A MULTIVARIATE VERSION OF STEIN'S IDENTITY WITH APPLICATIONS TO MOMENT CALCULATIONS AND ESTIMATION OF CONDITIONALLY SPECIFIED DISTRIBUTIONS

Abstract

Stein's identity, which involves integration by parts of E(g′(X)) when X has an exponential family distribution, is extended to a multivariate multiparameter setting. The resulting identity is useful for generating recurrence formulae for mixed moments and for deriving consistent moment based estimates of parameters. The technique is illustrated in a spectrum of one dimensional multiparameter cases and in a variety of bivariate conditionally specified distribution settings. Analogous identities can sometimes be derived in non-exponential family cases. An example is provided. Finally a simulation study is presented to illustrate the performance of the moment based estimates derived using the generalized Stein identity.

Keywords:

• Exponential families
• Exponential conditionals
• Normal conditionals
• Pareto conditionals
• Consistent estimates
• Recurrence relations for moments

ACKNOWLEDGMENTS

The authors are grateful to the Universities of Castilla-La Mancha and Cantabria, the Dirección General de Investigación Científica y Técnica (DGICYT) (project PB98-0421), and to Iberdrola for partial support of this research.