescence is mainly emitted by metabolically active bacteria and helps to quantify bacterial proliferation [START_REF] Duncan | Luminescence-based detection of activity of starved and viable but nonculturable bacteria[END_REF] but also allows a more direct measurement. Riedel et al. [START_REF] Riedel | Construction of p16Slux, a novel vector for improved bioluminescent labeling of gram-negative bacteria[END_REF] showed that luminescence produced by PAO1::p16Slux remained relatively stable throughout the entire growth curve, while luminescence in other species such as Listeria monocytogenes and other gram-positive bacteria decreased dramatically in stationary phase. In stationary phase, planktonic bacteria have lower metabolic activity and lower susceptibility to CIP comparable to those of certain phases of biofilm growth [START_REF] Mah | Mechanisms of biofilm resistance to antimicrobial agents[END_REF]. This suggests that, with enough sensitivity, luminescence could be used to monitor slow growing bacteria during bacterial biofilm infection. In the present study, an optimal bacteria/beads ratio was selected to inoculate the rats for achieving the best survival and a high luminescence signal (Figure 5A-B).

For the two lower inoculums (1.5 x 10 7 CFU and 6 x 10 6 CFU), after a sharp luminescence increase on day 2, the signals decreased on day 3 to values close to these obtained on day 1 and then were stable from day 5 to day 8. These profiles were very similar to those usually obtained by measuring the CFU numbers in the lungs [START_REF] Kukavica-Ibrulj | Animal models of chronic lung infection with Pseudomonas aeruginosa: useful tools for cystic fibrosis studies[END_REF][START_REF] Bragonzi | Nonmucoid Pseudomonas aeruginosa expresses alginate in the lungs of patients with cystic fibrosis and in a mouse model[END_REF]. However, the initial luminescence level seemed too low to accurately assess the effect of antibiotics. In addition, bacterial loads below 1.10 7 CFU per lung gave bioluminescence values that were close to the detection limit of the IVIS equipment. Thus, no correlation between bioluminescence and the number of CFU per lung measured on day 8 after infection could be obtained because most of the bacterial loads were less than 1.10 7 CFU per lung on that day (Fig. 7B). The use of a less virulent strain could help increase the inoculum used for infection and thereby increase the stable basal signal observed 4 days after infection. In addition, using a strain with a higher luminescence yield or increasing the exposure time could also increase the sensitivity of the method and make it possible to quantify the degree of infection according to the value of luminescence, not just to assess its spatial distribution.

Spray dried and micronized powders properties were suitable for inhalation

The RODOS unit has long been used to characterize powders for inhalation across a range of settings on the instrument. Over the past 15 years or so, many powders of different physical and chemical properties have been analyzed by different groups revealing some consistent trends. In particular, it has been generally shown that RODOS dispersion pressures above 3 kPa result in full dispersion and allows sizing of the primary particle sizes of a powder [START_REF] Jaffari | Rapid characterisation of the inherent dispersibility of respirable powders using dry dispersion laser diffraction[END_REF][START_REF] De Boer | Design and in vitro performance testing of multiple air classifier technology in a new disposable inhaler concept (Twincer®) for high powder doses[END_REF]. Similarly, dispersion pressures used in the RODOS of between 0.5-3 kPa can indicate the dispersion performance of a powder [START_REF] Jaffari | Rapid characterisation of the inherent dispersibility of respirable powders using dry dispersion laser diffraction[END_REF] agnostic to proprietary inhaler designs and even allow matching to novel inhaler dispersion performance [START_REF] De Boer | Design and in vitro performance testing of multiple air classifier technology in a new disposable inhaler concept (Twincer®) for high powder doses[END_REF]. Thus, in these preclinical in vivo feasibility studies we have characterized the powder particle size distributions to indicate how they will be dispersed as a powder and also specifically their performance for nose-only inhalation testing. Future studies will couple the powder with an inhaler system and be characterized using compendia methods to support eventual clinical testing.

CONCLUSION

Pulmonary delivery of ciprofloxacin-copper complex significantly reduced the P.

aeruginosa in a rat chronic lung infection model. Luminescent P. aeruginosa entrapped in agar beads were useful to spatially monitor the development of the lung infection in rats.

Figure 1 :

 1 Figure 1: Dosing regimen after i.t. inoculation of agar beads loaded with bioluminescent PA01. Ciprofloxacin doses were set at 1.65 mg of CIP (base) per animal.

Figure 2 :

 2 Figure 2: Characterization of spray dried CIP-Cu and jet milled CIP-HCl powders. A. SEM micrographs, B. Evolution of the PSD parameters D 10 , D 50, and D 90 versus dispersion pressure. Solid symbols are for spray dried CIP-Cu, and open symbols are for micronized CIP-HCl. Values are means ± SD (n=3).

Figure 3 :

 3 Figure 3: Size distribution of agar beads dispersed in PBS. Size distribution of blank beads was determined by laser diffraction using the Sympatec HELOS-R equipped with a Cuvette module. Values are means ± SD (n = 3).

Figure 4 :

 4 Figure 4: Time-luminescence curves obtained from bioluminescent P. aeruginosa-loaded beads. The beads were incubated with different CIP-base equivalent concentrations of CIP-HCl (A) and CIP-Cu (B). (Top: 0 -0.5 µg/mL, bottom: 2 -32 µg/mL). Luminescence (as relative light units (RLU)) was recorded every 20 min (n=3; mean).

  animals' body weight. By day 3, the CIP-HCl group had lost 11.8% of body weight and the CIP-Cu group had lost 6.9% relative to average body weight on day 1. By day 8, the average body weight of the groups was similar to the body weight on day 1, with CIP-HCl group reaching 99.7% of the initial weight and CIP-Cu reaching 97.6% of the initial body weight.

Figure 5 :

 5 Figure 5: In vivo chronic lung infection model. Sprague-Dawley rats were inoculated intratracheally with various ratios of luminescent P. aeruginosa entrapped in agar beads (CFU/beads ratios): 20 CFU/bead (6 x 10 6 CFU/animal), 250 CFU/bead (1.5 x 10 7 CFU/animal), and 500 CFU/bead (5 x 10 7 CFU/animal). A. Variation of luminescence (in photons s -1 cm -2 steradian -1 ) per day (n=2-4; mean ± S.E.M.). B. Luminescence imaging obtained in IVIS. C. Pictures of the infected lungs on day 8 and number of CFU recovered from the lungs (group 6 x 10 6 CFU/animal).

  the inhalation chamber using a Mercer-Style mini cascade impactor directly connected to the NOIES. Aerodynamic PSD of the powders' particles present in the inhalation chamber are presented in Figure6B. Both formulations had a narrow PSD and similar MMADs in the 1 to 3 µm range. Particles having an MMAD in this range generally give a same percentage of lung deposition in rats of approximately 20% as shown by Kuehl et al.[START_REF] Kuehl | Regional particle size dependent deposition of inhaled aerosols in rats and mice[END_REF]. Moreover, both formulation particles in the NOIES inhalation chamber have close GSD indicating a narrow dispersity of the particles from the MMAD value. Therefore, the same lung deposition should occur with both formulations. The other parameters evaluated before dosing the animals were the Total Inhaled Dose (TID) and the time needed to achieve the same TID in each group of animals (D in Eq. 1). The potency of CIP-Cu was lower than the CIP-HCl due to the presence of more excipients (CaCO 3 and hyaluronic acid). Thus, animals were dosed for 35 min with CIP-Cu and for 20 min with CIP-HCl to achieve comparable TID of around 1.65 mg. Using this TID, the aerodynamic properties of the powder particles evaluated with the Mercer-Style mini-cascade

Figure 6 :

 6 Figure 6: Microparticles aerodynamic properties measured using a Mercer-Style minicascade impactor connected to the nose-only inhalation exposure system (NOEIS). A) Animals treated in NOIES, B) Distribution of the aerodynamic size of the particles present in the NOIES measured using the Mercer-Style mini-cascade impactor (picture inserted). C) Aerodynamic properties parameters (MMAD: Mass median aerodynamic diameter, GSD: Geometric standard deviation, TID: Total Inhaled Dose).

6.


Elodie Brunel

Fabienne Comte
email: fabi-enne.comte@parisdescartes.fr.

