Families of weighted distributions with inverse-hazard size bias: theory and applications

Ahmed M. Gemeay; Yuri A. Iriarte; Ohud A. Alqasem; Fatma Masoud A. Zaghdoun; Manahil SidAhmed Mustafa; Ahmed M. Gemeay; Yuri A. Iriarte; Ohud A. Alqasem; Fatma Masoud A. Zaghdoun; Manahil SidAhmed Mustafa

doi:10.3934/math.2026464

AIMS Mathematics

2026, Volume 11, Issue 4: 11296-11321. doi: 10.3934/math.2026464

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Families of weighted distributions with inverse-hazard size bias: theory and applications

1.
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
2.
Departamento de Estadística y Ciencia de Datos, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Helwan University, Egypt
5.
Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia

Received: 22 February 2026 Revised: 03 April 2026 Accepted: 13 April 2026 Published: 22 April 2026
MSC : 62E15, 62E20, 62P12

We studied a new class of weighted distributions in which the weighting mechanism introduces a size bias inversely proportional to the hazard rate of the baseline model. This formulation, referred to as the inverse-hazard size-biased distribution family, provided a flexible framework that unifies and extends several well-known two-parameter models. When one-parameter baseline distributions were considered, classical models such as the gamma, generalized Rayleigh, and beta prime distributions arise as particular cases of the proposed class. Furthermore, two new members based on the Maxwell and half-normal baseline distributions were introduced and studied in detail. Analytical properties, including moments and parameter estimation methods, were derived, and simulation studies were conducted to assess the performance of the estimators. The practical applicability of the proposed models was also illustrated through empirical analyses using real data.
- weighted distributions,
- inverse-hazard weighting,
- size-biased models,
- flexible parametric models,
- moment properties,
- parameter estimation
Citation: Ahmed M. Gemeay, Yuri A. Iriarte, Ohud A. Alqasem, Fatma Masoud A. Zaghdoun, Manahil SidAhmed Mustafa. Families of weighted distributions with inverse-hazard size bias: theory and applications[J]. AIMS Mathematics, 2026, 11(4): 11296-11321. doi: 10.3934/math.2026464

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Abstract

We studied a new class of weighted distributions in which the weighting mechanism introduces a size bias inversely proportional to the hazard rate of the baseline model. This formulation, referred to as the inverse-hazard size-biased distribution family, provided a flexible framework that unifies and extends several well-known two-parameter models. When one-parameter baseline distributions were considered, classical models such as the gamma, generalized Rayleigh, and beta prime distributions arise as particular cases of the proposed class. Furthermore, two new members based on the Maxwell and half-normal baseline distributions were introduced and studied in detail. Analytical properties, including moments and parameter estimation methods, were derived, and simulation studies were conducted to assess the performance of the estimators. The practical applicability of the proposed models was also illustrated through empirical analyses using real data.

