Mathematical Science Syllabus Paper 2 Unit 22 To 28

Mathematical Science Syllabus Paper 2 Unit 22 To 28

22. Distribution Theory: Properties of distribution functions and characteristic functions; continuity theorem, inversion formula, Representation of distribution function as a mixture of discrete and continuous distribution functions; Convolutions, marginal and conditional distributions of bivariate discrete and continuous distributions.

Relations between characteristic functions and moments; Moment inequalities of Holder and Minkowski.

23. Statistical Inference and Decision Theory:

Statistical decision problem: non-randomized, mixed and randomized decision rules; risk function, admissibility, Bayes' rules, minimax rules, least favourable distributions, complete class, and minimal complete class. Decision problem for finite parameter space. Convex loss function. Role of sufficiency. Admissible, Bayes and minimax estimators; illustrations. Unbiasedness. UMVU estimators. Families of distributions with monotone likelihood property, exponential family of distributions. Test of a simple hypothesis against a simple alternative from decision-theoretic viewpoint. Tests with Neyman structure. Uniformly most powerful unbiased tests. Locally most powerful tests. Inference on location and scale parameters; estimation and tests. Equivariant estimators. Invariance in hypothesis testing.

24. Large sample statistical methods: Various modes of convergence. O_p and o_p, CLT, Sheffe's theorem, Polya's theorem and Slutsky's theorem. Transformation and variance stabilizing formula. Asymptotic distribution of function of sample moments. Sample quantiles. Order statistics and their functions. Tests on correlations, coefficients of variation, skewness and kurtosis. Pearson Chi-square, contingency Chi-square and likelihood ratio statistics. U-statistics. Consistency of Tests. Asymptotic relative efficiency.

25. Multivariate Statistical Analysis: Singular and non-singular multivariate distributions. Characteristics functions. Multivariate normal distribution; marginal and conditional distribution, distribution of linear forms, and quadratic forms, Cochran's theorem.

Inference on parameters of multivariate normal distributions: one-population and two-population cases. Wishart distribution. Hotellings T², Mahalanobis D². Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.

26. Linear Models and Regression: Standard Gauss-Markov models; Estimability of parameters; best linear unbiased estimates (BLUE); Method of least squares and Gauss-Markov theorem; Variance-covariance matrix of BLUES.

Tests of linear hypothesis; One-way and two-way classifications. Fixed, random and mixed effects models (two-way classifications only); variance components, Bivariate and multiple linear regression; Polynomial regression; use of orthogonal polynomials. Analysis of covariance. Linear and nonlinear regression. Outliers.

27. Sample Surveys: Sampling with varying probability of selection, Hurwitz-Thompson estimator; PPS sampling; Double sampling. Cluster sampling. Non-sampling errors: Interpenetrating samples. Multiphase sampling. Ratio and regression methods of estimation.

28. Design of Experiments: Factorial experiments, confounding and fractional replication. Split and strip plot designs; Quasi-Latin square designs; Youden square. Design for study of response surfaces; first and second order designs. Incomplete block designs; Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information; PBIBD with 2 associate classes. Analysis of series of experiments, estimation of residual effects. Construction of orthogonal-Latin squares, BIB designs, and confounded factorial designs. Optimality criteria for experimental designs.