The bound in the Berry-Esseen result for minimum contrast estimates

Summary

Let (X,

) be a measurable space, Θ⊂∔ an open interval,P _ϑ|

, ϑ∈Θ, a family of probability measures, and ϑ _n , n∈ℕ a sequence of minimum contrast estimates.Pfanzagl [1971] shows that — under certain regularity conditions — for every ϑ∈Θ there exists a constantc _ϑ such that for alln∈ℕt∈∔,

$$|p_\vartheta ^n \left\{ {x \in X^n :n^{1/2} \frac{{\vartheta _n (x) - \vartheta }}{{\beta (\vartheta )}}< t} \right\} - \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^t {\exp } \left[ {\frac{1}{2}r^2 } \right]dr| \leqq c_\vartheta n^{ - 1/2} .$$

Pfanzagl's proof uses compactness arguments and yields therefore only the existence ofc _ϑ, but gives no method to determine its value.

In this paper we shall show howc _ϑ can be estimated. This is done under an additional condition which, in the case of maximum likelihood estimates, reduces to the condition that the likelihood equation has one solution only.