## Error bounds for asymptotic expansion of

the conditional variance of the scale mixtures of

the 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**=*A*^{1/2}**G** be a scale mixture of a multivariate normal distribution with **X**, **G** \in **R**^{n}, **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(**X**_{2} | **X**_{1}), **X**_{1} \in **R**^{m}, *m* < *n*, under some moment expressions. A new representation form is also presented for conditional expectation of the scale variable on the random vector
**X**_{1} \in **R**^{m}, *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.

**Source**
( TeX )