Marie-Odile Soyer-Gobillard

Laura Gaspari

Charles Sultan

Evidence for link between mental disorders and in utero exposure to synthetic hormones: A long and crucial history

Keywords: Synthetic estrogens, diethylstilbestrol, ethinylestradiol, synthetic progestins, mental disorders, in utero exposure

The prenatal brain develops under the influence of the endogenous hormonal milieu (fetal, gonadal, adrenal, placental, maternal) and exogenous substances with hormonal activity. Somatic effects of diethylstilbestrol (DES) (genital malformations, infertility, cancers) on children exposed in utero have long been recognized. This is not the case for psychiatric disorders. Animal studies provide persuasive evidence of somatic and behavioral disorders.

Recent studies have reported psychiatric effects of synthetic estrogens on the brains of children exposed in utero, including mental disorders affecting the exposed children (psychotic disorders, schizophrenia, bipolar disorders, depression, eating disorders, suicides and suicide attempts), and have shown the epigenetic mechanism of the DES effect-a specific methylation of two genes playing important rôles in neuro-development: the ADAM TS9 and the ZFP 57 genes. The ADAM TS9 gene is involved in the control of the form of reproductive organs during development, the development of certain cancers; and (crucially) in controlling the development of the Central Nervous System ; methylation of ZFP 57 gene (located on chromosome 6), that is expressed very early in development as a transcription regulator, is highly suggesting to be associated with psychosis.

Progestins used in contraception and in hormone replacement therapy during menopause are known to affect the adult brain but no data on children existed before our recent paper on their effects after in utero exposure. Clinical data were collected from 1,200 families (1,934 children) of the HHORAGES cohort using a detailed medical questionnaire. Most families had children exposed to estrogens with or without progestin but 46 mothers (115 children) had at least one child exposed to only progestin. Our data show the presence of somatic disorders and a drastic increase of psychiatric disorders among children in utero exposed to progestin(s) (79,03 % vs 0% in unexposed first born and in children born after a previous maternal exposure). This result strongly suggests that synthetic progestin treatment during pregnancy may be detrimental for the development of children's brains. Mental disorders linked with in utero exposure to synthetic progestins are the same as pathologies provoked by in utero exposure to synthetic estrogens.

At the end of the 1990s, somewhere in France, an agricultural engineer, Mr. R.A. made the observation that his three children were suffering from various psychic and somatic pathologies, the elder suffering from bilateral cryptorchidism, micropenis, infertility due to azoospermia (no spermatozoa) and schizotypal character, the second of anxiety, depression and eating disorders (anorexia) coupled with small uterus and ovaries, and the third suffering from schizophrenia and severe depression associated with suicide attempts. He observed also the relationship with the fact that in all three cases his wife received medical treatment consisting of a synthetic hormone cocktail : Diethylstilbestrol (DES), Ethinyl Estradiol (EE), plus synthetic progestin delay during her three pregnancies after a previous miscarriage. He conducted research in the world literature on the subject, and came to the conclusion that not only was one of these products, Diethylstilbestrol (DES) already known for its misdeeds, but that it continued to be administered in France until 1977/82 : the product, inexpensive to make, was not patented and was manufactured and distributed by many pharmaceutical laboratories. The same goes for EE, which was banned for pregnant women in 1980, but remains the best-selling estrogen in the world because it is part of the contraceptive pill.

In 1998, one of us (MOS-G), concerned by the same kind of problems in her two children who were exposed in utero to the same cocktail of synthetic estrogens, lost her two children after psychiatric illnesses. Following the reading of a "Call to Families" printed in a newspaper, she met Mr. R. A., who had gathered around him about twenty French families concerned with their in utero exposed children suffering from psychiatric illnesses. He wanted to expose his observations and the results of his bibliographic researches at a meeting of patient families collecting observations on the genital malformations of girls exposed in utero to DES. Alas, he was condemned, rejected and disclaimed by doctors, mostly gynaecologists and psychiatrists as well as by associative members. They denied the existence of psychiatric disorders in exposed boys and girls. Discouraged, Mr R.A. died shortly afterwards. In order to continue his work and regroup the families concerned by the origin of the heavy psychiatric pathologies of their children, and despite the taboo surrounding such diseases as psychoses, we gathered several mothers concerned and created the Association of Patients Hhorages (Halt to Artificial Hormones for Pregnancies), in 2002. This Association which collected more than 1300 French spontaneous testimonies is now registered with the Epidemiology Portal of INSERM (French National Institute for Medical Research) as a French Health Database.

Despite various alerts published in the 1940s, after work on animals proving in particular its carcinogenic effect, and despite the work of Dieckmann et al. [START_REF] Dieckman | Does the administration of diethylstilbestrol during pregnancy have therapeutic value?[END_REF], initiated as early as 1953, demonstrating in a large cohort of pregnant women given Diethylstilbetrol (DES), a synthetic estrogen, versus placebo that the drug was inefficient in preventing miscarriages or premature births, this product has been widely distributed around the world, sowing a long list of misdeeds. After the discovery of cervico-vaginal cancers called "Clear Cells Adenoma (CCAD)" [START_REF] Herbst | Adenocarcinoma of the vagina. Association of maternal stilbestrol therapy with tumor appearance in young women[END_REF] in the "DES girls", DES was banned in the United States for pregnant women in 1971 but only in 1977 in France, where this recommendation disappeared from the "French Vidal book", but DES continued to be prescribed sporadically until 1982. Meanwhile another synthetic estrogen, steroidal, also synthesized on 1938, 17-alpha-ethinyl estradiol (EE) was often added to DES as a cocktail or later as a replacement, sometimes with the addition of synthetic-delay progestin. The idea that prevailed at the time was that women had a hormonal deficit that triggered a miscarriage, whereas now we know that the miscarriage itself causes this deficiency. These products were prescribed not only to women who had miscarriages, but also in comfort (« to have beautiful babies », according to an advertising), or even as a "morning after pill" or to cut milk after childbirth. DES and 17-alpha-EE, although belonging to different estrogenic and degrading categories, are, however, bound to the same ER beta-estrogen receptors.

Behavioral disorders demonstrated in animals (rats) exposed in utero [3-8].

Animal studies (on mice and rats) have, demonstrated the toxicity of these synthetic estrogens on the offspring, including the causing of behavioral disorders. Palanza et al. [START_REF] Palanza | Prenatal exposure to endocrine disrupting chemicals: effects on behavioral development[END_REF] demonstrated in particular that prenatal exposure to three different synthetic chemicals, DES and two pesticides, DDT and methoxychlor its analogue, affects the behavior of young suckling mice, showing increased aggression in males (increased numbers of attacks and decreased reaction time before the attack (Fig. 1). Doses of DES were 1000 times less than those of DDT, and caused much larger aggression responses, demonstrating the considerable effect of DES at very low doses. The treatment period for rodent mothers from day 11 to day 17 of pregnancy was also critical because it represents a key period in the differentiation of the reproductive system and brain development in these rodents in the early stages of pregnancy. Moreover, injection of 17-alpha-Estradiol (EE) in pregnant rats causes not only many abortions in mothers but also anxiety and depression disorders in offspring [START_REF] Dugard | Prenatal exposure to ethinylestradiol elicits behavioural abnormalities in the rat[END_REF][START_REF] Arabo | Administration of 17-alphaethinylestradiol during pregnancy elicits modifications of maternal behaviour and emotional alteration of the offspring in the rat[END_REF], the synthetic hormone having been administered at the same relative doses as in humans (15 g / kg-1 per day versus 19 g / kg-1 per day). At the cytological level of the brain, an alteration of the anterior part of the hippocampus in young rats exposed to EE in utero has been demonstrated in 2004 [START_REF] Sandner | Prenatal exposure of Long-Evans rats to 17alpha-ethinylestradiol modifies neither latent inhibition nor prepulse inhibition of the statle reflex but elicits minor deficiency in exploratory behaviour[END_REF]. The hippocampus is indeed a part of the brain that contains many estrogen receptors during the prenatal period. Ogiue-Ikeda et al [START_REF] Ogiue-Ikeda | Rapid modulation of synaptic plasticity by estrogens as well as endocrine disrupters in hippocampal neurons[END_REF] showed on 2008 that synaptic plasticity can be upset by estrogens or other Endocrine Disruptors (EDs). Later and unequivocally, Newbold demonstrated the validity of the rodent model transposed to humans [START_REF] Newbold | Lessons learned from perinatal exposure to diethylstilbestrol[END_REF].

Behavioral disorders, psychoses and depressions demonstrated in humans after in utero exposure to DES / EE.

