An introduction to multivariate statistics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
North-Holland
1979
|
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001501833&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XVI, 350 S. |
| ISBN: | 0444003029 |
Staff View
MARC
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| 100 | 1 | |a Srivastava, M. S. |d 1936- |e Verfasser |0 (DE-588)170217914 |4 aut | |
| 245 | 1 | 0 | |a An introduction to multivariate statistics |c M. S. Srivastava ; C. G. Khatri |
| 264 | 1 | |a New York [u.a.] |b North-Holland |c 1979 | |
| 300 | |a XVI, 350 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Analyse multivariée | |
| 650 | 7 | |a Multivariate analyse |2 gtt | |
| 650 | 4 | |a Multivariate analysis | |
| 650 | 0 | 7 | |a Multivariate Analyse |0 (DE-588)4040708-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Multivariate Analyse |0 (DE-588)4040708-1 |D s |
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| 700 | 1 | |a Khatri, C. G. |e Verfasser |0 (DE-588)170615529 |4 aut | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-001501833 | |
Record in the Search Index
| DE-BY-TUM_call_number | 0102 MAT 627f 2001 A 32036 |
|---|---|
| DE-BY-TUM_katkey | 381214 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020382363 |
| _version_ | 1821936913039228929 |
| adam_text | Contents
Preface xv
Chapter 1 Some Results on Matrices and Jacobians
of Transformations 1
1.1 Notation and Definitions 1
1.2 Matrix Operations 2
1.3 Determinants 3
1.3.1 Cofactors and Minors 6
1.4 Inverse of a Nonsingular Matrix 7
1.5 Rank of a Matrix 8
1.6 Generalized Inverse of a Matrix 12
1.7 Idempotent Matrices 14
1.8 Characteristic Roots and Characteristic Vectors 16
1.9 Hermitian and Positive Definite (or Semidefinite)
Matrices 18
1.10 Some Inequalities 20
1.11 Matrix Differentials and the Jacobian of
Transformations 26
Problems 32
Chapter 2 Multivariate Normal Distribution 41
2.1 Multivariate Normal Distribution 41
2.2 Some Properties of Multivariate
Normal Distributions 44
2.3 Some Probability Inequalities 48
2.4 Multiple Correlation 50
vii
Contents
2.5 Canonical Correlations 52
2.6 Partial Correlation 53
2.7 Random Sample from Np (0, 2) 54
2.8 Estimation of 0 and 2 if X~Np, N(6e , 2, IN) 55
2.8.1 Maximum Likelihood Estimates 56
2.9 Complex Multivariate Normal Distribution 56
2.10 Asymptotic Distributions 59
2.11 Results on Quadratic Forms 60
2.11.1 Distribution of Quadratic Forms 60
2.11.2 Necessary and Sufficient Conditions for Chi
Square and Independence 63
Problems 67
Chapter 3 Wishart Distributions 72
3.1 Introduction 72
3.2 Distribution of V= YY Where Y~Np n(0,2, /„) 72
3.3 Some Properties of Wishart Distributions 78
3.3.1 Distribution of Sample Covariance 78
3.3.2 Marginal Distribution 78
3.3.3 Additive Property 81
3.3.4 Distribution of AVA 81
3.3.5 Generalized Variance and Its Distribution 81
3.3.6 Characteristic Function 83
3.4 Distribution of the Characteristic Roots and
Vectors when 2 = / 86
3.4.1 Characteristic Roots 86
3.4.2 Characteristic Vectors 86
3.5 Distribution of Sample Correlations 87
3.5.1 Matrix of Correlations and Correlation in the
Bivariate Case 87
3.5.2 Multiple Correlation 89
3.5.3 Partial Correlation 91
3.6 Multivariate Beta Distributions 92
3.7 Complex Wishart Distribution 97
Problems 97
Chapter 4 Inference on Location; Hotelling s T2 104
4.1 Introduction 104
4.2 Union Intersection Principle 104
4.3 Test for /*=0, Where x~Np(p, 2) 110
4.3.1 Likelihood Ratio Test 110
4.3.2 Union Intersection Test Procedure 112
4.3.3 Distribution of T2 112
viii
Contents
4.3.4 Invariance 113
4.3.5 Optimality 114
4.3.6 Confidence Interval for p. 115
4.4 Test for a Subvector 115
4.4.1 Distribution of £ 117
4.4.2 Invariance 118
4.4.3 Optimum Property 118
AAA Other Invariant Tests 119
4.5 Comparing Means of Correlated Variables 119
4.5.1 Relation Between Means when 2 0 119
4.5.2 Intraclass Correlation Model 121
4.5.3 Test Procedure for Problem (i) 123
4.5.4 Likelihood Ratio Procedure for Problem (ii) 126
4.5.5 Solutions of Problems (a) to (f) 727
4.6 Two Sample Problem 128
4.7 Behrens Fisher Problem 129
Problems 130
Chapter 5 Linear Regression Models Estimation 132
