The heat equation and Stein's identity: Connections,applications |
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| Affiliation: | 1. University of Pennsylvania, USA;2. Department of Statistics, Purdue University, West Lafayette, IN 47907, USA;3. University of California, San Diego, CA, USA;4. Rutgers University, USA;1. McMaster University, Hamilton, Canada;2. University of Illinois at Chicago, USA;1. Mathematics Research Unit, University of Luxembourg, Campus Kirchberg, 6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Grand-Duchy of Luxembourg, Luxembourg;2. Faculté des Sciences de Tunis, Campus Universitaire 2092 - El Manar Tunis, Tunisie;1. University of Nova Gorica, Nova Gorica, Slovenia;2. Institute of Metals and Technology, Ljubljana, Slovenia;3. Taiyuan University of Technology, China;1. Departamento de Matemática Aplicada I, Universidad de Sevilla, Spain;2. Faculty of Mathematics and Sciences, University of Applied Sciences Zittau/Görlitz, Germany;3. Departamento de Análisis Matemático, Universidad de Alicante, Spain |
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| Abstract: | This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for moments, an improvement and a reversal of Jensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory. Some examples are also given. |
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