On the Differentiation of Certain Probabilities with Applications in Statistical Decision Theory

Abstract

Let P _λ be an absolutely continuous probability distribution on p-dimensional Euclidean space, dP _λ = f _x ,dx, where λ ≥ 0 is a parameter. Under certain smoothness conditions on fm, _λ and the boundary of B _λ, a formula is obtained for (∂/∂λ) P _λ (B _λ). This result is useful over a wide range of applications, and two examples appear in the following. First, we compute the UMVUE of θ from a N(θ, a θ + b) distribution (where a and b are known). In this case, the methodology can be viewed as a means of expediting Rao-Blackwell calculations. Another application is motivated by the papers of Hwang and Casella (1982) and (1984) who present improved confidence sets for the mean of a N _p (θ, I) distribution. In these separate papers, Hwang and Casella exploit two different methods of differentiating the coverage probability. The two ideas are subsumed (and unified) by our general result, and some new representations of the derivative are obtained as well. Finally, our method shows promise as being a useful technique for studying the important problem of finding improved confidence sets for the normal mean vector when the scale parameter is unknown.

The authors are grateful to Bruce Driver for some helpful suggestions which led to the formulation of Theorem 1.1. They are grateful, also, to Anirban DasGupta who found an error in the original formulation.

Research supported by NSF Grant DMS 89-01001.

Research supported by NIAMS 1 P60. AR40770.