Unbiased Estimators and their Applications: Volume 2: Multivariate CaseThis volume is a continuation of Unbiased Estimators and Their Applications, Vol. I: Univariate Case. It contains problems of parametric point estimation for multivariate probability distributions emphasizing problems of unbiased estimation. The volume consists of four chapters dealing, respectively, with some basic properties of multivariate continuous and discrete distributions, the general theory of point estimation in multivariate case, techniques for constructing unbiased estimators and applications of unbiased estimation theory in the multivariate case. These chapters contain numerous examples, many applications and are followed by a comprehensive Appendix which classifies and lists, in the form of tables, all known results relating to unbiased estimators of parameter functions for multivariate distributions. Audience: This volume will serve as a handbook on point unbiased estimation for researchers whose work involves statistics. It can also be recommended as a supplementary text for undergraduate and graduate students. |
Contents
BASIC REMARKS ON MULTIVARIATE PROBABILITY DISTRIBUTIONS | 1 |
12 Marginal and Conditional Distributions Independence | 3 |
13 Expected Values and Moments | 5 |
14 Characteristic Functions | 12 |
15 Multivariate Normal Distribution | 14 |
16 Some Multivariate Discrete Distributions | 22 |
17 Examples of Continuous Multivariate Distributions | 30 |
ELEMENTS OF THE THEORY OF POINT STATISTICAL ESTIMATION IN THE MULTIVARIATE CASE | 43 |
44 Testing Hypotheses for Small Samples in the Presence of Sufficient Statistics | 155 |
45 A Remark on Using of Conditional Distributions Gerigs Test | 167 |
46 A Remark on Analysis of Variance | 172 |
47 Unbiased estimators of the reliability | 178 |
48 A chisquare goodnessoffit test for several exponentials with a common shift | 185 |
TABLES OF UNBIASED ESTIMATORS Al MVUEs for Functions of Parameters of the Multivariate Normal Distribution | 191 |
A2 MVUEs for Finnlions of Parameters of the Multinomial Distribution | 202 |
A3 MVUEs for Functions of Parameters of the Trinomial Distribution | 204 |
22 Risk of Statistical Estimator Consistency Unbiasedness | 46 |
23 Sufficiency Completeness and Unbiased Estimators | 48 |
24 The RaoCramer Inequality | 51 |
25 The RaoBlackwellKolmogorov Theorem | 60 |
26 On Estimating Parameters of the Multivariate Normal Distribution | 62 |
27 The Stein Approach for Estimators Improvement | 74 |
TECHNIQUES FOR CONSTRUCTING UNBIASED ESTIMATORS | 96 |
32 Unbiased Estimators for the Multivariate Probability Density Functions | 110 |
33 Minimum Variance Unbiased Estimation for Multivariate Modified Power Series Distributions | 124 |
APPLICATIONS OF UNBIASED ESTIMATION | 137 |
42 A Statistical Group Classification of Normal Populations | 140 |
43 A Remark on the Chauvenet Criterion | 151 |
A4 MVUEs for Functions of Parameters of the Negative Multinomial Distribution | 205 |
A5 MVUEs for Functions of Parameters of the Left Truncated Multivariate Modified Power Series Distribution | 208 |
A6 MVUEs for Functions of Parameters of the Left Truncated Multivariate Exponential Distribution a is known | 212 |
A7 MVUEs for Functions of Parameters of the Left Truncated Multivariate Exponential Distribution | 215 |
A8 MVUEs for a Twodimensional Onetruncation Parameter Family | 219 |
ON EVALUATING SOME MULTIVARIABLE INTEGRALS | 222 |
PARTITIONS AND SOME MULTIVARIABLE STATISTICAL PROBLEMS | 228 |
| 241 | |
| 252 | |
| 258 | |
Other editions - View all
Unbiased Estimators and Their Applications: Volume 2: Multivariate Case V.G. Voinov,M.S. Nikulin No preview available - 1993 |
Common terms and phrases
A.N. PHILIPPOU a₁ binomial distribution bivariate exponential distribution Ch.A Charalambides classification rule complete sufficient statistic components conditional distribution considered construct covariance matrix decision rule defined Denote discrete distributions distribution function equivariant estimator evaluate example exponential distribution follows formula function F(x H₁ hypothesis independent integers James-Stein estimator joint probability density k-dimensional known loss function Lumel'sky M.S. Nikulin Math minimal minimum variance unbiased modified power series multinomial distribution multivariate exponential multivariate normal distribution MVU estimable noso obtain Olkin partitions positive definite power series distributions Prob probability density function probability distribution problem r₁ random variable random vector Remark risk sample Sankhyá Stat statistical estimation sufficient statistic T₁ THEOREM UMVUE unbiased estimator univariate variance unbiased estimation Voinov X(r₁ X₁ Y(ra Y₁ Σ Σ ΣΣ