On Randomized Rank Score Procedures of Bell and Doksum

Abstract

Bell and Doksum (1965) proposed a new class of nonparametric tests which have the advantage of possessing exact and well tabulated distributions under the null hypothesis. The basic departure from the usual nonparametric tests, suggested by the above authors is that of taking an additional sample from a known distribution such as normal, uniform, exponential, etc. and use those observations as the rank scores. Throughout this paper such procedures are called `Randomized Rank Score' (RRS) as opposed to the usual Rank Score (RS) procedures. Bell and Doksum (1965) have shown that the RRS tests have the same Pitman efficiency behaviour as the corresponding RS tests. One could interpret this result by saying that the effect of the superimposed noise of the additional sample dissipates near the null hypothesis as the sample size tends to infinity. However, as will be shown here, for the finite sample sizes the noise does create undesirable properties for the RRS tests. Firstly, in most of the familiar testing problems the power of the RRS test remains bounded away from unity as the parameter varies over the entire region of the alternative. Secondly, the confidence sets based on RRS procedure have the following peculiar property. With positive probability, the confidence set becomes the whole parameter space and the procedure completely disregards the observations of the experiment.