Applications of Mathematics, Vol. 66, No. 5, pp. 767-788, 2021


On lower bounds for the variance of functions of random variables

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran

Received Feburary 17, 2020.   Published online June 2, 2021.

Abstract:  In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable $X$, we find a characterization of the weighted function corresponding to density function $f(x)$, in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy and between variance past lifetime and dynamic cumulative past entropy. Moreover, we find lower bounds for the variance of functions of weighted random variables with specific weight functions applicable in reliability under suitable conditions.
Keywords:  variance bound; Chernoff inequality; size-biased distribution; reliability measure; dynamic cumulative residual entropy; dynamic cumulative past entropy
Classification MSC:  60E15


References:
[1] M. Asadi, Y. Zohrevand: On the dynamic cumulative residual entropy. J. Stat. Plann. Inference 137 (2007), 1931-1941. DOI 10.1016/j.jspi.2006.06.035 | MR 2323874 | Zbl 1118.62006
[2] T. Cacoullos: On upper and lower bounds for the variance of the function of a random variable. Ann. Probab. 10 (1982), 799-809. DOI 10.1214/aop/1176993788 | MR 0659549 | Zbl 0492.60021
[3] T. Cacoullos, V. Papathanasiou: On upper bounds for the variance of functions of random variables. Stat. Probab. Lett. 3 (1985), 175-184. DOI 10.1016/0167-7152(85)90014-8 | MR 0801687 | Zbl 0572.60021
[4] T. Cacoullos, V. Papathanasiou: Characterizations of distributions by variance bounds. Stat. Probab. Lett. 7 (1989), 351-356. DOI 10.1016/0167-7152(89)90050-3 | MR 1001133 | Zbl 0677.62012
[5] T. Cacoullos, V. Papathanasiou: A generalization of covariance identity and related characterization. Math. Methods Stat. 4 (1995), 106-113. MR 1324694 | Zbl 0831.62013
[6] T. Cacoullos, V. Papathanasiou: Characterizations of distributions by generalizations of variance bounds and simple proofs of the CLT. J. Stat. Plann. Inference 63 (1997), 157-171. DOI 10.1016/S0378-3758(97)00008-6 | MR 1491576 | Zbl 0922.62009
[7] L. H. Y. Chen: An inequality for the multivariate normal distribution. J. Multivariate Anal. 12 (1982), 306-315. DOI 10.1016/0047-259X(82)90022-7 | MR 0661566 | Zbl 0483.60011
[8] H. Chernoff: A note on an inequality involving the normal distribution. Ann. Probab. 9 (1981), 533-535. DOI 10.1214/aop/1176994428 | MR 0614640 | Zbl 0457.60014
[9] A. Di Crescenzo, L. Paolillo: Analysis and applications of the residual varentropy of random lifetimes. To appear in Probab. Eng. Inf. Sci. DOI 10.1017/S0269964820000133
[10] R. A. Fisher: The effects of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics 6 (1934), 13-25. DOI 10.1111/j.1469-1809.1934.tb02105.x
[11] F. Goodarzi, M. Amini, G. R. Mohtashami Borzadaran: On upper bounds for the variance of functions of random variables with weighted distributions. Lobachevskii J. Math. 37 (2016), 422-435. DOI 10.1134/S1995080216040089 | MR 3528019 | Zbl 1347.62030
[12] F. Goodarzi, M. Amini, G. R. Mohtashami Borzadaran: Characterizations of continuous distributions through inequalities involving the expected values of selected functions. Appl. Math., Praha 62 (2017), 493-507. DOI 10.21136/AM.2017.0182-16 | MR 3722901 | Zbl 06819518
[13] F. Goodarzi, M. Amini, G. R. Mohtashami Borzadaran: Some results on upper bounds for the variance of functions of the residual life random variables. J. Comput. Appl. Math. 320 (2017), 30-42. DOI 10.1016/j.cam.2017.01.001 | MR 3624716 | Zbl 1368.60021
[14] G. R. Mohtashami Borzadaran: A note on continuous exponential families. Thai J. Math. 8 (2010), 555-563. MR 2763677 | Zbl 1229.62011
[15] G. R. Mohtashami Borzadaran, D. N. Shanbhag: Further results based on Chernoff-type inequalities. Stat. Probab. Lett. 39 (1998), 109-117. DOI 10.1016/S0167-7152(98)00036-4 | MR 1652512 | Zbl 1094.62503
[16] N. U. Nair, K. K. Sudheesh: Characterization of continuous distributions by variance bound and its implications to reliability modeling and catastrophe theory. Commun. Stat., Theory Methods 35 (2006), 1189-1199. DOI 10.1080/03610920600629443 | MR 2328470 | Zbl 1105.62016
[17] N. U. Nair, K. K. Sudheesh: Characterization of continuous distributions by properties of conditional variance. Stat. Methodol. 7 (2010), 30-40. DOI 10.1016/j.stamet.2009.08.003 | MR 2744442 | Zbl 1232.62038
[18] A. K. Nanda, H. Singh, N. Misra, P. Paul: Reliability properties of reversed residual lifetime. Commun. Stat., Theory Methods 32 (2003), 2031-2042. DOI 10.1081/STA-120023264 | MR 2002004 | Zbl 1156.62360
[19] J. Navarro, Y. del Aguila, M. Asadi: Some new results on the cumulative residual entropy. J. Stat. Plann. Inference 140 (2010), 310-322. DOI 10.1016/j.jspi.2009.07.015 | MR 2568141 | Zbl 1177.62005
[20] G. Psarrakos, J. Navarro: Generalized cumulative residual entropy and record values. Metrika 76 (2013), 623-640. DOI 10.1007/s00184-012-0408-6 | MR 3078811 | Zbl 1307.62011
[21] C. R. Rao: On discrete distributions arising out of methods of ascertainment. Classical and Contagious Discrete Distributions. Statistical Publishing Society, Calcutta (1965), 320-332. MR 0214205
[22] M. Rao, Y. Chen, B. C. Vemuri, F. Wang: Cumulative residual entropy: A new measure of information. IEEE Trans. Inf. Theory 50 (2004), 1220-1228. DOI 10.1109/TIT.2004.828057 | MR 2094878 | Zbl 1302.94025
[23] M. Shaked, J. G. Shanthikumar: Stochastic Orders. Springer Series in Statistics. Springer, New York (2007). DOI 10.1007/978-0-387-34675-5 | MR 2265633 | Zbl 1111.62016

Affiliations:   Faranak Goodarzi (corresponding author), Department of Statistics, University of Kashan, 8731753153 Kashan, Iran, e-mail: f-goodarzi@kashanu.ac.ir; Mohammad Amini, Gholam Reza Mohtashami Borzadaran, Department of Statistics, Ordered Data, Reliability and Dependency Center of Excellence, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran, e-mail: m-amini@um.ac.ir, grmohtashami@um.ac.ir


 
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