In humans, the work concerning the appearance of behavioral disorders in children after in utero exposure to synthetic hormones is less numerous, but as early as 1977, June Reinisch [START_REF] Reinisch | Prenatal exposure of human foetuses to synthetic progestin and oestrogen affects personality[END_REF] published in Nature that prenatal exposure to estrogen and / or synthetic progestins could affect the personality of exposed children. More recently, in 2012, Kebir and Krebs [START_REF] Kebir | Diethylstilbestrol and risk of psychiatric disorders: A critical review and new insights[END_REF] have analyzed several epidemiological studies concerning the effects of DES on exposed children in utero and the occurrence of psychiatric disorders in these children. Their analysis shows that only three large epidemiological studies on the effects of DES were performed in 1952-53 (followed-up in 1983); in 2007 and in 2010. The first study (double-blinded) that supports the hypothesis of a link between psychiatric disorders and prenatal exposure to DES was performed by Vessey et al, in 1983 (11) from a clinical trial that had been performed thirty years earlier in 1953 in London by Dieckmann et al, on 700 women treated with DES versus 700 subjected to a placebo [START_REF] Dieckman | Does the administration of diethylstilbestrol during pregnancy have therapeutic value?[END_REF]. A doubling of depression and anxiety disorders has been demonstrated in the population exposed in utero. The second, published in 2007, and conducted by Verdoux et al. [START_REF] Verdoux | Serious psychiatric outcome of subjects prenatally exposed to diethylstilbestrol in the E3N cohort study[END_REF] from a cohort of women from the Mutuelle (Health) de l'Education Nationale (MGEN) concludes that there are no significant links between exposure to DES, suicides and / or psychiatric consultations or hospitalizations. A detailed analysis later, however, revealed a number of biases in this study [START_REF] Soyer-Gobillard | Behavioral and Somatic Disorders in Children exposed in utero to Synthetic Hormones : a Testimony-Case Study in a French Family Troop[END_REF]. The most recent Nurses Health Study was conducted by O'Reilly et al. 2010 [START_REF] O'reilly | Diethylstilbestrolexposure in utero and depression in women[END_REF] from 76,240 American women among whom 1,612 women were exposed to DES in utero. The statistical analysis shows that the later experienced an increase in depressive and anxiety disorders by a factor of 1.3. Kebir and Krebs [START_REF] Kebir | Diethylstilbestrol and risk of psychiatric disorders: A critical review and new insights[END_REF] emphasized the limitations of such epidemiological studies and noted in particular that, apart from depression and anxiety, other psychiatric disorders have not been studied. Post-adolescent behavioral disturbances reported for these two estrogens in exposed children were depression [START_REF] Ehrhardt | Psychopathology in prenatally DES-exposed females: current and lifetime adjustement[END_REF][START_REF] Pillard | Psychopathology and social functioning in men prenatally exposed to diethylstilbestrol (DES)[END_REF], anxiety [START_REF] Dieckman | Does the administration of diethylstilbestrol during pregnancy have therapeutic value?[END_REF][START_REF] Vessey | A randomized double-blind controlled trial of the value of stilboestrol therapy in pregnancy: long term follow-up of mothers and their offspring[END_REF][START_REF] Saunders | Physical and psychological problems associated with exposure to diethylstilbestrol (DES)[END_REF], schizophrenia-like [START_REF] Katz | Psychosis and prenatal exposure to diethylstilbestrol[END_REF][START_REF] Verdoux | Quelles sont les conséquences psychiatriques de l'exposition intra-utérine au diethylstilbestrol (DES)?[END_REF], anorexia, and bulimia nervosa [START_REF] Geary | The effect of estrogen on appetite[END_REF]. All these observations were synthesized by Pillard et al. [START_REF] Pillard | Psychopathology and social functioning in men prenatally exposed to diethylstilbestrol (DES)[END_REF] and Giusti [START_REF] Giusti | Diethylstilbestrol revisited: A review of the long term health effects[END_REF]. On 1987, Katz et al. [START_REF] Katz | Psychosis and prenatal exposure to diethylstilbestrol[END_REF] described the case of four male adults prenatally exposed to DES. It is in late adolescence that they develop psychotic disorders requiring neuroleptic treatment even though they have no family history of this type. He then hypothesizes that there may be a causal relationship between disruptions in neurodevelopment related to DES and the subsequent onset of psychotic disorders. Pillard et al. [START_REF] Pillard | Psychopathology and social functioning in men prenatally exposed to diethylstilbestrol (DES)[END_REF] showed that the frequency of recurrent major depressive episodes is significantly higher in the DES-exposed than in their unexposed siblings, which was confirmed in 2010 in the large cohort of DES girls by O'Reilly et al. [START_REF] O'reilly | Diethylstilbestrolexposure in utero and depression in women[END_REF].

Investigation of the causal link between exposure to these synthetic hormones in utero and severe psychotic disorders such as schizophrenia, bipolar disorders with or without eating disorders, schizoaffective disorders, occurring in post-adolescence in exposed children, was made possible thanks to the families of the Association Hhorages. Our database, constituted by these spontaneous families' testimonies, is based on responses to a detailed questionnaire, written by doctors and researchers, and accepted by the CNIL (French Center for the Protection of Data processing and Freedom). Our first global analysis (2004-2005 data) was based on 967 pregnancies from 470 mothers in collaboration with one of us (C.S.). The first results as well as the family questionnaires were detailed in 2012 in the chapter published in 2012 by Intech "Behavioral and somatic disorders in children exposed in utero to Synthetic Hormones" [START_REF] Soyer-Gobillard | Behavioral and Somatic Disorders in Children exposed in utero to Synthetic Hormones : a Testimony-Case Study in a French Family Troop[END_REF] in which we detailed somatic and psychiatric disorders, associated or not, in the exposed children of our cohort.

We have conducted a more recent analysis (2016) [START_REF] Soyer-Gobillard | Endocrine disruptors (ED) and psychiatric disorders in children exposed in utero: Evidence from a French cohort of 1002 prenatally exposed children and the example of diethylstilbestrol (DES) as a model for ED study[END_REF][START_REF] Soyer-Gobillard | Association between foetal DESexposure and psychiatric disorders in adolescence/adulthood: evidence from a French cohort of 1002 prenatally exposed children[END_REF] based on 1182 pregnancies from 529 mothers (Figure 2). Among the 740 (-20 stillborn) exposed children, 603 (exposed) +16 post-DES, (born without exposure but after a previous exposed pregnancy) are suffering from psychiatric disorders. The prevalence of the psychiatric disorders in comparison with the general population show a dramatic increase (Table 1). Prevalences of the psychological and/or psychiatric disorders in in utero exposed children to DES and/or EE and comparison with the general population [START_REF] Soyer-Gobillard | Association between foetal DESexposure and psychiatric disorders in adolescence/adulthood: evidence from a French cohort of 1002 prenatally exposed children[END_REF].

We were also interested by the effects on the brain of synthetic progestins on in uteroexposed children of our cohort (Figures 3,4). Currently, there is no research on this these effects of in utero exposure of children to progestins given alone during pregnancy. Our recent observations [START_REF] Soyer-Gobillard | Neurodevelopmental disorders in children exposed in utero to progestin treatment: study of a cohort of 95 children from the HHORAGES Association[END_REF][START_REF] Soyer-Gobillard | Neurodevelopmental disorders in children exposed in utero to synthetic Progestins: Analysis from the national cohort of the Hhorages Association[END_REF] were collected from 1200 families of the Hhorages cohort i.e.1934 children using always the same detailed questionnaire [START_REF] Soyer-Gobillard | Behavioral and Somatic Disorders in Children exposed in utero to Synthetic Hormones : a Testimony-Case Study in a French Family Troop[END_REF]. As previously shown, most families of our cohort had children exposed to estrogens or to estro-progestins, but only 46 families (115 children) had at least one child exposed to one or more progestins prescribed alone and representing 62 in utero-exposed children. 35 children were post-exposed (Figure 3). The prescribed progestins were : Among the 49 children affected by psychiatric disorders, 3 boys and 7 girls presented both somatic disorders in addition to psychiatric ones: -Boys (3): Hypospad (1), No urinary meatus (1), Bilateral cryptorchidia + sexual ambiguity (1); -Girls (7): Hormonal sterility (2), Hirsutism + enuresia (1), Enuresia (1), Hirsutism (1), Hermaphrodism (1, operated), Sexual ambiguity + Tight urethra (1). Among the 6 exposed children suffering from somatic disorders "only" we observed for boys (4): Mega bilateral ureter (1 grand-child), Unilateral cryptorchidia (1), Hypospadias with numerous interventions of reconstruction (1 grand-child), Sexual ambiguity (1); for girls (2): Hormonal sterility (2).

As shown in Figure 5, a comparison of synthetic estrogen and progestin exposures for girls and boys demonstrates that psychiatric disorders were of the same nature for progestin exposure as those observed after exposure to synthetic estrogens i.a. Behavioral disorders= 2, Eating disorders= 2, Schizophrenia= 29, Depression, bipolar disorders= 16, Suicides : Attempts : 7 series, Death : 1. The percentage of suicide attempts (11,29%) and death after suicide (1,6%) is proportionately lower after exposure to progestins than after exposure to synthetic estrogens.

An epigenetic mechanism.

Search for the molecular basis of the causal link between in utero-exposure to synthetic hormones and the appearance of psychoses as schizophrenia or bipolar disorder in children exposed in utero, has been achieved thanks to the partnership that unites Hhorages Patients' Association with the Inserm team of molecular psychiatrist Pr. M.-O. Krebs (St Anne Hospital, Paris, France, UMR S 894), which began in 2007. "It would have been" crazy "to miss the problem posed by the Association Hhorages to establish a causal link between taking artificial hormone (s) during pregnancy and appearance of psychiatric disorders of the type psychotic in exposed children, because diethylstilbestrol (Distilbene® or DES) has been given over a limited period of time and people who have taken this molecule are still there to testify. This is a case study that should not be missed " said Dr. Kebir (Center for Psychiatry and Neuroscience, UMR S 894), manager of these researches as part of the Krebs' team.

First, to document in utero exposure to synthetic estrogens Kebir and Krebs [START_REF] Kebir | Diethylstilbestrol and risk of psychiatric disorders: A critical review and new insights[END_REF] were able to analyze from our data a small number of family records that occurred in Hhorages testimonies, and studied a cohort of 472 exposed subjects. They account for 46.7% of mood disorders, 22.9% of psychotic disorders, 6.6% of anxiety disorders, 11% of eating disorders and 12.7% of others, which confirms their previous observations published on 2009 Second, genetic and epigenetic analyzes of Hhorages siblings have shown in patients suffering from psychotic disorders and exposed in utero to DES and / or EE that this prenatal exposure is associated with epigenetic processes. Starting from the fact that psychiatric diseases develop from a brain dysfunction during neuro development, and knowing that DES and EE are synthetic hormones (estrogens), endocrine disruptors, and confirming from the Hhorages data numerous cases of heavy psychiatric disorders in children exposed, the Krebs'team in association with Hhorages, designed ten yars ago, in 2007, a research project PICRI (Partnership Citizen Institution for Research and Innovation), funded by the Ile de France Region, net by the French National REsearch Agency (A.N.R.), that developed the hypothesis that the DES administered during pregnancies could be an environmental risk factor for the development of psychiatric disorders in impregnated children : the epigenome of the foetus could have been modified by in utero exposure to synthetic estrogens. The PICRI project was titled: "Influence of hormonal treatments on brain development during pregnancy: study of phenotypic, behavioral and biological changes in informative families". The families of Hhorages were called to perform peripheral blood sampling after thorough questioning. Many families volunteered to participate in the research: 31 families were selected, satisfying the rigorous inclusion criteria desired. Many more families had come forward during this study, but they could not be included because the psychotic patient refused to come to St. Anne's Hospital for blood sampling. In the selected families total siblings were composed of first born unexposed children, exposed and post exposed children, with first born unexposed serving as intrafamilial control. For the exploration of their epigenome, 485,000 cytosines by genome were studied and analysed, representing an immense work. To complete this study, a cohort of young adolescents with relational, emotional and social difficulties was followed for six months, some of whom had developed schizophrenic-type psychosis in these six months, although not exposed to DES. A comparison of their methylome, analyzed before and after the onset of the disease was performed. In this study [START_REF] Kebir | Methylomic changes during conversion to psychosis[END_REF], authors reported a global methylation of the psychotic patient genome.

