Wilks Theorem for penalized maximum likelihood estimators

This paper extends the famous statistical results like Fisher Theorem and Wilks phenomenon to the penalized maximum likelihood estimation with a quadratic penalization. Spokoiny (2013a) offered a novel approach which allows to study the properties of the quasi maximum likelihood estimators for finite samples and possible model misspecification. The results from Spokoiny (2013a) also apply for a growing parameter dimension p , however under the constraint “ p/n is small”, where n is the sample size. This paper shows that in the case of the penalized maximum likelihood estimation, the results can be applied to arbitrarily large or even infinite dimension p of the parameter space. The error bounds depend on the so called effective dimension ps which can be much smaller than the true dimension p of the parameter space. We particularly show for the i.i.d. case that the results apply under the condition “ p s /n is small”. AMS 2000 Subject Classification: Primary 62F10. Secondary 62J12,62F25,62H12