Preface......Page 6
Contents......Page 10
Symbols and Abbreviations......Page 16
Chapter1 Introduction......Page 18
References......Page 21
2.1 The Mathematical Model of Decision Making......Page 23
2.2 Minimax, Admissible, and Bayesian Families of Decision Rules......Page 24
2.3 The Bayesian Forecast Density......Page 29
2.4.1 The Mathematical Model......Page 33
2.4.2 Complete Prior Knowledge of {Οi,pi0(Β·)}......Page 34
2.4.3 Prior Uncertainty......Page 39
References......Page 45
3.1 Regression Models of Time Series......Page 46
3.2 Stationary Time Series Models......Page 48
3.3 ARIMA(p,d,q) Time Series Model......Page 50
3.4.1 A General Nonlinear Model......Page 54
3.4.3 Functional-Coefficient Autoregression Model FAR(p,d)......Page 55
3.4.5 Threshold Autoregression Model TAR(k)......Page 56
3.5.1 Multivariate Stationary Time Series......Page 57
3.5.2 Vector Autoregression Model VAR(p)......Page 58
3.5.3 Vector Moving Average Model VMA(q)......Page 60
3.5.4 VARMA(p,q) Model......Page 61
3.5.5 System of Simultaneous Equations (SSE) Model......Page 62
3.6.1 Markov Chains......Page 63
3.6.2 Markov Chains of Order s......Page 64
3.6.5 INAR(m) Model......Page 66
References......Page 67
4.1 A General Formulation of the Statistical Forecasting Problem......Page 69
4.2 The Risk Functional and Optimality of Forecasting Statistics......Page 72
4.3 Classification of Model Distortions......Page 79
4.4 Robustness Characteristics......Page 83
References......Page 86
5.1 Optimal Forecasting Under Complete Prior Information......Page 87
5.2.1 Bayesian Approach in the PU-P Setting......Page 90
5.2.2 Joint Estimation Using the Maximum Likelihood (ML) Principle......Page 91
5.2.3 Using the Plug-In Principle......Page 92
5.3 Logistic Regression Forecasting......Page 97
5.4 Nonparametric Kernel Regression Forecasting......Page 103
5.5 Nonparametric k-NN-Regression Forecasting......Page 112
5.6.1 Functional Series Expansions of Regression Functions......Page 115
5.6.2 Spline Smoothing......Page 117
References......Page 118
6.1.1 Formulation of the Problem......Page 119
6.1.2 The Hypothetical Regression Model and Its Functional Distortions......Page 120
6.1.3 Robustness Characteristics of Forecasting Algorithms......Page 122
6.1.4 Robustness Analysis of Least Squares Forecasting......Page 123
6.2.1 Mathematical Description of Model Distortions......Page 130
6.2.2 Robustness Evaluation of Least Squares Forecasting......Page 132
6.3 Robustness of Least Squares Forecasting Under Outliers......Page 134
6.4 Impact of Correlation Between Observation Errors on Forecast Risk......Page 137
6.5.1 Construction of a Robust Forecasting Algorithm......Page 140
6.5.2 Evaluation of the Constructed Robust Forecasting Algorithm......Page 143
6.5.3 Numerical Examples......Page 150
6.6 Robust Regression Forecasting Under Outliers Based on the Huber Estimator......Page 151
6.7.1 Description of the Method......Page 155
6.7.2 The Breakdown Point......Page 157
6.7.3 Probability Distribution of the LM Forecast......Page 159
6.7.4 Risk of the LM Forecast......Page 164
6.7.5 Robustness of LM Forecasting Compared to the Traditional Least Squares Method......Page 166
6.7.6 A Generalization of the LM Method for Multivariate Regression......Page 167
6.7.7 Numerical Results......Page 172
References......Page 175
7.1 Kolmogorov's Method......Page 177
7.2.1 The General Method for Stationary Time Series......Page 184
7.2.2 Forecasting Under the AR(p) Model......Page 187
7.2.3 Forecasting Under the MA(q) Model......Page 188
7.2.5 Forecasting Under the ARIMA(p,d,q) Model......Page 189
7.3.1 Plug-In Forecasting Algorithms Based on Covariance Function Estimators......Page 190
7.3.2 Plug-In Forecasting Algorithms Based on AR(p) Parameter Estimators......Page 192
7.3.3 Plug-In Forecasting Algorithms Based on Parameter Estimation of MA(q) Models......Page 194
7.3.4 Plug-In Forecasting Algorithms Based on ARMA(p,q) Parameter Estimators......Page 195
7.3.5 Plug-In Forecasting Algorithms Based on ARIMA(p,d,q) Parameter Estimators......Page 196
