Robust Estimation and Hypothesis Testing

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Robust Estimation and Hypothesis Testing
作者：Moti. L . Tiku
Copyright © 2004, New Age International (P) Ltd., Publishers

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关键词：HYPOTHESIS Estimation TESTING robust thesis

Preface (v)
1. ROBUSTNESS OF SOME CLASSICAL ESTIMATORS AND TESTS 1–21
1.1 Introduction ................................................................................................................ 1
1.2 Robustness of sample mean and variance ................................................................ 1
1.3 Asymptotic robustness of the t test ............................................................................ 4
1.4 Comparing several means .......................................................................................... 4
1.5 Robutness of t and F for small samples .................................................................... 5
1.6 Distribution of F under non-normality ...................................................................... 6
1.7 Distribution of the one-way classification variance ratio .......................................... 7
1.8 Non-normal power function ......................................................................................12
1.9 Effect of non-normality on the t statistic ..................................................................13
1.10 Testing equality of two variances ..............................................................................15
Appendix 1A ...............................................................................................................18
Appendix 1B ...............................................................................................................19
Appendix 1C ...............................................................................................................21
2. ESTIMATION OF LOCATION AND SCALE PARAMETERS 22–54
2.1 Introduction ...............................................................................................................22
2.2 Maximum likelihood ..................................................................................................22
2.3 Modified likelihood ....................................................................................................25
2.4 Estimating location and scale ...................................................................................28
2.5 Generalized logistic ...................................................................................................31
2.6 Extreme value distribution .......................................................................................33
2.7 Best linear unbiased estimators ................................................................................35
2.8 Numerical examples ..................................................................................................37
2.9 Aymptotic distributions of the MML estimators ......................................................40
2.10 Hypothesis testing .....................................................................................................43
2.11 Huber M-estimators ...................................................................................................45
Appendix 2A : Asymptotic equivalence .....................................................................50
Appendix 2B : Numerical comparison ......................................................................51
Appendix 2C : Asymptotic distribution .....................................................................53
Appendix 2D : The PSI-function ...............................................................................53
Appendix 2E : Estimators based on consored samples ............................................54
3. LINEAR REGRESSION WITH NORMAL AND NON-NORMAL ERROR
DISTRIBUTIONS 56–85
3.1 Introduction ...............................................................................................................56
3.2 Linear regression .......................................................................................................56
3.3 The Weibull distribution ............................................................................................59
3.4 Modified likelihood ....................................................................................................60
3.5 Least squares for the Weibull....................................................................................63
3.6 Short-tailed symmetric family ..................................................................................66
3.7 MML estimators for short-tailed family ...................................................................68
3.8 LS estimators for short-tailed family ........................................................................70
3.9 Long-tailed symmetric family ...................................................................................71
3.10 General linear model .................................................................................................74
3.11 Stochastic linear regression ......................................................................................75
3.12 MML estimators for the bivariate distribution ........................................................77
3.13 Hypothesis testing .....................................................................................................79
3.14 Numerical examples ..................................................................................................82
Appendix 3A : Information matrix ............................................................................84
Appendix 3B : Value of E{Z(i)} ....................................................................................85
4. BINARY REGRESSION WITH LOGISTIC AND NONLOGISTIC DENSITY
FUNCTIONS 87–107
4.1 Introduction ...............................................................................................................87
4.2 Link functions ............................................................................................................87
4.3 Modified likelihood estimators ..................................................................................89
4.4 Variances and covariances ........................................................................................90
4.5 Hypothesis testing .....................................................................................................91
4.6 Logistic density ..........................................................................................................92
4.7 Comparison of the ML and MML estimates .............................................................92
4.8 Nonlogistic density functions ....................................................................................93
4.9 Quadratic model ........................................................................................................95
4.10 Multiple covariates ....................................................................................................97
4.11 Stochastic covariates .................................................................................................99
4.12 Modified likelihood ..................................................................................................100
4.13 The MML estimators ...............................................................................................101
4.14 Asymptotic properties ..............................................................................................102
4.15 Symmetric family.....................................................................................................103
Appendix 4A : Density and cumulative density functions .....................................105
Appendix 4B .............................................................................................................106
Appendix 4C .............................................................................................................106
Appendix 4D : Elements of the information matrix ...............................................107