References

[1]	R. A. Fisher, The effect of methods of ascertainment upon the estimation of frequencies, Ann. Eugenics, 6 (1934), 13–25.
[2]	C. R. Rao, On discrete distributions arising out of methods of ascertainment, Sankhya Indian J. Stat. Ser. A, 27 (1965), 311–324.
[3]	M. Zelen, M. Feinleib, On the theory of screening for chronic diseases, Biometrika, 56 (1969), 601–614. https://doi.org/10.2307/2334668 doi: 10.2307/2334668
[4]	G. P. Patil, J. Ord, On size-biased sampling and related form-invariant weighted distributions, Sankhya Indian J. Stat. Ser. B, 38 (1976), 48–61.
[5]	G. P. Patil, C. R. Rao, The weighted distributions: a survey of their applications, Appl. Stat., 1977,383–405.
[6]	G. P. Patil, C. R. Rao, Weighted distributions and size-biased sampling with applications to wildlife populations and human families, Biometrics, 1978,179–189.
[7]	R. C. Gupta, J. P. Keating, Relations for reliability measures under length biased sampling, Scand. J. Stat., 13 (1986), 49–56.
[8]	B. O. Oluyede, On inequalities and selection of experiments for length biased distributions, Probab. Eng. Inf. Sci., 13 (1999), 169–185.
[9]	K. K. Das, T. D. Roy, On some length-biased weighted weibull distribution, Adv. Appl. Sci. Res., 2 (2011), 465–475.
[10]	G. Tzavelas, D. Panagiotakos, Statistical inference for the size-biased weibull distribution, J. Stat. Comput. Simul., 83 (2013), 1252–1265. https://doi.org/10.1080/00949655.2012.657197 doi: 10.1080/00949655.2012.657197
[11]	R. D. Gupta, D. Kundu, A new class of weighted exponential distributions, Statistics, 43 (2009), 621–634. https://doi.org/10.1080/02331880802605346 doi: 10.1080/02331880802605346
[12]	B. C. Arnold, R. J. Beaver, Hidden truncation models, Sankhya Indian J. Stat. Ser. A, 62 (2000), 23–35.
[13]	A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat., 12 (1985), 171–178.
[14]	H. J. Kim, On a class of two-piece skew-normal distributions, Statistics, 39 (2005), 537–553.
[15]	B. C. Arnold, H. W. Gómez, H. S. Salinas, On multiple constraint skewed models, Statistics, 43 (2009), 279–293. https://doi.org/10.1080/02331880802357914 doi: 10.1080/02331880802357914
[16]	D. Elal-Olivero, Alpha-skew-normal distribution, Proyecciones (Antofagasta), 29 (2010), 224–240. https://doi.org/10.4067/S0716-09172010000300006 doi: 10.4067/S0716-09172010000300006
[17]	H. Bolfarine, H. S. Salinas, H. W. Gómez, J. Arrué, A new class of skew-normal-Cauchy distribution, Stat. Oper. Res. Trans., 39 (2015), 35–50.
[18]	N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, John Wiley & Sons, 2 Eds., 1994.
[19]	V. G. Vodă, Inferential procedures on a generalized rayleigh variate. I, Appl. Math., 21 (1976), 395–412. https://doi.org/10.21136/am.1976.103663 doi: 10.21136/am.1976.103663
[20]	N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous univariate distributions, John wiley & Sons, 1995.
[21]	K. Soetaert, P. M. J. Herman, A practical guide to ecological modelling: using R as a simulation platform, Springer, 2009. https://doi.org/10.1007/978-1-4020-8624-3
[22]	K. Soetaert, rootsolve: nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations, R package version 1.6, 2009. Available from: https://CRAN.R-project.org/package = rootSolve.
[23]	R. H. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput., 16 (1995), 1190–1208. https://doi.org/10.1137/0916069 doi: 10.1137/0916069
[24]	W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech., 18 (1951), 293–297. https://doi.org/10.1115/1.4010337 doi: 10.1115/1.4010337
[25]	R. D. Gupta, D. Kundu, Theory & methods: generalized exponential distributions, Aust. N. Z. J. Stat., 41 (1999), 173–188. https://doi.org/10.1111/1467-842X.00072 doi: 10.1111/1467-842X.00072
[26]	J. Faraway, G. Marsaglia, J. Marsaglia, A. Baddeley, Goftest: classical goodness-of-fit tests for univariate distributions, R package version 1.2-3, 2021. Available from: https://CRAN.R-project.org/package = goftest.
[27]	H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control, 19 (1974), 716–723. https://doi.org/10.1109/TAC.1974.1100705 doi: 10.1109/TAC.1974.1100705
[28]	G. Schwarz, Estimating the dimension of a model, Ann. Stat., 6 (1978), 461–464. https://doi.org/10.1214/aos/1176344136 doi: 10.1214/aos/1176344136
[29]	R. Shanker, K. Shukla, On modeling of lifetime data using three-parameter generalized lindley and generalized gamma distributions, Biom. Biostat. Int. J., 4 (2016), 283–288. https://doi.org/10.15406/bbij.2016.04.00117 doi: 10.15406/bbij.2016.04.00117
[30]	M. Bader, A. Priest, Statistical aspects of fibre and bundle strength in hybrid composites, Progress Sci. Eng. Compos., 1982, 1129–1136.
[31]	G. Grimmett, D. Stirzaker, Probability and random processes, Oxford University Press, 2020.

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