After the analysis of the whole methylome of the selected Hhorages cohort, the team of Krebs-Kebir highlighted differential specific methylated regions (DMR) : in the zinc finger protein 57 gene ZFP57 and in the ADAM TS9gene, in young psychotic patients exposed in utero to DES/EE [START_REF] Rivollier | Methylomic changes in individuals with psychosis and prenatally exposed to Diethylstilbestrol[END_REF][START_REF] Rivollier | Methylomic changes in individuals with psychosis, prenatally exposed to endocrine disrupting compounds: Lessons from diethylstilbestrol[END_REF]. In this work, the authors observed that in exposed individuals, ZFP57 gene methylation may be associated with their psychosis. The ZFP57 gene (located on chromosome 6), is expressed very early in development. It is a transcription regulator, directly related to the phenomenon of methylation and neurodevelopment [START_REF] Quenneville | In embryonic stem cells, ZFP57/KAP1 recognize a methylated hexanucleotide to affect chromatin and DNA methylation of imprinting control regions[END_REF]. The ADAM TS9 gene is implicated in the control of organ shape, especially in the development and function of the uterus and reproductive organs [START_REF] Mittaz | Adamts-1 is essential for the development and function of the urogenital system[END_REF] which are often abnormal after in utero DES exposure as well as in the control of the CNS development [START_REF] Lemarchant | ADAMTS proteoglycanases in the physiological and pathological central nervous system[END_REF] and in several kinds of cancers [START_REF] Zhang | High-resolution melting analysis of ADAMTS9 methylation levels in gastric, colorectal, and pancreatic cancers[END_REF].

Discussion and conclusion

Very few studies have investigated the impact of prenatal exposure to DES, EE on psychiatric outcome. Animal studies on rats or mice allowed us to hypothesize that estrogenic hormones induce neurodevelopmental disturbances in exposed human subjects and may potentially mediate an increased risk of behavioral and psychiatric disorders. Our data therefore strongly suggest that DES/EE exposure during pregnancy is associated with high incidence of behavioral and/or psychiatric disorders. They illustrate a higher risk of schizophrenia, as this disease was 17 times more prevalent than in the general population, with sons being more affected than DES daughters. Regarding the existence of eating disorders (bulimia, anorexia), it should be noted that girls are much more affected than boys, and we often observed the association of eating disorders with bipolarity (manic-depressive disorders), anxiety and depression. With regard to suicides, our work clearly demonstrates a drastically increased risk of suicide attempts (65.4% versus 0 in the unexposed controls and 0.25% in general population) and suicides (3.4% versus 0 in the unexposed controls and 0.02% in general population). It could be noted that, as in the general population, DES sons commit more suicides than DES daughters, and the inverse for suicide attempts. Moreover, our data reveal that 50% of the sons who committed suicide suffered from schizophrenia. Psychiatric studies in general have shown that the percentage of suicides is generally higher in individuals with psychiatric disorders than in the general population. But to our knowledge, there is no information or specific studies concerning this association in the context of DES exposure. Sixteen subjects in Group 3 (post-DES children) (Figure 2, Table 1) had diagnosed psychiatric disorders. An explanation for this finding might be that DES, being a very lipophilic synthetic estrogen, remains in the mothers' fat after estrogenic impregnation in a previous pregnancy and is then released through the placental barrier during the next gestation.

No work had been reported on the impact of in utero exposure to synthetic progestin hormones administered alone on the occurrence of psychiatric disorders in exposed children before our first presentation in the European Congress of Gynaecology in 2017 [START_REF] Soyer-Gobillard | Neurodevelopmental disorders in children exposed in utero to progestin treatment: study of a cohort of 95 children from the HHORAGES Association[END_REF]. For the first time we described psychiatric disorders that can affect children exposed in utero to progestins. Previously, and during many years, synthetic progestogens were not considered as dangerous during pregnancy or during replacement or contraceptive treatment. Moreover, they have been suggested to exert neuroprotective effects in several animal models of neurological disease [START_REF] Melcangi | Levels and actions of progesterone and its metabolites in the nervous system during physiological and pathological conditions[END_REF]. Negative mood symptoms have been reported by Andreen et al. (36) in some women as a result of progesterone during the luteal phase of menstrual cycles. This is believed to be mediated via the action of allopregnegnolone on the GABA-A system. A reduction of allopregnanolone circulating levels that correlates to depressive symptoms has been recently reported [START_REF] Slopien | Correlation between allopregnanolone levels and depressive symptoms during late menopausal transition and early postmenopause[END_REF] and conversely, healthy women reported increased anxiety and mood disorders after long-acting subdermal implant of progestogens. In a group of 236 schizophrenic patients at onset, an elevated concentration of progesterone has been found, and authors suggested that steroid hormones may influence brain function, underlying schizophrenia and major depressive disorders [START_REF] Guest | Altered levels of circulating insulin and other neuroendocrine hormones associated with the onset of schizophrenia[END_REF]. Moreover, Buoli et al (2016) found high DHAS levels in patients with a history of psychotic symptoms, suggesting a role of steroids in the aetiology of psychosis and mood disorders [START_REF] Buoli | Sex Steroids and Major Psychoses: Which Role for DHEA-S and Progesterone?[END_REF]. Progestins are known to induce GABA receptor activity/neural activation before birth, it is likely that GABAergic system could contribute to schizophrenia, anxiety, depression, panic disorders, epilepsy, autism and others [START_REF] Schmidt | Neurodevelopment, GABA System dysfunction, and schizophrenia[END_REF]. Although some progestins have been banned from the market, others are not: our data demonstrated that caution should be taken with regard to the use of these progestins during pregnancy and even outside these periods (contraception or hormone replacement therapy).

The brain is a very vulnerable organ because its development covers a very broad period extending from the early prenatal stage (3rd week of pregnancy) to end around the age of 20. During its development, there are times when its vulnerability is even greater than others; these periods are called "shooting windows", during which the environment can impact the normal process of development. Abdomaleky et al [START_REF] Abdolmaleky | Methylomics in psychiatry: modulation of gene-environment interactions may be through DNA methylation[END_REF] as early as 2005 had developed the hypothesis that gene-environment modulations could be performed via DNA methylations. The Krebs' team put forward the hypothesis that DES-induced changes in epigenetic background and alteration of DNA functioning (methylations) could be significant factors to demonstrate a possible origin of psychotic disorders and a link with in utero DES exposure of the children suffuring from these illnesses [START_REF] Rivollier | Methylomic changes in individuals with psychosis and prenatally exposed to Diethylstilbestrol[END_REF][START_REF] Rivollier | Methylomic changes in individuals with psychosis, prenatally exposed to endocrine disrupting compounds: Lessons from diethylstilbestrol[END_REF].

Numerous studies have shown that, for example in the rat, early maternal separation or the fact of causing significant stress to the mother changes the methylation signals of certain genes of the rat directly related to the regulation of anxiety. It has also been discovered that the proper environment for changing the methylation signals may be chemical. This is the case of DES recognized by the scientific community as an endocrine disruptor, and banned for pregnant women. This change in level of methylation caused in utero by DES has been demonstrated for uro-genital malformations of girls and boys as well as for cancers. On 2015, Harlid et al.published in a pioneer work the first study for evaluation of possible effects of in utero DES exposure on genome-wide DNA methylation in humans [START_REF] Harlid | In utero exposure to diethylstilbestrol and blood DNA methylation in women ages 40-59 years from the sister study[END_REF]. They studied whole blood DNA methylation in one hundred 40-59 years old women reporting in utero exposure, compared to 100 unexposed women. They did not find any differential methylation, but the DMR approach was not used in their recent work (2015). On the other hand, in 2017 Rivollier et al. [START_REF] Rivollier | Methylomic changes in individuals with psychosis, prenatally exposed to endocrine disrupting compounds: Lessons from diethylstilbestrol[END_REF] described specific Differential Methylated Regions (DMR) on two genes implicated in neurodevelopment (ZFP57 and ADAMTS9). Surprisingly, they cautiously claim that these DMR are « supposedly » associated with prenatal exposure to DES in young psychotic patients in utero exposed to DES/EE. Nevertheless, these modifications of methylation are really specific because they do not exist in the methylome of young psychotic patients not exposed to DES [START_REF] Kebir | Methylomic changes during conversion to psychosis[END_REF] in which global methylation of the genome was observed. Moreover authors have compared exposed subjects to their unexposed siblings which do not present these specific methylations although they shared environmental and genetic factors.

In conclusion, the citizen work carried out between the French Hhorages Patient Association and two major medical research laboratories has provided convincing scientific results 1-on the detection and confirmation of the existence of psychiatric disorders (accompanied or not of somatic disorders) in children exposed in utero to synthetic hormones, 2-on the mechanisms of action of these synthetic hormones administered to pregnant mothers on the brain of their offspring. The effects of these endocrine disruptors in humans through what is becoming a public health scandal, denied for a long time by doctors, especially psychiatrists, scientists and specialized journalists are thus better known. The fact that these synthetic products do not degrade in the human body in the same way as the natural hormones [START_REF] Soyer-Gobillard | Behavioral and Somatic Disorders in Children exposed in utero to Synthetic Hormones : a Testimony-Case Study in a French Family Troop[END_REF], and act on the functioning of genes implicated in neurodevelopment during the foetal life, following an epigenetic mechanism, is a real time bomb. Indeed, this mechanism induces a transgenerational effect already partially demonstrated in the Hhorages cohort at the 3rd generation level for hypospadias, a specific genital malformation. So far, only a few third-generation children suffering psychiatric illness are documented in the Hhorages testimonies. This is understandable because third generation exposed children are still too young (excepted in some cases) to present psychiatric disorders as schizophrenia which is not the case for hypospads that are detectable from birth in male children and grand-children [START_REF] Kalfa | Incidence of hypospadias in grandsons of DES-exposed women during pregnancy: a multigenerational national cohort study[END_REF]. In contrast, psychiatric disorders usually appear in postadolescence, 18-20 years, and sometimes later.

Figure 1 .

