-norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotellingʹs generalized

This paper is concerned with the distribution of a multivariate scale mixture variate X = ( X 1 , … , X p ) ′ with X i = S i Z i , where Z 1 , … , Z p are i . i . d . random variables, S i > 0 ( i = 1 , … , p ) , and { S 1 , … , S p } is independent of { Z 1 , … , Z p } . First we obtain L 1 -norm error bounds for an asymptotic expansion of the density function of X in the multivariate case as well as in the univariate case. Then the results are applied in obtaining error bounds for asymptotic expansions of the null distribution of Hotellingʹs generalized T 0 2 -statistic. The special features of our results are that our error bounds are given in explicit and computable forms. Further, their orders are the same as ones of the usual order estimates, and hence the paper provides a new proof for validity of the asymptotic expansions.