Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation

Publication date: 7 November 2020

Abstract: Let

X^{(n)}

be an observation sampled from a distribution

P_{h e t a}^{(n)}

with an unknown parameter

h e t a,

h e t a

being a vector in a Banach space

E

(most often, a high-dimensional space of dimension

d

). We study the problem of estimation of

f (h e t a)

for a functional

f : E m a p s t o m a t h b b R

of some smoothness

s > 0

based on an observation

X^{(n)} s i m P_{h e t a}^{(n)} .

Assuming that there exists an estimator

h a t h e t a_{n} = h a t h e t a_{n} (X^{(n)})

of parameter

h e t a

such that

s q r t n (h a t h e t a_{n} - h e t a)

is sufficiently close in distribution to a mean zero Gaussian random vector in

E,

we construct a functional

g : E m a p s t o m a t h b b R

such that

g (h a t h e t a_{n})

is an asymptotically normal estimator of

f (h e t a)

with

s q r t n

rate provided that

s > f r a c 1 1 - a l p h a

and

d l e q n^{a l p h a}

for some

a l p h a i n (0, 1) .

We also derive general upper bounds on Orlicz norm error rates for estimator

g (h a t h e t a)

depending on smoothness

s,

dimension

d,

sample size

n

and the accuracy of normal approximation of

s q r t n (h a t h e t a_{n} - h e t a) .

In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.

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