Advances on Theoretical and Methodological Aspects of Probability and Statistics

Advances on Theoretical and Methodological Aspects of Probability and Statistics......Page 3
CONTENTS......Page 5
PREFACE......Page 18
LIST OF CONTRIBUTORS......Page 20
LIST OF TABLES......Page 25
LIST OF FIGURES......Page 27
Part I: Stochastic Processes and Inference......Page 29
1.1 INTRODUCTION......Page 30
1.2 PRELIMINARIES......Page 32
1.3 STOCHASTIC DELAY DIFFERENTIAL EQUATIONS......Page 34
1.4 THE FILTERING PROBLEM......Page 47
1.5 ZAKAI EQUATION AND UNIQUENESS......Page 58
REFERENCES......Page 62
2.1 SOME CLASSES OF NONSTATIONARY PROCESSES......Page 64
2.2 SIGMA OSCILLATORY PROCESSES......Page 68
2.3 DETERMINATION OF THE EVOLUTIONARY SPECTRA......Page 71
REFERENCES......Page 73
3.1 INTRODUCTION......Page 75
3.2 INCREMENTAL PROCESSES......Page 77
3.3 MOMENTS OF HARMONIZABLE PROCESSES......Page 78
3.4 VIRILE REPRESENTATIONS......Page 80
REFERENCES......Page 82
4.1 INTRODUCTION......Page 83
4.2 GALTON-WATSON BRANCHING PROCESS: BACKGROUND......Page 84
LAMN Conditions......Page 85
4.4 G-W BRANCHING PROCESS AS A PROTO-TYPE EXAMPLE OF A LAMN MODEL......Page 86
4.5 ESTIMATION EFFICIENCY......Page 87
4.6 TEST EFFICIENCY......Page 88
4.7 CONFIDENCE BOUNDS......Page 90
4.8 CONDITIONAL INFERENCE......Page 91
4.9 PREDICTION AND TEST OF FIT......Page 92
4.10 QUASILIKELIHOOD ESTIMATION......Page 93
4.11 BAYES AND EMPIRICAL BAYES ESTIMATION......Page 95
REFERENCES......Page 96
Part II: Distributions and Characterizations......Page 98
5.1 INTRODUCTION......Page 99
5.2 THE DISTRIBUTION OF X GIVEN X=Y CAN BE ALMOST ANYTHING......Page 100
5.3 DEPENDENT VARIABLES......Page 102
5.4 RELATED EXAMPLES......Page 103
REFERENCES......Page 105
6.1 INTRODUCTION......Page 106
6.2.1 The Mean and the Variance of N......Page 108
6.2.2 Behavior for Large c: Almost Sure Limits......Page 112
6.3 CHARACTERIZATION RESULTS......Page 114
REFERENCES......Page 118
7.1 INTRODUCTION......Page 119
7.1.1 Some Basic Results from Linear Algebra......Page 120
(b) The content of a simplex......Page 121
7.1.2 Some Basic Results on Jacobians of Matrix Transformations......Page 124
7.1.3 Some Practical Situations......Page 126
(c) Travel distance in a triangular city core......Page 127
(d) Sylvester problem......Page 128
7.2 DISTRIBUTION OF THE VOLUME OR CONTENT OF A RANDOM PARALLELOTOPE IN Rn......Page 129
7.2.1 Matrix-Variate Type-1 Beta Distribution......Page 131
7.2.2 Matrix-Variate Type-2 Beta Density......Page 132
7.3 SPHERICALLY SYMMETRIC DISTRIBUTIONS......Page 133
7.4 ARRIVAL OF POINTS BY A POISSON PROCESS......Page 135
REFERENCES......Page 136
8.1 INTRODUCTION AND NOTATION......Page 138
8.2 A REPRESENTATION OF THE DENSITY FUNCTION OF ELLIPTICAL VECTORS......Page 140
8.3 THE EXACT DISTRIBUTION OF QUADRATIC FORMS......Page 141
8.4 MOMENTS AND APPROXIMATE DISTRIBUTION......Page 144