HAZARD REGRESSION WITH NON COMPACTLY SUPPORTED BASES

Keywords: Hazard rate, Laguerre basis, Least squares regression, Nonparametric estimation, Projection estimator

In this paper, we consider the problem of nonparametric hazard rate estimation in presence of right-censored observations. We provide a generalized risk bound for a regression type nonparametric estimator of the hazard function of interest. Under adequate integrability conditions, our bound is a generalization to non necessarily compactly supported bases, of strategies which were specific to compact support of estimation. We show that it encompasses those previous compact-support results. We discuss the model selection method which comes out from the new terms of the risk bounds, and compare the performance of the new estimator to previous ones, when using a non compact Laguerre basis. A real data example is also presented.

Introduction

Consider the model where the observations are (1)

Z i = X i ∧ C i , δ i = 1 {X i ≤C i } ,
where the sequences (X i ) i and (C i ) i are two independent sequences of i.i.d. nonnegative random variables. The function of interest is λ = f /S where f is the density of X 1 and S its survival function, called hazard rate. The Z i 's are called right-censored observations, and the δ i 's are non-censoring indicators. This type of model is most commonly used in reliability or survival analysis: more precisely, we consider here lifetimes (or failure times) of some individuals in presence of right-censoring. This occurs for instance when some of the individuals are not observed until the end (death, remission, recovery) of the study; only a lower bound on their lifetime is observed.

There are different nonparametric methods used in the literature to estimate hazard rate, most of them rely on quotient strategies. Indeed, let S C and S Z denote the survival function of the C i 's and Z i 's: S C (x) = P(C 1 > x), S Z (x) = P(Z 1 > x). Then, the hazard rate can be written as λ = f S C /S Z , where the function f S C is often called subdensity. This function can be estimated using censored observations, and S Z has an obvious empirical counterpart, namely the empirical survival function of all Z i 's. This idea is used in Blum and Susarla (1980), Mielniczuk (1986), [START_REF] Diehl | Kernel density and hazard function estimation in the presence of censoring[END_REF], [START_REF] Lo | Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator[END_REF], [START_REF] Uzunoḡullari | A comparison of hazard rate estimators for left truncated and right censored data[END_REF], who propose kernel estimators of the numerator. Note that bandwidth selection is an important issue in this context and practical methods are suggested. [START_REF] Antoniadis | Density and hazard rate estimation for rightcensored data by using wavelet methods[END_REF] consider both subdensity and hazard estimators via wavelet methods, and the optimal wavelet resolution depends on the unknown function. [START_REF] Brunel | Penalized contrast estimation of density and hazard rate with censored data[END_REF] build projection estimators based on these ideas, and propose model selection methods to determine in a data driven way, the relevant dimension for the projection space; they prove that their strategy ensures an automatic squared-bias/variance tradeoff.

Nonparametric estimators of the hazard rate have also been constructed by convolving with a kernel some cumulative hazard estimator such as the Nelson-Aalen or the Kaplan-Meier estimators, see [START_REF] Tanner | The estimation of the hazard function from randomly censored data by the kernel method[END_REF], [START_REF] Ramlau-Hansen | Smoothing counting process intensities by means of kernel functions[END_REF] and [START_REF] Yandell | Nonparametric inference for rates with censored survival data[END_REF]. [START_REF] Müller | Hazard rate estimation under random censoring with varying kernels and bandwidths[END_REF] propose estimators with varying kernels and data-adaptive bandwidths and more recently [START_REF] Bouezmarni | Gamma kernel estimators for density and hazard rate of right-censored data[END_REF] study a Gamma kernel estimator. Later on, [START_REF] Wu | Nonparametric estimation of hazard functions by wavelet methods[END_REF], proposed a wavelet-type estimator also based on the transform of a Nelson-Aalen cumulative hazard estimator. [START_REF] Kooperberg | The L2 rate of convergence for hazard regression[END_REF] study the L 2 convergence rate of a hazard rate estimator in a context of tensor product splines. Dölher and Rüschendorf (2002) introduce an adaptive sieved maximum likelihood method. Reynaud-Bouret (2002) obtains adaptive results and minimax rates for penalized projection estimators of the Aalen multiplicative intensity process. Lastly, [START_REF] Brunel | Penalized contrast estimation of density and hazard rate with censored data[END_REF] consider penalized contrast estimator using the Kaplan-Meier cumulative hazard estimator and a large variety of models.

In this work, we consider a direct regression strategy already described in [START_REF] Plancade | Model selection for hazard rate estimation in presence of censoring[END_REF], or which can be obtained as a particular case of [START_REF] Comte | Adaptive estimation of the conditional intensity of marker-dependent counting processes[END_REF]. The generalization comes from the fact that we no longer assume that the estimation support is a compact set: this assumption was crucial in previous works, and we no longer require it. The ideas are inspired by those of [START_REF] Cohen | On the stability and acuracy of least squares approximations[END_REF][START_REF] Cohen | Correction to: On the stability and accuracy of least squares approximations[END_REF] and Comte and Genon-Catalot (2019) for standard regression, but hazard rate regression has specificities, both in theory (e.g. the stability constraint given by (9) in section 2.2 is different from standard regression) and in practice. We have in mind that survival analysis is a context where the Laguerre basis (see Section 2.4), which is R + -supported, is specifically well suited for estimation: the resulting estimators are general combinations of Gamma-type distributions. As many parametric models involve Gamma densities, projection estimators in the Laguerre basis are a relevant generalization of these densities and allow a lot of flexibility.

The plan of the paper is as follows. The estimator and the assumptions are given in section 2, bounds for empirical and L 2 -risk are then stated, and examples of compact and non compact settings are given. Then, Section 3 describes a model selection procedure; the new risk bound suggests an easy-to-compute penalty function, which avoids to estimate unconvenient quantities, like upper or lower bounds of unknown functions. The method is applied through simulation experiments to previous examples of the literature, for comparison. A real data example is considered in section 4, and illustrates the relevance and the flexibility of our procedure. A short concluding section 5 ends the presentation. Proofs are postponed in Section 6.

Hazard rate estimation in presence of right censoring

Let us start with preliminary notations. For a function u, we denote by u 2 := u 2 (x)dx, by u 2 S Z := u 2 (x)S Z (x)dx and for two square integrable functions u 1 and u 2 , by u 1 , u 2 and u 1 , u 2 S Z the associated scalar products. The corresponding spaces for square-integrable A-supported functions are denoted by L 2 (A, dx) and L 2 (A, S Z (x)dx). For a m-dimensional vector v with coordinates (v 1 , . . . , v m ), we denote by v 2 2,m := m j=1 v 2 j its euclidean norm. For a matrix M , we define the operator norm M op as the square-root of the largest (nonnegative) eigenvalue of M t M , where t M is the transpose of M . When M is symmetric, it coincides with its largest eigenvalue in absolute value. The so-called Frobenius norm is also defined by

M 2 F = i,j [M ]
γ n (t) = t 2 n - 2 n n i=1 δ i t(Z i ),
where the empirical scalar product and its associated empirical norm are defined by

t 2 n = 1 n n i=1 t 2 (x)1 {Z i >x} dx , s, t n = 1 n n i=1 s(x)t(x)1 {Z i >x} dx. (3)
Let us explain why this contrast is related to our hazard rate estimation problem. First note that E(1 {Z 1 >x} ) = P(Z 1 > x) = S Z (x) = S C (x)S(x) with S Z and S C denoting the survival functions of Z 1 and C 1 . Secondly, we have

E(δ 1 t(Z 1 )) = E(1 {X 1 ≤C 1 } t(X 1 )) = E(S C (X 1 )f (X 1 )).
Therefore, we find that

E(γ n (t)) = t 2 (x)S Z (x)dx -2 t(x)S C (x)f (x)dx = (t(x) -λ(x)) 2 S Z (x)dx -λ 2 (x)S Z (x)dx.
Thus, minimizing γ n for large n, should provide a function t minimizing (t(x)-λ(x)) 2 S Z (x)dx, that is a weighted L 2 -distance to λ. Therefore, we should estimate the L 2 orthogonal projection of λ w.r.t the S Z -weighted scalar product on a subspace S m of functions over which the minimization is performed. Let A ⊆ R + and let (ϕ j , j = 0, • • • , m -1) be an orthonormal system of functions supported on A belonging to L 2 (A, dx), i.e. such that ϕ j , ϕ k = δ j,k , 0 ≤ j, k ≤ m -1. We define S m as the space linearly spanned by the functions ϕ j : S m = span(ϕ 0 , • • • , ϕ m-1 ). The space S m has thus finite dimension m. We define the matrix

Ψ m,Z = ( ϕ j , ϕ k n ) 0≤j,k≤m-1 = ϕ j (x)ϕ k (x) S Z,n (x)dx 0≤j,k≤m-1
where

S Z,n (x) = 1 n n i=1 1 {Z i >x}
and the matrix

Ψ m,Z := ϕ j (x)ϕ k (x)S Z (x)dx 0≤j,k≤m-1
.

