On length and area-biased Maxwell distributions

ABSTRACT This article addresses the various properties and different methods of estimation of the unknown parameter of length and area-biased Maxwell distributions. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of length and area-biased Maxwell distributions (such as moments, moment-generating function (mgf), hazard rate function, mean residual lifetime function, residual lifetime function, reversed residual life function, conditional moments and conditional mgf, stochastic ordering, and measures of uncertainty) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimator, moments estimator, least-square and weighted least-square estimators, maximum product of spacings estimator and compare them using extensive numerical simulations. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using inverted gamma prior for the scale parameter. Furthermore, Bayes estimators and their respective posterior risks are computed and compared using Markov chain Monte Carlo (MCMC) algorithm. Also, bootstrap confidence intervals using frequentist approaches are provided to compare with Bayes credible intervals. Finally, a real dataset has been analyzed for illustrative purposes.

[1]  A. P. Dawid,et al.  Invariant Prior Distributions , 2006 .

[2]  Ouajdi Korbaa,et al.  SPARE PARTS INVENTORY SYSTEMS UNDER AN INCREASING FAILURE RATE DEMAND INTERVAL DISTRIBUTION , 2011 .

[3]  Umesh Singh,et al.  The generalized inverse Lindley distribution: A new inverse statistical model for the study of upside-down bathtub data , 2016 .

[4]  Ronald E. Glaser,et al.  Bathtub and Related Failure Rate Characterizations , 1980 .

[5]  C. R. Rao,et al.  On discrete distributions arising out of methods of ascertainment , 1965 .

[6]  G. P. Patil,et al.  Weighted distributions , 2001 .

[7]  S. Lalley.  RENEWAL THEORY , 2014 .

[8]  Russell C. H. Cheng,et al.  Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin , 1983 .

[9]  Umesh Singh,et al.  Product Spacings as an Alternative to Likelihood for Bayesian Inferences , 2014 .

[10]  J. Mandel,et al.  Reproducibility of electromigration measurements , 1987, IEEE Transactions on Electron Devices.

[11]  Min-Tsai Lai,et al.  Optimum number of minimal repairs for a system under increasing failure rate shock model with cumulative repair-cost limit , 2013 .

[12]  Francisco Louzada-Neto,et al.  The complementary exponential power lifetime model , 2011, Comput. Stat. Data Anal..

[13]  Hassan S. Bakouch,et al.  An extended Maxwell distribution: Properties and applications , 2017, Commun. Stat. Simul. Comput..

[14]  R. Fisher.  THE EFFECT OF METHODS OF ASCERTAINMENT UPON THE ESTIMATION OF FREQUENCIES , 1934 .

[15]  Sanku Dey,et al.  Bayesian Estimation of the Parameter of Maxwell Distribution under Different Loss Functions , 2010 .

[16]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[17]  Claude E. Shannon,et al.  Prediction and Entropy of Printed English , 1951 .

[18]  Vasilis P. Koutras,et al.  Two-Level Software Rejuvenation Model with Increasing Failure Rate Degradation , 2011 .

[19]  Lewis Randolph Grosenbaugh.  Point-Sampling and Line-Sampling: Probability Theory, Geometric Implications, Synthesis , 2019 .

[20]  Ganapati P. Patil,et al.  On discrete weighted distributions and their use in model choice for observed data , 1986 .

[21]  Hare Krishna,et al.  Reliability estimation in Maxwell distribution with progressively Type-II censored data , 2012 .

[22]  L. Kaiser,et al.  Unbiased Estimation in Line-Intercept Sampling , 1983 .

[23]  Dhaifalla K. Al-Mutairi,et al.  Two types of generalized Wiebull distributions and their applications under different enviromental conditions , 1999 .

[24]  Jeffrey H. Gove,et al.  Moment and maximum likelihood estimators for Weibull distributions under length- and area-biased sampling , 2003, Environmental and Ecological Statistics.

[25]  M. E. Ghitany,et al.  A two-parameter weighted Lindley distribution and its applications to survival data , 2011, Math. Comput. Simul..

[26]  Suchismita Das.  On weighted generalized entropy , 2017 .

[27]  Sanku Dey,et al.  Weighted Weibull Distribution: Properties and Estimation , 2015 .

[28]  G. Crooks.  On Measures of Entropy and Information , 2015 .

[29]  J. Cossman,et al.  Drug Development and the FDA's Critical Path Initiative , 2007, Clinical pharmacology and therapeutics.

[30]  James Clerk Maxwell.  On the Dynamical Theory of Gases , 2003 .

[31]  Hare Krishna,et al.  Reliability estimation in Maxwell distribution with Type‐II censored data , 2009 .

[32]  D. Lezotte,et al.  The generalized inverse , 1976, APLQ.