Smooth Tests

Abstract

In this chapter we discuss smooth tests. This class of tests actually dates back from Neyman (1937), who developed a smooth test as a score test for which he proved some optimality properties. Although smooth tests are considered as nonparametric tests, they are actually constructed by first considering a k-dimensional smooth family of alternativesin which the hypothesised distribution is embedded. These smooth alternatives are the subject of Section 4.1. The tests are given in Section 4.2. The power of the test depends on how well the true distribution is approximated by the k-dimensional smooth alternative. In particular, for each data-generating distribution there exists an optimal order k. In Section 4.3 we discuss adaptive smooth tests of which the order is estimated from the data so that often the power is improved. Sections 4.1 up to 4.3 are limited to continuous distributions; smooth tests for discrete distributions are the topic of Section 4.4, and in Section 4.5 we show how smooth tests may be viewed from within a semiparametric framework. Finally, in Section 4.7 we give a brief summary from a practical viewpoint