Examples of inconsistency of the likelihood ratio statistic

Bahadur, R. R. (1972) Examples of inconsistency of the likelihood ratio statistic Sankhya, 34 (1). pp. 81-84. ISSN 0972-7671

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In testing a null hypothesis on the basis of independent and identically distributed observations let Tˆn be the normalized likelihood ratio statistic and let Lˆn be the level attained by Tˆn when the sample size is n. It is known (cf. Bahadur (1965)) that in typical cases Tˆn→J and n-1log Lˆn→-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 Lˆn does not even tend to zero. The examples show that some of the regularity assumptions of Bahadur (1965) are essential in the general case.

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