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UID:6529@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20210222T140000
DTEND;TZID=Europe/Paris:20210222T160000
DTSTAMP:20241120T201732Z
URL:https://www.i2m.univ-amu.fr/evenements/christopher-fragneau-estimation
 -dans-le-modele-de-regression-single-index-en-grande-dimension/
SUMMARY: (...): Christopher Fragneau (université Paris-Nanterre) : High di
 mensional estimation in the monotone single-index model.
DESCRIPTION:: \n\n\n\nRésumé :\n\n&nbsp\;\n\nHigh dimensional estimation 
 in the monotone single-index model\n\nFadoua Balabdaoui\, C ́ecile Durot 
 and Christopher Fragneau\n\nAbstract\n\nI study the monotone single-index 
 model which assumes that a real variable Y is linked to a d dimensional re
 al vector X through the relationship E[Y |X] = Ψ0(α0T X) a.s.\, where th
 e real monotonic function Ψ0 and α0 are unknown. This model is well-know
 n in economics\, medecine and biostatistics where the monotonicity of Ψ0 
 appears naturally. Given n replications of (X\, Y ) and assuming that α0 
 belongs to S\, the d unit dimensional sphere\, my aim is to estimate (α0 
 \, Ψ0 ) in the high dimensional context\, where d is allowed to depend on
  n and to grow to innity with n.\n\nTo address this issue\, I consider two
  different M-estimation procedures: a least-squares procedure\, and a vari
 ant\, which consist in minimizing an appropriate criterion over general cl
 asses K = F × C\, where F is a given closed subset of S\, and C is the se
 t of all non- decreasing real valued functions. The facts that the unknown
  index α0 is bundled into the unknown ridge function\, and that no smooth
 ness assumption is made on the ridge function\, make the estimation proble
 m very challenging.\n\nWith the goal of studying the asymptotic behavior o
 f both M-estimation procedures\, I first consider the population least-squ
 ares criterion\n\n(α\, Ψ) → M(α\, Ψ) := E[(Ψ0(α0T X) − Ψ(αT X)
 )2].\n\nI establish the pointwise convergence over K of the least-squares 
 criterion\, as the sample size goes to infinity\, to the population least-
 squares criterion. Moreover\, I prove existence of minimizers of the popul
 ation least-squares criterion over K and I study the direction of variatio
 n of this criterion in order to describe the minimizers.\n\nSecond\, I foc
 us on constrained least-squares estimators over K. In a setting where d de
 pends on n and the distribution of X is either bounded or sub-Gaussian\, I
  establish the rates of convergence of the estimators of Ψ0(α0T )\, α0 
 and Ψ0 in case where (α0\, Ψ0) ∈ K\, as well as the consistency of es
 timators of Ψ0(α0T )\, otherwise. A simulation study of the estimators o
 f Ψ0(α0T )\, α0 on simulated data in case where F is the set of vector 
 of S with few nonzero components\, has shown good performance\, particular
 ly in terms of support recovery of α0.\n\nThird\, I consider an estimatio
 n method of (α0\, Ψ0) when X is assumed to be a Gaussian vector. This me
 thod fits a mispecified linear model\, and estimates its parameter vector 
 thanks to the de-sparcified Lasso method of Zhang and Zhang (2014). I show
  that the resulting estimator divided by its Euclidean norm is Gaussian an
 d converges to α0\, at parametric rate. I provide estimators of Ψ0(α0T 
 ) and Ψ0\, and I establish their rates of convergence. The advantage of t
 his estimator as compared to the previous one is that it is less computati
 onaly expensive and it requires the choice of a tuning parameter and X is 
 assumed to be Gaussian. A simulation study of the estimators of Ψ0(α0T )
 \, α0 from both two M-estimation procedures on simulated data has shown g
 ood performance\, particularly in terms of support recovery of α0.\n\n\n\
 n
CATEGORIES:Séminaire,Statistique
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