A probability inequality for elliptically contoured densities with applications in order restricted inference. (English) Zbl 0609.62086

Mukerjee, Hari; Robertson, Tim; Wright, F. T.

Ann. Stat. 14, 1544-1554 (1986).

Let f(x) be an elliptically contoured density on \({\mathbb{R}}^ k\), let \(\mu \in {\mathbb{R}}^ k\), and define \(h(\delta)=\int_{A-\delta \mu}f(x)dx\), for real-valued \(\delta\), to be the translate of the integral of f(\(\cdot)\) over A in the direction \(\mu\). For a certain class of asymmetric convex sets A, which includes the closed convex cones, it is shown that h(\(\delta)\) is nonincreasing in \(| \delta |\). This result is then used to establish certain monotonicity properties for the power function in the likelihood ratio test for testing whether or not the mean \(\mu\) of a multivariate normal distribution satisfies a specified ordering.

Reviewer: J.Kent

MSC:

62H15	Hypothesis testing in multivariate analysis
62F03	Parametric hypothesis testing
60E15	Inequalities; stochastic orderings

Keywords:

order restricted tests; Anderson’s inequality; elliptically contoured density; asymmetric convex sets; closed convex cones; monotonicity properties; power function; likelihood ratio test; multivariate normal distribution