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Approximating moments of continuous functions of random variables using Bernstein polynomials |
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| Affiliation: | 1. Department of Statistics, University of Florida, P.O. Box 118545, Gainesville, FL 32611-8545, USA;2. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India;3. Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY, USA;1. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran;2. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran;1. Department of Mathematics, Netaji Subhash Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India;2. Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece;1. Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka, 572-8508, Japan;2. Research Institute of Capital Formation, Development Bank of Japan, 9-7, Otemachi 1-chome, Chiyoda-ku, Tokyo 100-8178, Japan;3. Institute for Research in Contemporary Political and Economic Affairs, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan |
| Abstract: | Bernstein polynomials have many interesting properties. In statistics, they were mainly used to estimate density functions and regression relationships. The main objective of this paper is to promote further use of Bernstein polynomials in statistics. This includes (1) providing a high-level approximation of the moments of a continuous function of a random variable , and (2) proving Jensen’s inequality concerning a convex function without requiring second differentiability of the function. The approximation in (1) is demonstrated to be quite superior to the delta method, which is used to approximate the variance of with the added assumption of differentiability of the function. Two numerical examples are given to illustrate the application of the proposed methodology in (1). |
| Keywords: | Balanced and unbalanced data Delta method Heritability function Jensen’s inequality Polynomial approximation Tchebycheff polynomials Uniform convergence Weierstrass approximation theorem |
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