Linear statistical inference and its applications:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1973
|
| Edition: | 2. ed. |
| Series: | Wiley series in probability and mathematical statistics : Probability and mathematical statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 625 S. |
| ISBN: | 0471708232 |
Staff View
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| 245 | 1 | 0 | |a Linear statistical inference and its applications |c C. Radhakrishna Rao |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1973 | |
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Record in the Search Index
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| adam_text | Contents
CHAPTER 1
Algebra of Vectors and Matrices 1
la. Vector Spaces 2
la.l Definition of Vector Spaces and Subspaces, Ja.2 Basis of a
Vector Space, la.3 Linear Equations, la.4 Vector Spaces with
an Inner Product
Complements and Problems 11
lb. Theory of Matrices and Determinants 14
ib.I Matrix Operations, lb.2 Elementary Matrices and Diagonal
Reduction of a Matrix, lb.3 Determinants, lb.4 Transformations
lb.5 Generalized Inverse of a Matrix, Ib.6 Matrix Representation,
of Vector Spaces,Bases, etc., lb.7 IdempotentMatrices, lb.8 Spe¬
cial Products of Matrices
Complements and Problems 30
lc. Eigenvalues and Reduction of Matrices 34
lc. I Classification and Transformation of Quadratic Forms, lc.2 Roots
of Determinantal Equations, lc.3 Canonical Reduction of Matrices,
lc.4 Projection Operator, 1 c.5 Further Results on g-Inverse, Ic.6
Restricted Eigenvalue Problem
Id. Convex Sets in Vector Spaces 51
Id. I Definitions, ld.2 Separation Theorems for Convex Sets
xiii
xiv CONTENTS
le. Inequalities 53
le.l Cauchy-Schwarz (C-S) Inequality, le.2 Holder s Inequality,
le.3 Hadamard s Inequality, le.4 Inequalities Involving Moments,
le.5 Convex Functions and Jensen s Inequality, le.6 Inequalities in
Information Theory, le.7 Stirling s Approximation
If. Extrema of Quadratic Forms 60
lf.1 General Results, If.2 Results Involving Eigenvalues and Vectors
If.3 Minimum Trace Problems
Complements and Problems 67
CHAPTER 2
Probability Theory, Tools and Techniques 79
2a. Calculus of Probability 80
2a.l The Space of Elementary Events, 2a.2 The Class of Subsets
(Events), 2a.3 Probability as a Set Function, 2a.4 Borel Field
(a-field) and Extension of Probability Measure, 2a.5 Notion of a
Random Variable and Distribution Function, 2a.6 Multidimensional
Random Variable, 2a.7 Conditional Probability and Statistical Inde¬
pendence, 2a.8 Conditional Distribution of a Random Variable
2b. Mathematical Expectation and Moments of Random Variables 92
2b.1 Properties of Mathematical Expectation, 2b.2 Moments, 2b.3
Conditional Expectation, 2b.4 Characteristic Function (c./.), 2b.5
Inversion Theorems, 2b.6 Multivariate Moments
2c. Limit Theorems 108
2c.1 Kolmogorov Consistency Theorem, 2c.2 Convergence of a
Sequence of Random Variables, 2c.3 Law of Large Numbers,
2c.4 Convergence of a Sequence of Distribution Functions, 2c.5
Central Limit Theorems, 2c.6 Sums of Independent Random Variables
2d. Family of Probability Measures and Problems of Statistics 130
2d. 1 Family of Probability Measures, 2d.2 The Concept of a Sufficient
Statistic, 2d.3 Characterization of Sufficiency
Appendix 2A. Stieltjes and Lebesgue Integrals 132
Appendix 2B. Some Important Theorems in Measure Theory
and Integration 134
CONTENTS xv
Appendix 2C. Invariance 138
Appendix 2D. Statistics, Subfields, and Sufficiency 139
Appendix 2E. Non-Negative Definiteness of a Characteristic
Function 141
Complements and Problems 142
CHAPTER 3
Continuous Probability Models 155
3a. Univariate Models 158
3a.1 Normal Distribution, 3a.2 Gamma Distribution, 3a.3 Beta
Distribution, 3a.4 Cauchy Distribution, 3a.5 Student s t Distri¬
