A model for perturbed production or measurement processes involving compound normal distributions

摘要

To solve an easy-to-understand real world problem, we invite the reader to travel through many useful concepts related to the important models provided by the compound normal distributions. This class of distributions is a subclass of the multivariate elliptical distributions that allows a clearcut interpretation of a data generating mechanism useful in certain situations. A multivariate elliptical distribution is characterized by the existence of a particular function φ. In the special case of a compound normal distribution, the function φ appears to be simply the (one-sided) Laplace-Stieltjes transform (LST) of the mixing distribution. This fact allows in particular to easily express the covariance matrix and the kurtosis parameter of the compound normal in terms of derivatives, evaluated at the origin, of the LST of the mixing distribution. An example of application of the results is the asymptotic theory for canonical correlation analysis. The asymptotic distributions of the sample canonical correlation coefficients (and of statistics 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 kurtosis parameter. Here we focus our attention on the inference on the simple correlation coefficient ϱ of the components of bivariate compound normal distributions. We illustrate with a simple example in industrial engineering how inference on ϱ is affected by the choice of the associated mixing distribution.