David Harris (Monash University)

Title: The Role of the Support in Semi-Parametric Hypothesis Testing: the Case of Testing Independence in Count Data

Abstract

This paper looks at some issues that arise when testing low counts for independence in a semi-parametric framework. When treating distributions non-parametrically little or no attention is typically given to the nature of the support on which probabilities are defined. In the case of looking at low counts it is perfectly feasible that data might be generated by a binomial distribution, say, where both parameters are unknown. In this case the support is finite with an unknown upper bound. It turns out that classical likelihood based tests, derived under the assumption of an infinite support, have a size of either zero or one should the support be finite or contain gaps. The reason for this phenomenon is that, with this type of restriction, the support of the observations now depends on the value of the parameter under test ie. the support of the null is potentially different from the support under the alternative. The effective score test is robust to these types of restrictions on the support and remains asymptotically normal when standardised at the usual T^{-1/2} rate. Remarkably though, the effective score is found to have power against local alternatives that shrink to the null at the rate T⁻¹ when the support has gaps.