Consistency of Estimators and Order of Consistency

Summary

Suppose that X₁, X₂, …, X_n, … is a sequence of independent identically distributed (i.i.d.) random variables. We assume that a parameter space (H) is an open subset in a Euclidean p-space. In the textbook discussion of asymptotic theory, it is usually shown that the asymptotically best (in some sense or other) estimator ${\rm{\hat \theta }}_{\rm{n}}^*$ has an asymptotic distribution of order $\sqrt {\rm{n}}$ , in the sense that the distribution of $\sqrt {\rm{n}} ({\rm{\hat \theta }}_{\rm{n}}^* - {\rm{\theta }})$ converges to some probability law (in most cases normal). There are sporadic examples where the distribution of $\sqrt {\rm{n}} ({\rm{\hat \theta }}_{\rm{n}}^* - {\rm{\theta }})$ or $\sqrt {{\rm{n log n}}} ({\rm{\hat \theta }}_{\rm{n}}^* - {\rm{\theta }})$ converges to some law (Woodroofe [57]) when X_i’s are i.i.d. random variables with an uniform distribution or a truncated distribution. The purpose of this chapter is to give a systematic treatment to the problem of whether for a given sequence $\{ {{\rm{c}}_{\rm{n}}}\} ,{{\rm{c}}_{\rm{n}}}({\rm{\hat \theta }}_{\rm{n}}^* - {\rm{\theta }})$ converges to some law, and what is the possible bound for such a sequence. In the location parameter case it will be shown that such a bound can be explicitely given. The asymptotic distribution of ${{\rm{c}}_{\rm{n}}}(\hat \theta _{\rm{n}}^* - \theta )$ and the bound for it in non-regular cases is discussed by Akahira [2]. Also some results in terms of the asymptotic distribution of estimators are given in Takeuchi [42]. Asymptotic sufficiency of consistent estimators is discussed by Akahira [5] in non-regular cases.