Powerful Mathematica Codes for Goodness-of-Fit Tests for Censored Data

Abstract

In reliability studies of energy and electrical systems life data are often censored, because life tests are terminated, and life data are analyzed before the failure of all sample units. The most important task to accomplish a successful reliability analysis is to choose, through statistical goodness-of-fit tests, the correct or nearly correct probability distribution to describe the failure mechanism of given experimental data. However, due to censoring, this task would not be as easy as testing complete samples. Unfortunately, the built-in functions and codes of the available computation programs are not valid to test for incomplete or censored samples and give completely wrong results if they are used for that purpose, even on the most sophisticated ones like Mathematica and MATLAB. On the other hand, there is a high chance to slip up when trying to perform this type of tests by someone with humble probabilistic and mathematical background. Correct performance of such tests requires a deep knowledge in how to treat the estimating equations of the candidate distribution’s parameters from a censored sample. This type of equations is usually implicit, which often needs a careful numerical treatment to be successfully solved. Also, we should keep in mind that the test statistics formulas of censored samples are different from those of complete samples. The corresponding critical value of the test must be modified according to the type of the distribution nominated, the degree of censoring, and the complete sample size. Therefore, there is a crucial need to have codes that safely run the tests and give reliable results. This book chapter is devoted to introducing efficient Mathematica codes for two of the best goodness-of-fit tests for censored data, the Cramér–von Mises and Anderson-Darling tests for Weibull and lognormal distributions, which are useful in a great variety of applications in energy studies, particularly as models for product life. The codes are presented together with some practical examples extracted from the literature in various topics of energy systems and related fields.