New Estimations for Varextropy under Complete Data and Uniformity Testing

Document Type : Original Article

Authors
Faranak Goodarzi 1
Raheleh Zamini 2
1 Faculty of Mathematics, Department of Statistics, University of Kashan
2 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.
10.22034/jirss.2025.2049691.1092
Abstract
Recently Alizadeh Noughabi and Shafaei Noughabi (2024) introduced some estimators for the varextropy of an absolutely continuous random variable. In this paper, we propose other nonparametric estimators for the varextropy function. Additionally, we prove asymptotic properties of two estimators given in Alizadeh Noughabi
and Shafaei Noughabi (2024). Asymptotic properties of the proposed estimators are established under suitable regularity conditions. Moreover, a simulation study is performed to compare the performance of the proposed estimators based on mean squared error (MSE) and bias. Furthermore, by using the proposed estimators some tests are constructed for uniformity. It is shown that the varextropy-based test proposed in this paper performs well in terms of power when compared to other uniformity hypothesis tests. Real datasets are utilized to evaluate the performance of the varextropy estimators.
Keywords
Varextropy Estimator
Consistency
Goodness-of-fit Test
Testing Uniformity
Monte Carlo Simulation
Subjects
Alizadeh Noughabi, H., and Jarrahiferiz, J. (2019), On the estimation of extropy. Journal of Nonparametric Statistics, 31(1), 88-99.
Alizadeh Noughabi, H., and Shafaei Noughabi, M. (2023). Varentropy estimators with applications in testing uniformity. Journal of Statistical Computation and Simulation, 93, 2582-2599.
Alizadeh Noughabi, H., and Shafaei Noughabi, M. (2024), On the estimation of varextropy. Statistics, 58, 827-841.
Al-Labadi, L., and Berry, S. (2022), Bayesian estimation of extropy and goodness of fit tests. Journal of Applied Statistics, 49, 357-370.
Alshenawy, R. (2020), A new one parameter distribution: properties and estimation with applications to complete and type II censored data. Journal of Taibah University for Science, 14(1), 11-18.
Becerra, A., de la Rosa, J. I., González, E., Pedroza, A. D., and Escalante, N. I. (2018), Training deep neural networks with non-uniform frame-level cost  function for automatic speech recognition. Multimedia Tools and Applications, 77, 27231-27267.
Correa, J.C. (1995), A new estimator of entropy. Communication in Statistics-Theory and Methods, 24(10), 2439-2449.
Ebrahimi, N., Pflughoeft, K., and Soofi, E. (1994), Two measures of sample entropy. Statistics and Probability Letters, 20(3), 225-234.
Fradelizi, M., Madiman, M., andWang, L. (2016), Optimal concentration of information content for logconcave densities. In C. Houdr´e, D. Mason, P. Reynaud-Bouret & J. Rosi ´ nski (eds.), High Dimensional Probability VII. Progress in Probability, vol. 71, Cham, Springer, pp. 45-60.
Fuller, E.J., Frieman, S., Quinn, J., Quinn, G., Carter, W. (1994), Fracture mechanics approach to the design of glass aircraft windows. A case study.  International Society for Optics and Photonics, 419-430.
Furuichi, S., and Mitroi, F.C. (2012), Mathematical inequalities for some divergences. Physica A, 391, 388-400.
Goodarzi, F. (2024), Results on conditional variance in parallel system and lower bounds for varextropy.Communication in Statistics-Theory and Methods, 53, 1590-1610.
Gneiting, T., and Raftery, A.E. (2007), Strictly proper scoring rules, prediction, and estimation. Journal of the American statistical Association, 102(477), 359-378.
Grzegorzewski, P., and Wieczorkowski, R. (1999), Entropy-based goodness-of-fit test for exponentiality. Communications in StatisticsTheory and Methods, 28, 1183-1202.
Illowsky, B., and Dean, S. (2018), Introductory statistics. Houston: OpenStax. Jiang, J., Wang, R., Pezeril, M., and Wang, Q.A. (2011), Application of varentropy as a measure of probabilistic uncertainty for complex networks. Science Bulletin, 56, 3677-3682.
Kolmogorov, A.N. (1933), Sulla determinazione empirica di una legge didistibuziane. Giornale Z. dellIstitutaItaliano degli Attuari, 4, 83-91.
Lad, F., Sanfilippo, G., and Agr`o, G. (2015), Extropy: Complementary Dual of Entropy. Statistical Science, 30, 40-58.
Li, J., Fradelizi, M., and Madiman, M. (2016), Information concentration for convex measures. IEEE International Symposium on Information Theory, Barcelona, 1128-1132.
Madiman, M., and Barron, A. (2007), Generalized entropy power inequalities and monotonicity properties of information. IEEE Transactions on Information Theory, 53, 2317-2329.
Mahmoud, M.R., and Mandouh, R.M. (2013), On the transmuted frechet distribution. Journal of Applied Sciences Research, 9(10), 5553-5561.
Shannon, C.E. (1948), A mathematical theory of communication. The Bell System Technical Journal, 27, 379-432.
Silverman, B.W. (1978),Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. The annals of statistics, 6, 177-184.
Smirnov, N. (1939), Estimate of derivation between empirical distribution functions in two independent samples (in Russian). Bulletin Moscow University, 2, 316.
Soni, P., Dewan, I., and Jain, K. (2012), Nonparametric estimation of quantile density function. Computational Statistics and Data Analysis, 56(12), 3876-3886.
Stephens, M.A. (1974), EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347), 730-737.
Subhash, S., Sunoj, S.M., Sankaran, P.G., and Rajesh, G. (2023), Nonparametric estimation of quantile-based entropy function. Communications in Statistics - Simulation and Computation, 52, 1805-1821.
Qiu, G., and Jia, K. (2018a), Extropy estimators with applications in testing uniformity. Journal of Nonparametric Statistics, 30(1), 182-196.
Qiu, G., and Jia, K. (2018b), The residual extropy of order statistics. Statistics and Probability Letters, 133, 15-22.
van Es, B. (1992), Estimating functionals related to a density by a class of statistics based on spacings. Scandinavian Journal of Statistics, 19(1), 61-72.
Vaselabadi, N.M., Tahmasebi, S., Kazemi, M.R., and Buono, F. (2021), Results on varextropy measure of random variables. Entropy, 23, 356.
Vasicek, O. (1976), A test for normality based on sample entropy. Journal of the Royal Statistical Society: Series B, 38, 54-59.
Vontobel, P.O. (2013), The Bethe permanent of a nonnegative matrix. IEEE Transactions on Information Theory, 59, 1866-1901.
Yang, J., Xia, W., and Hu, T. (2019), Bounds on extropy with variational distance constraint. Probability in the Engineering and Informational Sciences, 33(2), 186-204.
Zamanzade, E. (2015), Testing uniformity based on new entropy estimators. Journal of Statistical Computation and Simulation, 85(16), 3191-3205.
Zhang, J. (2002), Powerful goodness-of-fit tests based on the likelihood ratio. Journal of the Royal Statistical Society Series B, 64(2), 281-294.
Journal of the Iranian Statistical Society
Volume 24, Issue 1
June 2025
Pages 11-34

  • Receive Date 03 January 2025
  • Revise Date 26 April 2025
  • Accept Date 09 June 2025