Finite sample analysis of profile M-estimation in the Single Index model

TL;DR

This paper provides a finite sample analysis of profile M-estimators in the single index model, demonstrating convergence, confidence set construction, and finite-sample properties using wavelet approximations.

Contribution

It extends profile M-estimation theory to finite samples in the single index model with wavelet link function approximation, including convergence and confidence set results.

Findings

Wilks phenomenon and Fisher Theorem established in finite samples

Alternating maximization converges to the global maximum

Finite-sample bounds and confidence sets derived

Abstract

We apply the results of Andresen A. and Spokoiny V. on profile M-estimators and the alternating maximization procedure to analyse a sieve profile quasi maximum likelihood estimator in the single index model with linear index function. The link function is approximated with \(C^3\)-Daubechies-wavelets with compact support. We derive results like Wilks phenomenon and Fisher Theorem in a finite sample setup. Further we show that an alternation maximization procedure converges to the global maximizer and assess the performance of a projection pursuit procedure in that context. The approach is based on showing that the conditions of Andresen A. and Spokoiny V. on profile M-estimators and the alternating maximization procedure can be satisfied under a set of mild regularity and moment conditions on the index function, the regressors and the additive noise. This allows to construct nonasymptotic confidence sets and to derive asymptotic bounds for the estimator as corollaries.