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<p>Let (<i>X</i>,&#xa0;<i>Y</i>) denote a two-dimensional random vector with a standard bivariate normal distribution. Here we consider the case in which <i>X</i> is unobserved and <i>Y</i> is only observed if <i>X</i> takes values in some specified interval. Then the distribution of the observed values of <i>Y</i> is the conditional distribution of <i>Y</i> given that <i>X</i> 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 <i>Y</i> given that <i>X</i> is in some specified interval is given; this result can be used to find the conditional moments of <i>Y</i>. An expansion of the conditional expectation of a function of <i>Y</i> in terms of Hermite polynomials is derived and this result is used to derive expansions for the conditional distribution function of <i>Y</i>, for the distribution function of the skew-normal distribution and for Owen’s T-function.</p>

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

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  • 发表日期：

    2026年1月14日

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  • Thomas A. Severini

摘要

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.