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Tables of Contents for Unbiased Estimators and Their Applications
Chapter/Section Title
Page #
Page Count
Preface
vii
Basic Remarks on Multivariate Probability Distributions
Cumulative Distribution Function and Probability Density Function
1
2
Marginal and Conditional Distributions. Independence
3
2
Expected Values and Moments
5
7
Characteristic Functions
12
2
Multivariate Normal Distribution
14
8
Some Multivariate Discrete Distributions
22
8
Examples of Continuous Multivariate Distributions
30
13
Elements of the Theory of Point Statistical Estimation in the Multivariate Case
Samples, Statistics, Estimators and Problems of Statistical Estimation
43
3
Risk of Statistical Estimator. Consistency. Unbiasedness
46
2
Sufficiency, Completeness and Unbiased Estimators
48
3
The Rao-Cramer Inequality
51
9
The Rao-Blackwell-Kolmogorov Theorem
60
2
On Estimating Parameters of the Multivariate Normal Distribution
62
12
The Stein Approach for Estimators Improvement
74
22
Techniques for Constructing Unbiased Estimators
Techniques for Constructing MVUEs
96
14
Unbiased Estimators for the Multivariate Probability Density Functions
110
14
Minimum Variance Unbiased Estimation for Multivariate Modified Power Series Distributions
124
13
Applications of Unbiased Estimators
On an Estimating the Momentum of a Relativistic Particle
137
3
A Statistical Group Classification of Normal Populations
140
11
A Remark on the Chauvenet Criterion
151
4
Testing Hypotheses for Small Samples in the Presence of Sufficient Statistics
155
12
A Remark on Using of Conditional Distributions. Gerig's Test
167
5
A Remark on Analysis of Variance
172
6
Unbiased estimators of the reliability
178
7
A Chi-Square Goodness-of-Fit Test for Several Exponentials With a Common Shift
185
6
Appendix 1. Tables of Unbiased Estimators
A1. MVUEs for Functions of Parameters of the Multivariate Normal Distribution
191
11
A2. MVUEs for Functions of Parameters of the Multinomial Distribution
202
2
A3. MVUEs for Functions of Parameters of the Trinomial Distribution
204
1
A4. MVUEs for Functions of Parameters of the Negative Multionomial Distribution
205
3
A5. MVUEs for the Left Truncated Multivariate Modified Power Series Distribution
208
4
A6. MVUEs for Functions of Parameters of the Left Truncated Multivariate Exponential Distribution (σ is known)
212
3
A7. MVUEs for Functions of Parameters of the Left Truncated Multivariate Exponential Distribution
215
4
A8. MVUEs for a two-dimensional one-truncation parameter family
219
3
Appendix 2. On Evaluating Some Multivariable Integrals
222
6
Appendix 3. Partitions and Some Multivariable Statistical Problems
228
13
References
241
11
Subject Index
252
6
Author Index
258