Exact goodness-of-fit tests for shape-scale families and type II censoring

Abstract

A survey of statistical methods for validation of shape-scale families of probability distributions from type II censored samples is given. We propose “integrated likelihood ratio tests” which are modifications of Zhang’s tests from complete to type II censored data. We also give modifications of Cramér–von-Mises and Anderson–Darling tests using integration with respect to non-parametric estimators of the cumulative distribution function. Explicit formulas for modified chi-squared tests from censored data with data driven choice of partitioning are given. Powers of tests against most used alternatives to the Weibull, loglogistic and lognormal distribution are compared.

Appendices

Appendix 1: Critical values of test statistics

We use notation $p=r/n$ (Tables 5, 6, 7, 8, 9, 10).

Table 5 Critical values $ILR_{r,n}^{(1)}(\alpha )$ and $ILR_{r,n}^{(1)}(\alpha )$ (in parentheses) under Weibull distribution

Table 6 Critical values $AD_{r,n}^{(2)}(\alpha )$ and $CM_{r,n}^{(2)}(\alpha )$ (in parentheses) under Weibull distribution

Table 7 Critical values $ILR_{r,n}^{(1)}(\alpha )$ and $ILR_{r,n}^{(1)}(\alpha )$ (in parentheses) under loglogistic distribution

Table 8 Critical values $AD_{r,n}^{(2)}(\alpha )$ and $CM_{r,n}^{(2)}(\alpha )$ (in parentheses) under loglogistic distribution

Table 9 Critical values $ILR_{r,n}^{(1)}(\alpha )$ and $ILR_{r,n}^{(1)}(\alpha )$ (in parentheses) under lognormal distribution

Table 10 Critical values $AD_{r,n}^{(2)}(\alpha )$ and $CM_{r,n}^{(2)}(\alpha )$ (in parentheses) under lognormal distribution

Appendix 2: Powers of tests (significance level $0.05$)

See Tables 11, 12, 13, 14, 15, 16, 17, 18, and 19.

Table 11 Powers of tests for the Weibull model against lognormal distribution $LN(0, 5)$

Table 12 Powers of tests for the Weibull model against loglogistic distribution $LL(1, 3)$

Table 13 Powers of tests for the Weibull model against the gamma distribution $G(8,1)$

Table 14 Powers of tests for the loglogistic model against lognormal distribution $LN(0, 2.5)$

Table 15 Powers of tests for the loglogistic model against Weibull distribution $W(1, 3)$

Table 16 Powers of tests for the loglogistic model against the gamma distribution $G(0.5,1)$

Table 17 Powers of tests for the lognormal model against loglogistic distribution $LL(1, 2)$

Table 18 Powers of tests for the lognormal model against Weibull distribution $W(1, 3)$

Table 19 Powers of tests for the lognormal model against the gamma distribution $G(2,1)$