5. AUTOREGRESSIVE MODELS IN NORMAL AND NON-NORMAL SITUATIONS
5.1 Introduction .............................................................................................................108
5.2 A simple autoregressive model................................................................................109
5.3 Gamma distribution ................................................................................................109
5.4 Modified likelihood ..................................................................................................110
5.5 Asymptotic covariance matrix .................................................................................112
5.6 Least squares ...........................................................................................................113
5.7 Hypothesis testing for gamma ................................................................................114
5.8 Short-tailed symmetric distributions ......................................................................116
5.9 Asymptotic covariance matrix .................................................................................117
5.10 Hypothesis testing for STS distributions ................................................................118
5.11 Long-tailed symmetric distributions .......................................................................120
5.12 Hypothesis testing for LTS distributions................................................................123
5.13 Time series model ....................................................................................................124
5.14 Asymptotic properties ..............................................................................................125
5.15 Unit root problem ....................................................................................................128
5.16 Unknown location ....................................................................................................130
5.17 Generalization to AR(q) models ..............................................................................132
Appendix 5A : Expected Value and Variance of yt ..................................................133
6. ANALYSIS OF VARIANCE IN EXPERIMENTAL DESIGN 134–152
6.1 Introduction .............................................................................................................134
6.2 One-way classification .............................................................................................134
6.3 Generalized logistic .................................................................................................136
6.4 Modified likelihood ..................................................................................................137
6.5 Testing block effects.................................................................................................139
6.6 The Weibull family ..................................................................................................140
6.7 Two-way classification and interaction ...................................................................141
6.8 Effects under non-normality ...................................................................................142
6.9 Variance ratio statistics ...........................................................................................143
6.10 Box-Cox data ............................................................................................................144
6.11 Linear contrasts .......................................................................................................146
6.12 Non-normal distributions ........................................................................................148
6.13 Non-identical error distributions ............................................................................149
6.14 Linear contrast with non-identical distributions ...................................................150
6.15 Normal theory test with non-identical blocks ........................................................152

7. CENSORED SAMPLES FROM NORMAL AND NON-NORMAL DISTRIBUTIONS
7.1 Introduction .............................................................................................................155
7.2 Estimation of parameters ........................................................................................155
7.3 Censored samples from normal distribution ..........................................................159
7.4 Symmetric censoring ...............................................................................................162
7.5 Censored samples in experimental design .............................................................167
7.6 Inliers in normal samples ........................................................................................169
7.7 Rayleigh distribution ...............................................................................................173
7.8 Censored samples from LTS distributions .............................................................176
7.9 Variances and covariances ......................................................................................179
7.10 Hypothesis testing ...................................................................................................181
7.11 Type I censoring .......................................................................................................182
7.12 Progressively censored samples ..............................................................................184
7.13 Truncated normal distribution ................................................................................186
7.14 Experimental design with truncated normal .........................................................189
Appendix 7A : Exponential Sample Spacings .........................................................193
Appendix 7B .............................................................................................................193
Appendix 7C : The coefficients for censored normal samples ................................194
8. ROBUSTNESS OF ESTIMATORS IN TESTS 195–220
8.1 Introduction .............................................................................................................195
8.2 Robust estimators of location and scale ..................................................................195
8.3 Comparison for skew distributions .........................................................................199
8.4 Comparison with Tukey and Tiku estimators ........................................................200
8.5 Hypothesis testing ...................................................................................................200
8.6 Robustness for STS distributions ............................................................................202
8.7 Robustness of regression estimators .......................................................................204
8.8 Robustness of estimators in binary regression .......................................................209
8.9 Robustness of autoregression estimators ...............................................................212
8.10 Robustness in experimental design ........................................................................217
Appendix 8A : Simulated means, variances and efficiencies .................................220
9. GOODNESS-OF-FIT AND DETECTION OF OUTLIERS 225–265
9.1 Introduction .............................................................................................................225
9.2 Q-Q plots ..................................................................................................................225
9.3 Goodness of fit tests .................................................................................................229
9.4 Directional test for any distribution .......................................................................232
9.5 Omnibus tests ..........................................................................................................238
9.6 Shapiro-Wilk test .....................................................................................................240
9.7 Filliben and Smith-Bain statistic ............................................................................241
9.8 Tiku statistics based on spacings ............................................................................241
9.9 Extension to non-exponential distributions ............................................................243
9.10 The omnibus U and U* tests ...................................................................................245
9.11 Extreme value distribution .....................................................................................247
9.12 Multi-sample situations ...........................................................................................248
9.13 Sample entropy statistics ........................................................................................250
9.14 Censored samples ....................................................................................................250
9.15 Outlier detection ......................................................................................................252
9.16 Testing for outliers ..................................................................................................253
9.17 Tiku approach to outlier detection ..........................................................................255
9.18 Power of the outlier tests.........................................................................................258
9.19 Testing the sample for outliers ...............................................................................261
Appendix 9A .............................................................................................................264
Appendix 9B .............................................................................................................265
10. ESTIMATION IN SAMPLE SURVEY 266–280
10.1 Introduction .............................................................................................................266
10.2 Super-population model ..........................................................................................266
10.3 Symmetric family.....................................................................................................267
10.4 Finite population model...........................................................................................269
10.5 Sample size determination ......................................................................................271
10.6 Stratified sampling ..................................................................................................272
10.7 Skew distributions in sample survey ......................................................................273
10.8 Estimating the mean ...............................................................................................278
Appendix 10A ...........................................................................................................280
11. APPLICATIONS 281–331
11.1 Introduction .............................................................................................................281
11.2 Estimators of location and scale parameters ..........................................................281
11.3 Determination of shape parameters .......................................................................288
11.4 Estimation in linear regression models ..................................................................289
11.5 Multiple linear regression .......................................................................................294
11.6 Autoregression .........................................................................................................297
11.7 Experimental design ................................................................................................299
Appendix 11A : MMLE for the beta distribution ....................................................304
Appendix 11B ...........................................................................................................305
Appendix 11C ...........................................................................................................306
BIBLIOGRAPHY 308
INDEX 331

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