Bahadur efficiency

The large sample study of test statistics in a given hypotheses testing problem is commonly based on the following concept of asymptotic Bahadur efficiency [a1], [a2] (cf. also Statistical hypotheses, verification of). Let and be the parametric sets corresponding to the null hypothesis and its alternative, respectively. Assume that large values of a test statistic (cf. Test statistics) based on a random sample give evidence against the null hypothesis. For a fixed and a real number , put and let . The random quantity is the -value corresponding to the statistic when is the true parametric value. For example, if , the null hypothesis is rejected at the significance level . If for with -probability one,

then is called the Bahadur exact slope of . The larger the Bahadur exact slope, the faster the rate of decay of the -value under the alternative. It is known that for any , , where is the information number corresponding to and . A test statistic is called Bahadur efficient at if

The concept of Bahadur efficiency allows one to compare two (sequences of) test statistics and from the following perspective. Let , , be the smallest sample size required to reject at the significance level on the basis of a random sample when is the true parametric value. The ratio gives a measure of relative efficiency of to . To reduce the number of arguments , and , one usually considers the random variable which is the limit of this ratio, as . In many situations this limit does not depend on , so it represents the efficiency of against at with the convenient formula

where and are the corresponding Bahadur slopes.

To evaluate the exact slope, the following result ([a2], Thm. 7.2) is commonly used. Assume that for any with -probability one as , and the limit exists for taking values in an open interval and is a continuous function there. Then the exact slope of at has the form . See [a4] for generalizations of this formula.

The exact Bahadur slopes of many classical tests have been found. See [a3].

References

[a1]	R.R. Bahadur, "Rates of convergence of estimates and tests statistics" Ann. Math. Stat. , 38 (1967) pp. 303–324
[a2]	R.R. Bahadur, "Some limit theorems in statistics" , Regional Conf. Ser. Applied Math. , SIAM (1971)
[a3]	Ya.Yu. Nikitin, "Asymptotic efficiency of nonparametric tests" , Cambridge Univ. Press (1995)
[a4]	L.J. Gleser, "Large deviation indices and Bahadur exact slopes" Statistics and Decision , 1 (1984) pp. 193–204