## Error bounds for asymptotic expansion ofthe conditional variance of the scale mixtures ofthe multivariate normal distribution

### Stergios B. Fotopoulos and Lijian He

Department of Management and Systems, and the Program in Statistics,
Washington State University, Pullman, WA 99164-4726, U.S.A.

(Received July 7, 1997; revised February 9, 1998)

Abstract.    Let X=A1/2G be a scale mixture of a multivariate normal distribution with X, G \in Rn, G is a multivariate normal vector, and A is a positive random variable independent of the multivariate random vector G. This study presents asymptotic results of the conditional variance-covariance, Cov(X2 | X1), X1 \in Rm, m < n, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector X1 \in Rm, m < n. Both the asymptotic expression and the representation are manageable and in computable form. Finally, an example is presented to illustrate how the computations are carried out.

Key words and phrases:    Heteroscedasticity, orthogonal polynomials, Laguerre polynomials, Laplace transform.

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