5.1 Introduction 132
5.2 Growth Curve Model 133
5.3 Estimability Criterion 136
5.3.1 Necessary and Sufficient Condition with V
Nonsingular and No Restrictions on ft 136
5.3.2 Necessary and Sufficient Condition with
V Singular 137
5.4 Estimation of /3 in the Ordinary Linear Regression
Model 138
5.4.1 Solution of Eq. (5.4.11) when W Is
Nonsingular 140
5A.2 Solution of Eq. (5.4.11) when TV Is Singular 140
5.5 BLUE Under Two Different Models 142
5.6 Estimation of Location Parameters for the
Growth Curve Model 144
5.7 Best Quadratic Unbiased Estimator of the Variance 145
5.7.1 Uniformly Minimum Variance Quadratic Unbiased
Estimate with Invariance (UMVIQUE) 146
5.7.2 UMVQUE (Without Invariance) 149
5.8 Estimation of Variance Components 152
5.8.1 MINQUE Theory (Without Invariance) 153
5.8.2 MINQUE with Invariance 156
5.8.3 Examples 157
5.8.4 Locally Uniformly Minimum Variance
Unbiased Estimate (LUMVUE) 159
Problems 162
ix
Contents
Chapter 6 Linear Models—Testing of Hypotheses for
Regression Parameters 166
6.1 Introduction 166
6.2 Tests on the Regression Parameter ft 166
6.2.1 Likelihood Ratio Test for C/J 0 when
V Is P.D. 166
6.2.2 Union Intersection Test Procedure when V Is
Nonsingular and p(C)= I 168
6.2.3 A Test Procedure Based on BLUE when
V Is Singular 169
6.3 Multivariate Linear Model 170
6.3.1 Likelihood Ratio Test Procedure when
V Is Nonsingular 171
6.3.2 Union Intersection Test Procedure when
V Is Nonsingular and p(.C) l 172
6.3.3 Testing Equality of k Mean Vectors 173
6.3.4 A Test Procedure Based on BLUE when
V Is Singular 174
6.3.5 Moments ofX Under H: fiC O 175
6.3.6 Exact Distribution of An 177
6.3.7 Asymptotic Distribution 182
6.3.8 Distribution of ch^S^S, 1) Under H 186
6.3.9 Confidence Bounds on Estimable
Functions PC 187
6.3.10 Canonical Reduction for the Noncentral
Distributions 189
6.3.11 Other Tests 190
6.4 Generalized Linear Models (Growth Curve Models) 191
6.4.1 Introduction 191
6.4.2 Maximum Likelihood Estimates of £ and 2 191
6.4.3 Canonical Reduction of the Model 192
6.4.4 Likelihood Ratio Test for £=0 193
6.4.5 hth Moment of X Under H 194
Problems 195
Chapter 7 Inference on Covariances 198
7.1 Introduction 198
7.2 Testing That the Covariance Matrix Is an
Identity Matrix 198
7.2.1 Invariance 199
7.2.2 Moments of X* 199
7.2.3 Asymptotic Expansion of the cdf of — 21n *
Under H 200
7.2.4 Asymptotic Expansion of the cdf of 21nX*
Under Alternatives Close to the
Hypothesis H 201
x
Contents
7.2.5 Union Intersection Test Procedure and
Confidence Bounds 204
7.2.6 Some Other Tests 206
7.3 Testing That the Covariance Matrix Is an
Identity Matrix and the Mean
Vector Is Zero 206
7.3.1 Invariance 207
7.4 Test for Sphericity 207
7.4.1 Invariance 208
7.4.2 Moments of A* Under H 208
7.4.3 Exact Distribution Under H 209
7.4.4 Asymptotic Expansion of the cdf Under H 210
7.4.5 An Alternative Test 211
7.4.6 Union Intersection Test Procedures 212
7.4.7 Simultaneous Confidence Bounds 215
7.5 Test for Independence 217
7.5.1 Invariance 218
7.5.2 Moments of A* Under H 219
7.5.3 Exact Distribution Under H 220
7.5.4 Asymptotic Distribution Under H 221
7.5.5 An Alternative Test 222
7.5.6 Union Intersection Test Procedure
when*=2 223
7.6 Test for the Equality of Covariances 224
7.6.1 Invariance 225
7.6.2 Moments of A Under H 226
7.6.3 Asymptotic cdf of mlnA* Under H 227
7.6.4 An Alternative Test 229
7.6.5 Union—Intersection Test Procedure and
Confidence Bounds 230
7.7 Testing 2 = / in the Growth Curve Model 233
7.7.1 Ath Moment of A* Under H 235
Problems 236
Chapter 8 Classification and Discrimination 241
8.1 Introduction 241
8.2 The Problem of Classifying into Two Known
Normals with Common Covariance 242
8.2.1 Errors of Misclassification 242
8.2.2 Fisher s Discriminant Function 243
8.3 Classifying into Two Normals with Known
Common Covariance 244
8.3.1 Errors of Misclassification 246
8.3.2 Monotonicity of the Errors of
Misclassification 247
8.3.3 An Estimator of the Error of
Misclassification, el 251
xi
Contents
8.4 Classifying into Two Completely Unknown
Normals 252
8.4.1 Errors of Misclassification 254
8.4.2 Monotonicity of the Errors of
Misclassification 256
8.4.3 An Estimate of e, 257