 1 Figure 1. Measurement of aggressiveness in young male mice after prenatal exposure to Diethylstilbestrol (DES) (0.02 and 0.2 μg / kg [in gray], DDT [20 and 200 μg / kg] [in black], with a control without DES [in white]. There is an increase in the number of attacks and a decrease of the reaction time before the attack although the doses of DES are 1000 times less than those of DDT. According to Palanza et al. [3], with the permission of Elsevier (License No : 2514671323242).

Figure 2 .

 2 Figure 2. Total number of psychological/psychiatric disorders among the 982 (1002 total with-20 stillborns) DES-exposed and post-DES unexposed children. First born children (intrafamilial control) unexposed are not ill.

Figure 3 .Figure 4 .

 34 Figure 3. Total HHORAGES cohort of 1934 children in which three groups can be observed.

Figure 5 .

 5 Figure 5. Comparative diagrams representing psychiatric disorder cases presented respectively by girls (a, c) and boys (b, d) exposed either to synthetic estrogens DES, Diethylstilbestrol, EE, Ethinyl estradiol, (c, d) or to synthetic progestins (a, b).

  and 2010 at the 7th & 8th Congresses of the Encephalon in Paris (26, 27) on 43 exposed children highlighting clinical pictures with atypical associations.

Acknowledgements.

Mr René Alexandre, Dr Henri Pézerat, Pfr Jean Caston, Mr Michel Datry, Dr Charles Zelwer worked tirelessly to demonstrate the harmful role of the endocrine disruptors, and in particular of synthetic hormones, before passing away: we shall never forget their efficient help. This work could not continue without the daily ongoing support of the HHORAGES board especially Mrs Geneviève A., Mauricette P., Yette B., Annie R., Aimée I., Sylvette C., Denise H., Marie-Noelle P. and Pierre D. We ackowledge Mr André Cicolella, President of the French Environment-Health network (RES) for his constant support. We are very grateful to Pfr Michael Dolan, Professor at the University of Massachusetts, Amherst (USA) for his help and critical reading during the preparation of the manuscript.

Disclaimer

The authors declare they have no competitive financial interests; the financing of HHORAGES-France association comes exclusively from the donations of families and of sympathetic individuals.

Author details

Marie-Odile Soyer-Gobillard, PH.D. Email : c-sultan@chu-montpellier.fr

Emeritus Honorary Research


E A Pchelintsev

Improved model selection method for an adaptive estimation in semimartingale regression models *

Keywords: Improved non-asymptotic estimation, Least squares estimates, Robust quadratic risk, Non-parametric regression, Semimartingale noise, Ornstein-Uhlenbeck-Lévy process, Model selection, Sharp oracle inequality, Asymptotic efficiency

This paper considers the problem of robust adaptive efficient estimating of a periodic function in a continuous time regression model with the dependent noises given by a general square integrable semimartingale with a conditionally Gaussian distribution. An example of such noise is the non-Gaussian Ornstein-Uhlenbeck-Lévy processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Under some conditions on the noise distribution, sharp oracle inequality for the robust risk has been proved and the robust efficiency of the model selection procedure has been established. The numerical analysis results are given.

Introduction

Consider a regression model in continuous time dy t = S(t)dt + dξ t , 0 ≤ t ≤ n , (1.1) where S is an unknown 1-periodic R → R function, S ∈ L 2 [0, 1], (ξ t ) t≥0 is an unobservable noise which is a square integrated semimartingale with the values in the Skorokhod space D[0, n] such that, for any function f from L 2 [0, n], the stochastic integral

I n (f ) = n 0 f (s)dξ s (1.2)
has the following properties

E Q I n (f ) = 0 and E Q I 2 n (f ) ≤ κ Q n 0 f 2 (s)ds . (1.3)
Here E Q denotes the expectation with respect to the distribution Q of the noise process (ξ t ) 0≤t≤n on the space D[0, n], κ Q > 0 is some positive constant depending on the distribution Q. The noise distribution Q is unknown and assumed to belong to some probability family Q n specified below. Note that the firstly semimartingale regression models in continuous time were introduced by Konev and Pergamenshchikov in [START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities // Vestnik Tomskogo Universiteta[END_REF][START_REF] Konev | Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency[END_REF] for the signal estimation problems. It should be noted also, that, the class of the noise processes (ξ t ) t≥0 satisfying conditions (1.3) is rather wide and comprises, in particular, the Lévy processes which are used in different applied problems (see [START_REF] Bertoin | Lévy Processes[END_REF], for details). Moreover, as is shown in Section 2, non-Gaussian Ornstein-Uhlenbeck-based models enter this class.

The problem is to estimate the unknown function S in the model (1.1) on the basis of observations (y t ) 0≤t≤n . In this paper we use the quadratic risk, i.e. for any estimate S we set R Q ( S, S) := E Q,S S -S 2 and f 2 :=

1 0 f 2 (t)dt , (1.4)
where E Q,S stands for the expectation with respect to the distribution P Q,S of the process (1.1) with a fixed distribution Q of the noise (ξ t ) 0≤t≤n and a given function S. Moreover, in the case when the distribution Q is unknown we use also the robust risk

R * n ( S, S) = sup Q∈Q n R Q ( S, S) . (1.5)
The goal of this paper is to develop the adaptive robust efficient model selection method for the regression (1.1) with dependent noises having conditionally Gaussian distribution using the improved estimation approach. This paper proposes the shrinkage least squares estimates which enable us to improve the non-asymptotic estimation accuracy. For the first time such idea was proposed by Fourdrinier and Pergamenshchikov in [START_REF] Fourdrinier | Improved selection model method for the regression with dependent noise // Annals of the Institute of Statistical Mathematics[END_REF] for regression models in discrete time and by Konev and Pergamenshchikov in [START_REF] Konev | General model selection estimation of a periodic regression with a Gaussian noise[END_REF] for Gaussian regression models in continuous time. We develop these methods for the general semimartingale regression models in continuous time. It should be noted that for the conditionally Gaussian regression models we can not use the well-known improved estimators proposed in [START_REF] Stein | Estimation with quadratic loss[END_REF] for Gaussian or spherically symmetric observations. To apply the improved estimation methods to the non-Gaussian regression models in continuous time one needs to use the modifications of the well-known James -Stein estimators proposed in [START_REF] Konev | Estimation of a regression with the pulse type noise from discrete data // Theory Probab[END_REF][START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF] for parametric problems. We develop the new analytical tools which allow one to obtain the sharp non-asymptotic oracle inequalities for robust risks under general conditions on the distribution of the noise in the model (1.1). This method enables us to treat both the cases of dependent and independent observations from the same standpoint, it does not assume the knowledge of the noise distribution and leads to the efficient estimation procedure with respect to the risk (1.5). The validity of the conditions, imposed on the noise in the equation (1.1) is verified for a non-Gaussian Ornstein-Uhlenbeck process.

The rest of the paper is organized as follows. In the next Section 2, we describe the Ornstein-Uhlenbeck process as the example of a semimartingale noise in the model (1.1). In Section 3 we construct the shrinkage weighted least squares estimates and study the improvement effect. In Section 4 we construct the model selection procedure on the basis of improved weighted least squares estimates and state the main results in the form of oracle inequalities for the quadratic risk (1.4) and the robust risk (1.5). In Section 5 it is shown that the proposed model selection procedure for estimating S in (1.1) is asymptotically efficient with respect to the robust risk (1.5). In Section 6 we illustrate the performance of the proposed model selection procedure through numerical simulations. Section 7 gives the proofs of the main results.

Ornstein-Uhlenbeck-Lévy process

Now we consider the noise process (ξ t ) t≥0 in (1.1) defined by a non-Gaussian Ornstein-Uhlenbeck process with the Lévy subordinator. Such processes are used in the financial Black-Scholes type markets with jumps (see, for example, [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics[END_REF], and the references therein). Let the noise process in (1.1) obeys the equation dξ t = aξ t dt + du t , ξ 0 = 0 ,

where

u t = 1 w t + 2 z t and z t = R x * (µ -µ) t . (2.2)
Here (w t ) t≥0 is a standard Brownian motion, µ(ds dx) is the jump measure with the deterministic compensator µ(ds dx) = dsΠ(dx), Π(•) is the Lévy measure on R * = R \ {0}, (see, for example in [START_REF] Cont | Financial Modelling with Jump Processes[END_REF]), such that Π(x 2 ) = 1 and Π(x 8 ) < ∞ .

(2.3)

We use the notation Π(|x| m ) = R * |z| m Π(dz). Moreover, we assume that the nuisance parameters a ≤ 0, 1 and 2 satisfy the conditions

-a max ≤ a ≤ 0 , 0 < ≤ 2 1 and σ Q = 2 1 + 2 2 ≤ ς * , (2.4)
where the bounds a max , and ς * are functions of n, i.e. a max = a max (n), = n and ς * = ς * n , such that for any δ > 0

lim n→∞ a max (n) n = 0 , lim inf n→∞ n n > 0 and lim n→∞ n -ς * n = 0 . (2.5)
We denote by Q n the family of all distributions of process (1.1) -(2.1) on the Skorokhod space D[0, n] satisfying the conditions (2.4) - (2.5).

It should be noted that in view of Corollary 7.2 in [START_REF] Pchelintsev | Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time[END_REF] the condition (1.3) for the process (2.1) holds with

κ Q = 2 * .
(2.6)

Note also that the process (2.1) is conditionally-Gaussian square integrated semimartingale with respect to σ-algebra G = σ{z t , t ≥ 0} which is generated by jump process (z t ) t≥0 .

Improved estimation

For estimating the unknown function S in (1.1) we will consider it's Fourier expansion. Let (φ j ) j≥ 1 be an orthonormal basis in L 2 [0, 1]. We extend these functions by the periodic way on R, i.e. φ j (t)=φ j (t + 1) for any t ∈ R. B 1 ) Assume that the basis functions are uniformly bounded, i.e. for some constant φ * ≥ 1, which may be depend on n,

sup 0≤j≤n sup 0≤t≤1 |φ j (t)| ≤ φ * < ∞ . (3.1)
B 2 ) Assume that there exist some d 0 ≥ 7 and ǎ ≥ 1 such that

sup d≥d 0 1 d 1 0 Φ * d (v) dv ≤ ǎ , (3.2)
where Φ * d (v) = max t≥v d j=1 φ j (t) φ j (t -v) . For example, we can take the trigonometric basis defined as Tr 1 ≡ 1 and Tr j (x) = √ 2 cos( j x) for even j and Tr j (x) = √ 2 cos( j x) for odd j ≥ 2, where the frequency j = 2π[j/2] and [x] denotes integer part of x. In Lemma A1 in [START_REF] Pchelintsev | Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time[END_REF] it is shown that these functions satisfy the condition B 2 ) with d 0 = inf{d ≥ 7 : 5 + ln d ≤ ǎd} and ǎ = (1 -e -a max )/(4a max ).