bution, 3a.6 Distributions Describing Equilibrium States in Statistical
Mechanics, 3a.7 Distribution on a Circle
3b. Sampling Distributions 179
3b.1 Definitions and Results, 3b.2Sumof Squares of Normal Variables,
3b.3 Joint Distribution of the Sample Mean and Variance, 3b.4 Distri¬
bution of Quadratic Forms, 3b.5 Three Fundamental Theorems of the
Least Squares Theory, 3b.6 The p-Variate Normal Distribution,
3b.7 The Exponential Family of Distributions
3c. Symmetric Normal Distribution 197
3c.1 Definition, 3c.2 Sampling Distributions
3d. Bivariate Normal Distribution 201
3d.l General Properties, 3d.2 Sampling Distributions
Complements and Problems 209
CHAPTER 4
The Theory of Least Squares and Analysis of Variance 220
4a. Theory of Least Squares (Linear Estimation) 221
4a. 1 Gauss-Markoff Setup (Y, Xfi, a21), 4a.2 Normal Equations and
Least Squares (l.s.) Estimators, 4a.3 g-Inverse and a Solution of the
Normal Equation, 4a.4 Variances and Covariances of l.s. Estimators,
4a.5 Estimation of a2, 4a.6 Other Approaches to the l.s. Theory
(Geometric Solution), 4a.7 Explicit Expressions for Correlated
Observations, 4a.8 Some Computational Aspects of the l.s. Theory,
4a.9 Least Squares Estimation with Restrictions on Parameters,
xvi CONTENTS
4a. 10 Simultaneous Estimation of Parametric Functions, 4a.11 Least
Squares Theory when the Parameters Are Random Variables, 4a. 12
Choice of the Design Matrix
4b. Tests of Hypotheses and Interval Estimation 236
4b.1 Single Parametric Function {Inference), 4b.2 More than One
Parametric Function (Inference), 4b.3 Setup with Restrictions
4c. Problems of a Single Sample 243
4c.1 The Test Criterion, 4c.2 Asymmetry of Right and Left Femora
(Paired Comparison)
4d. One-Way Classified Data 244
4d.l The Test Criterion, 4d.2 An Example
4e. Two-Way Classified Data 247
4e.l Single Observation in Each Cell, 4e.2 Multiple but Equal Num¬
bers in Each Cell, 4e.3 Unequal Numbers in Cells
4f. A General Model for Two-Way Data and Variance Components 258
4f.l A General Model, 4f.2 Variance Components Model, 4f.3
Treatment of the General Model
4g. The Theory and Application of Statistical Regression 263
4g.l Concept of Regression (General Theory), 4g.2 Measurement of
Additional Association, 4g.3 Prediction of Cranial Capacity (a
Practical Example), 4g.4 Test for Equality of the Regression Equations,
4g.5 The Test for an Assigned Regression Function, 4g.6 Re¬
stricted Regression
4h. The General Problem of Least Squares with Two Sets of Parameters 288
4h.l Concomitant Variables, 4h.2 Analysis of Covariance, 4h.3
An Illustrative Example
4i. Unified Theory of Linear Estimation 294
4i.l A Basic Lemma on Generalized Inverse, 4i.2 The General Gauss-
Markoff Model (GGM), 4L3 The Inverse Partitioned Matrix (IPM)
Method, 4i.4 Unified Theory of Least Squares
4j. Estimation of Variance Components 302
4J.1 Variance Components Model, 4j.2 MINQUE Theory, 4J.3
Computation under the Euclidian Norm
CONTENTS xvii
4k. Biased Estimation in Linear Models 305
4k.1 Best Linear Estimator (BLE), 4k.2 Best Linear Minimum Bias
Estimation (BLIMBE)
Complements and Problems 308
CHAPTER 5
Criteria and Methods of Estimation 314
5a. Minimum Variance Unbiased Estimation 315
5a. 1 Minimum Variance Criterion, 5a.2 Some Fundamental Results on
Minimum Variance Estimation, 5a.3 The Case of Several Parameters,
5a.4 Fisher s Information Measure, 5a.5 An Improvement of Un¬
biased Estimators
5b. General Procedures 334
5b.1 Statement of the General Problem {Bayes Theorem), 5b.2 Joint
d.f. of (6, x) Completely Known, 5b.3 The Law of Equal Ignorance,
5b.4 Empirical Bayes Estimation Procedures, 5b.5 Fiducial Proba¬
bility, 5b.6 Minimax Principle, 5b.7 Principle of Invariance
5c. Criteria of Estimation in Large Samples 344
5c.l, Consistency, 5c.2 Efficiency
