Author | Read, Timothy R. C. author |
---|---|
Title | Goodness-of-Fit Statistics for Discrete Multivariate Data [electronic resource] / by Timothy R. C. Read, Noel A. C. Cressie |
Imprint | New York, NY : Springer New York, 1988 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4578-0 |
Descript | XII, 212 p. online resource |
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 Akaikeโs 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: Pearsonโs 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 Birchโs 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 ? ? ยฑ ? (Section 6.2) -- Author Index