We write the Fourier expansion of the unknown function S in the form

S(t) = ∞ j=1 θ j φ j (t),
where the corresponding Fourier coefficients

θ j = (S, φ j ) = 1 0 S(t) φ j (t) dt (3.3)
can be estimated as

θ j,n = 1 n n 0 φ j (t) dy t . (3.4)
We replace the differential S(t)dt by the stochastic observed differential dy t .

In view of (1.1), one obtains

θ j,n = θ j + 1 √ n ξ j,n , ξ j,n = 1 √ n I n (φ j ) (3.5)
where I n (φ j ) is given in (1.2). As in [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincaré[END_REF] we define a class of weighted least squares estimates for S(t) as

S γ = n j=1 γ(j) θ j,n φ j , (3.6)
where the weights γ = (γ(j)) 1≤j≤n ∈ R n belong to some finite set Γ from [0, 1] n for which we set

ν = card(Γ) and |Γ| * = max γ∈Γ n j=1 γ(j) , (3.7)
where card(Γ) is the number of the vectors γ in Γ. In the sequel we assume that all vectors from Γ satisfies the following condition.

D 1 ) Assume that for for any vector γ ∈ Γ there exists some fixed integer d = d(γ) such that their first d components equal to one, i.e. γ(j) = 1 for 1 ≤ j ≤ d for any γ ∈ Γ.

D 2 ) There exists n 0 ≥ 1 such that for any n ≥ n 0 there exists a σ -field G n for which the random vector ξ d,n = (ξ j,n ) 1≤j≤d is the G n -conditionally Gaussian in R d with the covariance matrix

G n = E ξ i,n ξ j,n |G n ) 1≤i,j≤d (3.8)
and for some nonrandom constant

l * n > 0 inf Q∈Q n (tr G n -λ max (G n )) ≥ l * n a.s. , (3.9)
where λ max (A) is the maximal eigenvalue of the matrix A.

As it is shown in Proposition 7.11 in [START_REF] Pchelintsev | Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time[END_REF] the condition D 2 ) holds for the non-

Gaussian Ornstein-Uhlenbeck-based model (1.1) -(2.1) with l * n = n (d - 6)/2 and d ≥ d 0 .
Further, for the first d Fourier coefficients in (3.5) we will use the improved estimation method proposed for parametric models in [START_REF] Pchelintsev | Improved estimation in a non-Gaussian parametric regression[END_REF]. To this end we set θ n = ( θ j,n ) 1≤j≤d . In the sequel we will use the norm

|x| 2 d = d j=1
x 2 j for any vector x = (x j ) 1≤j≤d from R d . Now we define the shrinkage estimators as

θ * j,n = (1 -g(j)) θ j,n , (3.10)
where

g(j) = (c n /| θ n | d )1 {1≤j≤d} , c n = l * n r * n + dκ * /n n and κ * = sup Q∈Q n κ Q .
The positive parameter r * n is such that

lim n→∞ r * n = ∞ and lim n→∞ r * n n δ = 0 (3.11)
for any δ > 0. Now we introduce a class of shrinkage weighted least squares estimates for S as

S * γ = n j=1 γ(j)θ * j,n φ j . (3.12)
We denote the difference of quadratic risks of the estimates (3.6) and (3.12) as

∆ Q (S) := R Q (S * γ , S) -R Q ( S γ , S
) . For this difference we obtain the following result.

Model selection

This Section gives the construction of a model selection procedure for estimating a function S in (1.1) on the basis of improved weighted least square estimates and states the sharp oracle inequality for the robust risk of proposed procedure.

The model selection procedure for the unknown function S in (1.1) will be constructed on the basis of a family of estimates (S * γ ) γ∈Γ . The performance of any estimate S * γ will be measured by the empirical squared error

Err n (γ) = S * γ -S 2 . In order to obtain a good estimate, we have to write a rule to choose a weight vector γ ∈ Γ in (3.12). It is obvious, that the best way is to minimise the empirical squared error with respect to γ. Making use the estimate definition (3.12) and the Fourier transformation of S implies

Err n (γ) = n j=1 γ 2 (j)(θ * j,n ) 2 -2 n j=1 γ(j)θ * j,n θ j + n j=1 θ 2 j . (4.1)
Since the Fourier coefficients (θ j ) j≥1 are unknown, the weight coefficients (γ j ) j≥1 can not be found by minimizing this quantity. To circumvent this difficulty one needs to replace the terms θ * j,n θ j by their estimators θ j,n . We set θ j,n = θ * j,n θ j,n -

σ n n (4.2)
where σ n is the estimate for the limiting variance of E Q ξ 2 j,n which we choose in the following form

σ n = n j=[ √ n]+1 t 2 j,n , t j,n = 1 0 Tr j (t)dy t . (4.3)
For this change in the empirical squared error, one has to pay some penalty. Thus, one comes to the cost function of the form

J n (γ) = n j=1 γ 2 (j)(θ * j,n ) 2 -2 n j=1 γ(j) θ j,n + ρ P n (γ) (4.4)
where ρ is some positive constant, P n (γ) is the penalty term defined as

P n (γ) = σ n |γ| 2 n n . (4.5)
Substituting the weight coefficients, minimizing the cost function

γ * = argmin γ∈Γ J n (γ) , (4.6)
in (3.6) leads to the improved model selection procedure

S * = S * γ * . (4.7)
It will be noted that γ * exists because Γ is a finite set. If the minimizing sequence in (4.6) γ * is not unique, one can take any minimizer.

To prove the sharp oracle inequality, the following conditions will be needed for the family Q n of distributions of the noise (ξ t ) t≥0 in (1.1).

We need to impose some stability conditions for the noise Fourier transform sequence (ξ j,n ) 1≤j≤n introduced in [START_REF] Pchelintsev | Oracle inequalities for the stochastic differential equations[END_REF]. To this end for some parameter σ Q > 0 we set the following function

L 1,n (Q) = n j=1 E Q ξ 2 j,n -σ Q . (4.8)
In [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincaré[END_REF] the parameter σ Q is called proxy variance. C 1 ) There exists a proxy variance σ Q > 0 such that for any > 0

lim n→∞ L 1,n (Q) n = 0 .
Moreover, we define

L 2,n (Q) = sup |x|≤1 E Q   n j=1 x j ξ j,n   2 and ξ j,n = ξ 2 j,n -E Q ξ 2 j,n . C 2 ) Assume that for any > 0 lim n→∞ L 2,n (Q) n = 0 .
R Q (S * , S) ≤ 1 + 5ρ 1 -ρ min γ∈Γ R Q (S * γ , S) + B n (Q) ρn , (4.9)
where

B n (Q) = U n (Q) 1 + |Γ| * E Q | σ n -σ Q | and the coefficient U n (Q) is such that for any > 0 lim n→∞ U n (Q) n = 0 . (4.10)
In the case, when the value of σ Q in C 1 ) is known, one can take σ n = σ Q and

P n (γ) = σ Q |γ| 2 n n (4.11)
and then we can rewrite the oracle inequality (4.9) with

B n (Q) = U n (Q).
E Q | σ n -σ Q | ≤ κ n (Q)(1 + Ṡ 2 ) √ n ,
where the term κ n (Q) possesses the property (4.10) and Ṡ is the derivative of the function S.

To obtain the oracle inequality for the robust risk (1.5) we need some additional condition on the distribution family Q n . We set As is shown in [START_REF] Pchelintsev | Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time[END_REF], both the conditions C * 1 ) and C * 2 ) hold for the model (1.1) with Ornstein-Uhlenbeck noise process (2.1). Using Proposition 4.2 from [START_REF] Pchelintsev | Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time[END_REF] we can obtain the following result Theorem 4.3. Assume that the conditions C * 1 )-C * 2 ) hold. Then the robust risk (1.5) of the estimate (4.7) for continuously differentiable function S(t) satisfies for any n ≥ 2 and 0 < ρ < 1/2 the oracle inequality

ς * = ς * n = sup Q∈Q n σ Q and L * n = sup Q∈Q n (L 1,n (Q) + L 2,n (Q)) . ( 4
R * n (S * , S) ≤ 1 + 5ρ 1 -ρ min γ∈Γ R * n (S * γ , S) + 1 ρn B * n (1 + Ṡ 2 ) ,
where the term B * n satisfies the property (4.10). Now we specify the weight coefficients (γ(j)) j≥1 as it is proposed in [START_REF] Galtchouk | Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models[END_REF][START_REF] Galtchouk | Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression[END_REF] for a heteroscedastic regression model in discrete time. Firstly, we define the normalizing coefficient v n = n/ς * . Consider a numerical grid of the form

A n = {1, . . . , k * } × {r 1 , . . . , r m } ,
where r i = iε and m = [1/ε 2 ]. Both parameters k * ≥ 1 and 0 < ε ≤ 1 are assumed to be functions of n, i.e. k * = k * (n) and ε = ε(n), such that for any δ > 0

     lim n→∞ k * (n) = +∞ , lim n→∞ k * (n) ln n = 0 , lim n→∞ ε(n) = 0 and lim n→∞ n δ ε(n) = +∞ .
One can take, for example, ε(n) = 1/ ln(n + 1) and k * (n) = ln(n + 1). For each α = (β, r) ∈ A n we introduce the weight sequence γ α = (γ α (j)) j≥1 as

γ α (j) = 1 {1≤j≤d} + 1 -(j/ω α ) β 1 {d<j≤ωα} (4.13)
where

d = d(α) = [ω α / ln(n + 1)], ω α = τ β r v n 1/(2β+1)
and

τ β = (β + 1)(2β + 1) π 2β β .
We set Γ = {γ α , α ∈ A n } . (4.14)

It will be noted that such weight coefficients satisfy the condition D 1 ) and in this case the cardinal of the set Γ is ν = k * m. Moreover, taking into account that τ β < 1 for β ≥ 1 we obtain for the set (4.14)

|Γ| * ≤ 1 + sup α∈A ω α ≤ 1 + (υ n /ε) 1/3 .
Remark 4.1. These weight coefficients are used in [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincaré[END_REF][START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data // Stochastic processes and their applications[END_REF] to show the asymptotic efficiency for model selection procedures.