8.5 A NUMERICAL EXAMPLE......Page 146
REFERENCES......Page 147
CHAPTER 9: INVERSE NORMALIZING TRANSFORMATIONS AND AN EXTENDED NORMALIZING TRANSFORMATION......Page 151
9.1 INTRODUCTION......Page 152
9.2 DERIVATION OF THE TRANSFORMATIONS......Page 153
9.3 NUMERICAL ASSESSMENT......Page 156
9.4 ESTIMATION......Page 157
9.5 CONCLUSIONS......Page 159
REFERENCES......Page 160
10.1 DEFINITION OF THE GAUSSIAN CURVATURE......Page 166
10.2 EXAMPLES......Page 170
10.3 SOME BASIC PROPERTIES OF GAUSSIAN CURVATURE......Page 172
10.4 APPLICATIONS OF THE GAUSS EQUATIONS......Page 176
REFERENCES......Page 177
Part III: Inference......Page 179
11.1 INTRODUCTION......Page 180
11.2 A CONVEXITY RESULT......Page 181
11.3 ASYMPTOTIC MINIMAXITY......Page 183
APPENDIX......Page 187
REFERENCES......Page 188
12.1 INTRODUCTION......Page 189
12.2 LINEAR AND NONLINEAR FILTERS......Page 191
12.2.1 Optimal Combination Extension......Page 193
Notes......Page 194
12.3.1 Linear State Space Models......Page 195
12.3.3 Robust Estimation Filtering Equations......Page 196
12.3.4 Censored Autocorrelated Data......Page 197
REFERENCES......Page 198
13.1 INTRODUCTION......Page 200
13.2 THE SETUP......Page 202
13.3 THE ‘CONVENTIONAL’ APPROACHES......Page 203
13.4 THE NEW CONDITIONAL SEQUENTIAL TEST......Page 206
13.5 AN APPLICATION......Page 208
REFERENCES......Page 211
14.1 INTRODUCTION......Page 213
14.2 A GENERAL FRAMEWORK......Page 216
14.3.1 The Sufficiency Principle......Page 218
14.3.2 Weak Conditionality......Page 220
14.4 THE LIKELIHOOD PRINCIPLE......Page 221
14.5 DISCUSSION......Page 224
REFERENCES......Page 225
15.1 INTRODUCTION......Page 227
15.2 A SHEWHART CHART AND A CUSUM SCHEME......Page 228
15.3 NONCENTRAL t-STATISTICS BASED CUSUM PROCEDURES......Page 230
15.4 SIMULATIONS......Page 235
REFERENCES......Page 237
16.1 INTRODUCTION......Page 238
16.2.1 Change in kappa, mu Fixed and Known......Page 240
16.2.2 Change in kappa, mu Fixed but Unknown......Page 241
16.2.3 Change in mu or kappa or Both......Page 242
16.3 SIMULATION RESULTS......Page 243
16.4 POWER COMPARISONS......Page 244
16.5 AN EXAMPLE......Page 245
REFERENCES......Page 246
17.1 INTRODUCTION......Page 252
17.2.1 Ancova Estimator......Page 253
17.2.2 Adjustment for Nonnegativeness......Page 254
17.4 AN ESTIMATOR DERIVED FROM THE MIVQUE PROCEDURE......Page 255
17.5 COMPARISON OF THE ESTIMATORS......Page 256
REFERENCES......Page 258
18.1 INTRODUCTION......Page 259
18.2 GENERAL FORMULATION AND MAIN RESULTS......Page 261
18.3 PROOFS OF THE MAIN RESULTS......Page 267
18.3.1 Proof of Theorem 18.2.1......Page 268
18.3.2 Auxiliary Lemmas......Page 269
18.3.4 Proof of Theorem 18.2.3......Page 276
18.4 APPLICATIONS OF THE MAIN RESULTS......Page 277
18.4.1 Negative Exponential Location Estimation......Page 278
Fixed-size confidence region......Page 280
Minimum risk point estimation......Page 281
18.4.3 Linear Regression Parameters Estimation......Page 282
