Goodness-of-fit statistics for discrete multivariate data

Abstract

1 Introduction to the Power-Divergence Statistic.- 1.1 A Unified Approach to Model Testing.- 1.2 The Power-Divergence Statistic.- 1.3 Outline of the Chapters.- 2 Defining and Testing Models: Concepts and Examples.- 2.1 Modeling Discrete Multivariate Data.- 2.2 Testing the Fit of a Model.- 2.3 An Example: Time Passage and Memory Recall.- 2.4 Applying the Power-Divergence Statistic.- 2.5 Power-Divergence Measures in Visual Perception.- 3 Modeling Cross-Classified Categorical Data.- 3.1 Association Models and Contingency Tables.- 3.2 Two-Dimensional Tables: Independence and Homogeneity.- 3.3 Loglinear Models for Two and Three Dimensions.- 3.4 Parameter Estimation Methods: Minimum Distance Estimation.- 3.5 Model Generation: A Characterization of the Loglinear, Linear, and Other Models through Minimum Distance Estimation.- 3.6 Model Selection and Testing Strategy for Loglinear Models.- 4 Testing the Models: Large-Sample Results.- 4.1 Significance Levels under the Classical (Fixed-Cells) Assumptions.- 4.2 Efficiency under the Classical (Fixed-Cells) Assumptions.- 4.3 Significance Levels and Efficiency under Sparseness Assumptions.- 4.4 A Summary Comparison of the Power-Divergence Family Members.- 4.5 Which Test Statistic?.- 5 Improving the Accuracy of Tests with Small Sample Size.- 5.1 Improved Accuracy through More Accurate Moments.- 5.2 A Second-Order Correction Term Applied Directly to the Asymptotic Distribution.- 5.3 Four Approximations to the Exact Significance Level: How Do They Compare?.- 5.4 Exact Power Comparisons.- 5.5 Which Test Statistic?.- 6 Comparing the Sensitivity of the Test Statistics.- 6.1 Relative Deviations between Observed and Expected Cell Frequencies.- 6.2 Minimum Magnitude of the Power-Divergence Test Statistic.- 6.3 Further Insights into the Accuracy of Large-Sample Approximations.- 6.4 Three Illustrations.- 6.5 Transforming for Closer Asymptotic Approximations in Contingency Tables with Some Small Expected Cell Frequencies.- 6.6 A Geometric Interpretation of the Power-Divergence Statistic.- 6.7 Which Test Statistic?.- 7 Links with Other Test Statistics and Measures of Divergence.- 7.1 Test Statistics Based on Quantiles and Spacings.- 7.2 A Continuous Analogue to the Discrete Test Statistic.- 7.3 Comparisons of Discrete and Continuous Test Statistics.- 7.4 Diversity and Divergence Measures from Information Theory.- 8 Future Directions.- 8.1 Hypothesis Testing and Parameter Estimation under Sparseness Assumptions.- 8.2 The Parameter ? as a Transformation.- 8.3 A Generalization of Akaike's Information Criterion.- 8.4 The Power-Divergence Statistic as a Measure of Loss and a Criterion for General Parameter Estimation.- 8.5 Generalizing the Multinomial Distribution.- Historical Perspective: Pearson's X2 and the Loglikelihood Ratio Statistic G2.- 1. Small-Sample Comparisons of X2 and G2 under the Classical (Fixed-Cells) Assumptions.- 2. Comparing X2 and G2 under Sparseness Assumptions.- 3. Efficiency Comparisons.- 4. Modified Assumptions and Their Impact.- Appendix: Proofs of Important Results.- A1. Some Results on Rao Second-Order Efficiency and Hodges-Lehmann Deficiency (Section 3.4).- A2. Characterization of the Generalized Minimum Power-Divergence Estimate (Section 3.5).- A3. Characterization of the Lancaster-Additive Model (Section 3.5).- A4. Proof of Results (i), (ii), and (iii) (Section 4.1).- A5. Statement of Birch's Regularity Conditions and Proof that the Minimum Power-Divergence Estimator Is BAN (Section 4.1).- A6. Proof of Results (i*), (ii*), and (iii*) (Section 4.1).- A7. The Power-Divergence Generalization of the Chernoff-Lehmann Statistic: An Outline (Section 4.1).- A8. Derivation of the Asymptotic Noncentral Chi-Squared Distribution for the Power-Divergence Statistic under Local Alternative Models (Section 4.2).- A9. Derivation of the Mean and Variance of the Power-Divergence Statistic for ? > -1 under a Nonlocal Alternative Model (Section 4.2).- A10. Proof of the Asymptotic Normality of the Power-Divergence Statistic under Sparseness Assumptions (Section 4.3).- A12. Derivation of the Second-Order Terms for the Distribution Function of the Power-Divergence Statistic under the Classical (Fixed-Cells) Assumptions (Section 5.2).- A13. Derivation of the Minimum Asymptotic Value of the Power-Divergence Statistic (Section 6.2).- A14. Limiting Form of the Power-Divergence Statistic as the Parameter ? ? +- ? (Section 6.2).- Author Index.