7.4.1 The General Case......Page 197
7.4.2 Stationary Time Series Forecasting Under Misspecification of Covariance Functions......Page 198
7.4.3 Forecasting of AR(p) Time Series Under Misspecification of Autoregression Coefficients......Page 200
7.5 Robustness Under Functional Innovation Process Distortions in the Mean Value......Page 204
7.6.1 The Mathematical Model......Page 210
7.6.2 Presence of a Specification Error......Page 211
7.6.3 Least Squares Estimation of ΞΈ0......Page 214
7.7 Robustness of Autoregression Time Series Forecasting Under IO-Outliers......Page 217
7.8 Robustness of Autoregression Time Series Forecasting Under AO Outliers......Page 222
7.9.2 The Bilinear Model and Its Stationarity Conditions......Page 231
7.9.3 First and Second Order Moments in Stationary Bilinear Time Series Models......Page 232
7.9.4 Robustness of Autoregression Forecasting Under Bilinear Distortion......Page 236
7.9.5 Robustness Analysis of Autoregression Forecasting......Page 240
7.9.6 Numerical Results......Page 242
References......Page 244
8.1 VAR Time Series Models Under Missing Values......Page 245
8.2 The Optimal Forecasting Statistic and Its Risk......Page 248
8.3 Robustness of the Optimal Forecasting Statistic Under Specification Errors......Page 251
8.4 Modified Least Squares Estimators Under Missing Values......Page 252
8.5 Least Squares Forecasting and Its Risk Under Missing Values......Page 262
8.6 Results of Computer Experiments......Page 265
8.6.1 Performance of the Estimator B......Page 266
8.6.2 Experimental Evaluation of the Forecast Risk......Page 267
8.7.1 A Mathematical Model of Simultaneous Distortion by Outliers and Missing Values......Page 269
8.7.2 A Family of Robust Estimators for Correlations Based on the Cauchy Probability Distribution......Page 271
8.7.3 Minimizing Asymptotic Variance of Ο-Estimators......Page 274
8.7.4 Robust Estimators of Autoregression Coefficients......Page 278
8.7.5 Estimation of the Contamination Level......Page 279
8.7.6 Simulation-Based Performance Evaluation of the Constructed Estimators and Forecasting Algorithms......Page 280
References......Page 285
9.1.1 SSE Model......Page 287
9.1.2 Example of an SSE: Klein's Model I......Page 290
9.1.3 The Optimal Forecasting Statistic Under the SSE Model......Page 292
9.2 Robustness of SSE-Based Forecasting Under Specification Errors......Page 293
9.3 Plug-In Forecasting Statistics in the SSE Model......Page 295
9.4.1 Drifting Coefficient Models for SSEs......Page 297
9.4.2 LS Parameter Estimators Under Parameter Drift......Page 299
9.5 Sensitivity of Forecast Risk to Parameter Drift......Page 311
9.6 Numerical Results for the Ludeke Econometric Model......Page 314
References......Page 317
10.1.1 The Time Series Model......Page 318
10.1.2 The Bayesian Decision Rule and Its Properties......Page 319
10.1.3 The Plug-In Decision Rule and Its Risk......Page 324
10.1.4 An Asymptotic Expansion of the PBDR Risk......Page 327
10.2.1 Likelihood Functions for HMCs with Missing Values......Page 332
10.2.2 The Decision Rule for Known {Ο(l),P(l)}......Page 335
10.2.3 The Case of Unknown Parameters......Page 336
10.3.1 The Beta-Binomial Model, Its Properties and Distortions......Page 337
10.3.2 Robustness of MM Estimators......Page 339
10.3.3 Robustness of MLEs......Page 340
10.3.4 MM Estimators Under Distortion of A Priori Known Levels......Page 342
10.3.5 Joint Estimation of Probability Distribution Parameters and Distortion Levels......Page 343
10.3.6 Robustness of the Classical Bayesian Predictor......Page 345
10.3.7 Robust Forecasting in the Beta-Binomial Model......Page 347
10.3.8 Experimental Results......Page 348
10.4 Forecasting of HMCs......Page 350
10.5.1 Optimal Forecasting Under HMC(s) Models......Page 354
10.5.2 Identification of the JacobsβLewis Model......Page 356
10.5.3 Identification of Raftery's MTD Model......Page 357
10.5.4 Identification of the MC(s,r) Model......Page 359
References......Page 364
Index......Page 366