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UID:6720@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20210222T140000
DTEND;TZID=Europe/Paris:20210222T160000
DTSTAMP:20210217T170954Z
URL:https://www.i2m.univ-amu.fr/events/christopher-fragneau-estimation-dan
s-le-modele-de-regression-single-index-en-grande-dimension/
SUMMARY:Christopher Fragneau (université Paris-Nanterre) : High dimensiona
l estimation in the monotone single-index model. -
DESCRIPTION:\n\n\n\nRésumé :\n\n \;\n\nHigh dimensional estimation in
the monotone single-index model\n\nFadoua Balabdaoui\, C ́ecile Durot an
d Christopher Fragneau\n\nAbstract\n\nI study the monotone single-index mo
del which assumes that a real variable Y is linked to a d dimensional real
vector X through the relationship E[Y |X] = Ψ0(α0T X) a.s.\, where the
real monotonic function Ψ0 and α0 are unknown. This model is well-known
in economics\, medecine and biostatistics where the monotonicity of Ψ0 ap
pears naturally. Given n replications of (X\, Y ) and assuming that α0 be
longs 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 d
ifferent M-estimation procedures: a least-squares procedure\, and a varian
t\, which consist in minimizing an appropriate criterion over general clas
ses K = F × C\, where F is a given closed subset of S\, and C is the set
of all non- decreasing real valued functions. The facts that the unknown i
ndex α0 is bundled into the unknown ridge function\, and that no smoothne
ss assumption is made on the ridge function\, make the estimation problem
very challenging.\n\nWith the goal of studying the asymptotic behavior of
both M-estimation procedures\, I first consider the population least-squar
es 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 cr
iterion\, as the sample size goes to infinity\, to the population least-sq
uares criterion. Moreover\, I prove existence of minimizers of the populat
ion least-squares criterion over K and I study the direction of variation
of this criterion in order to describe the minimizers.\n\nSecond\, I focus
on constrained least-squares estimators over K. In a setting where d depe
nds on n and the distribution of X is either bounded or sub-Gaussian\, I e
stablish the rates of convergence of the estimators of Ψ0(α0T )\, α0 an
d Ψ0 in case where (α0\, Ψ0) ∈ K\, as well as the consistency of esti
mators of Ψ0(α0T )\, otherwise. A simulation study of the estimators of
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\, particularly
in terms of support recovery of α0.\n\nThird\, I consider an estimation
method of (α0\, Ψ0) when X is assumed to be a Gaussian vector. This meth
od fits a mispecified linear model\, and estimates its parameter vector th
anks to the de-sparcified Lasso method of Zhang and Zhang (2014). I show t
hat the resulting estimator divided by its Euclidean norm is Gaussian and
converges to α0\, at parametric rate. I provide estimators of Ψ0(α0T )
and Ψ0\, and I establish their rates of convergence. The advantage of thi
s estimator as compared to the previous one is that it is less computation
aly expensive and it requires the choice of a tuning parameter and X is as
sumed to be Gaussian. A simulation study of the estimators of Ψ0(α0T )\,
α0 from both two M-estimation procedures on simulated data has shown goo
d performance\, particularly in terms of support recovery of α0.\n\n\n\n
CATEGORIES:Séminaire Statistique
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