8.5 Classifying into k Normals with Common
Covariance 258
8.6 Classifying into k Normals with the Population
Means Are Linearly Restricted 259
8.7 Classifying into Two Normals with Unequal
Covariances 261
8.8 Classifying into k Normals with Known
Unequal Covariances 266
8.9 A Sequential Approach to Classification 267
8.10 The Problem of Minimum Distance 268
8.11 Remarks 270
Problems 270
Chapter 9 Principal Component Analysis 272
9.1 Introduction 272
9.2 Definition 273
9.2.1 Effects of Units of Measurement 274
9.3 Maximum Likelihood Estimates 275
9.4 Asymptotic Distribution of the Roots and Vectors
of a Wishart Matrix 276
9.4.1 Asymptotic Distribution up to O(n~x) 277
9.4.2 An Asymptotic Expansion of /,• when
Is Simple 283
9A3 Asymptotic Expansion up to O(n~3/2) when the
Roots of 2 Are Simple 284
9.4.4 Variances and Covariances of Unordered
Characteristic Roots when the Population
Roots Are Equal (Girshick) 291
9.5 Inference on Characteristic Roots 292
9.6 Asymptotic Test for a Given Principal Component
(Anderson) 294
9.7 Exact Test for a Given Principal Component 295
Problems 296
Chapter 10 Monotonicity and Unbiasedness of
Some Power Functions 298
10.1 Introduction 298
10.2 Some Probability Inequalities 298
10.3 Monotonicity of Some Tests for MANOVA 301
10.3.1 Bibliographical Note 304
xii
Contents
10.4 Monotonicity of Tests for Covariance 305
10.4.1 Monotonicity of the Modified Likelihood Ratio
Test for 2 = 7 305
10.4.2 Monotonicity of the Likelihood Ratio Tests for
Sphericity and Homogeneity of Variances 307
10.4.3 Unbiasedness of the Likelihood Ratio Test
for /»=0 and 2 = 7 ill
10.4.4 Some Monotonicity and Unbiasedness Results
for the Modified Likelihood Ratio Test for
2,=22 312
10.4.5 Monotonicity of Some One Sided Invariant Tests
for the Equality of Two Covariances 317
10.5 Monotonicity of Some Tests for the Independence of
Two Vectors 320
10.5.1 Canonical Reduction and Invariance 320
10.5.2 Monotonicity of Some Invariant Tests 321
Problems 322
Bibliography 324
Author Index 341
Subject Index 345
xiii
|
| any_adam_object | 1 |
| author | Srivastava, M. S. 1936- Khatri, C. G. |
| author_GND | (DE-588)170217914 (DE-588)170615529 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-search | 519.5/3 |
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| discipline | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV002285155 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:48:53Z |
| institution | BVB |
| isbn | 0444003029 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001501833 |
| oclc_num | 4494668 |
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| physical | XVI, 350 S. |
| publishDate | 1979 |
| publishDateSearch | 1979 |
| publishDateSort | 1979 |
| publisher | North-Holland |
| record_format | marc |
| spellingShingle | Srivastava, M. S. 1936- Khatri, C. G. An introduction to multivariate statistics Analyse multivariée Multivariate analyse gtt Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| subject_GND | (DE-588)4040708-1 |
| title | An introduction to multivariate statistics |
| title_auth | An introduction to multivariate statistics |
| title_exact_search | An introduction to multivariate statistics |
| title_full | An introduction to multivariate statistics M. S. Srivastava ; C. G. Khatri |
| title_fullStr | An introduction to multivariate statistics M. S. Srivastava ; C. G. Khatri |
| title_full_unstemmed | An introduction to multivariate statistics M. S. Srivastava ; C. G. Khatri |
| title_short | An introduction to multivariate statistics |
| title_sort | an introduction to multivariate statistics |
| topic | Analyse multivariée Multivariate analyse gtt Multivariate analysis Multivariate Analyse (DE-588)4040708-1 gnd |
| topic_facet | Analyse multivariée Multivariate analyse Multivariate analysis Multivariate Analyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001501833&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT srivastavams anintroductiontomultivariatestatistics AT khatricg anintroductiontomultivariatestatistics |
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| Call Number: |
0102 MAT 627f 2001 A 32036
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