Note that the matrix Ψ m,Z = E Ψ m,Z is the matrix of the scalar products ϕ j , ϕ k S Z (with associated L 2 -weighted norm . S Z ) and Ψ m,Z is its empirical counterpart with ϕ j , ϕ k n for 1 ≤ j, k ≤ m. Then we define

λ m = arg min t∈Sm γ n (t).
Setting the gradient of γ n (t) to zero and standard algebra calculations give, provided that Ψ m,Z is a.s. invertible,

λ m = m-1 j=0 âj ϕ j with â(m) =    â0 . . . âm-1    = 1 n Ψ -1 m,Z t Φ m δ, (4)
where Φ m = (ϕ j (Z i )) 1≤i≤n,0≤j≤m-1 and δ = t (δ 1 , . . . , δ n ).

Remark 2.1. We can compare with the regression model: Y i = b(Z i )+ε i where (Z i , Y i ) are observed, ε i is a centered unobserved noise and the (X i ) i and the (ε i ) i are i.i.d. independent sequences. To estimate the regression function b, estimators of the m first coefficients of b in the basis are (

1/n) Ψ -1 m t Φ m Y where Y = t (Y 1 , . . . , Y n ) and Ψ m = (1/n) t Φ m Φ m .
Here the fact that the same matrix Φ m appears in all terms is very important and convenient. This is what makes an important difference with hazard rate estimation. Here, Ψ m,Z is not directly related to Φ m . Formula (4) provides an easy way to compute our projection estimator λ m provided that Ψ m,Z is a.s. invertible. So, to guarantee it is always satisfied, we define the trimmed estimator by :

λm = λ m if Ψ -1 m,Z op ≤ c n log(n) 0 otherwise ( 5
)
where c is a constant defined further (see Proposition 2.1).

Convention. We set Ψ -1 m,Z op = +∞ if Ψ m,Z is not invertible.

2.2.

Bounds for the empirical risk and the integrated risk of one estimator. We consider a general context where the estimation support A is such that A ⊆ R + and (6)

A λ 2 (x)S Z (x)dx < +∞.
Condition ( 6) is fulfilled for most classical models. Indeed as S Z ≤ S, the condition holds if the distribution of X is such that A λ 2 S < +∞. We shall denote λ A = λ1 A .

Examples of models satisfying

R + λ 2 S = R + f 2 /S < +∞.
(1) Exponential density:

f exponential E(θ), θ > 0, S(x) = exp(-θx)1 {x≥0} , λ(x) = θ 1 {x≥0} , (2) Weibull model, λ(x) = αθ α x α-1 1 {x≥0} , S(x) = exp(-(θx) α )1 {x≥0} , α > 1/2, θ > 0 (3) Gamma model, f (x) = θ ν x ν-1 e -θx /Γ(ν)1 {x≥0} , ν > 1/2, θ > 0, (4) Gompertz-Makeham, λ(x) = γ 0 + γ 1 e γ 2 x , S(x) = e -γ 0 x-(γ 2 /γ 1 )(e γ 2 x -1) 1 {x≥0} , for real numbers γ 0 , γ 1 , γ 2 > 0, (5) Log-logistic, λ(x) = θνx ν-1 /(1 + θx ν )1 {x≥0} , ν > 1/2, θ > 0, S(x) = 1/(1 + θx ν )1 {x≥0} , (6) Log-normal λ(x) = (1/(xσ))φ ((ln x -µ)/σ) / [1 -Φ ((ln x -µ)/σ)] 1 {x≥0}
, where φ(x) and Φ(x) are respectively the density and the cumulative distribution function of a standard gaussian, µ ∈ R, σ > 0. In addition, we assume that the basis (ϕ j ) j is such that Then, we can prove risk bounds with respect to the empirical risk first, and to the integrated S Z -weighted risk in a second time. In the regression setting, the risk bound obtained for the empirical risk is rather straightforward and relies on projection arguments; it is interesting to see that the problem here also involves linear algebra but still, is more involved.

Proposition 2.1. Assume that Ψ m,Z is invertible, that condition [START_REF] Comte | Nonparametric Estimation[END_REF] holds and that

(8) A λ 2 (z) S Z (z)dz < +∞.
Then, for any m such that L(m) ≤ n and

(9) Ψ -1 m,Z op ≤ c 2 n log(n) , c = 3 log(3/2) -1 10 ,
we have

E λm -λ A 2 n ≤ inf t∈Sm t -λ A 2 S Z + 2 Tr(Ψ -1 m,Z Ψ m,λS Z ) n + C 1 n . ( 10
)
where C 1 is a positive constant and

Ψ m,λS Z = ϕ j (x)ϕ k (x)λ(x)S Z (x)dx 0≤j,k≤m-1 . ( 11
)
Note that, as S Z (x) ≤ 1, condition [START_REF] Comte | Adaptive estimation of the conditional intensity of marker-dependent counting processes[END_REF] implies that A λ 2 (z)S Z (z)dz < +∞ and thus condition [START_REF] Cohen | Correction to: On the stability and accuracy of least squares approximations[END_REF] 6) and ( 7) hold. Then for any m such that L(m) ≤ n and ( 9) holds, we have

E λm -λ A 2 S Z ≤ 1 + 8 c log(n) inf t∈Sm t -λ A 2 S Z + 8 Tr(Ψ -1 m,Z Ψ m,λS Z ) n + C 2 n , ( 12
)
where c is defined in ( 9) and C 2 is a positive constant.

Equations ( 10) and ( 12) provide empirical and integrated risk bounds involving three terms: a negligible one of order 1/n, a variance term of order Tr(Ψ -1 m,Z Ψ m,λS Z )/n and a bias term inf t∈Sm t -λ A 2 S Z . It is noteworthy that the coefficient in front of the bias term is exactly 1 in the first case and of order 1 for large n in the second one. Clearly, this term is decreasing if the space S m grows when m increases (with nested collection, m ≤ m =⇒ S m ⊂ S m ). On the other hand, the true novelty stands in the variance bound Tr(Ψ -1 m,Z Ψ m,λS Z )/n obtained in Propositions 2.1 and 2.2, which is more general than in previous works. The result holds without constraint on the support. Moreover, even it is not obvious at first sight, we can prove that m → Tr(Ψ -1 m,Z Ψ m,λS Z ) is increasing.

Lemma 2.1. Let the collection S m be nested, then m → Tr(Ψ -1 m,Z Ψ m,λS Z ) is increasing. Therefore, both bounds in [START_REF] Döhler | Adaptive estimation of hazard functions[END_REF] and ( 12) lead to the same conclusion that a compromise has to be found for the choice of m, making a tradeoff between bias and variance. In the next section, we illustrate that, in the standard case of compact support A, we can obtain a more explicit upper bound on the variance, and recover previous results.

Specific cases of compact A. Let us assume here that A is compact and show in what extent our new results encompass previous ones.

We can consider a trigonometric basis on A = [0, a]:

ϕ 0 (x) = (1/ √ a)1 [0,a] (x), ϕ 2j-1 (x) = 2/a cos(2πjx/a)1 [0,a] (x), ϕ 2j (x) = 2/a sin(2πjx/a)1 [0,a] (x), j ≥ m. Clearly, in that case, L(m) ≤ (2/a)m and L(m) = m/a if m is even.
We may also choose the histogram basis on A = [0, a], we set ϕ j (x) = √ ma1 [ja/m,(j+1)a/m[ for j = 0, . . . , m -1. We can consider more general piecewise polynomials with given degree r, by rescaling Q 0 , . . . , Q r the Legendre basis on each sub-interval [ja/m, (j + 1)a/m[, j = 0, . . . , m -1. In that case, we have L(m) = a m for histograms and L(m) ≤ (r + 1)a m for piecewise polynomials (see Comte (2017, chap.2)). Consequently, condition ( 7) is satisfied for these bases, and L(m) ≤ c 2 ϕ m, where c 2 ϕ is a known constant depending on the basis and not on m.

