The Fréchet distribution: Estimation and application - An overview

Abstract In this article we consider the problem of estimating the parameters of the Fréchet distribution from both frequentist and Bayesian points of view. First we briefly describe different frequentist approaches, namely, maximum likelihood, method of moments, percentile estimators, L-moments, ordinary and weighted least squares, maximum product of spacings, maximum goodness-of-fit estimators and compare them using extensive numerical simulations. Next, we consider the Bayesian infer- ence approach using reference priors. Five real data sets related to the minimum flow of water on Piracicaba river in Brazil are used to illustrate the applicability of the discussed procedures.

[1]  Devendra Kumar,et al.  Exponentiated Chen distribution: Properties and estimation , 2017, Commun. Stat. Simul. Comput..

[2]  T. Wesche,et al.  Status of Instream Flow Legislation and Practices in North America , 1989 .

[3]  Bo Ranneby,et al.  The Maximum Spacing Method. An Estimation Method Related to the Maximum Likelihood Method , 2016 .

[4]  Kevin P. Murphy,et al.  Machine learning - a probabilistic perspective , 2012, Adaptive computation and machine learning series.

[5]  Mohammad Z. Raqab,et al.  Generalized Rayleigh Distribution , 2011, International Encyclopedia of Statistical Science.

[6]  Jorge Alberto Achcar,et al.  Bayesian analysis of the generalized gamma distribution using non-informative priors , 2017 .

[7]  F. Louzada,et al.  The Long Term Fr\'echet distribution: Estimation, Properties and its Application , 2017, 1709.07593.

[8]  John H. K. Kao.  A Graphical Estimation of Mixed Weibull Parameters in Life-Testing of Electron Tubes , 1959 .

[9]  Donald L. Tennant.  Instream Flow Regimens for Fish, Wildlife, Recreation and Related Environmental Resources , 1976 .

[10]  COMPARISON OF ESTIMATION METHODS FOR INVERSE WEIBULL PARAMETERS , 2017 .

[11]  Juliette Gardner.  Genesis , 1985 .

[12]  John H. K. Kao.  Computer Methods for Estimating Weibull Parameters in Reliability Studies , 1958 .

[13]  Debasis Kundu,et al.  Generalized exponential distribution: different method of estimations , 2001 .

[14]  R. Calabria,et al.  Confidence limits for reliability and tolerance limits in the inverse Weibull distribution , 1989 .

[15]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

[16]  Debasis Kundu,et al.  Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data , 2010, Comput. Stat. Data Anal..

[17]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[18]  G. Casella,et al.  The Effect of Improper Priors on Gibbs Sampling in Hierarchical Linear Mixed Models , 1996 .

[19]  S. Nadarajah,et al.  Extreme Value Distributions: Theory and Applications , 2000 .

[20]  Keewhan Choi,et al.  An Estimation Procedure for Mixtures of Distributions , 1968 .

[21]  Order statistics from inverse Weibull distribution and characterizations , 2015 .

[22]  Debasis Kundu,et al.  Generalized Rayleigh distribution: different methods of estimations , 2005, Comput. Stat. Data Anal..

[23]  E. Kay,et al.  Methods for statistical analysis of reliability and life data , 1974 .

[24]  J. Hosking.  L‐Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics , 1990 .

[25]  Yincai Tang,et al.  Analysis of Frechet Distribution Using Reference Priors , 2015 .

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

[27]  F. Louzada,et al.  Poisson–exponential distribution: different methods of estimation , 2018 .

[28]  Creasy Problem,et al.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[29]  J. Bernardo.  Reference Analysis , 2005 .

[30]  F. Louzada,et al.  The generalized weighted Lindley distribution: Properties, estimation, and applications , 2016, 1601.07410.

[31]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[32]  Sajid Ali,et al.  Two-parameter Maxwell distribution: Properties and different methods of estimation , 2016 .

[33]  Francisco Louzada,et al.  Comparison of estimation methods for the parameters of the weighted Lindley distribution , 2013, Appl. Math. Comput..

[34]  James O. Berger,et al.  Overall Objective Priors , 2015, 1504.02689.

[35]  M. Maswadah,et al.  Conditional confidence interval estimation for the inverse weibull distribution based on censored generalized order statistics , 2003 .

[36]  S. M. Hoseini,et al.  Comparison of estimation methods for the Weibull distribution , 2013 .

[37]  Saikat Mukherjee,et al.  Statistical properties and different methods of estimation of transmuted Rayleigh distribution , 2017 .

[38]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[39]  Mohammad Z. Raqab,et al.  Estimation of the generalized logistic distribution parameters: Comparative study , 2009 .

[40]  Debasis Kundu,et al.  Two-Parameter Rayleigh Distribution: Different Methods of Estimation , 2014 .

[41]  R. Calabria,et al.  Bayes 2-sample prediction for the inverse weibull distribution , 1994 .

[42]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[43]  L. Wasserman,et al.  The Selection of Prior Distributions by Formal Rules , 1996 .

[44]  Dennis D. Boos,et al.  Minimum Distance Estimators for Location and Goodness of Fit , 1981 .