Exploring Improved Inference of Birnbaum-Saunders Distribution Based on Modified Moment Estimation

Waqas  Makhdoom; Muahammad  Kashif Ali Shah; Nighat  Zahra; Syed  Ejaz Ahmed

Authors

Waqas Makhdoom Department of Statistics, Government College University, Lahore, Punjab, Pakistan
Muahammad Kashif Ali Shah Department of Statistics, Government College University, Lahore, Punjab, Pakistan
Nighat Zahra Department of Statistics, Government College University, Lahore, Punjab, Pakistan
Syed Ejaz Ahmed Department of Mathematics and Statistics, Brock University, St. Catharines, Ontario, Canada

Keywords:

Asymptotic mean sqaure error, Birnbaum-Saunders distribution, improved estimation, moment estimation, Shrinkage preliminary test estimator

Abstract

The Birnbuam-Saunders distribution is widely well-known lifetime distribution having shape and scale parameters. In this study, we employed improved estimation methodologies for the both parameters of the Birnbaum-Saunders distribution considering for one parameter and keeping other known. We have used modified moment estimators instead of maximum likelihood estimators of Birnbaum Saunders distribution while integrating uncertain prior information with the sample information. The three improved estimators are considered and their performance is compared with their respective unrestricted modified moment estimator. The Walds test statistics are also suggested to test the available uncertain prior information. The asymptotic theoretical results of the estimators also provided. The performance of the proposed estimators is evaluated on the basis of asymptotic mean square error. The simulation study of the estimators with their graphical presentation are furnished with real life data application. This study concludes that our proposed improved estimators for both cases of parameters have appealing performance.

References

Aldeni M, Wagaman J, Amezziane M, Ahmed SE. Pretest and shrinkage estimators for log-normal means. Comput Stat. 2023; 38: 1555-1578.

Ahmed SE, Budsaba K, Lisawadi S, Volodin A. Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a new parametrization. Thail Stat. 2008; 6(2): 213-240.

Ahmed SE. Penalty, shrinkage and pretest strategies variable selection and estimation. 1st ed. Springer; 2014.

Arru J, Arellano-Valle RB, Geraldo HG, Leiva V. On a new type of Birnbaum-Saunders models and its inference and application to fatigue data. J App Stat. 2019; 47(13-15): 2690-2710.

Balakrishnan N, Kundu D. A review of models, analysis, and application. Appl Stoch Models Bus Ind. 2019; 35(1): 4-49.

Baklizi A, Ahmed SE. On the estimation of reliability function in a Weibull lifetime distribution. Statistics. 2008; 42(4): 351-362.

Birnbaum ZW, Saunders SC. A new family of lifetime distribution. J Appl Probab. 1969; 6: 319-327.

Birnbaum ZW, Saunders SC. Estimation for a family of life distributions with applications to fatigue. J Appl Probab. 1969; 6: 328-347.

Costa E, Santos-Neto M, Leiva V. Optimal sample size for the Birnbaum-Saunders distribution under decision theory with symmetric and asymmetric loss functions. Symmetry. 2021; 13(6): 926.

Engelhardt M, Bain LJ, Wright FT. Inferences on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation. Technometrics. 1981; 23(2): 251-255.

Khan S, Saleh AKME. On the comparison of the pre-test and shrinkage estimators for the univariate normal mean. Stat Pap. 2001; 23: 451-473.

Lemonte JA, Cribari-Neto F, Klaus LP, Vasconcellos. Improved statistical inference for the two parameter Birnbaum-Saunders distribution. Comput Stat Data Anal. 2006; 51(9): 4656-4681.

Leiva V. The Birnbaum-Saunders distribution. 1st ed. Elsevier Science; 2015.

Liu K, Balakrishnan N, He M, Xie L. Likelihood inference for the Birnbaum-Saunders frailty model with an application to bone marrow transplant data. J Stat Comput Simul. 2023; 93(13): 2158-2175.

Makhdoom W, Shah MKA, Zahra N, Ahmed SE. Optimizing robust shape parameter: improved methodologies for Birnbaum-Saunders distribution. J Stat Theory Pract. 2024; 18(38).

Makhdoom W, Shah MKA, Zahra N, Ahmed SE. Efficient estimation strategies for estimating the shape parameter of the Birnbaum-Saunders distribution using preliminary test type estimators. Lobachevskii J Math. 2024; 45(9): 4207-4215.

Ng HKT, Kundu D, Balakrishnan N. Modified moment estimation for the two-parameter Birnbaum-Saunders distribution. Comput Stat Data Anal. 2003; 43(3): 283-298.

Park C, Wang M. bsgof: Birnbaum-Saunders goodness-of-fit test. R package version 0.23.8; 2023.

Salman AN, Ali AH, Salman MD. Preliminary test single stage shrinkage estimator for the scale parameter of gamma distribution. Am J Appl Math Stat. 2014; 4(3): 131-136.

Shah MKA, Lisawadi S, Ahmed SE. Merging data from multiple sources: pretest and shrinkage perspectives. J Stat Comput Simul. 2017; 87(8): 1577-1592.

Santos-Neto M, Francisco JAC, Leiva V, Barros M. A reparameterized Birnbaum-Saunders distribution and its moments, estimation and applications. Revstat Stat J. 2014; 12(3): 247-272.

Thangjai W, Niwitpong S, Niwitpong S, Somkhuean R. Confidence intervals for function of percentiles of Birnbaum-Saunders distributions containing zero values with application to wind speed modelling. Modelling. 2025; 6(1): 16.

Zhang J, Chen W, Yang R. Birnbaum-Saunders parameters estimation using simple random sampling and ranked set sampling. Commun Stat Simulat. 2024; 1-20.