An Introduction to Probability and Statistics, 3rd Edition

13NONPARAMETRIC STATISTICAL INFERENCE

13.1 INTRODUCTION

In all the problems of statistical inference considered so far, we assumed that the distribution of the random variable being sampled is known except, perhaps, for some parameters. In practice, however, the functional form of the distribution is seldom, if ever, known. It is therefore desirable to devise methods that are free of this assumption concerning distribution. In this chapter we study some procedures that are commonly referred to as distribution-free or nonparametric methods. The term “distribution-free” refers to the fact that no assumptions are made about the underlying distribution except that the distribution function being sampled is absolutely continuous. The term “nonparametric” refers to the fact that there are no parameters involved in the traditional sense of the term “parameter” used thus far. To be sure, there is a parameter which indexes the family of absolutely continuous DFs, but it is not numerical and hence the parameter set cannot be represented as a subset of _n, for any . The restriction to absolutely continuous distribution functions is a simplifying assumption that allows us to use the probability integral transformation (Theorem 5.3.1) and the fact that ties occur with probability 0.

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