For these bases with specifically compact supports, we can assume that:

(13) ∀x ∈ A, S Z (x) ≥ S 0 > 0 and λ(x) ≤ λ A ∞ < +∞.
Note that S Z is lower bounded on A if both S and S C are; moreover, given the first part, the second part of ( 13) can be obtained if f is bounded on A, as λ(x) ≤ f (x)/S 0 . However, if condition λ ∞ < +∞ generally holds for compact A, it is not the case for A = R + , see the Weibull (2) or the Gompertz-Makeham ( 4) examples.

Lemma 2.2. Let A be a compact set and consider a basis such that L(m) ≤ c 2 ϕ m. Under (13), condition ( 8) is fulfilled. Moreover,

(i) Ψ -1 m,Z op ≤ 1/S 0 , (ii) 0 ≤ Tr(Ψ -1 m,Z Ψ m,λS Z ) ≤ m λ A ∞ (iii) 0 ≤ Tr(Ψ -1 m,Z Ψ m,λS Z ) ≤ c 2 ϕ m/S 0 .
Bound (i) shows that condition ( 9) is automatically fulfilled for n large enough: this is why this condition does not appear in a compact setting. Moreover, with (ii) and (iii) we recover the variance bounds proposed in Plancade (2011), see equation ( 11) and Theorem 1 therein, see also [START_REF] Comte | Adaptive estimation of the conditional intensity of marker-dependent counting processes[END_REF], Theorem 1 for bound (ii), in presence of covariates.

Example of non compact

A. The Laguerre basis on A = R + is defined by: ( 14)

P j (x) = j k=0 (-1) k j k x k k! , ϕ j (x) = √ 2P j (2x)e -x 1 x≥0 , j ≥ 0.
The P j are called Laguerre polynomials (P j ) and the ϕ j Laguerre functions. The collection (ϕ j ) j≥0 is a complete orthonormal system on L 2 (R + ), such that (see Abramowitz and Stegun (1964

)) ∀j ≥ 0, ∀x ∈ R + , |ϕ j (x)| ≤ √ 2.
Therefore L(m) ≤ 2m and condition ( 7) is satisfied.

However, condition [START_REF] Klein | Survival Analysis : Techniques for Censored and Truncated Data[END_REF] does not hold anymore, and Lemma 2.2 has to be stated neither:

Lemma 2.3. If µ(R + ∩ Supp(S Z ))
> 0 where µ is the Lebesgue measure and Supp(S Z ) = {x ∈ R + , S(x) > 0} is the support of S Z , then Ψ m,Z is invertible. Moreover, there exists c > 0 such that, for m large enough,

Ψ -1 m,Z op ≥ c √ m.
Lemma 2.3 shows clearly that in the context of the Laguerre basis, bound (i) of Lemma 2.2 is not true. So, the order of the variance is not obvious.

Note that if X ∼ E(β) i.e. f (x) = βe -βx 1 R + (x) and S(x) = e -βx 1 R + (x), then λ(x) = β. Therefore Ψ m,Z = (1/β)Ψ m,λS Z and Tr(Ψ -1 m,Z Ψ m,λS Z ) = βTr(Id m ) = β m.
Thus, the variance term can remain of order m/n in the non-compact setting as well. Numerical experiments support the conjecture that that the quantity Tr(Ψ -1 m,Z Ψ m,λS Z ) is generally of order c m with c a constant that can be evaluated (see Figure 1). However, in the same examples, the term Ψ -1 m,Z op can grow very fast, so that bounding

Tr(Ψ -1 m,Z Ψ m,λS Z ) by Ψ -1/2 m,Z 2 op Ψ 1/2 m,λS Z 2 F = Ψ -1 m,Z op Tr(Ψ m,λS Z )
is not a good strategy in the non-compact setting.

Model selection and simulations

3.1. Procedure. In this section, we propose a practical procedure for model selection. A theoretical study has been made for a similar proposal in the case of nonparametric

f (x) = βe -βx 1 x≥0 , β = 1/3 f (x) = 3/(1 + x) 4 1 x≥0 f (x) = xe -x 2 /2 1 x≥0 â = -0.11, b = 0.36 â = 4.56, b = 1.31 â = -3.31, b = 2.21 Figure 1. Plots of m → Tr( Ψ -1 m,Z Ψ m,λS Z ) for m = 1, . . . , 20
, from n = 10000 observations with no censoring, in blue. In bold dotted red, the best approximating line y = â + bx, with value of the coefficients in each case. regression function estimation, see Comte and Genon-Catalot (2020), and we refer the reader to this paper for technicalities, which are numerous. For now, let us describe it.

An important preliminary remark is that, as λS Z = f S C , the matrix Ψ m,λS Z can easily be estimated by [START_REF] Lo | Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator[END_REF] Ψ m,λS

Z = 1 n n i=1 δ i ϕ j (Z i )ϕ k (Z i ) 0≤j,k≤m-1
.

Now, let M n be the theoretical collection of models defined by

M n = {m ∈ {1, . . . , n}, Ψ -1 m,Z op ≤ c 2 n log(n) }
and its empirical version

M n = {m ∈ {1, . . . , n}, Ψ -1 m,Z op ≤ c n log(n) }.
Then we select m = arg min

m∈ Mn (-λ m 2 n + pen(m)), pen(m) = κ Tr( Ψ -1 m,Z Ψ m,λS Z ) n .
Indeed, it is easy to check that γ n ( λm ) = -λm 2 n and this term is taken as an estimate of the squared bias term. The penalty is the empirical version of the variance order. The criterion is thus an empirical version of the bias variance decomposition. The constant κ is numerical and from a theoretical point of view, it depends neither on λ nor on n; it has to be calibrated once and for all on a set of preliminary simulations.

3.2.

Simulations. The constant κ is calibrated through preliminary experiments and we take κ = 2. Then we apply the procedure with only one major change: the set M n is too small in most experiments. To be able to consider more models with larger dimension, we replace it by M n = {m ∈ {1, . . . , n}, Ψ -1 m,Z op ≤ n 5/2 }. This is much more than expected from the theory, and still a true limitation since Ψ -1 m,Z op grows really very fast with m.

The matrix Ψ m,λS Z is straightforward from formula [START_REF] Lo | Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator[END_REF], and matrix Ψ m,Z is computed by writing its coefficients (1/n) n i=1 Z i 0 ϕ j (x)ϕ k (x)dx and Riemann discretization of the integrals over [0, Z i ] with 200 steps.

• Comparison with [START_REF] Antoniadis | Density and hazard rate estimation for rightcensored data by using wavelet methods[END_REF] and others. First, we consider two cases, which have been studied in previous papers: (a) The first set of simulations is called in the following the "Gamma case". The X i 's are generated from a Gamma distribution with shape parameter 5 and scale 1 and the independent C i 's from an exponential distribution with mean 6. (b) The second set is called "the bimodal case". The X i 's have a bimodal density defined by

f = 0.8u + 0.2v
where u is the density of exp(Y /2) with Y ∼ N (0, 1) and v = 0.17Y + 2. The C i s are generated from an exponential distribution with mean 2.5.

Examples These authors give the mean squared errors of their estimator computed over T = 200 replications of samples of size n = 200 and n = 500. The error is computed over K regularly spaced points t k , k = 1, . . . , K, of the interval in which the X i 's fall ([0, max X i ]), as the mean over the replications j of

MSE j = 1 K K k=1 (λ(t k ) -λj (t k )) 2
where λj is the estimate of λ for the sample number j, j = 1, . . . , T .

In order to take into account the sparsity of the observations at the end of the interval, (P(X > 6) = 0.25 in the Gamma case and P(X > 2) = 0.16 in the bimodal case), they also compute an error MSE2 defined by the same kind of mean squared error but with a truncated mean over the t k 's less than 6 in the Gamma case and less than 2 in the bimodal case.

Reynaud We remark that the MSE of our new estimator is always substantially smaller than the one of all previous estimators. A contrario, the value of our MSE2 is slightly larger in all cases. This means that locally on this part of the interval, our new estimator is not better, but that, considered on the whole domain, it is globally much more performing. Let us add that the relevant cut for the MSE to compute restricted MSE2 is in general unknown, so that the only reliable result is related to the complete observation interval.

