By R. R. Bahadur

A dialogue of a few issues within the thought of enormous deviations equivalent to moment-generating features and Chernoff's theorem, and of facets of estimation and trying out in huge samples, akin to distinctive slopes of try out information.

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**Extra info for Some Limit Theorems in Statistics (CBMS-NSF Regional Conference Series in Applied Mathematics)**

**Example text**

31) that so {T n ] is asymptotically efficient. Suppose now that g(0) is a continuously differentiate function over («, b) and g'(0) ^ 0 for a < 0 < b. Let Un = g(Tn). It can be shown by an elaboration of the preceding argument that { U n } is an asymptotically efficient estimate of g. 26 R. R. BAHADUR Notes. 1 is contained in unpublished work of Stem. The first published statement of the lemma seems to be in [C2]. 1 is based on the proof in [B9]. 1 is due to Bahadur [B6]. 7. Exact slopes of test statistics.

41 (1970), 1695-1699. [R2] C. R. 139-148. [R3] J. S. RAO, Bahadur efficiencies of some tests of uniformity on the circle, Ann. Math. , to appear. [SI] I. N. SANOV, On the probability of large deviations of random variables, Sel. Trans. Math. Statist. , 1 (1957), pp. 213-244. [S2] J. SETHURAMAN, On the probability of large deviations of families of sample means, Ann. Math. , 35 (1964), pp. 1304-1316. [S3] G. L. , 40 (1969), pp. 19081921. [Wl] A. WALD, Sequential Analysis, John Wiley, New York, 1947.

Since / is continuous and e is arbitrary we conclude that lim,,^ n~1 log Ln = —f(b). Remark 1. Suppose 9 is a point in 0 0 . 3) holds wit c(9) — 0. 5 below and Ln ^ 1. Remark 2. 2 it may well be the case that {T*} does, where, for each n, T* is equivalent to Tn in the sense that T* = (pn(Tn), where