Selecting the best normal population......Page 283
Selecting the best negative exponential population......Page 285
18.5 CONCLUDING THOUGHTS......Page 286
18.6 REFERENCES......Page 287
19.1 INTRODUCTION......Page 290
19.2 TWO-STAGE PROCEDURE......Page 292
19.3 ASYMPTOTIC PROPERTIES......Page 293
REFERENCES......Page 297
CHAPTER 20: THE ELUSIVE AND ILLUSORY MULTIVARIATE NORMALITY......Page 299
20.1 INTRODUCTION......Page 300
Tests based on Mahalanobis distance......Page 301
Tests based on skewness and kurtosis......Page 302
Tests based on adaptation of the univariate procedures......Page 303
Cork data set [Rao (1948)]......Page 304
Iris data (Edgar Anderson’s data [Fisher (1936)])......Page 305
Peruvian Indian data......Page 307
REFERENCES......Page 308
Part IV: Bayesian Inference......Page 312
21.1 INTRODUCTION......Page 313
21.2 TAILFREE PRIORS......Page 314
21.3 NEUTRAL TO RIGHT PRIORS......Page 318
21.4 NR PRIORS FROM CENSORED OBSERVATIONS......Page 321
REFERENCES......Page 323
22.1 INTRODUCTION......Page 325
22.2.1 Bayes Estimation......Page 326
22.2.2 Bayes Testing......Page 327
22.3.1 Empirical Bayes Estimator......Page 328
22.4 ASYMPTOTIC OPTIMALITY OF THE EMPIRICAL BAYES ESTIMATOR......Page 329
22.5 ASYMPTOTIC OPTIMALITY OF THE EMPIRICAL BAYES TESTING RULE......Page 333
REMARKS......Page 335
REFERENCES......Page 336
23.1 INTRODUCTION......Page 339
23.2 THE EMPIRICAL BAYES ESTIMATORS......Page 341
23.3 ASYMPTOTIC OPTIMALITY......Page 343
REFERENCES......Page 349
Part V: Selection Methods......Page 351
CHAPTER 24: ON A SELECTION PROCEDURE FOR SELECTING THE BEST LOGISTIC POPULATION COMPARED WITH A CONTROL......Page 352
24.1 INTRODUCTION......Page 353
24.2 FORMULATION OF THE SELECTION PROBLEM WITH THE SELECTION RULE......Page 354
24.3 ASYMPTOTIC OPTIMALITY OF THE PROPOSED SELECTION PROCEDURE......Page 359
24.4 SIMULATIONS......Page 369
REFERENCES......Page 370
25.1 INTRODUCTION......Page 377
25.2 SOME PRELIMINARY RESULTS......Page 379
25.3 INDIFFERENCE ZONE FORMULATION: KNOWN COMMON VARIANCE......Page 380
25.4 SUBSET SELECTION FORMULATION: KNOWN COMMON VARIANCE......Page 381
25.5 INDIFFERENCE ZONE FORMULATION: UNKNOWN COMMON VARIANCE......Page 382
25.6 SUBSET SELECTION FORMULATION: UNKNOWN COMMON VARIANCE......Page 384
25.7 AN INTEGRATED FORMULATION......Page 385
25.8 SIMULTANEOUS SELECTION OF THE EXTREME POPULATIONS: INDIFFERENCE ZONE FORMULATION AND KNOWN COMMON VARIANCE......Page 386
25.9 SIMULTANEOUS SELECTION OF THE EXTREME POPULATIONS: SUBSET SELECTION FORMULATION AND KNOWN COMMON VARIANCE......Page 390
25.10 CONCLUDING REMARKS......Page 392
REFERENCES......Page 393
26.1 INTRODUCTION......Page 396
26.2 THE SELECTION PROCEDURE......Page 397
26.3 TABLE, SIMULATION STUDY AND AN EXAMPLE......Page 401
REFERENCES......Page 403
Part VI: Regression Methods......Page 409
27.1 INTRODUCTION......Page 410
