A MODEL FOR PERTURBED PRODUCTION OR MEASUREMENT PROCESSES INVOLVING COMPOUND NORMAL-DISTRIBUTIONS

To solve an easy-to-understand real world problem, we invite the reade r to travel through many useful concepts related to the important mode ls provided by the compound normal distributions. This class of distri butions is a subclass of the multivariate elliptical distributions tha t allows a clearcut interpretation of a data generating mechanism usef ul in certain situations. A multivariate elliptical distribution is ch aracterized by the existence of a particular function phi. In the spec ial case of a compound normal distribution, the function phi appears t o be simply the tone-sided) Laplace-Stieltjes transform (LST) of the m ixing distribution. This fact allows in particular to easily express t he covariance matrix and the kurtosis parameter of the compound normal in terms of derivatives, evaluated at the origin, of the LST of the m ixing distribution. An example of application of the results is the as ymptotic theory for canonical correlation analysis. The asymptotic dis tributions of the sample canonical correlation coefficients land of st atistics used for testing hypotheses about the population coefficients ) have very simple forms in the case of compound normal distributions. They depend on the LST of the mixing distribution through the kurtosi s parameter. Here we focus our attention on the inference on the simpl e correlation coefficient rho of the components of bivariate compound normal distributions. We illustrate with a simple example in industria l engineering how inference on rho is affected by the choice of the as sociated mixing distribution. (C) 1998 Elsevier Science Inc. All right s reserved.