Simplified Likelihood Based Goodness-of-fit Tests for the Weibull Distribution

The aim of this paper is to present new likelihood based goodness-of-fit tests for the two-parameter Weibull distribution. These tests consist in nesting the Weibull distribution in three-parameter generalized Weibull families and testing the value of the third parameter by using the Wald, score, and likelihood ratio procedures. We simplify the usual likelihood based tests by getting rid of the nuisance parameters, using three estimation methods. The proposed tests are not asymptotic. A comprehensive comparison study is presented. Among a large range of possible GOF tests, the best ones are identified. The results depend strongly on the shape of the underlying hazard rate.

[1]  I. W. Burr.  Cumulative Frequency Functions , 1942 .

[2]  E. Stacy.  A Generalization of the Gamma Distribution , 1962 .

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  L. J. Bain,et al.  A Property of Maximum Likelihood Estimators of Location and Scale Parameters , 1969 .

[5]  K. S. Kölbig,et al.  Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints , 1972 .

[6]  N. Mann,et al.  A men goodness-of-fit test for the two-parameter wetbull or extreme-value distribution with unknown parameters , 1973 .

[7]  J. Lawless.  Inference in the Generalized Gamma and Log Gamma Distributions , 1980 .

[8]  M. Tiku,et al.  Testing the two parameter weibull distribution , 1981 .

[9]  J. Hosking.  Testing whether the shape parameter is zero in the generalized extreme-value distribution , 1984 .

[10]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

[11]  Steven Ascher.  A survey of tests for exponentiality , 1990 .

[12]  G. S. Mudholkar,et al.  Exponentiated Weibull family for analyzing bathtub failure-rate data , 1993 .

[13]  Min Xie,et al.  Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function , 1996 .

[14]  G. S. Mudholkar,et al.  A Generalization of the Weibull Distribution with Application to the Analysis of Survival Data , 1996 .

[15]  I. Olkin,et al.  A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families , 1997 .

[16]  Min Liao,et al.  A new goodness-of-fit test for type-I extreme-value and 2-parameter weibull distributions with estimated parameters , 1999 .

[17]  D. N. Prabhakar Murthy,et al.  A modified Weibull distribution , 2003, IEEE Trans. Reliab..

[18]  Gordon Johnston,et al.  Statistical Models and Methods for Lifetime Data , 2003, Technometrics.

[19]  Necip Doganaksoy,et al.  Weibull Models , 2004, Technometrics.

[20]  N. Henze,et al.  Recent and classical tests for exponentiality: a partial review with comparisons , 2005 .

[21]  A. Quiroz,et al.  Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions , 2005 .

[22]  M. Nikulin,et al.  A Chi-Squared Test for the Generalized Power Weibull Family for the Head-and-Neck Cancer Censored Data , 2006 .

[23]  G. Celeux,et al.  An alternative competing risk model to the Weibull distribution for modelling aging in lifetime data analysis , 2006, Lifetime data analysis.

[24]  Hoang Pham,et al.  On Recent Generalizations of the Weibull Distribution , 2007, IEEE Transactions on Reliability.

[25]  Horst Rinne,et al.  The Weibull Distribution: A Handbook , 2008 .

[26]  C. Caroni.  Testing for the Marshall–Olkin extended form of the Weibull distribution , 2010 .

[27]  Gauss M. Cordeiro,et al.  A Log-Linear Regression Model for the Beta-Weibull Distribution , 2011, Commun. Stat. Simul. Comput..

[28]  O. Gaudoin,et al.  Review and comparison of goodness-of-fit tests for the exponential and Weibull distributions , 2012 .

[29]  R. Kay.  The Analysis of Survival Data , 2012 .

[30]  H. A. Noughabi,et al.  General treatment of goodness-of-fit tests based on Kullback–Leibler information , 2013 .

[31]  Gauss M. Cordeiro,et al.  The compound class of extended Weibull power series distributions , 2012, Comput. Stat. Data Anal..

[32]  Saad J. Almalki,et al.  A new modified Weibull distribution , 2013, Reliab. Eng. Syst. Saf..

[33]  Natalya Pya,et al.  Goodness-of-Fit Tests for the Power-Generalized Weibull Probability Distribution , 2013, Commun. Stat. Simul. Comput..