Asymptotic efficiency

In order to study the asymptotic efficiency we define the following functional Sobolev ball

W k,r = {f ∈ C k p [0, 1] : k j=0 f (j) 2 ≤ r} ,
where r > 0 and k ≥ 1 are some unknown parameters, C k p [0, 1] is the space of k times differentiable 1 -periodic R → R functions such that f (i) (0) = f (i) (1) for any 0 ≤ i ≤ k -1. To study the asymptotic efficiency we denote by Σ n all estimators S n i.e. any σ{y t , 0 ≤ t ≤ n} mesurable functions. In the sequel we denote by Q * the distribution of the process (y t ) 0≤t≤n with ξ t = ς * w t , i.e. white noise model with the intensity ς * . Theorem 5.1. Assume that Q * ∈ Q n . The robust risk (1.5) admits the following lower bound

lim inf n→∞ inf S n ∈Σ n v 2k/(2k+1) n sup S∈W k,r R * n ( S n , S) ≥ l k (r) ,
where

l k (r) = ((1 + 2k)r) 1/(2k+1) (k/π(k + 1)) 2k/(2k+1) .
We show that this lower bound is sharp in the following sense.

Theorem 5.2. The quadratic risk (1.2) for the estimating procedure (4.7) has the following asymptotic upper bound

lim sup n→∞ v 2k/(2k+1) n sup S∈W k,r R * n (S * , S) ≤ l k (r) .
It is clear that these theorems imply the following efficient property.

Monte Carlo simulations

In this section we give the results of numerical simulations to assess the performance and improvement of the proposed model selection procedure (4.6). We simulate the model (1.1) with 1-periodic function S of the form

S(t) = t sin(2πt) + t 2 (1 -t) cos(4πt) (6.1)
on [0, 1] and the Lévy noise process ξ t is defined as

dξ t = -ξ t dt + 0.5 dw t + 0.5 dz t , z t = N t j=1 Y j ,
where N t is a homogeneous Poisson process with intensity λ = 1 and (Y j ) j≥1 is i.i.d. N (0, 1) sequence (see, for example, [START_REF] Konev | Robust model selection for a semimartingale continuous time regression from discrete data // Stochastic processes and their applications[END_REF]).

We use the model selection procedure (4.6) with the weights (4.13) in which k * = 100 + ln(n + 1), r i = i/ ln(n + 1), m = [ln 2 (n + 1)], ς * = 0.5 and ρ = (3 + ln n) -2 . We define the empirical risk as

R(S * , S) = 1 p p j=1 E∆ 2 n (t j ) and E∆ 2 n (t) = 1 N N l=1 ∆ 2 n,l (t) ,
where ∆ n (t) = S * n (t) -S(t) and ∆ n,l (t) = S * n,l (t) -S(t) is the deviation for the lth replication. In this example we take p = 100001 and N = 1000. 1 gives the values for the sample risks of the improved estimate (4.6) and the model selection procedure based on the weighted LSE (3.15) from [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincaré[END_REF] for different numbers of observation period n. Table 2 gives the values for the sample risks of the the model selection procedure based on the weighted LSE (3.15) from [START_REF] Konev | Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincaré[END_REF] and it's improved version for different numbers of observation period n. Remark 6.1. Figures 12show the behavior of the procedures (3.6) and (4.6) depending on the values of observation periods n. The bold line is the function (6.1), the continuous line is the model selection procedure based on the least squares estimators S and the dashed line is the improved model selection procedure S * . From the Table 2 for the same γ with various observations numbers n we can conclude that theoretical result on the improvement effect (3.13) is confirmed by the numerical simulations. Moreover, for the proposed shrinkage procedure, Table 1 and Figures 12, we can conclude that the benefit is considerable for non large n. Consider the quadratic error of the estimate (3.12)

S * γ -S 2 = n j=1 (γ(j)θ * j,n -θ j ) 2 = d j=1 (γ(j)θ * j,n -θ j ) 2 + n j=d+1 (γ(j) θ j,n -θ j ) 2 = n j=1 (γ(j) θ j,n -θ j ) 2 + c 2 n -2c n d j=1 ( θ j,n -θ j ) θ j,n θ n d = S γ -S 2 + c 2 n -2c n d j=1 ( θ j,n -θ j )ι j ( θ n ) ,
where ι j (x) = x j / x d for x = (x j ) 1≤j≤d ∈ R d . Therefore, we can represent the risk for the improved estimator S * γ as

R Q (S * γ , S) = R Q ( S γ , S) + c 2 n -2c n E Q,S d j=1 ( θ j,n -θ j ) I j,n ,
where I j,n = E(ι j ( θ n )( θ j,n -θ j )|G n ). Now, taking into account that the vector θ n = ( θ j,n ) 1≤j≤d is the G n conditionally Gaussian vector in R d with mean θ = (θ j ) 1≤j≤d and covariance matrix n -1 G n , we obtain

I j,n = R d ι j (x)(x -θ j )p(x|G n )dx .
Here p(x|G n ) is the conditional distribution density of the vector θ n , i.e.

p(x|G n ) = 1 (2π) d/2 det G n exp - (x -θ) G -1 n (x -θ) 2 .
Changing the variables by u = G -1/2 n (x -θ), one finds that

I j,n = 1 (2π) d/2 d l=1 g j,l R d ιj,n (u)u l exp - u 2 d 2 du , (7.1)
where ι j,n (u) = ι j (G 1/2 n u + θ) and g ij denotes the (i, j)-th element of G 1/2 n . Furthermore, integrating by parts, the integral I j,n can be rewritten as

I j,n = d l=1 d k=1 E g jl g kl ∂ι j ∂u k (u)| u= θ n |G n .
Now taking into account that z Az ≤ λ max (A) z 2 and the condition D 2 ) we obtain that

∆ Q (S) = c 2 n -2c n n -1 E Q,S trG n θ n d - θ n G n θ n θ n 3 ≤ c 2 n -2c n l * n n -1 E Q,S 1
θ n d .
Recall, that the denotes the transposition. Moreover, in view of the Jensen inequality we can estimate the last expectation from below as

E Q,S ( θ n d ) -1 = E Q,S ( θ + n -1/2 ξ n d ) -1 ≥ ( θ d + n -1/2 E Q,S ξ n d ) -1 .
Note now that the condition through the inequality (1.3) we obtain

E Q,S ξ n 2 d ≤ κ Q d . So, for S 2 ≤ r * n E Q,S θ n -1 ≥ r * n + dκ Q /n -1
and, therefore,

∆ Q (S) ≤ c 2 n -2c n l * n r * n + dκ * /n n = -c 2 n .
Hence Theorem 3.1.

σ n n -θ * j,n θ j + S 2 -ρ P n (γ) . (7.2) Now we set L(γ) = n j=1 γ(j), B 1,n (γ) = n j=1 γ(j)(E Q ξ 2 j,n -σ Q ) , B 2,n (γ) = n j=1 γ(j) ξ j,n , M (γ) = 1 √ n n j=1 γ(j)θ j ξ j,n and B 3,n (γ) = 1 √ n n j=1 γ(j)g(j) θ j,n ξ j,n .
Taking into account the definition (4.5), we can rewrite (7.2) as

Err n (γ) = J n (γ) + 2 σ Q -σ n n L(γ) + 2 M (γ) + 2 n B 1,n (γ)
+ 2 P n (γ) B 2,n (γ) √ σ Q n -2B 3,n (γ) + S 2 -ρ P n (γ) (7.3)
with γ = γ/|γ| n . Let γ 0 = (γ 0 (j)) 1≤n be a fixed sequence in Γ and γ * be as in (4.6). Substituting γ 0 and γ * in (7.3), we consider the difference

Err n (γ * ) -Err n (γ 0 ) ≤ 2 σ Q -σ n n L(x) + 2M (x) + 2 n B 1,n (x) + 2 P n (γ * ) B 2,n (γ * ) √ σ Q n -2 P n (γ 0 ) B 2,n (γ 0 ) √ σ Q n -2B 3,n (γ * ) + 2B 3,n (γ 0 ) -ρ P n (γ * ) + ρ P n (γ 0 ) , where x = γ * -γ 0 . Note that |L(x)| ≤ 2|Γ| * and |B 1,n (x)| ≤ L 1,n (Q).
Applying the elementary inequality

2|ab| ≤ εa 2 + ε -1 b 2 (7.4)
with any ε > 0, we get

2 P n (γ) B 2,n (γ) √ σ Q n ≤ εP n (γ) + B 2 2,n (γ) εσ Q n ≤ εP n (γ) + B * 2 εσn ,
where

B * 2 = max γ∈Γ B 2 2,n (γ) + B 2 2,n (γ 2 )
with γ 2 = (γ 2 j ) 1≤j≤n . Note that from definition the function L 2,n (Q) in the condition C 2 ) we obtain that

E Q B * 2 ≤ γ∈Γ E Q B 2 2,n (γ) + E Q B 2 2,n (γ 2 ) ≤ 2νL 2,n (Q) . (7.5)
Moreover, by the same method we estimate the term B 3,n . Note that

n j=1 g 2 γ (j) θ 2 j = c 2 n ≤ c * n n , (7.6)
where c * n = n max γ∈Γ c 2 n . Therefore, through the Cauchy-Schwarz inequality we can estimate the term B 3,n (γ) as

|B 3,n (γ)| ≤ |γ| n √ n c n   n j=1 γ 2 (j) ξ 2 j   1/2 = |γ| n √ n c n σ Q + B 2,n (γ 2 ) 1/2 .
So, applying the elementary inequality (7.4) with some arbitrary ε > 0, we have

2|B 3,n (γ)| ≤ εP n (γ) + c * n εσ Q n (σ Q + B * 2 ) .

