A note on critical dimensions in profile semiparametric estimation

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).