27.2 TOLERANCE INTERVALS, SIMULTANEOUS TOLERANCE INTERVALS AND A MARGINAL PROPERTY......Page 413
27.3 NUMERICAL RESULTS......Page 417
27.3.1 The Simulation of (27.2.19) and (27.2.20)......Page 419
27.3.2 An Example......Page 423
27.4 CALIBRATION......Page 424
27.5 CONCLUSIONS......Page 425
APPENDIX A: SOME FITTED FUNCTIONS k(c)......Page 426
REFERENCES......Page 428
28.1 INTRODUCTION......Page 429
28.2 THE MODELS AND THE METHODOLOGIES—A GENERAL DISCUSSION......Page 430
28.2.1 Linear Regression and Related Models......Page 431
28.2.2 Sequential and Multistage Methodologies......Page 432
28.2.3 Sequential Inference in Regression: A Motivating Example......Page 434
28.3 FIXED-PRECISION INFERENCE IN DETERMINISTIC REGRESSION MODELS......Page 435
28.3.1 Confidence Set Estimation......Page 436
28.3.2 Point Estimation......Page 438
28.4 SEQUENTIAL SHRINKAGE ESTIMATION IN REGRESSION......Page 440
28.5 BAYES SEQUENTIAL INFERENCE IN REGRESSION......Page 441
28.6 SEQUENTIAL INFERENCE IN STOCHASTIC REGRESSION MODELS......Page 442
28.7 SEQUENTIAL INFERENCE IN INVERSE LINEAR REGRESSION AND ERRORS-IN-VARIABLES MODELS......Page 443
28.8 SOME MISCELLANEOUS TOPICS......Page 444
REFERENCES......Page 445
29.1 INTRODUCTION......Page 452
29.2 GIBBS SAMPLER......Page 454
29.3 BAYESIAN PRELIMINARIES AND THE NONLINEAR CHANGE-POINT MODEL......Page 457
29.4 BAYESIAN INFERENTIAL METHODS......Page 458
29.5 IMPLEMENTATION AND THE RESULTS......Page 461
APPENDIX......Page 464
REFERENCES......Page 466
CHAPTER 30: CONVERGENCE TO TWEEDIE MODELS AND RELATED TOPICS......Page 473
30.1 INTRODUCTION......Page 474
30.2 SPECIAL CASES: INVERSE GAUSSIAN AND GENERALIZED INVERSE GAUSSIAN DISTRIBUTIONS......Page 478
30.3 CRITICAL POINTS IN THE FORMATION OF LARGE DEVIATIONS......Page 480
30.4 DIFFERENT MECHANISMS OF RUIN IN NON-LIFE INSURANCE......Page 483
REFERENCES......Page 486
Part VII: Methods in Health Research......Page 490
CHAPTER 31: ESTIMATION OF STAGE OCCUPATION PROBABILITIES IN MULTISTAGE MODELS......Page 491
31.1 INTRODUCTION......Page 492
31.2 THE FRACTIONAL RISK SET ESTIMATORS......Page 493
31.3 VARIANCE ESTIMATION......Page 498
31.4 EXTENSION TO MULTISTAGE MODELS......Page 500
REFERENCES......Page 501
32.1 INTRODUCTION......Page 504
32.2.1 The Strictly Parallel Model......Page 506
Definition of reliability coefficient......Page 507
CAC and principal components analysis......Page 508
Step by step procedure to select items with CAC......Page 509
32.3 MODERN PSYCHOMETRIC THEORY......Page 511
32.3.1 The Rasch Model......Page 512
Specific objectivity......Page 513
Parameter estimation......Page 514
Goodness of fit tests......Page 516
Assessment of reliability......Page 518
Example, the communication scale of the SIP......Page 519
REFERENCES......Page 520
ANNEX 1: COMMUNICATION DIMENSION OF THE SIP (9 ITEMS)......Page 524
ANNEX 2: SOCIAL INTERACTION DIMENSION OF THE SIP (20 ITEMS)......Page 525