Smooth Tests of Goodness of Fit

Contents......Page 14
1.1 The Problem Defined......Page 18
1.2 A Brief History of Smooth Tests......Page 21
1.3 Monograph Outline......Page 25
1.4 Examples......Page 27
2.2 Foundations......Page 35
2.3 The Pearson's X[sup(2)] Test—an Update......Page 37
2.3.1 Notation, definition of the test, and class construction......Page 38
2.3.2 Power related properties......Page 40
2.4 X[sup(2)] Tests of Composite Hypotheses......Page 42
2.5 Examples......Page 43
3.1 Introduction......Page 48
3.2 The Likelihood Ratio, Wald, and Score Tests for a Simple Null Hypothesis......Page 49
3.3 The Likelihood Ratio, Wald, and Score Tests for Composite Null Hypotheses......Page 52
3.4 Properties of the Asymptotically Optimal Tests......Page 56
4.1 Neyman's Ψ[sup(2)] Test......Page 60
4.2 Neyman Smooth Tests for Uncategorized Simple Null Hypotheses......Page 63
4.3 Effective Order and Power Comparisons......Page 66
4.4 Examples......Page 67
5.1 Smooth Tests for Completely Specified Multinomials......Page 72
5.2 X[sup(2)] Effective Order......Page 76
5.3 Components of X[sup(2)][sub(p)]......Page 78
5.4 Examples......Page 81
5.5 A More Comprehensive Class of Tests......Page 86
5.6 Overlapping Cells Tests......Page 88
6.1 Neyman Smooth Tests for Uncategorized Composite Null Hypotheses......Page 92
6.2 Smooth Tests for the Univariate Normal Distribution......Page 96
6.3 Smooth Tests for the Exponential Distribution......Page 102
6.4 Smooth Tests for the Poisson Distribution......Page 106
6.5 Smooth Tests for the Geometric Distribution......Page 111
6.6 Smooth Tests for the Multivariate Normal Distribution......Page 113
6.7 Components of the Rao–Robson X[sup(2)] Test......Page 124
7.1 Neyman Smooth Tests for Composite Multinomials......Page 126
7.2 Components of the Pearson–Fisher Statistic......Page 130
7.3 Composite Overlapping Cells and Cell Focusing X[sup(2)] Tests......Page 131
7.4 A Comparison Between the Pearson–Fisher and Rao–Robson X[sup(2)] Tests......Page 136
8.2.1 Density estimation......Page 139
8.2.2 Outlier detection......Page 144
8.2.3 Normality testing of several samples......Page 146
8.2.4 Residuals......Page 149
8.3 Concluding Example......Page 150
8.4 Closing Remarks......Page 152
Appendix 1 Orthogonal Polynomials......Page 155
Appendix 2 Computer Program for Implementing the Smooth Tests of Fit for the Uniform, Normal, and Exponential Distributions......Page 157
Appendix 3 Explicit Formulas for the Components V[sup(2)][sub(1)], and V[sup(2)][sub(2)] of X[sup(2)][sub(p)]......Page 162
References......Page 163
G......Page 172
T......Page 173
W......Page 174
F......Page 175
R......Page 176
W......Page 177