On the second order local comparison between perturbed maximum likelihood estimators and Rao's statistic as test statistics

A bias-adjusted maximum likelihood estimator (mle), which has been shown to possess certain optimality criteria as an estimate of [theta] (see Ghosh, J. K., Sinha, B. K., and Joshi, S. N. (1982). In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, Eds.), Vol. 1, pp. 403-456. Academic Press, Orlando, FL) is compared with Rao's statistic as a test statistic for the standard two-sided testing problem. It is shown that Rao's statistic is locally superior to any bias-adjusted mle in the sense of Chandra and Joshi (1983, Sankhya Ser. A 45 226-246). A second interpretation of a conjecture of C. R. Rao is proposed and Rao's statistic is shown to be superior as a test statistic according to the new interpretation as well. The last fact provides an interesting supplement to the results of Chandra and Joshi (1983, Sankhya Ser. A 45 226-246). Furthermore, a partial reason for the inferiority of the likelihood ratio and Wald's statistic to Rao's statistic is supplied and certain regularity assumptions of the last paper are eliminated. Finally, the local powers of certain modified versions of Rao's and Wald's statistics (see Skovgaard, I. M. (1985). Ann. Statist. 13 534-551) are studied.