5d. Some Methods of Estimation in Large Samples 351
5d.l Method of Moments, 5d.2 Minimum Chi-Square and Associated
Methods, 5d.3 Maximum Likelihood
5e. Estimation of the Multinomial Distribution 355
5e.l Nonparametric Case, 5e.2 Parametric Case
5f. Estimation of Parameters in the General Case 363
5f.l Assumptions and Notations, 5f.2 Properties of m.l. Equation
Estimators
5g. The Method of Scoring for the Estimation of Parameters 366
Complements and Problems 374
CHAPTER 6
Large Sample Theory and Methods 382
6a. Some Basic Results 382
6a.l Asymptotic Distribution of Quadratic Functions of Frequencies,
6a.2 Some Convergence Theorems
xviii CONTENTS
6b. Chi-Square Tests for the Multinomial Distribution 390
6b.1 Test of Departure from a Simple Hypothesis, 6b.2 Chi-Square
Test for Goodness of Fit, 6b.3 Test for Deviation in a Single Cell,
6b.4 Test Whether the Parameters Lie in a Subset, 6b.5 Some
Examples, 6b.6 Test for Deviations in a Number of Cells
6c. Tests Relating to Independent Samples from Multinomial
Distributions 398
6c. 1 General Results, 6c.2 Test of Homogeneity of Parallel Samples,
6c.3 An Example
6d. Contingency Tables 403
6d.l The Probability of an Observed Configuration and Tests in Large
Samples, 6d.2 Tests of Independence in a Contingency Table, 6d.3
Tests of Independence in Small Samples
6e. Some General Classes of Large Sample Tests 415
6e.l Notations and Basic Results, 6e.2 Test of a Simple Hypothesis,
6e.3 Test of a Composite Hypothesis
6f. Order Statistics 420
6f.l The Empirical Distribution Function, 6f.2 Asymptotic Distribu¬
tion of Sample Fractiles
6g. Transformation of Statistics 426
6g.l A General Formula, 6g.2 Square Root Transformation of the
Poisson Variate, 6g.3 Sin 1 Transformation of the Square Root of
the Binomial Proportion, 6g.4 Tanh~ * Transformation of the Correla¬
tion Coefficient
6h. Standard Errors of Moments and Related Statistics 436
6h.l Variances and Covariances of Raw Moments, 6h.2 Asymptotic
Variances and Covariances of Central Moments, 6h.3 Exact Ex¬
pressions for Variances and Covariances of Central Moments
Complements and Problems 439
CHAPTER 7
Theory of Statistical Inference 444
7a. Testing of Statistical Hypotheses 445
7a.l Statement of the Problem, 7a.2 Neyman-Pearson Fundamental
Lemma and Generalizations, 7a.3 Simple Ho against Simple H,
CONTENTS xix
7a.4 Locally Most Powerful Tests, 7a.5 Testing a Composite Hypothe¬
sis, 7a.6 Fisher-Behrens Problem, 7a.7 Asymptotic Efficiency of
Tests
7b. Confidence Intervals 470
7b.1 The General Problem, 7b.2 A General Method of Constructing a
Confidence Set, 7b.3 Set Estimators for Functions of 6
7c. Sequential Analysis 474
7c.l Wald s Sequential Probability Ratio Test, 7c.2 Some Properties
of the S.P.R. T, 7c3 Efficiency of the S.P.R. T, 7c A An Example
of Economy of Sequential Testing, 7c.5 The Fundamental Identity of
Sequential Analysis, 7c.6 Sequential Estimation, 7c.7 Sequential
Tests with Power One
7d. Problem of Identification—Decision Theory 491
7d. 1 Statement of the Problem, 7d.2 Randomized and Nonrandomized
Decision Rules, 7d.3 Bayes Solution, 7d.4 Complete Class of
Decision Rules, 7d.5 Minimax Rule
7e. Nonparametric Inference 497
7e.l Concept of Robustness, 7e.2 Distribution-Free Methods, 7e.3
Some Nonparametric Tests, 7e.4 Principle of Randomization
7f. Ancillary Information 505
Complements and Problems 506
CHAPTER 8
Multivariate Analysis 516
8a. Multivariate Normal Distribution 517
8a.l Definition, 8a.2 Properties of the Distribution, 8a.3 Some
Characterizations of Np, 8a.4 Density Function of the Multivariate
Normal Distribution, 8a.5 Estimation of Parameters, 8a.6 Np as a
Distribution with Maximum Entropy
8b. Wishart Distribution 533
8b. 1 Definition and Notation, 8b.2 Some Results on Wishart Distribu¬
tion
8c. Analysis of Dispersion 543
