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

A note on critical dimensions in profile semiparametric estimation