Examples of inconsistency of the likelihood ratio statistic

Abstract

In testing a null hypothesis on the basis of independent and identically distributed observations let T^ˆ_n be the normalized likelihood ratio statistic and let L^ˆ_n be the level attained by T^ˆ_n when the sample size is n. It is known (cf. Bahadur (1965)) that in typical cases T^ˆ_n→J and n^-1log L^ˆ_n→-J as n→∞, where J is a positive constant defined in terms of the Kullback-Leibler information numbers. This note gives examples where these properties of likelihood ratios do not hold; in some example L^ˆ_n does not even tend to zero. The examples show that some of the regularity assumptions of Bahadur (1965) are essential in the general case.