1. Introduction. Several authors have recently investigated the power func-tion of the frequency x2-test. Eisenhart [1] and Patnaik [21 have obtained large sample expressions for the power of the simple goodness of fit x2-test (ie where the class probabilities are completely specified by the null hypothesis). The more complicated case, in which the parameters occurring in the expression for class probabilities require tp be estimated, has not received a unified treatment, although the problem has been treated in a number of specific situations by dif-ferent authors, including, Patnaik [3], Sillito [4], Stevens [5], Pearson and Mer-rington [6], Poti [7], Chiang [8] and Taylor [9]. Due to difficulties in obtaining the power function of the frequency X2-test in the usual manner, Cochran, in an expository article [10] has suggested the derivation of its Pitman limiting power [11], and he illustrated it in the case of the simple goodness of fit test. The colncept of asymptotic power suggested by Pitman has also been extensively used in various other areas like nonparametric inference (see eg Hoeffding and Rosenblatt [12]) and seems to be a useful tool for comparing alternative consistent tests or alternative designs for experimenta-tion, with regard to their performance in the immediate neighbourhood of the null hypothesis.

The consistency of the frequency X2-test has already been established by Neyman [13]. The object of the present paper is to obtain the Pitman limiting power of this test when the unknown parameters occurring in the specification of class probabilities are estimated from the sample by an asymptotically efficient method like the method of maximum likelihood, minimum x2 etc. In section 5, we discuss a few applications of the Pitman limiting power for fre-quency x2-tests.