)
consider a hazard rate estimator built as a quotient of a Gammakernel density estimator divided by a Kaplan-Meier survival function estimator. The bandwidth selection method is not clearly specified. The authors consider two models A and B. Model A corresponds to an exponential distribution with parameter 1 for X and a uniform density on [0, c] for C, where c is chosen to ensure the desired censoring rate. In Model B, X follows a Weibull distribution with scale parameter b = 2 and shape parameter a = 1.2, and C a Weibull distribution with shape parameter a and scale parameter given by b = ((1 -p)/p) 1/a . This ensures that the degree of censoring is equal to p.

Table 3 presents the results obtained by [START_REF] Bouezmarni | Gamma kernel estimators for density and hazard rate of right-censored data[END_REF] in column G, by Müller and Wang (1994) in column MW, and by our estimator in columns MS. The column MS 99% presents the MSE computed on an interval corresponding to 99% of the observations and MSE 85% on an interval corresponding to 85% of the observations. We can see that the performances of our estimator is in the range of the two others for n = 125 and sometimes better for n = 250. The performances on the smaller interval are clearly better but of course the comparison is unfair. The sample sizes here are quite small (with possibly 50% of censoring) for nonparametric methods, which makes the resulting performances hardly reliable.

• Comparison with Barbeito and Cao (2018). Barbeito and Cao (2018) consider hazard rate estimation in a model without censoring. Their estimator is a quotient of a standard kernel estimator divided by an integrated version of it, and they concentrate on the bandwidth selection problem, for which they propose two strategies: double one sided cross validation denoted 'DO' and a bootstrap method (the best for those three models) 'Boot2'. We recall in Table 4 their results and compare to our estimator, for three of their models corresponding to nonnegative X:

a n = 125 n =

Real data example

We study a real dataset from the National Longitudinal Survey of Youth of the U.S. Bureau of Labor Statistics (https://www.nlsinfo.org/content/cohorts/nlsy79). In the survey, women, aged 14 to 21 in 1979, have been interviewed yearly from 1983 through 1988.

They were asked about any pregnancies and breast feeding. This data set consists of the information from n = 927 first-born children to mothers who chose to breast feed their children. The lifetime in the data set is the duration of breast feeding in weeks, followed by an indicator of whether the breast feeding was completed (i.e. time to weaning of breastfed newborns). The data was restricted to children born after 1978 and whose gestation was between 20 and 45 weeks and it is available from the KMsurv package.

We have 892 events over 927 observed data which correspond to 96% of uncensored lifetimes. The minimum observed duration to weaning is one week and the maximum one is 192 weeks with a median at 12 weeks. See also Section 5.4 in Klein and Moeshberger (2003) for a brief analysis of the dataset. Our estimator with Laguerre basis is applied to observations rescaled on the interval [0, 3] in order to deal with the high time values of the duration which may cause numerical error in the digital process. The rescaled observations (Z i ) 1≤i≤n are obtained by applying the transformation t → (t -min(Z i ))/(max(Z i ) -min(Z i ))/b), with b = 3 to the original observations (Z i ) 1≤i≤n . Then, the estimator is plotted in its original scale.

In Figure 2 we present the collection of estimators λm defined in ( 5), for

m ∈ M n = {m ∈ {1, • • • , D max }, Ψ -1
m,Z op ≤ n 5/2 }. Setting κ = 2 as in the simulation studies, and D max = 7 our selection procedure chooses the model m opt = 5 with D max = 7 but only models with m ≤ 6 are allowed by the numerical constraint required with the set M n . As these estimators are not necessarily positive, we take the positive part max( λm (x), 0). The corresponding estimator is displayed in Figure 3 (Left) along with the Müller and Wang kernel estimators and in Figure 3 (Right) with classical parametric models whose parameters were adjusted by maximum likelihood. The kernel estimators in Figure 3 (Left) are built using the function muhaz available in the muhaz package with a local optimal bandwidth computed at a grid point obtained by minimizing the local MSE and with the Epanechnikov kernel. We can see that the kernel estimator needs to be corrected at the end of the interval whereas our estimator is not affected by boundary effects. We have fitted an exponential hazard rate λ1 (x) = 0.059, a log-logistic hazard rate λ2 (x) = θνx ν-1 /(1+ θx ν ) with ν = 1.44 and θ = 0.037 and a log-normal hazard

λ3 (x) = 1 xσ φ ln x - μ σ / 1 -Φ ln x - μ σ
with μ = 2.24 and σ = 1.18 and φ(x) and Φ(x) are respectively the density and the cumulative distribution function of a standard gaussian.

All these parametric models cannot capture correctly the shape of the hazard rate. We can observe that the shape of the nonparametric estimator makes sense since the risk of stopping breast feeding is high at the very beginning, then the curve is decreasing and achieves a first local minimum between the 12th and 18th week. Then the curve is increasing and after the week 50, only 43 women keep going on with breast feeding. These remaining women stop between week 50 and 100, and finally over the week 100, only 3 women are still breastfeeding, so the curve is increasing with large slope. This is corroborated by the aspect of the cumulative hazard estimators in Figure 4 where the cumulative hazard rate estimators are displayed. Since the cumulative hazard rate Λ(x) = x 0 λ(u)du is the hazard rate primitive, we can obtain the integrated Laguerre estimator by using the following formula giving the primitive of the Laguerre basis L j (x) = x 0 ϕ j (u)du L 0 (x) = ϕ 0 (0) -ϕ 0 (x) and L j (x) = -L j-1 (x) -ϕ j (x) + ϕ j-1 (x) for j ≥ 1. Our integrated Laguerre estimator (plain blue), Nelson-Aalen estimator (dashed red), -ln( ŜKM (x)) (plain black) and parametric cumulative hazard curves : exponential (orange plain), log-logistic (cyan dotted) and log-normal (dashed magenta)

Concluding remarks

Our study presents a generalization of risk bounds for nonparametric least-squares estimator of the hazard rate, which allows to consider non compactly supported bases. This is very useful to propose developments of the hazard rate in the Laguerre basis, which can be seen as a combination of gamma-type functions. We show that our new result encompasses the ones obtained in [START_REF] Plancade | Model selection for hazard rate estimation in presence of censoring[END_REF] or [START_REF] Comte | Adaptive estimation of the conditional intensity of marker-dependent counting processes[END_REF]. We show on simulations that the performance of the new estimator are comparable to or better than previous kernel or wavelet proposals and we also illustrate that it can be used successfully to analyze real data. Further simulation may certainly be conducted, in particular to improve numerical stability of the computation of Laguerre functions. Indeed in practice, we rescale the real data to a smaller range to avoid numerical problems with the Laguerre basis. A theoretical study including the choice of the range from the data may be conducted: adding a range parameter in the definition of the Laguerre basis is possible and a selection procedure for this parameter may be developed. We chose to compare our estimator to previous results, but sample sizes in these examples are sometimes quite small. We believe that such nonparametric method requires a rather large data set, and empirical experiments maybe be conducted to explored this aspect. Lastly, from theoretical point of view, the model selection procedure has to be studied; it is beyond the scope of the present work, as it would require quite lengthy developments; therefore it is left for further work.

Ω m = ∀t ∈ S m , t 2 n t 2 S Z -1 ≤ 1 2 (16)
Λ m = Ψ -1 m,Z op ≤ c n log(n) (17)
The following Lemma provides preliminary results which are the main ingredients to bound the empirical risk and the integrated risk of one estimator. Lemma 6.1. Under the assumptions of Proposition 2.1,

P(Ω c m ) ≤ 2/n 4 and P(Λ c m ) ≤ 2/n 4 (i) Decomposition of the empirical risk. λm -λ 2 n = λm -λ 2 n 1 Ωm∩Λm + λm -λ 2 n 1 Ω c m ∩Λm + λ 2 n 1 Λ c m := T 1 + T 2 + T 3
We bound successively the expectation of the three terms.

(ii) Study of T 1 , main term in the decomposition. On Λ m , it holds that λm = λ m , and we get

T 1 = λ m -λ 2 n 1 Ωm∩Λm = λ m -Π m λ 2 n + inf t∈Sm t -λ 2 n 1 Ωm∩Λm with Π m λ the orthogonal projection for norm . n of λ on S m , that is Π m λ = m-1 j=0 b j ϕ j is such that λ -Π m λ, ϕ j n = 0, for j = 1, • • • , m. Taking the expectation, E[T 1 ] ≤ E λ m -Π m λ 2 n 1 Ωm∩Λm + E inf t∈Sm t -λ 2 n ≤ E λ m -Π m λ 2 n 1 Ωm∩Λm + inf t∈Sm t -λ 2 S Z (18)
and this corresponds to the classical variance/squared bias decomposition.

