An invariance property of marginal density and tail probability approximations for smooth functions

Thomas J. DiCiccio, Michael A. Martin and G. Alastair Young

Statistics & Probability Letters, 1991, vol. 12, issue 3, 249-255

Abstract: Asymptotic approximations of marginal densities and tail probabilities for smooth functions of a continuous random vector, developed by Tierney, Kass and Kadane (1989) and DiCiccio and Martin (1991), respectively, in general fail to be invariant under transformations of the underlying random vector. This lack of invariance raises the complex issue of which choice of variables yields the most accurate approximations. However, in this paper, we show that the density and tail probability approximations remain invariant under transformations of the underlying variables if the joint density of the transformed variables is specified appropriately. The invariance property allows approximations to be computed using a convenient choice of variables, thereby avoiding the need for explicit nonlinear transformation of the underlying joint density. The utility of the invariance property is illustrated in an example involving the Studentized mean.

Keywords: Asymptotic; approximations; distribution; function; Laplace; approximation; non-linear; transformation; normal; approximation; saddlepoint; methods; Studentized; mean (search for similar items in EconPapers)
Date: 1991
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