8c. 1 The Gauss-Markoff Setup for Multiple Measurements, 8c.2 Esti¬
mation of Parameters, 8c.3 Tests of Linear Hypotheses, Analysis of
xx CONTENTS
Dispersion (A.D.), 8c.4 Test for Additional Information, 8c.5 The
Distribution of A, 8c.6 Test for Dimensionality (Structural Relation¬
ship), 8c.7 Analysis of Dispersion with Structural Parameters (Growth
Model)
8d. Some Applications of Multivariate Tests 562
8d.l Test for Assigned Mean Values, 8d.2 Test for a Given Structure
of Mean Values, 8d.3 Test for Differences between Mean Values of
Two Populations, 8d.4 Test for Differences in Mean Values between
Several Populations, 8d.5 Barnard s Problem of Secular Variations in
Skull Characters
8e. Discriminatory Analysis (Identification) 574
8e.l Discriminant Scores for Decision, 8e.2 Discriminant Analysis in
Research Work, 8e.3 Discrimination between Composite Hypotheses
8f. Relation between Sets of Variates 582
8fl Canonical Correlations, 8f.2 Properties of Canonical Variables,
8f.3 Effective Number of Common Factors, 8f.4 Factor Analysis
8g. Orthonormal Basis of a Random Variable 587
8g.l The Gram-Schmidt Basis, 8g.2 Principal Component Analysis
Complements and Problems 593
Publications of the Author 605
Author Index 615
Subject Index 618
|
| any_adam_object | 1 |
| author | Rao, Calyampudi Radhakrishna 1920-2023 |
| author_GND | (DE-588)119285924 |
| author_facet | Rao, Calyampudi Radhakrishna 1920-2023 |
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| ctrlnum | (OCoLC)611271434 (DE-599)BVBBV025904937 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| indexdate | 2024-07-09T22:14:43Z |
| institution | BVB |
| isbn | 0471708232 |
| language | English |
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| series2 | Wiley series in probability and mathematical statistics : Probability and mathematical statistics |
| spelling | Rao, Calyampudi Radhakrishna 1920-2023 Verfasser (DE-588)119285924 aut Linear statistical inference and its applications C. Radhakrishna Rao 2. ed. New York [u.a.] Wiley 1973 XVIII, 625 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and mathematical statistics : Probability and mathematical statistics Methode (DE-588)4038971-6 gnd rswk-swf Statistische Schlussweise (DE-588)4182963-3 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Statistik (DE-588)4056995-0 s DE-604 Methode (DE-588)4038971-6 s Statistische Schlussweise (DE-588)4182963-3 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019150589&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rao, Calyampudi Radhakrishna 1920-2023 Linear statistical inference and its applications Methode (DE-588)4038971-6 gnd Statistische Schlussweise (DE-588)4182963-3 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4038971-6 (DE-588)4182963-3 (DE-588)4056995-0 (DE-588)4151278-9 |
| title | Linear statistical inference and its applications |
| title_auth | Linear statistical inference and its applications |
| title_exact_search | Linear statistical inference and its applications |
| title_full | Linear statistical inference and its applications C. Radhakrishna Rao |
| title_fullStr | Linear statistical inference and its applications C. Radhakrishna Rao |
| title_full_unstemmed | Linear statistical inference and its applications C. Radhakrishna Rao |
| title_short | Linear statistical inference and its applications |
| title_sort | linear statistical inference and its applications |
| topic | Methode (DE-588)4038971-6 gnd Statistische Schlussweise (DE-588)4182963-3 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Methode Statistische Schlussweise Statistik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019150589&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT raocalyampudiradhakrishna linearstatisticalinferenceanditsapplications |