Let us bound the variance term. Recall that Φ m = (ϕ j (Z i )) 1≤i≤n,0≤j≤m-1 , and set,

v = 1 n t Φ m δ -( λ, ϕ j n ) 0≤j≤m-1
Note that E( v) = 0 and remember that λ -Π m λ, ϕ j n = 0 for j = 0, • • • , m -1 so that :

λ, ϕ j n = Π m λ, ϕ j n = m-1 k=0 b k ϕ k , ϕ j n with Π m λ = m-1 j=0 b j ϕ j .
Thus, we find that ( λ, ϕ j n

) 0≤j≤m-1 = Ψ m,Z b (m) with b (m) = t (b 0 , • • • , b m-1 ) and we can write Ψ -1 m,Z v = â(m) -b (m) . Then, λ m -Π m λ 2 n = t ( â(m) -b (m) ) Ψ m,Z ( â(m) -b (m) ) = t ( Ψ -1 m,Z v) Ψ m,Z Ψ -1 m,Z v = t v Ψ -1 m,Z v = t v Ψ -1/2 m,Z Ψ 1/2 m,Z Ψ -1 m,Z Ψ 1/2 m,Z Ψ -1/2 m,Z v ≤ λ max (Ψ 1/2 m,Z Ψ -1 m,Z Ψ 1/2 m,Z ) t vΨ -1 m,Z v
But on Ω m , we have λ max (Ψ

1/2 m,Z Ψ -1 m,Z Ψ 1/2 m,Z ) ≤ 2 since Ψ -1/2 m,Z Ψ m,Z Ψ -1/2 m,Z -Id m op ≤ 1/2 and we get (19) E λ m -Π m λ 2 n 1 Ωm∩Λm ≤ 2 E[ t v Ψ -1 m,Z v]
The study of the variance term will be complete as soon as we have computed

E[ t v Ψ -1 m,Z v]. E[ t v Ψ -1 m,Z v] = E   j,k v j v k [Ψ -1 m,Z ] j,k   with v j = (1/n) n i=1 δ i ϕ j (Z i ) -λ(x)ϕ j (x)1 {Z i >x} dx the j-th coordinate of v and E[v j ] = 0. Note also that E δ 1 ϕ j (Z 1 ) -λ(x)ϕ j (x)1 {Z 1 >x} dx δ 1 ϕ k (Z 1 ) -λ(x)ϕ k (x)1 {Z 1 >x} dx = E(δ 1 ϕ j (Z 1 )ϕ k (Z 1 )) -λ(x)ϕ k (x)E(δ 1 ϕ j (Z 1 )1 {Z 1 >x} )dx -λ(x)ϕ j (x)E(δ 1 ϕ k (Z 1 )1 {Z 1 >x} )dx + λ(x)λ(y)S Z (x ∨ y)ϕ j (x)ϕ k (y)dxdy (20) = E(δ 1 ϕ j (Z 1 )ϕ k (Z 1 )). Indeed λ(x)ϕ k (x)E(δ 1 ϕ j (Z 1 )1 {Z 1 >x} )dx = λ(x)ϕ k (x)E(S C (X 1 )ϕ j (X 1 )1 {X 1 >x} )dx = λ(x)ϕ k (x) ϕ j (y)S C (y)f (y)1 {y>x} dydx = λ(x)λ(y)S Z (y)1 {y>x} ϕ j (x)ϕ k (y)dxdy
so that the sum of the two middle terms in [START_REF] Ramlau-Hansen | Smoothing counting process intensities by means of kernel functions[END_REF] cancel with the last one. Then,

E[ t v Ψ -1 m,Z v] = j,k [Ψ -1 m,Z ] j,k E [v j v k ] = j,k [Ψ -1 m,Z ] j,k 1 n 2 n i=1 E [δ i ϕ j (Z i )ϕ k (Z i )] = 1 n j,k [Ψ -1 m,Z ] j,k E [δ 1 ϕ j (Z 1 )ϕ k (Z 1 )] = 1 n j,k [Ψ -1 m,Z ] j,k ϕ j (x)ϕ k (x)f (x)S C (x)dx = 1 n j,k [Ψ -1 m,Z ] j,k ϕ j (x)ϕ k (x)λ(x)S Z (x)dx
Finally, with Ψ m,λS Z defined by [START_REF] Gàmiz | Double one-sided crossvalidation of local linear hazards[END_REF], we can see that:

(21) E[ t v Ψ -1 m,Z v] = 1 n j,k [Ψ -1 m,Z ] j,k [Ψ m,λS Z ] j,k = 1 n Tr(Ψ -1 m,Z Ψ m,λS Z ).
Thus, plugging this in ( 19) yields

(22) E λ m -Π m λ 2 n 1 Ωm∩Λm ≤ 2 n Tr(Ψ -1 m,Z Ψ m,λS Z )
and with [START_REF] Patil | Kernel based estimation of ratio functions[END_REF], we have

E(T 1 ) ≤ inf t∈Sm t -λ 2 S Z + 2 n Tr(Ψ -1 m,Z Ψ m,λS Z ).
(iii) Residual terms.

T 2 = λm -λ 2 n 1 Ω c m ∩Λm ≤ 2 λ m 2 n 1 1 Ω c m ∩Λm + 2 λ 2 n 1 Ω c m
We write

λ m 2 n = t â(m) Ψ m,Z â(m) = 1 n 2 t δ Φ m Ψ -1 m,Z Ψ m,Z Ψ -1 m,Z Φ m δ ≤ 1 n 2 Ψ -1 m,Z op t δ Φ t m Φ m δ = 1 n 2 Ψ -1 m,Z 2 op m-1 j=0 n i=1 δ i ϕ j (Z i ) 2 ≤ 1 n 2 Ψ -1 m,Z op m-1 j=0 n i=1 δ 2 i n i=1 ϕ 2 j (Z i ) (Cauchy-Schwarz) ≤ 1 n 2 Ψ -1 m,Z op   m-1 j=0 ϕ 2 j (Z i )   × n 2 ≤ Ψ -1 m,Z op L(m) ≤ cL(m) n log(n) ,
since on Λ m , Ψ -1 m,Z op ≤ cn/ log(n). Thus, we obtain, by using Lemma 6.1 and

L(m) ≤ n, that E[ λ m 2 n 1 1 Ω c m ∩Λm ] ≤ cL(m) n log(n) P(Ω c m ) ≤ c n .
Second,

E[ λ 2 n 1 Ω c m ] ≤ E[ λ 4 n ] 1/2 P(Ω c m ) with E[ λ 4 n ] = 1 n 2 i,k λ 2 (x)λ 2 (y)E(1 Z i >x 1 Z k >y )dxdy ≤ 1 n 2 i,k λ 2 (x)λ 2 (y)E 1/2 (1 Z i >x )E 1/2 (1 Z k >y )dxdy = λ 2 (x) S Z (x)dx 2 < +∞
by using assumption [START_REF] Comte | Adaptive estimation of the conditional intensity of marker-dependent counting processes[END_REF]. Finally,

E[T 3 ] = E[ λ 2 n 1 Λ c m ] ≤ E[ λ 4 n ]
1/2 P(Λ c m ) ≤ c/n by using Lemma 6.1. 2 6.2. Proof of Lemma 6.1. First, note that

P(Ω c m ) = P sup t∈Sm, t S Z =1 1 n n i=1 t 2 (x)1 Z i >x dx -1 > 1 2
and then sup

t∈Sm, t S Z =1 1 n n i=1 t 2 (x)1 Z i >x dx -1 = Ψ -1/2 m,Z ( Ψ m,Z -Id m )Ψ -1/2 m,Z op .
To apply Matrix Chernoff Inequality given in Theorem 1.1 of Tropp (2012), we denote by

K m (Z i ) = Ψ -1/2 m,Z ϕ j (x)ϕ k (x)1 Z i >x dx j,k Ψ -1/2 m,Z , for i = 1, • • • , n. We have Ψ -1/2 m,Z Ψ m,Z Ψ -1/2 m,Z = 1 n n i=1 K m (Z i ), and
E n i=1 K m (Z i ) = nId m .
This yields that µ min = µ max = n in Tropp's notations. Moreover

