University of Rajasthan 2007 M.Sc Statistics M..A./ - Question Paper

Paper VIII: MULTIVARIATE ANALYSIS AND STATISTICAL INFERENCE
3 hrs. duration 100 Marks

Section-A
Multivarate normal distribution and its properties. Marginal and conditional distribution, characteristics function distribution of Quadratic form.

Random sampling from a multivariate normal distribution. Maximum likelihood estimators of Parameters. Distribution of sample mean vector.

Wishart matrix - its distribution and properties. Distribution of sample generalized variance. Null and non-Null distribution of simple correlation coefficient. Null distribution of Partial and multiple correlation coefficient. distribution of sample regression coefficients. Application in testing and interval estimation.

Distribution of sample intra-class correlation - coefficient in a random sample from a symmetric multivariate normal distribution. Application in testing and interval estimation.

Null Distribution of Hotelling's T2 statistic. Application in tests on mean vector for 1 and more multivariate normal populations and also on equality of the components of a mean vector in a multivariate normal population.

Multivariate linear regression model- estimation of parameters, tests of linear hypotheses about regression coefficients. likelihood ratio test criterion. Multivariate Analysis of Variance (MANOVA) 1 and two-way classified data.

Classification and discrimination producers for discrimination ranging from 2 multivariate normal populations - sample discriminant function, tests associated with discriminant funcions, probabilities of misclassification and their estimation, classification into more than 2 multivariate normal populations.

Principal components, Dimension reduction, Canonical variables and canonical correlation - definition use, estimation and calculation.

Section-B
Review of convergence in proability and convergence in distribution. Cramer and Slutsky's theorems.

Consistent Estimation o real and vector valued parameters. Invariance of Consistent estimator under continuous transformation. Consistency of estimators by method of moments and method of percentiles. Mean squared fault criterion. Asymptotic relative efficiency. fault probabilities and their rates of convergence. Minimum sample size needed to attain provided level of accuracy.

Consistent Asymprotic Normal (CAN) estimators. Invariance of CAN estimator under differentiable transformation. CAN property of estimators found by moments and percentiles. CAN estimators found by moments and MLE method in 1 parameter exponential family. Extension to multiparameter exponential family. Examples of consistent but not asymptotically normal from Pitman family. Method of maximum likelihood. CAN estimators for 1 parameter Cramer family. Cramer- Huzurbazar theorem. Solution of likelihood equations. Method of scoring. Newton-Raphson and other iterative procedures. Fisher Lower Bound to asymptotic variance, extension to multiparameter case (without proof). Multinomial distribution with cell probabilities depending on a parameter

MLE in pitman family and double Exponent distribution. MLE in censored and truncated distributions. UMVUE of probability densities (univariate case only)

Likelihood ratio test (LRT). Asymptotic distribution of LRT statistic wald test. Rao's Score test. Pearon chi-square test for Goodness of fit. Bartlett's test for hamogeneity of variances, Large sample tests. Consistency of Large Sample Tests. Asymptotic power of large sample tests.

References:
Fergusion,T.S.: A Course on large sample theory. Chapman & Hall
Rao,C.R.: Linear Statistical Inference
Rohatgi,V.K.: An Introduction to Probability and Statistics
Kale,B.K.: A 1st course of parametric inference
Lehmann,E.L.: Testing Statistical Hypothesis
Lehmann,E.L.: Theory of Point Estimation
Anderson,T.W.: An Introduction to Multivariate Statistical Analysis, Wiley
Giri,N.C.: Multivariate Statistical Inference
Morison,D.F.: Multivariate Statistical methods, McGraw Hill.
Sharma,S.(1996): Applied Multivariate Techniques, Wiley.
Srivastava,M.S. and Khatri,C.G.(1979): An Introduction to Multivariate Statistics.
Reference Books for Section-A:
Rao,C.R.: Linear Statistical Inference
Reference Books for Section-B:
Rao,C.R.: Linear Statistical Inference
Kendall and stuart: Advanced Theory of Statistics, Vol.II
Hogg and Craig; Introduction to Mathematical Statistics

Earning:   Approval pending.