A note on critical dimensions in profile semiparametric estimation

TL;DR

This paper explores the critical dimensions in profile semiparametric estimation, providing examples that clarify the necessity of smoothness constraints and analyzing how critical ratios behave under different conditions.

Contribution

It offers new insights into the role of smoothness constraints and the behavior of critical ratios in profile semiparametric models, extending previous results.

Findings

Smoothness constraint is necessary for the critical ratio bound.

Critical ratio for Fisher type results remains constant when target dimension is proportional to full dimension.

Critical ratio for Wilks phenomenon scales with the square root of the full dimension.

Abstract

This paper complements the results of Andresen et. al "Critical dimension in profile semiparametric estimation" (2014) on profile estimators in semiparametric models. We present two examples. One that illustrates that the smoothness constraint on the expected value of the contrast functional used to define the profile M-estimator is necessary for the bound derived for the critical ratio of dimension to sample size. A second one to show that in the case that the target dimension is proportional to the full dimension the critical ratio for the Fisher type result stays the same while for the Wilks phenomenon it is multiplied with the square root of the full dimension, just as in the upper bound in Andresen et. al (2014).