New robust tests for the parameters of the Weibull distribution for complete and censored data

Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown is examined. In simulation studies the behavior in finite sample size and in contaminated data is analyzed and the new method is compared to existing ones. Here it is shown that the new tests based on likelihood depth give quite good results compared to standard methods and are robust against contamination. They are also robust in right-censored data in contrast to existing methods like the method of medians.

[1]  W. Weibull A Statistical Distribution Function of Wide Applicability , 1951 .

[2]  D G Hoel,et al.  A representation of mortality data by competing risks. , 1972, Biometrics.

[3]  J. Tukey Mathematics and the Picturing of Data , 1975 .

[4]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[5]  A curtailed test for the shape parameter of the Weibull distribution , 1982 .

[6]  D. Ruppert Robust Statistics: The Approach Based on Influence Functions , 1987 .

[7]  R. Y. Liu,et al.  On a notion of simplicial depth. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Regina Y. Liu On a Notion of Data Depth Based on Random Simplices , 1990 .

[9]  D. G. Simpson,et al.  Breakdown robustness of tests , 1990 .

[10]  Stefun D. Leigh U-Statistics Theory and Practice , 1992 .

[11]  C. W. Coakley,et al.  THE MAXIMUM RESISTANCE OF TESTS , 1994 .

[12]  Jian Zhang,et al.  The sample breakdown points of tests , 1996 .

[13]  W. Kahle Estimation of the parameters of the Weibull distribution for censored samples , 1996 .

[14]  Zhenmin Chen,et al.  Statistical inference about the shape parameter of the Weibull distribution , 1997 .

[15]  B. Price,et al.  Robust Planning and Analysis of Experiments , 1997 .

[16]  P. Rousseeuw,et al.  Constructing the bivariate Tukey median , 1998 .

[17]  W K Fung,et al.  Method of medians for lifetime data with Weibull models. , 1999, Statistics in medicine.

[18]  Regina Y. Liu,et al.  Regression depth. Commentaries. Rejoinder , 1999 .

[19]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[20]  Structural properties and convergence results for contours of sample statistical depth functions , 2000 .

[21]  K. Mosler "Multivariate Dispersion, Central Regions, and Depth": The Lift Zonoid Approach , 2002 .

[22]  I. Mizera On depth and deep points: a calculus , 2002 .

[23]  K. Mosler Multivariate Dispersion, Central Regions, and Depth , 2002 .

[24]  J. Lawless Statistical Models and Methods for Lifetime Data , 2002 .

[25]  K. Mosler Central Regions and Dependency , 2003 .

[26]  C. Müller,et al.  Location–Scale Depth , 2004 .

[27]  C. Müller Depth estimators and tests based on the likelihood principle with application to regression , 2005 .

[28]  Minghua Chen,et al.  Robust estimating equation based on statistical depth , 2006 .

[29]  Stanislav Katina,et al.  Calculation of simplicial depth estimators for polynomial regression with applications , 2007, Comput. Stat. Data Anal..

[30]  Jun Li,et al.  Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions , 2008, 0806.2970.

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

[32]  Christine H. Müller,et al.  Distribution-free tests for polynomial regression based on simplicial depth , 2009, J. Multivar. Anal..

[33]  Mario Romanazzi Data depth, random simplices and multivariate dispersion , 2009 .

[34]  J. Romo,et al.  On the Concept of Depth for Functional Data , 2009 .

[35]  Aurora Torrente,et al.  Robust depth-based tools for the analysis of gene expression data. , 2010, Biostatistics.

[36]  Christine H. Müller,et al.  Depth notions for orthogonal regression , 2010, J. Multivar. Anal..

[37]  Liesa Denecke,et al.  Estimators and Tests based on Likelihood-Depth with Application to Weibull Distribution, Gaussian and Gumbel Copula , 2010 .

[38]  Christine H. Müller,et al.  Tests for multiple regression based on simplicial depth , 2010, J. Multivar. Anal..

[39]  Christine H. Müller,et al.  Robust estimators and tests for bivariate copulas based on likelihood depth , 2011, Comput. Stat. Data Anal..

[40]  Kris Boudt,et al.  Robust explicit estimators of Weibull parameters , 2011 .

[41]  Yonggang Hu,et al.  Generalized Mahalanobis depth in the reproducing kernel Hilbert space , 2011 .

[42]  C. Müller,et al.  Consistency and robustness of tests and estimators based on depth , 2012 .

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