Chi-squared goodness of fit tests with applications / V. Voinov, KIMEP University; Institute for Mathematics and Mathematical Modeling of the Ministry of Education and Science, Almaty, Kazakhstan, M. Nikulin, University Bordeaux-2, Bordeaux, France, N. Balakrishnan, McMaster University, Hamilton, Ontario, Canada.

Format:

Book

Author/Creator:

Voinov, Vassiliy.

Balakrishnan, N., 1956- author.

Nikulin, M. S. (Mikhail Stepanovich), author.

Series:

Gale eBooks

Language:

English

Subjects (All):

Chi-square test.

Distribution (Probability theory).

Physical Description:

1 online resource (xii, 229 pages) : illustrations

Edition:

1st edition

Place of Publication:

Amsterdam : Academic Press, 2013

Waltham, MA : Academic Press, 2013.

Language Note:

English

System Details:

text file

Summary:

"If the number of sample observations n ! 1, the statistic in (1.1) will follow the chi-squared probability distribution with r-1 degrees of freedom. We know that this remarkable result is true only for a simple null hypothesis when a hypothetical distribution is specified uniquely (i.e., the parameter is considered to be known). Until 1934, Pearson believed that the limiting distribution of the statistic in (1.1) will be the same if the unknown parameters of the null hypothesis are replaced by their estimates based on a sample; see, for example, Baird (1983), Plackett (1983, p. 63), Lindley (1996), Rao (2002), and Stigler (2008, p. 266). In this regard, it is important to reproduce the words of Plackett (1983, p. 69) concerning E. S. Pearson's opinion: "I knew long ago that KP (meaning Karl Pearson) used the 'correct' degrees of freedom for (a) difference between two samples and (b) multiple contingency tables. But he could not see that in curve fitting should be got asymptotically into the same category." Plackett explained that this crucial mistake of Pearson arose from to Karl Pearson's assumption "that individual normality implies joint normality." Stigler (2008) noted that this error of Pearson "has left a positive and lasting negative impression upon the statistical world." Fisher (1924) clearly showed 1 2 CHAPTER 1. A HISTORICAL ACCOUNT that the number of degrees of freedom of Pearson's test must be reduced by the number of parameters estimated from the sample"-- Provided by publisher.

Contents:

Half Title; Title Page; Copyright; Dedication; Contents; Preface; A Historical Account; Pearson's Sum and Pearson-Fisher Test; 2.1 Pearson's chi-squared sum; 2.2 Decompositions of Pearson's chi-squared sum; 2.3 Neyman-Pearson classes and applications of decompositions of Pearson's sum; 2.4 Pearson-Fisher and Dzhaparidze-Nikulin tests; 2.5 Chernoff-Lehmann theorem; 2.6 Pearson-Fisher test for random class end points; Wald's Method and Nikulin-Rao-Robson Test; 3.1 Wald's method; 3.2 Modifications of Nikulin-Rao-Robson Test; 3.3 Optimality of Nikulin-Rao-Robson Test

3.4 Decomposition of Nikulin-Rao-Robson Test3.5 Chi-Squared Tests for Multivariate Normality; 3.5.1 Introduction; 3.5.2 Modified chi-squared tests; 3.5.3 Testing for bivariate circular normality; 3.5.4 Comparison of different tests; 3.5.5 Conclusions; 3.6 Modified Chi-Squared Tests for The Exponential Distribution; 3.6.1 Two-parameter exponential distribution; 3.6.2 Scale-exponential distribution; 3.7 Power Generalized Weibull Distribution; 3.7.1 Estimation of parameters; 3.7.2 Modified chi-squared test; 3.7.3 Evaluation of power

3.8 Modified chi-Squared Goodness of Fit Test for Randomly Right Censored Data3.8.1 Introduction; 3.8.2 Maximum likelihood estimation for right censored data; 3.8.3 Chi-squared goodness of fit test; 3.8.4 Examples; 3.9 Testing Normality for Some Classical Data on Physical Constants; 3.9.1 Cavendish's measurements; 3.9.2 Millikan's measurements; 3.9.3 Michelson's measurements; 3.9.4 Newcomb's measurements; 3.10 Tests Based on Data on Stock Returns of Two Kazakhstani Companies; 3.10.1 Analysis of daily returns; 3.10.2 Analysis of weekly returns; Wald's Method and Hsuan-Robson-Mirvaliev Test

4.1 Wald's method and moment-type estimators4.2 Decomposition of Hsuan-Robson-Mirvaliev test; 4.3 Equivalence of Nikulin-Rao-Robson and Hsuan-Robson-Mirvaliev tests for exponential family; 4.4 Comparisons of some modified chi-squared tests; 4.4.1 Maximum likelihood estimates; 4.4.2 Moment-type estimators; 4.5 Neyman-Pearson classes; 4.5.1 Maximum likelihood estimators; 4.5.2 Moment-type estimators; 4.6 Modified chi-squared test for three-parameter Weibull distribution; 4.6.1 Parameter estimation and modified chi-squared tests; 4.6.2 Power evaluation; 4.6.3 Neyman-Pearson classes

4.6.4 Discussion4.6.5 Concluding remarks; Modifications Based on UMVUEs; 5.1 Tests for Poisson, binomial, and negative binomial distributions; 5.2 Chi-squared tests for one-parameter exponential family; 5.3 Revisiting Clarke's data on flying bombs; Vector-Valued Tests; 6.1 Introduction; 6.2 Vector-valued tests: an artificial example; 6.3 Example of Section 2.3 revisited; 6.4 Combining nonparametric and parametric tests; 6.5 Combining nonparametric tests; 6.6 Concluding comments; Applications of Modified Chi-Squared Tests

7.1 Poisson versus binomial: Appointment of judges to the US Supreme Court

Notes:

Description based upon print version of record

Includes bibliographical references and index.

ISBN:

9780123977830

0123977835

9781299196148

1299196144

OCLC:

868597603