Group invariance in statistical inference

In applied and pure sciences, the structural properties of groups are increasingly utilised to find better solutions in statistical sciences. Modern computers make statistical methods with large numbers of variables feasible. Invariance is a mathematical term for symmetry, and many statistical problems exhibit such properties. In statistical analysis with large numbers of variables, the invariance approach is becoming increasingly popular and useful because of its ability and usefulness in deriving better statistical procedures. In this book, Multivariate Statistical Inference is presented through Invariance.

Ch. 0. Group invariance. 0.0. Introduction. 0.1. Examples -- ch. 1. Matrices, groups and Jacobians. 1.0. Introduction. 1.1. Matrices. 1.2. Groups. 1.3. Homomorphism, isomorphism and direct product. 1.4. Topological transitive groups. 1.5. Jacobians -- ch. 2. Invariance. 2.0. Introduction. 2.1. Invariance of distributions. 2.2. Invariance of testing problems. 2.3. Invariance of statistical tests and maximal invariant. 2.4. Some examples of maximal invariants. 2.5. Distribution of maximal invariant. 2.6. Applications. 2.7. The distribution of a maximal invariant in the general case. 2.8. An important multivariate distribution. 2.9. Almost invariance, sufficiency and invariance. 2.10. Invariance, type D and E regions -- ch. 3. Equivariant estimation in curved models. 3.1. Best equivariant estimation of [symbol] with [symbol] known. 3.2. A particular case. 3.3. Best equivariant estimation in curved covariance models -- ch. 4. Some best invariant tests in multinormals. 4.0. Introduction. 4.1. Tests of mean vector. 4.2. The classification problem (two populations). 4.3. Tests of multiple correlation. 4.4. Tests of multiple correlation with partial information -- ch. 5. Some minimax test…