Using the bounds above, one has

Err

n (γ * ) ≤ Err n (γ 0 ) + 4|Γ| n | σ n -σ Q | n + 2M (x) + 2 n L 1,n (Q) + 2 ε c * nσ Q (σ Q + B * 2 ) + 2 ε B * 2 nσ Q + 2εP n (γ * ) + 2εP n (γ 0 ) -ρ P n (γ * ) + ρ P n (γ 0 ) .
The setting ε = ρ/4 and the estimating where this is possible ρ by 1 in this inequality imply

Err n (γ * ) ≤ Err n (γ 0 ) + 5|Γ| n | σ n -σ Q | n + 2M (x) + 2 n L 1,n (Q) + 16(c * n + 1)(σ Q + B * 2 ) ρnσ Q - ρ 2 P n (γ * ) + ρ 2 P n (γ 0 ) + ρ P n (γ 0 ) .
Moreover, taking into account here that

| P n (γ 0 ) -P n (γ 0 )| ≤ |Γ| n | σ n -σ Q | n and that ρ < 1/2, we obtain that Err n (γ * ) ≤ Err n (γ 0 ) + 6|Γ| n | σ n -σ Q | n + 2M (x) + 2 n L 1,n (Q) + 16(c * n + 1)(σ Q + B * 2 ) ρnσ Q - ρ 2 P n (γ * ) + 3ρ 2 P n (γ 0 ) . (7.7)
Now we examine the third term in the right-hand side of this inequality. Firstly we note that

2|M (x)| ≤ ε S x 2 + Z * nε , (7.8)
where S x = n j=1 x j θ j φ j and

Z * = sup x∈Γ 1 nM 2 (x) S x 2 .
We remind that the set Γ 1 = Γ -γ 0 . Using Proposition ?? we can obtain that for any fixed

x = (x j ) 1≤j≤n ∈ R n E M 2 (x) = E I 2 n (S x ) n 2 = σ Q S x 2 n = σ Q n n j=1
x 2 j θ 2 j (7.9) and, therefore,

E Q Z * ≤ x∈Γ 1 nM 2 (x) S x 2 ≤ σ Q ν . ( 7
S * γ -S * γ 0 2 = n j=1 (x(j) + β(j)) 2 θ 2 j ≥ S x 2 + 2 n j=1 x(j)β(j) θ 2 j ,
where β(j) = γ 0 (j)g j (γ 0 ) -γ(j)g j (γ). Therefore, in view of (3.5)

S x 2 -S * γ -S * γ 0 2 ≤ S x 2 -S x 2 -2 n j=1 x(j)β(j) θ 2 j ≤ -2M (x 2 ) -2 n j=1
x(j)β(j) θ j θ j -2 √ n Υ(x) ,

where Υ(γ) = n j=1 γ(j)β(j) θ j ξ j . Note that the first term in this inequality we can estimate as 2M (x 2 ) ≤ ε S Note that, similarly to (7.10) we can estimate the last term as

E Q Z * 1 ≤ σ Q ν .
From this it follows that for any 0 < ε < 1 x(j)β(j) θ j θ j ≤ ε S x 2 + 4c * εn .

S x 2 ≤ 1 1 -ε S * γ -S *
To estimate the last term in the right hand of (7.11) we use first the Cauchy -Schwarz inequality and then the bound (7.12), i.e. The inequality (7.6) implies the bound (7.13). Hence Lemma 7.1.

Theorem 3 . 1 .

 31 Assume that the conditions D 1 ) -D 2 ) hold. Then for any n ≥ n 0 supQ∈Q n sup S ≤r * n ∆ Q (S) < -c 2 n . (3.13) Remark 3.1. The inequality (3.13) means that non asymptotically, i.e. for any n ≥ n 0 the estimate (3.12) outperforms in mean square accuracy the estimate (3.6).

Theorem 4 . 1 .

 41 If the conditions C 1 ) and C 2 ) hold for the distribution Q of the process ξ in (1.1), then, for any n ≥ 1 and 0 < ρ < 1/2, the risk (1.4) of estimate (4.7) for S satisfies the oracle inequality

.12) C * 1 ) 2 )

 12 Assume that the conditions C 1 )-C 2 ) hold and for any > Now we impose the conditions on the set of the weight coefficients Γ. C * Assume that the set Γ is such that for any >

Corollary 5 . 3 .

 53 The model selection procedure (4.7) is efficient, i.e. * , S) = l k (r) . Theorem 5.2 and Theorem 5.1 are shown by the same way as Theorems 1 and 2 in [9].

Figure 1 :

 1 Figure 1: Behavior of the regression function and its estimates for n = 500.

Figure 2 :

 2 Figure 2: Behavior of the regression function and its estimates for n = 1000.

7. 2 1 Substituting ( 4 . 4 )

 2144 Proof of Theorem 4.in (4.1) yields for any γ ∈ Γ Err n (γ) = J n (γ) + 2 n j=1 γ(j) θ * j,n θ j,n -

. 10 )

 10 Moreover, the norm S * γ * -S * γ 0 can be estimated from below as

1 nM 2 (x 2 ) S x 2 .

 1222

  )β(j) θ j θ jthat |x(j)| ≤ 1 and using the inequality (7.4), we get that for any ε > 0 2 n j=1

≤ 7 . 3 Lemma 7 . 1 . 1 P 2 ≥

 737112 εP n (γ * ) + εP n (γ 0 ) + 2c * (σ Q + B * 2 ) nεσ Q. Property of Penalty term For any n ≥ 1, γ ∈ Γ and 0 < ε < By the definition of Err n (γ) one hasErr n (γ) = n j=1 (γ(j)θ * j,n -θ j ) 2 = n j=1 γ(j)(θ * j,n -θ j ) + (γ(j) -1)θ j 2 ≥ n j=1 γ(j) 2 (θ * j,n -θ j ) 2 + 2 n j=1 γ(j)(γ(j) -1)θ j (θ * j,n -θ j ).Taking into account the condition B 2 ) and the definition (3.10) we obtain that the last term in tho sum can be replaced asn j=1 γ(j)(γ(j) -1)θ j (θ * j,n -θ j ) = n j=1 γ(j)(γ(j) -1)θ j ( θ j,n -θ j ) ,i.e. E n j=1 γ(j)(γ(j) -1)θ j (θ * j,n -θ j ) = 0 and, therefore, taking into account the definition (4.11) we obtain thatE Err n (γ) ≥ n j=1 γ(j) 2 E (θ * j,n -θ j ) 2 = n j=1 γ(j) 2 E ξ j,n√ n -g γ (j) θ j P n (γ) -) 2 g γ (j) θ j,n ξ j ≥ (1 -ε) P n (γ) -

  Now we study the estimate (4.3).

Proposition 4.2. Let in the model (1.1) the function S(•) is continuously differentiable. Then, for any n ≥ 2,

Table 1 :

 1 The sample quadratic risks for different optimal γ

	n	100	200	500	1000
	R(S * γ * , S)	0.0289 0.0089 0.0021 0.0011
	R( S γ , S)	0.0457 0.0216 0.0133 0.0098
	R( S γ , S)/R(S * γ * , S)	1.6	2.4	6.3	8.9

Table 2 :

 2 The sample quadratic risks for the same optimal γ

	n	100	200	500	1000
	R(S * γ , S)	0.0391 0.0159 0.0098 0.0066
	R( S γ , S)	0.0457 0.0216 0.0133 0.0098
	R( S γ , S)/R(S * γ , S)	1.2	1.4	1.3	1.5
	Table

Acknownledgements. This work was partially supported by the research project no. 2.3208.2017/4.6 (the Ministry of Education and Science of the Russian Federation) and RFBR Grant 16-01-00121. The work of the second author was partially supported by the Russian Federal Professor program (project no. 1.472.2016/1.4, the Ministry of Education and Science of the Russian Federation).

This work is supported by RSF, Grant no 17-11-01049.

So, using all these bounds in (7.11), we obtain that

Using in the inequality (7.8) this bound and the estimate

Choosing here ε ≤ ρ/2 < 1/2 we obtain that

From here and (7.7), it follows that

Choosing here ε = ρ/3 and estimating (1 -ρ) -1 by 2 where this is possible, we get

Taking the expectation and using the upper bound for

where ǓQ,n = 4L 1,n (Q) + 56(1 + c * n )(2L 2,n (Q)ν + 1) + 2c * n . The inequality holds for each γ 0 ∈ Λ, this implies Theorem 4.1.


Bruno Guillet
email: bruno.guillet@unicaen.fr

Stéphane Flament
email: stephane.flament@ensicaen.fr

Shuang Liu
email: shuang.liu@unicaen.fr

Olivier Rousseau
email: olivier.rousseau@unicaen.fr

Ammar Aryan
email: ammar.aryan@unicaen.fr

Marc Lam
email: marc.lam@ensicaen.fr

Chok Sing

Laurence Méchin
email: laurence.mechin@ensicaen.fr

Jean-Marc Routoure
email: jean-marc.routoure@unicaen.fr

Guiherme Luiz

Sylvain Enger

Lebargy

Guillet Bruno

Flament Stéphane

Liu Shuang

Rousseau Olivier

Aryan Ammar

Lam Chok

Sing Marc

Méchin Laurence

Routoure Jean-Marc

Enger Luiz Guilherme
email: luiz-guilherme.enger@ensicaen.fr

Lebargy Sylvain
email: sylvain.lebargy@unicaen.fr

Nascimento Vanuza Marques
email: marques-donascimento@unicaen.fr

Wu Sheng
email: sheng.wu@unicaen.fr

Low-frequency noise considerations for sensors based on manganites

Keywords: low frequency noise, LSMO, bolometers, magnetoresistances

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Among manganites, the ferromagnetic composition La0.33Sr0.67MnO3 (LSMO) with a Curie temperature of 360 K is particularly promising for sensors since the metal-toinsulator transition accompanying the ferromagnetic-toparamagnetic transition occurs above room temperature, thus allowing room temperature applications [START_REF] Coll | Towards Oxide Electronics: A Roadmap[END_REF].

The objective of the present paper is to show that before using LSMO thin films for sensors, we had to optimize the deposition conditions, the patterned geometries and the electronic readout in order to obtain very low electrical noise, as summarized in section II. In addition, we show the effect of film thickness on the structural and electrical properties. Section III and section IV briefly present two examples of high signal-to-noise ratio sensors based on LSMO thin films: uncooled bolometers on buffered silicon substrates [START_REF] Liu | La0.7Sr0.3MnO3 suspended microbridges for uncooled bolometers made using reactive ion etching of the silicon substrates[END_REF][START_REF] Liu | Freestanding La0.7Sr0.3MnO3 suspended micro-bridges on buffered silicon substrates showing undegraded low frequency noise properties[END_REF], and uncooled low-field magnetic sensors on SrTiO3 (STO) substrates [START_REF] Perna | Large Anisotropic Magnetoresistance in La0.7Sr0.3MnO3 Films at Room Temperature[END_REF].

