STATISTICS
PAPER-I
Probability : Sample space and events, probability measure and probability space, random variable as a measurable function, distribution function of a random variable, discrete and continuous-type random variable, probability mass function, probability density function, vector-valued random variable, marginal and conditional distributions, stochastic independence of events and of random variables, expectation and moments of a random variable, conditional expectation, convergence of a sequence of random variable in distribution, in probability, in pth mean and almost everywhere, their criteria and inter-relations, Borel-Cantelli lemma, Chebyshev’s and Khinchine’s weak laws of large numbers, strong law of large numbers and Kolmogorov’s theorems, Glivenko-Cantelli theorem, probability generating function, characteristic function, inversion theorem, Laplace transform, related uniqueness and continuity theorems, determination of distribution by its moments. Linderberg and Levy forms of central limit theorem, standard discrete and continuous probability distributions, their interrelations and limiting cases, simple properties of finite Markov chains.
Statistical Inference : Consistency, unbiasedness, efficiency, sufficiency, minimal sufficiency, completeness, ancillary statistic, factorization theorem, exponential family of distribution and its properties, uniformly minimum variance unbiased (UMVU) estimation, Rao-Blackwell and Lehmann- Scheffe theorems, Cramer-Rao inequality for single and several-parameter family of distributions, minimum variance bound estimator and its properties, modifications and extensions of Cramer-Rao inequality, Chapman-Robbins inequality, Bhattacharya’s bounds, estimation by methods of moments, maximum likelihood, least squares, minimum chisquare and modified minimum chi-square properties of maximum likelihood and other estimators, idea of asymptotic efficiency, idea of prior and posterior distributions, Bayes’, estimators. Non-randomised and randomised tests, critical function, MP tests, Neyman- Pearson lemma, UMP tests, monotone likelihood ratio, generalised Neyman- Pearson lemma, similar and unbiased tests, UMPU tests for single and severalparameter families of distributions, likelihood rotates and its large sample properties, chi-square goodness of fit test and its asymptotic distribution. Confidence bounds and its relation with tests, uniformly most accurate (UMA) and UMA unbiased confidence bounds. Kolmogorov’s test for goodness of fit and its consistency, sign test and its optimality, Wilcoxon signed-ranks test and its consistency, Kolmogorov-Smirnov twosample test, run test, Wilcoxon-Mann- Whitney test and median test, their consistency and asymptotic normality. Wald’s SPRT and its properties, OC and ASN functions, Wald’s fundamental identity, sequential estimation.
Linear Inference and Multivariate Analysis : Linear statistical models, theory of least squares and analysis of variance, Gauss- Markoff theory, normal equations, least squares estimates and their precision, test of significance and interval estimates based on least squares theory in one-way, two-way and three-way classified data, regression analysis, linear regression, curvilinear regression and orthogonal polynomials, multiple regression, multiple and partial correlations, regression diagnostics and sensitivity analysis, calibration problems, estimation of variance and covariance components, MINQUE theory, multivariate normal distribution, Mahalanobis; D2 and Hotelling’s T2 statistics and their applications and properties, discriminant analysis, canonical correlations, one-way MANOVA, principal component analysis, elements of factor analysis.
Sampling Theory and Design of Experiments: An outline of fixed-population and superpopulation approaches, distinctive features of finite population sampling, probability sampling designs, simple random sampling with and without replacement, stratified random sampling, systematic sampling and its efficacy for structural populations, cluster sampling, two-stage and multi-stage sampling, ratio and regression, methods of estimation involving one or more auxiliary variables, two-phase sampling, probability proportional to size sampling with and without replacement, the Hansen-Hurwitz and the Horvitz-Thompson estimator, nonnegative variance estimation with reference to the Horvitz-Thompson estimators, non-sampling errors, Warner’s randomised response technique for sensitive characteristics. Fixed effects model (two-way classification) random and mixed effects models (two-way classification with equal number of observation per cell), CRD, RBD, LSD and their analysis, incomplete block designs, concepts of orthogonality and balance, BIBD, missing plot technique, factorial designs: 2n, 32 and 33, confounding in factorial experiments, splitplot and simple lattice designs.
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