Chi-squared Goodness-of-fit Tests for Censored Data

Mikhail S. Nikulin, Ekaterina V. Chimitova

• Publisher: Wiley-ISTE
• Year: 2017
• Print ISBN: 9781786300003
• eBook ISBN: 9781119427636
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book is devoted to the problems of construction and application of chi-squared goodness-of-fit tests for complete and censored data. Classical chi-squared tests assume that unknown distribution parameters are estimated using grouped data, but in practice this assumption is often forgotten. In this book, we consider modified chi-squared tests, which do not suffer from such a drawback. The authors provide examples of chi-squared tests for various distributions widely used in practice, and also consider chi-squared tests for the parametric proportional hazards model and accelerated failure time model, which are widely used in reliability and survival analysis. Particular attention is paid to the choice of grouping intervals and simulations.

This book covers recent innovations in the field as well as important results previously only published in Russian. Chi-squared tests are compared with other goodness-of-fit tests (such as the Cramer-von Mises-Smirnov, Anderson-Darling and Zhang tests) in terms of power when testing close competing hypotheses.

1
Chi-squared Goodness-of-fit Tests for Complete Data

Let X = (X₁, X₂, …, X_n)^T be a sample of independent identically distributed random variables, where X_i ∈ R¹, belongs to the distribution defined by the cumulative distribution function:

[1.1]

We can say also that we have complete data X₁, X₂, …, X_n.

In this chapter, we consider the problem of testing a composite goodness-of-fit hypothesis:

[1.2]

If the value of parameter θ is known from prior experiments, then H₀ is a simple hypothesis.

1.1. Classical Pearson’s chi-squared test

Now we consider a partition of the real line by the points a_i

into k (k > s + 2) intervals (a_j−1, a_j] for grouping the data. Following Cramer [CRA 46], we suppose that:

1. 1) There exists a positive number c such that:
   
2. 2) The functions p_j(θ) have continuous first- and second-order partial derivatives on the set Θ;
3. 3) The matrix
   [1.3]

   
4. has rank s.

Furthermore, let ν = (ν₁, …, ν_k)^T be the result of grouping the random variables X_1...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Introduction
• 1 Chi-squared Goodness-of-fit Tests for Complete Data
• 2 Chi-squared Test for Censored Data
• 3 Comparison of the Chi-squared Goodness-of-fit Test with Other Tests
• 4 Chi-squared Goodness-of-fit Tests for Regression Models
• Appendices
• Bibliography
• Index
• End User License Agreement