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Contents
Basic Notation 1
Introduction 3
Chapter I
The Problem of Statistical Estimation 10
I The Statistical Experiment 10
2 Formulation of the Problem of Statistical Estimation 16
3 Some Examples 24
4 Consistency. Methods for Constructing Consistent Estimators 30
5 Inequalities for Probabilities of Large Deviations 41
6 Lower Bounds on the Risk Function 58
7 Regular Statistical Experiments. The Cramer-Rao Inequality 62
8 Approximating Estimators by Means of Sums of Independent Random
Variables 82
9 Asymptotic Efficiency 90
10 Two Theorems on the Asymptotic Behavior of Estimators 103
Chapter II
Local Asymptotic Normality of Families of Distributions 113
Independent Identically Distributed Observations 113
2 Local Asymptotic Normality (LAN) 120
3 Independent Nonhomogeneous Observations 123
4 Corollaries for the Model H Signal Plus Noise" 131
5 Observations with an .... Almost Smooth" Density 133
6 Multidimensional Parameter Set 139
7 Observations in Gaussian White Noise 143
8 Some Properties of Families of Distributions Admitting the LAN
Condition 147
9 Characterization of Limiting Distributions of Estimators 150
I 0 Anderson's Lemma 155
11 Asymptotic Efficiency under LAN Conditions 158
12 Asymptotically Minimax Risk Bound 162
13 Some Corollaries. Superefficient Estimators 169
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VI
Contents
Chapter III
Properties of Estimators in the Regular Case
I Maximum Likelihood Estimator
2 Bayesian Estimators
3 Independent Identically Distributed Observations
4 Independent Nonhomogeneous Observations
5 Gaussian White Noise
173
173
178
184
190
199
Cha pter I V
Some Applications to Nonparametric Estimation
I A Minimax Bound on Risks
2 Bounds on Risks for Some Smooth Functionals
3 Examples of Asymptotically Efficient Estimators
4 Estimation of Unknown Density
5 Minimax Bounds on Estimators for Density
214
214
220
229
232
237
Chapter V
Independent Identically Distributed Observations.
Densities with Jumps
241
241
246
260
266
276
I Basic Assumptions
2 Convergence of Marginal Distributions of the Likelihood Ratio
3 Convergence in the Space Do
4 The Asymptotic Behavior of Estimators
5 Locally Asymptotic Exponential Statistical Experiments
Cha pter VI
Independent Identically Distributed Observations.
Classification of Singularities
I Assumptions. Types of Singularities
2 Limiting Behavior of the Likelihood Ratio
3 1;/1 Processes. Singularities of the First and Third Type
4 l':" Processes. Singularities of the Second Type
5 Proofs of Theorems 2.1-2.3
6 Properties of Estimators
281
281
288
297
303
309
312
Chapter VI I
Several E timation Problems in a Gaussian White Noise
321
321
329
338
345
354
I Frequency Modulation
2 Estimation of Parameters of Discontinuous Signals
3 Calculation of Efficiency of Maximum Likelihood Estimators
4 Nonparameteric Estimation of an Unknown Signal
5 Lower Bounds on Nonparametric Estimators
Contents
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Vll
Appendix I
Some Limit Theorems of Probability Theory
I Convergence of Random Variables and Distributions in R k
2 Some Limit Theorems for Sums of Independent Random Variables
3 Weak Convergence on Function Spaces
4 Conditions for the Density of Families of Distributions in C(F) and
Co(R k ) and Criteria for Uniform Convergence
5 A Limit Theorem for Integrals of Random Functions
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380
Appendix II
Stochastic Integrals and Absolute Continuity of Measures
I Stochastic I ntegrals over h(t)
2 Some Definitions and Theorems of Measure Theory
3 Stochastic Integrals over Orthogonal Random Measure
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387
Remarks
389
Bibliography
395
Index
401
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