λ max (K m (Z i )) = sup x 2,m =1, x∈R m t (Ψ -1/2 m,Z x)( ϕ j (z)ϕ k (z)1 Z i >z dz) j,k (Ψ -1/2 m,Z x)
and setting y = Ψ -1/2 m,Z x,

t x K m (Z i ) x =   m-1 j=0 y j ϕ j (z)   2 1 Z i >z dz ≤   m-1 j=0 y j ϕ j (z)   2 dz = y 2 2,m .
So,

λ max (K m (Z i )) ≤ sup x 2,m =1, x∈R m Ψ -1/2 m,Z x 2 2,m = Ψ -1 m,Z op .
Therefore in Tropp (2012)'s notation, we get R = Ψ -1 m,Z op . Now, applying Matrix Chernoff Inequality as stated in Tropp (2012) (Theorem 1.1), we get

P(Ω c m ) ≤ 2m exp -c(1/2) n Ψ -1 m,Z op provided that Ψ m,Z is invertible and with c(u) = u + (1 -u) log(1 -u) for 0 < u < 1.
Under condition [START_REF] Diehl | Kernel density and hazard function estimation in the presence of censoring[END_REF], as c(1/2) = (3 log(3/2) -1)/2, we obtain

P(Ω c m ) ≤ 2m exp(-5 log(n)) ≤ 2 n 4 ,
which is our first statement. Now, we turn to

P(Λ c m ). Under (9), Ψ -1 m,Z op ≤ (c/2)(n/ log(n)) and on Λ c m , Ψ -1 m,Z op > c(n/ log(n)). So, c n log(n) < Ψ -1 m,Z op ≤ Ψ -1 m,Z -Ψ -1 m,Z op + Ψ -1 m,Z op ≤ Ψ -1 m,Z -Ψ -1 m,Z op + c 2 n log(n) and thus Ψ -1 m,Z -Ψ -1 m,Z op > c 2 n log(n)
.

At the end

Ψ -1 m,Z -Ψ -1 m,Z op > Ψ -1 m,Z op . Therefore, under (9), Λ c m ⊂ Ψ -1 m,Z -Ψ -1 m,Z op > Ψ -1 m,Z
-λ 2 S Z = λm -λ 2 S Z 1 Ωm∩Λm + λm -λ 2 S Z 1 Ω c m ∩Λm + λ 2 S Z 1 Λ c m := T 1 + T 2 + T 3
We bound successively the expectation of the three terms.

Clearly,

E( T 3 ) = λ 2 S Z P(Λ c m ) ≤ c/n.
For T 2 , we write

T 2 ≤ 2( λm 2 S Z + λ 2 S Z )1 Ω c m ∩Λm . Obviously, E( λ 2 S Z 1 Ω c m ∩Λm ) ≤ λ 2 S Z P(Ω c m ) ≤ c/n.
For the other term, we note that

Ψ m,Z op = λ max (Ψ m,Z ) = sup t x x=1 t x Ψ m,Z x = sup t x x=1   m-1 j=0 x j ϕ j (u)   2 S Z (u)du ≤ sup t x x=1   m-1 j=0 x j ϕ j (u)   2 du = sup t x x=1 m-1 j=0 x 2 j = 1.
Then we write as for λm

2 n previously, λ m 2 S Z = t ( â(m) )Ψ m,Z â(m) ≤ Ψ m,Z op t ( â(m) ) â(m) ≤ 1 n 2 t δ Φ m Ψ -2 m,Z t Φ m δ ≤ 1 n 2 Ψ -2 m,Z op t δ Φ m t Φ m δ = 1 n 2 Ψ -2 m,Z 2 op m-1 j=0 n i=1 δ i ϕ j (Z i ) 2 ≤ Ψ -2 m,Z op L(m) ≤ c 2 L(m) n log(n) 2 , since on Λ m , Ψ -1 m,Z op ≤ cn/ log(n). Thus, we obtain E[ λ m 2 S Z 1 1 Ω c m ∩Λm ] ≤ c 2 L(m) n log(n) 2 P(Ω c m ) ≤ c n
under assumption [START_REF] Comte | Nonparametric Estimation[END_REF] and L(m) ≤ n. As a consequence, E( T 2 ) ≤ c/n.

To study T 1 , we introduce λ

(S Z ) m
the orthogonal projection on S m of λ w.r.t. the scalar product weighted by S Z and g := λ -λ

(S Z ) m . We write λ m -λ A 2 S Z = λ m -Π m λ + Π m λ -λ 2 S Z and λ -Π m λ = λ -λ (S Z ) m -Π m (λ -λ (S Z ) m ) = g -Π m g, as Π m λ (S Z ) m = λ (S Z ) m . Thus λ m -λ A 2 S Z = λ m -Π m λ + Π m g -g 2 S Z = λ m -Π m λ + Π m g 2 S Z + g 2 S Z as g is orthogonal in L 2 (A, S Z (x)dx) to any function in S m . Therefore E( T 1 ) ≤ g 2 S Z + 2E( Π m g 2 S Z 1 Ωm∩Λm ) + 2E λ m -Π m λ 2 S Z 1 Ωm∩Λm ) ≤ inf t∈Sm t -λ 2 S Z + 2E( Π m g 2 S Z 1 Ωm∩Λm ) + 4E λ m -Π m λ 2 n 1 Ωm∩Λm , by using that λ m -Π m λ 2 S Z ≤ 2 λ m -Π m λ 2 n on Ω m (
E( Π m g 2 S Z 1 Ωm∩Λm ) ≤ 4 c log(n) g 2 S Z = 4 c log(n) inf t∈Sm t -λ 2 S Z .
Thus, we get

E( T 1 ) ≤ (1 + 8 c log(n) ) inf t∈Sm t -λ 2 S Z + 8 n Tr(Ψ -1 m,Z Ψ m,λS Z ),
and we obtain the bound [START_REF] Huber | Lower bounds for estimating a hazard[END_REF]. 2

Proof of Lemma 6.2. Let ( φj ) 0≤j≤m-1 be an orthonormal basis w.r.t.

S Z scalar prod- uct. If φj = m-1 k=0 α j,k ϕ k and A m = (α j,k ) 0≤j,k≤m-1 , then Id m = ( φj φk S Z ) j,k = t A m Ψ m,Z A m so that A m = Ψ -1/2 m,Z . Let G m = ( φj , φk n ) j,k = t A Ψ m,Z A m = Ψ -1/2 m,Z Ψ m,Z Ψ -1/2 m,Z . Therefore, on Ω m , G -1 m op ≤ 2, as G m -Id m op ≤ 1/2. Now if Π m g = m-1 j=0 β j φj , as g -Π m g, φj n = 0 for j = 0, 1, . . . , m -1, we get g, φj n = Π m g, φj n = m-1 k=0 β k φk , φj n so that G m β m = ( g, φj n ) 0≤j≤m-1 := d m . Therefore (23) Π m g 2 S Z = β m 2 2,m = G -1 m d m 2 2,m ≤ G -1 m 2 op d m 2 2,m ≤ 4 m-1 j=0 g, φj 2
n .

Now, we recall that g, φj We remark that, for t = m-1 j=0 a j ϕ j , then ν n (t) = m-1 j=0 a j v j , where v j = ν n (ϕ j ). Moreover, This implies that for fixed m, the trace term is increasing with m if the S m 's are nested (and this increasing with m in the inclusion sense). 2 x j ϕ j (u)

  2 S Z (u)du = λ A ∞ x Ψ m,Z x.
This implies Tr(Ψ -1 m,Z Ψ m,λS Z ) ≤ λ A ∞ Tr(Id m ) = m λ A ∞ . Inequality (iii) can be obtained from [START_REF] Uzunoḡullari | A comparison of hazard rate estimators for left truncated and right censored data[END_REF], as t 2 S Z = 1 ≥ S 0 t 2 , so that

E sup t∈Sm, t S Z =1 [ν n (t)] 2 ≤ 1 S 0 E( t v v) = 1 nS 0 m-1 j=0 E(δ 1 ϕ 2 j (X 1 ))
by using previous computations of E(v 2 j ). We get that Tr(Ψ -1 m,Z Ψ m,λS Z ) ≤ L(m)/S 0 ≤ c 2 ϕ m/S 0 . 2 6.6. Proof of Lemma 2.3. Let u = (u 0 , u 1 , . . . , u m-1 ) be a vector such that Ψ m,Z u = 0. Then u Ψ mZ u = 0 = t 2 S Z for t = m-1 j=0 u j ϕ j . This implies, under λ(R + ∩Supp(S Z )) > 0, that the function t is null on a set with positive Lebesgue measure. Therefore, x → P (x) = t(x)e x also; as P is a polynomials of degree m -1 with an infinity of zeros, it is null and thus u j = 0, for j = 0, . . . , m -1.

