Non-parametric sequential tests for symmetry

Abstract

Let X1, X2,... be a sequence of independent and identically distributed random variables (i.i.d.r.v.) with a continuous distribution function (d.f.) F. For testing the hypothesis of symmetry several non-parametric sequential tests have been proposed. We shall mention only a few articles which are next in spirit to the present paper. Assuming Lehmann alternatives Weed, Bradley and Govindarajulu (1974) constructed an invariant sequential probability ratio test (SPRT). No approximations to the operating characteristic (OC) or the average sample number (ASN) seem to be known. Sen and Ghosh (1974), motivated by a procedure by Cox, developed a general class of sequential signed-rank tests for the location model. Limiting expressions for the OCand the ASN-curve are derived and asymptotic relative efficiencies are computed. Nevertheless, these tests suffer from two drawbacks. First, at every stage an unknown parameter has to be estimated. Second, the kind of Wiener approximation applied requires an increasing initial sample size. Modifying the Pyke-Shorack technique, Braun (1976) proved a functional limit theorem for two-sample linear rank statistics and heuristically derived the concept of Pitman efficiency in the sequential case. Hall and Loynes (1977) extended Le Cain's theory to a sequential analysis setting, presented applications to various tests and announced further results in this direction. Meanwhile the present author learnt from a yet unpublished paper by Lai (1977) in which a sequential version of the classical PitmanNoether theorem together with a uniform functional limit theorem and a result on certain large deviation probabilities have been derived. In this paper we consider the construction of sequential tests by means of the familiar linear signed-rank statistics. Along with the preliminary notations SPRT-type tests and some of their properties are studied in Section 2. In Section 3 the necessary distribution theory is derived. This includes a Chernoff-Savage representation cf. Sen and Ghosh (1973) and Sen (1975), and a "uniform"