A Note on the Application of Hermite Polynomials to Hidden Truncation Models

Let ( X , Y ) denote a two-dimensional random vector with a standard bivariate normal distribution. Here we consider the case in which X is unobserved and Y is only observed if X takes values in some specified interval. Then the distribution of the observed values of Y is the conditional distribution of Y given that X falls in that interval. In this note, the use of Hermite polynomials in analyzing the distributional properties of such models is considered. A general expression for the conditional expectation of polynomial functions of Y given that X is in some specified interval is given; this result can be used to find the conditional moments of Y . An expansion of the conditional expectation of a function of Y in terms of Hermite polynomials is derived and this result is used to derive expansions for the conditional distribution function of Y , for the distribution function of the skew-normal distribution and for Owen’s T-function.