Preface ....................................................... vii
Introduction .................................................... 1
1 Diffusion Processes and Statistical Problems ................. 17
1.1 Stochastic Differential Equation ......................... 17
1.1.1 Stochastic Integral ................................ 17
1.1.2 Diffusion Process .................................. 23
1.1.3 Local Time ......................................... 27
1.1.4 Likelihood Ratio ................................... 33
1.2 Limit Theorems ........................................... 39
1.2.1 Law of Large Numbers ............................... 39
1.2.2 Central Limit Theorem .............................. 43
1.3 Statistical Inference .................................... 50
1.3.1 Parameter Estimation ............................... 51
1.3.2 Nonparametric Estimation ........................... 83
1.3.3 Hypotheses Testing ................................ 103
2 Parameter Estimation ........................................ 111
2.1 Maximum Likelihood and Bayesian Estimators .............. 111
2.1.1 Lower Bound ....................................... 112
2.1.2 Maximum Likelihood Estimator ...................... 116
2.1.3 Bayesian Estimators ............................... 129
2.1.4 Examples .......................................... 134
2.2 Minimum Distance Estimators ............................. 139
2.2.1 First MDE ......................................... 140
2.2.2 Examples .......................................... 147
2.2.3 Second MDE ........................................ 149
2.3 Trajectory Fitting Estimator ............................ 158
2.3.1 Properties of TFE ................................. 159
2.3.2 Example ........................................... 169
2.4 Estimator of the Method of Moments ...................... 170
2.4.1 Properties of EMM ................................. 171
2.4.2 Examples .......................................... 174
2.5 One-Step MLE ............................................ 176
2.5.1 Parameter Estimation .............................. 176
2.5.2 Distribution Function Estimation .................. 182
2.5.3 Density Estimation ................................ 186
2.6 Miscellaneous ........................................... 190
2.6.1 No True Model ..................................... 191
2.6.2 Too Many True Models .............................. 204
2.6.3 Null Fisher Information ........................... 208
2.6.4 Optimal Observation Window ........................ 211
2.6.5 Asymptotic Expansions ............................. 216
2.6.6 Recursive Estimation .............................. 223
3 Special Models .............................................. 227
3.1 Partially Observed Systems .............................. 227
3.1.1 Kalman-Bucy Filter ................................ 228
3.1.2 Properties of Estimators .......................... 230
3.1.3 Examples .......................................... 236
3.2 Cusp Estimation ......................................... 238
3.2.1 Model ............................................. 238
3.2.2 Properties of Estimators .......................... 238
3.2.3 Discussions ....................................... 251
3.3 Delay Estimation ........................................ 253
3.3.1 SDE with Delay .................................... 253
3.3.2 Properties of MLE and BE .......................... 255
3.3.3 Discussion ........................................ 268
3.4 Change-Point Estimation ................................. 269
3.4.1 Simple Switching .................................. 269
3.4.2 General Case ...................................... 274
3.4.3 Examples .......................................... 282
3.4.4 Discussion ........................................ 283
3.4.5 Contaminated Switching ............................ 290
3.5 Non Ergodic Processes ................................... 293
3.5.1 Null Recurrent Process ............................ 294
3.5.2 Polynomial Growth Process ......................... 302
3.5.3 Exponential Growth Process ........................ 303
4 Nonparametric Estimation .................................... 309
4.1 Distribution Function Estimation ........................ 309
4.1.1 Lower Bound ....................................... 310
4.1.2 EDF ............................................... 315
4.1.3 Example ........................................... 318
4.1.4 Other Metrics ..................................... 321
4.2 Density Estimation ...................................... 322
4.2.1 Lower Bound ....................................... 322
4.2.2 Local-Time Estimator .............................. 326
4.2.3 Other Estimators .................................. 334
4.3 Semiparametric Estimation ............................... 337
4.3.1 Lower Bound ....................................... 338
4.3.2 Empirical Estimator ............................... 341
4.3.3 Remarks ........................................... 343
4.3.4 Integral Type Risk ................................ 345
4.4 Density Derivative Estimation ........................... 351
4.4.1 Global Risk ....................................... 353
4.4.2 Local Risk ........................................ 373
4.5 Trend Coefficient Estimation ............................ 383
4.5.1 Optimal Rate ...................................... 384
4.5.2 Lower Bound ....................................... 389
4.5.3 Efficient Estimator ............................... 390
4.5.4 Adaptive Estimator ................................ 393
4.6 Second Order Efficiency ................................. 396
4.6.1 Problem ........................................... 396
4.6.2 Lower Bound ....................................... 399
4.6.3 Efficient Estimator ............................... 408
4.6.4 Discussion ........................................ 417
5 Hypotheses Testing .......................................... 421
5.1 Simple Hypothesis and Alternative ....................... 421
5.1.1 Large Deviations Principle ........................ 422
5.1.2 Asymptotic Behavior of Errors ..................... 424
5.2 One-Sided Parametric Alternatives ....................... 427
5.2.1 Score Function Test ............................... 430
5.2.2 Likelihood Ratio and Bayesian Tests ............... 438
5.3 One-Sided Nonparametric Alternative ..................... 451
5.3.1 Local Alternatives ................................ 451
5.3.2 Asymptotically Optimal Test ....................... 453
5.4 Goodness-of-fit Test .................................... 458
Historical Remarks ............................................ 461
References .................................................... 467
Index ......................................................... 479
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