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Expectation identities from integration by parts for univariate continuous random variables with applications to high-order moments

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Abstract

Inspired by the Conjugate Variables Theorem in physics, we provide a general expectation identity for univariate continuous random variables by utilizing integration by parts. We then apply the general expectation identity to some common univariate continuous random variables (normal, gamma (including chi-square and exponential), beta, double exponential, F, inverse gamma, logistic, lognormal, Pareto, t, uniform, and Weibul) and obtain their specific expectation identities from the general expectation identity. After that, we use the specific expectation identities to derive high-order moments of the corresponding univariate continuous random variables.

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Funding

The research was supported by the Ministry of Education (MOE) Project of Humanities and Social Sciences on the West and the Border Area (Grant No. 20XJC910001), the National Social Science Fund of China (Grant No. 21XTJ001), and the National Natural Science Foundation of China (Grant Nos. 11671060; 12001068; 72071019).

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Authors and Affiliations

  1. Department of Statistics and Actuarial Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Hong-Jiang Wu & Ying-Ying Zhang

  2. Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing University, Chongqing, China

    Ying-Ying Zhang & Han-Yu Li

  3. Department of Information and Computing Science, College of Mathematics and Statistics, Chongqing University, Chongqing, China

    Han-Yu Li

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  1. Hong-Jiang Wu
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  2. Ying-Ying Zhang
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Correspondence to Ying-Ying Zhang.

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Hong-Jiang Wu and Ying-Ying Zhang are co-first authors.

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Wu, HJ., Zhang, YY. & Li, HY. Expectation identities from integration by parts for univariate continuous random variables with applications to high-order moments. Stat Papers 64, 477–496 (2023). https://doi.org/10.1007/s00362-022-01329-5

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  • DOI: https://doi.org/10.1007/s00362-022-01329-5

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