No we turn to the lower bound on Ψ -1 m,Z . First, following the line of the proof of Lemma 8.2 in Comte and Genon-Catalot (2018), we get that, if E(Z) < 0, there exists a constant c 0 > 0 such that +∞ 0 ϕ 2 j (x)dx ≤ c 0 / √ j. This is due to the fact that s Z is bounded, +∞ 0 u -1/2 S Z (u)du = E( √ Z) and +∞ 0 S Z (u)du = E(Z). Then, the conclusion follows as in the proof of Proposition 8 in Comte and Genon-Catalot (2018). 2.

2 j

 2 (x) < +∞ For most bases, we have L(m) ≤ c 2 ϕ m where c ϕ is a constant depending on the bases (see examples in Sections 2.3 and 2.4 below).

  (a) and (b) have been studied by Antoniadis et al. (1999) (wavelet estimator with selection of the coefficients by cross-validation), Reynaud-Bouret (2006) (histogram and Fourier estimators of the Aalen intensity) and Brunel and Comte (2005) (two ratio strategies for projection estimators on compact support). Antoniadis et al. (1999) and Brunel and Comte (2005) estimate both the subdensity f S C and the hazard rate λ, whereas Reynaud-Bouret (2006) estimates λ only.

Figure 2 .Figure 3 .

 23 Figure 2. Collection of projection estimators with Laguerre basis : m = 1 (solid black), m = 2 (dashed red), m = 3 (dotted green), m = 4 (dotdashed blue), m opt = 5 (plain blue), m = 6 (twodashed magenta)

Figure 4 .

 4 Figure 4. Our integrated Laguerre estimator (plain blue), Nelson-Aalen estimator (dashed red), -ln( ŜKM (x)) (plain black) and parametric cumulative hazard curves : exponential (orange plain), log-logistic (cyan dotted) and log-normal (dashed magenta)

6 . Proofs 6 . 1 .

 661 Proof of Proposition 2.1. Two sets are of interest in the sequel:

1 2 n 1 1 2 = E A m u 2 2 2 op≤This is the announced result. 2 6. 4 .

 121122224 {Z i >x} g(x)dx 2 and E φj (x)1 {Z i >x} g(x)dx = φj (x)g(x)S Z (x)dx = φj , g S Z = 0 as g, ϕ j S Z = 0. ThusE[ g, φj {Z i >x} g(x)dx = 1 n Var φj (x)1 {Z 1 >x} g(x)dx and {Z 1 >x} g(x)dx ,m where u = ( ϕ j (x)1 {Z 1 >x} g(x)dx) 0≤j≤m-1 . As A m Ψ -1 m,Z op E Proj Sm (g1 {Z 1 >x} 2 ≤ Ψ -1 m,Z op E g 2 (x)1 {Z 1 >x} dx . Proof of Lemma 2.1. The proof relies on the notations and computations of the proof of Proposition 2.1. Let ν n (t) = 1 n n i=1 δ i t(X i ) -λ(x)t(x)1 Z i >x dx .

  sup t∈Sm, t S Z =1 [ν n (t)] 2 = sup t aΨ m,Z a=1 ] ,k v j v = t vΨ -1 m,Z v. Therefore, it follows from (21) in the proof of Proposition 2.2, that (24) E sup t∈Sm, t S Z =1 [ν n (t)] 2 = E( t vΨ -1 m,Z v) = Tr(Ψ -1 m,Z Ψ m,λS Z ) n .

6. 5 .(x(x

 5 Proof of Lemma 2.2. For (i), it follows from the remark: forx ∈ R m such that x 2,m = 1, x Ψ m,Z x = A j ϕ j (u)) 2 S Z (u)du ≥ S 0 A j ϕ j (u)) 2 du = S 0 . Now the trace is nonnegative since Tr(Ψ -1 m,Z Ψ m,λS Z ) = Tr(Ψ -1/2 m,Z Ψ m,λS Z Ψ -1/2 m,Z ) where Ψ -1/2 m,Z is a symmetric square root of Ψ -1 m,Z . As the matrix Ψ -1/2 m,Z Ψ m,λS Z Ψ -1/2 m,Z is nonnegative (x Ψ -1/2 m,Z Ψ m,λS Z Ψ -1/2m,Z x ≥ 0 for all m-dimensional vector x), we get the result.To prove (ii), let ε 0 , . . . , ε m-1 be independent centered random variables with unit variance and write:Tr(Ψ -1 m,Z Ψ m,λS Z ) = Tr(Ψ

  2 i,j = Tr(M t M ) where Tr denotes the trace. 2.1. Definition of the estimator. The following contrast has been considered in Comte et al. (2011) and in Plancade (2011). Let s, t : A → R be two square integrable functions from A ⊆ R

+ into R and (2)

  holds. It is also fulfilled in the examples listed above. Assume that Ψ m,Z is invertible, that conditions (

	regression entails a different constraint, namely m Ψ -1 m op ≤ (c/2)(n/ log(n)) where Ψ m =
	E( Ψ m ) and Ψ m is defined in Remark 2.1.
	Following ideas developped in Cohen et al. (2013), we can obtain a risk bound on the
	integrated weighted risk, with coefficient in front of the squared bias nearly 1 for large n.
	Proposition 2.2.
	Condition (9) corresponds to what Cohen et al. (2013) call a stability condition in the
	classical regression setting described in Remark 2.1. It is expressed in function of similar
	matrices in Comte and Genon-Catalot (2018), but is is noteworthy that the standard

  -Bouret (2006)'s results, those of Antoniadis et al. (1999) and those of Brunel and Comte (2005) are recalled in Table 1, while ours are given in Table 2.

Table 1 .

 1 Results of Antoniadis et al. (1999, Table 2), of the Fourier strategy in Reynaud-Bouret (2006) and of the ratio strategy of Brunel and Comte (2005), for the estimation of λ, T = 200 replications

			Estimator of			Estimator of			Estimator of
			Antoniadis et al.		Reynaud-Bouret		Brunel-Comte
	Model	Gamma	Bimodal	Gamma	Bimodal	Gamma	Bimodal
	n	200	500	200	500	200	500	200	500	200	500	200	500
	10 MSE	1.12 0.995 20.80 19.70 0.55 0.579 12.59 11.22 0.857 0.900 9.02 7.06
	10 MSE2 0.025 0.016 0.48 0.32 0.032 0.012 1.50 0.51 0.023 0.013 1.068 0.408
						λ m				λ m (larger n)
			Model		Gamma	Bimodal	Gamma	Bimodal
			n =	200	500	200	500	1000	2000 1000 2000
			10 MSE 0.275 0.084 6.287 4.87 0.032 0.019 3.726 3.069
			10 MSE2 0.035 0.014 1.268 0.969 0.0067 0.0035 0.697 0.557

Table 2 .

 2 Results for hazard-regression estimators of λ with Laguerre basis, T = 200 replications, κ = 2. • Comparison with Bouezmarni et al. (2011) and Müller and Wang (1994). Bouezmarni et al. (2011

Table 3

 3

	250

. 100*MSE (100*variance of ISE), comparison with Bouezmarni et al. (2011) and Müller and Wang (1994)

Weibull with scale parameter α = 2 and shape parameter λ = π, a χ 2 (2) and a χ 2 (3) density. Here again, the sample size is n = 100 and is rather small for nonparametric estimation. However, we can see that our estimator performs analogously to Barbeito and Cao (2018)'s.

Table 4 .

 4 Comparison withBarbeito and Cao (2018), n = 100, 500 replications.

  op .

	It follows from Proposition 2.4 (ii) in Comte and Genon-Catalot (2018) that the last set
	is a subset of Ω c m . Thus P(Λ c m ) ≤ P(Ω c m ) ≤ 2/n 4 . 2
	6.3. Proof of Proposition 2.2. We start with a risk decomposition of the same type as
	in empirical case
	λm

  all terms are in S m ). For this last term, we can use the bound obtained w.r.t the empirical norm given by 2Tr(Ψ -1 m,Z Ψ m,λS Z )/n, see[START_REF] Plancade | Model selection for hazard rate estimation in presence of censoring[END_REF]. Now we have the following Lemma, inspired from Cohen et al. (2013, 2019): Lemma 6.2. Under the assumptions of Proposition 2.2,