A new definition of form-invariance matrix variate distributions

Abstract

Form-invariance is the most practical and desired property of weighted distributions. Weighted distributions commonly arise as univariate probability models in many fields of applications, for example, survival data analysis in medical science. However, the explicit global definition of "form-invariance" seems to be absent (to our knowledge) in spite of the fact that there is a growing use of such distributions during the last three decades. Therefore, in this note an effort is made to provide a new definition of form-invariance with reference to the matrix variate distributions. Obviously this definition and related implications are applicable to the multivariate and univariate distributions (as it should be), even though not necessarily to all families of probability distributions that exist, but definitely to those that are commonly used in practice. To demonstrate the validity of the last statement certain results, based on our new definition of form-invariance, are proved as the characterizations of the class of matrix elliptical distributions. Also considered in this note are certain extensions of our results, as may seem appropriate within the scope of the theory of matrix variate distributions. The concept of form-invariance is extended to marginal and conditional structures, as well as to the sum of independent matrix variates from elliptical distributions