Group Invariance in Statistical Inference

by Giri, Narayan C.

• ISBN13: 9789810218751
• ISBN10: 9810218753
• eBook ISBN(s): 9789812831705
• Format: Hardcover
• Copyright: 1997-05-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Table of Contents

Chapter 0. GROUP INVARIANCE	1	(6)
0.0. Introduction	1	(2)
0.1. Examples	3	(4)
Chapter 1. MATRICES, GROUPS AND JACOBIANS	7	(9)
1.0. Introduction	7	(1)
1.1. Matrices	7	(3)
1.1.1. Characteristic Roots and Vectors	8	(1)
1.1.2. Factorization of Matrices	9	(1)
1.1.3. Partitioned Matrices	9	(1)
1.2. Groups	10	(2)
1.3. Homomorphism, Isomorphism and Direct Product	12	(1)
1.4. Topological Transitive Groups	12	(1)
1.5. Jacobians	13	(20)
Chapter 2. INVARIANCE	16	(33)
2.0. Introduction	16	(1)
2.1. Invariance of Distributions	16	(9)
2.1.1. Transformation of Variable in Abstract Integral	22	(3)
2.2. Invariance of Testing Problems	25	(1)
2.3. Invariance of Statistical Tests and Maximal Invariant	26	(2)
2.4. Some Examples of Maximal Invariants	28	(3)
2.5. Distribution of Maximal Invariant	31	(2)
2.5.1. Existence of an Invariant Probability Measure on O(p) (Group of p x p Othogonal Matrices)	33	(1)
2.6. Applications	33	(3)
2.7. The Distribution of a Maximal Invariant in the General Case	36	(1)
2.8. An Important Multivariate Distribution	37	(7)
2.9. Almost Invariance, Sufficiency and Invariance	44	(1)
2.10. Invariance, Type D and E Regions	45	(4)
Chapter 3. EQUIVARIANT ESTIMATION IN CURVED MODELS	49	(19)
3.1. Best Equivariant Estimation of with Known	50	(3)
3.1.1. Maximum Likelihood Estimators	53	(1)
3.2. A Particular Case	53	(7)
3.2.1. An Application	58	(1)
3.2.2. Maximum Likelihood Estimator	58	(2)
3.3. Best Equivariant Estimation in Curved Covariance Models	60	(8)
3.3.1. Characterization of Equivariant Estimators of	61	(2)
3.3.2. Characterization of Equivariant Estimators of	63	(5)
Chapter 4. SOME BEST INVARIANT TESTS IN MULTINORMALS	68	(23)
4.0. Introduction	68	(1)
4.1. Tests of Mean Vector	68	(7)
4.2. The Classification Problem (Two Populations)	75	(7)
4.3. Tests of Multiple Correlation	82	(3)
4.4. Tests of Multiple Correlation with Partial Information	85	(6)
Chapter 5. SOME MINIMAX TESTS IN MULTINORMALES	91	(46)
5.0. Introduction	91	(2)
5.1. Locally Minimax Tests	93	(13)
5.2. Asymptotically Minimax Tests	106	(5)
5.3. Minimax Tests	111	(26)
5.3.1. Hotelling's T2 Test	112	(12)
5.3.2. R2-Test	124	(11)
5.3.3. E-Minimax Test (Linnik, 1966)	135	(2)
Chapter 6. LOCALLY MINIMAX TESTS IN SYMMETRICAL DISTRIBUTIONS	137	(25)
6.0. Introduction	137	(1)
6.1. Elliptically Symmetric Distributions	137	(3)
6.2. Locally Minimax Tests in Ep ( , )	140	(3)
6.3. Examples	143	(19)
Chapter 7. TYPE D AND E REGIONS	162