II. LSMO MATERIAL PROPERTIES AND CONTACT DESIGN

LSMO films of various thicknesses in the 5-150 nm range were epitaxially grown either on STO (100) substrates by Pulsed Laser Deposition (PLD) or on STO buffered Si (001) substrates by reactive Molecular Beam Epitaxy (MBE) [START_REF] Méchin | Epitaxial La0.7Sr0.3MnO3 Thin Films Grown on SrTiO3 Buffered Silicon Substrates by Reactive Molecular-Beam-Epitaxy[END_REF]. The lattice mismatch between epitaxial films and substrates can significantly influence the electrical and magnetic properties (DC properties, noise level, etc.). The structural and electrical properties were systematically checked using -2 out-of-plane X-ray Diffraction (XRD) and electrical resistivity versus temperature measurements, respectively, as shown in Fig. 1. From the out-of-plane lattice parameter of LSMO, named cLSMO, we could measure the LSMO out-of-plane lattice deformation ε[0 0 1] defined as

ε[0 0 1] = (cLSMO -cLSMOpc) / cLSMOpc ( 1
)
where cLSMOpc is the pseudo-cubic lattice parameter of bulk LSMO (0.3873 nm). ε[0 0 1] is negative for the whole film thickness range, as expected in case of tensile strain, when grown on STO material, of lattice parameter equal to 0.3905 nm. As shown in [START_REF] Méchin | 1⁄f noise in patterned La2⁄3Sr1⁄3MnO3 thin films in the 300 400K range[END_REF] noise spectra typically consist of two parts: the low-frequency noise that depends on the bias current and the frequency, and the white (or Johnson) noise that is equal to 4kBTR, where kB is the Boltzmann constant, T is the temperature and R is the sample electrical resistance. We used the semi-empirical Hooge relation to describe the low frequency noise in our samples [START_REF] Hooge | 1/ƒ noise is no surface effect[END_REF]:

S V/V 2 = H/(n f)
where SV is the voltage noise spectral density (V 2 Hz -1 ), V is the sample voltage (V), αH is the Hooge parameter (dimensionless), n is the charge carrier concentration (m -3 ).

The electrical resistivity at 300 K and the temperature Tpeak at which the electrical resistivity is maximum are other parameters of interest. ε[0 0 1], the electrical resistivity at 300 K and Tpeak are then plotted in In order to measure the low frequency noise, a four-probe configuration was used and home-made electronics was designed. The voltage instrumentation amplifier circuit was build using 3 low noise operational amplifiers AD743, leading to a noise floor of 20×10 -18 V²•Hz -1 in 1 Hz -100 kHz bandwidth and negligible input current noise. A quasi-ideal DC current source was specifically designed, and showed high output impedance and negligible output current noise [START_REF] Routoure | A Low-Noise and Quasi-Ideal DC Current Source Dedicated to Four-Probe Low-Frequency Noise Measurements[END_REF].

In addition to dedicated low noise electronics, a specific attention has been paid on the electrical contacts [START_REF] Liu | La0.7Sr0.3MnO3 suspended microbridges for uncooled bolometers made using reactive ion etching of the silicon substrates[END_REF], as well as on the contact geometry in order to minimize parasitic effects, due to non-homogeneity in the current density distribution. Samples were patterned with different designs. More details will be given in a coming paper. The mask designs are made to avoid any deviation of the current lines. When the geometry is not well chosen, up to 3 orders of magnitude difference in H/n values have been measured on the same sample.

Finally, we showed that a clear correlation exists between the 1/f noise level, the temperature of the metal-to-insulator transition and the thickness of the LSMO thin film as previously published in [START_REF] Mechin | Experimental evidence of correlation between 1/f noise level and metal-to-insulator transition temperature in epitaxial La0.7Sr0.3MnO3 thin films[END_REF]. In case of well matched crystalline substrate (for example STO), higher noise was measured in films thinner than 10 nm, where higher electrical resistivity and higher lattice deformation were also measured. In the 20 -80 nm thickness range, LSMO films are very probably relaxed and therefore show low noise values. At about 80 nm, which can be close to the critical thickness value for epitaxial films on STO, one can observe an increase of the noise levels, which can be related to the presence of inhomogeneous strain as stated above, or even cracks for these higher thickness values. On STO/Si substrate, the 1/f noise level can be larger by up to one order of magnitude but stayed at very low values. They do not show a clear dependence versus thickness as it was seen on STO. LSMO films on STO/Si were used for bolometers, in order to allow the use of standard micromachining techniques of silicon. LSMO films on STO were used for anisotropic magnetoresistances. In that case film thickness was kept below 25 nm so as to get a uniaxial magnetic anisotropy [START_REF] Chaluvadi | Thickness and angular dependent magnetic anisotropy of La0.67Sr0.33MnO3 thin films by vectorial magneto optical kerr magnetometry[END_REF].

Figure 3 shows the dependence of the normalized Hooge parameters with the electrical resistivity of LSMO thin films of various thicknesses deposited on STO substrates and on buffered Si substrates. The normalized Hooge parameter could be lower than 10 -30 m 3 in a large number of cases.

III. PERFORMANCE OF LSMO DEVICES

A. LSMO uncooled bolometers

A bolometer measures the power of incident electromagnetic radiation via the heating of a material with a temperature-dependent sensor. The classical principle is based on the variation of a electrical resistance due to absorbed electromagnetic radiation. The bolometers are usually used in infrared radiation measurements since optical responsivity depends on the absorbed wavelength only through absorption. The performance of a bolometer is mainly determined by Temperature Coefficient of Resistance (TCR) and low thermal conduction.

LSMO is a promising candidate for fabricate uncooled bolometers thanks to the large TCR (around 2-3 % K -1 ) close to room temperature and a low-noise level compared with other resistive materials such as semiconductors (a-Si, a-Si:H, a-Ge, poly SiGe) and other oxide materials (semiconducting YBCO, VOx, etc.). An example of the electrical resistivity and the derivative of R versus T noted dR/dT is shown in figure 4. Non-suspended and free-standing LSMO bolometers have been investigated (Fig. 5) [START_REF] Liu | La0.7Sr0.3MnO3 suspended microbridges for uncooled bolometers made using reactive ion etching of the silicon substrates[END_REF][START_REF] Liu | Freestanding La0.7Sr0.3MnO3 suspended micro-bridges on buffered silicon substrates showing undegraded low frequency noise properties[END_REF][START_REF] Aryan | Measurement of thermal conductance of La0.7Sr0.3MnO3 thin films deposited on SrTiO3 and MgO substrates[END_REF]. The substrate below the active area can be removed by using standard micromachining techniques, which reduce thermal conductance and therefore enables to fabricate sensitive and fast sensors, while keeping very low intrinsic electrical noise. The thermal conductance can be reduced by 3 orders of magnitude (down to 10 -7 W•K -1 ) thus increasing the sensitivity by a factor of 1000. Thanks to the very low 1/f noise, and with appropriate geometries, ultra low Noise Equivalent Power (NEP) values around pW•Hz -1/2 could be obtained at 300 K. An example of an ultra low NEP value is given in Fig. 6 for a phonon noise limited bolometer.

B. LSMO uncooled low-field magnetoresistances

One major problem with low-field magnetic sensors is their excess noise at low frequency. Therefore, thanks to their low 1/f noise, LSMO may find important applications in magnetoresistive sensors. Moreover, LSMO exhibits a Curie temperature of about 350 K, which allows room temperature applications. An example of measured differential voltage of a Wheatstone bridge patterned in LSMO and relative magnetic field sensitivity as a function of the magnetic field μ H are shown in Fig. 7. The observed magnetoresistance is due to anisotropic effect in a ferromagnetic film showing uniaxial anisotropy. showing an uniaxial anisotropy, and (right axis) relative magnetic field sensitivity as a function of the magnetic field μ0H.

Some growth parameters such as the nature of the substrate, and the direction of the patterned stripes with respect to the easy magnetization axis, were modified in order to optimize the magnetic detectivity of the realized devices. Making use of uniaxial anisotropy in thin LSMO films on STO (001) and the Anisotropic MagnetoResistive effect (AMR), we could measure sensitivity of about 400%•T -1 .

Promising results were obtained on Wheatstone bridge geometries patterned in LSMO thin films. Magnetic noise was finally measured to be in the order of 200 pT•Hz -1/2 at 310 K in a frequency range corresponding to neuron activities (Fig. 8).

Fig. 1 .

 1 Fig. 1. -2 out-of-plane X-ray Diffraction (XRD) and resistivity versus temperature measurements, respectively, for (a) and (c) LSMO / STO (001); (b) and (d) LSMO/STO/Si (001).

Fig. 2 .

 2 Fig. 2. ε[0 0 1], electrical resistivity at 300 K, Tpeak and H/n versus LSMO film thickness in case of STO and STO/Si substrates.

  Fig 2.a), b) and c) as a function of film thickness, together with the normalized Hooge parameters H/n, which gives an indication of the 1/f noise level of the material, in Fig 2.d).

Fig. 3 .

 3 Fig. 3. Correlation between normalised Hooge parameter and electrical resistivity of LSMO thin films of various thicknesses on STO substrates and buffered Si substrates.

Fig. 4 .

 4 Fig. 4. Electrical resistivity (left axis) and derivative of the electrical resistance (right axis) versus temperature of a suspended 150 μm long and 4 μm wide bridges patterned in a 75 nm thick LSMO thin films.

Fig. 5 .

 5 Fig. 5. Examples of non-suspended meander of width 18 μm and freestanding LSMO bolometers of width 4 μm and length 50 μm.

Fig. 6 .

 6 Fig. 6. NEP of a 75 nm thick 2 μm wide suspended bolometer with different length.

Fig. 7 .

 7 Fig. 7. (left axis) Example of differential voltage of a Wheatstone bridge (current perpendicular to applied magnetic field, bias current = 200 μA, R=8100showing an uniaxial anisotropy, and (right axis) relative magnetic field sensitivity as a function of the magnetic field μ0H.

Fig. 8 .

 8 Fig. 8. Example of detectivity versus frequency at 0.2 mT and at different bias voltage (current perpendicular to applied magnetic field, R=8100 IV. CONCLUSIONS We have shown that LSMO thin films, associated with home-made readout and specific mask design, could be useful to fabricate state of the art uncooled bolometers and magnetoresistances.

ACKNOWLEDGMENT

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 737116 (byAxon)