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
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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 p-th 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 inter-relations 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, Bhattacharyya�s bounds, estimation by methods of moments, maximum
likelihood, least squares, minimum chi-square 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 several-parameter 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 two-sample test, run test,
Wilcoxon-Mann-Whiltney 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 modesl, theory of least
squares and analysis of variance, Gauss-Markoff theory, normal equations, least
squares estimates and their precision, test of signficance 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 distributin, 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
super-population 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 estimators, non-negative variance estimation with reference to
the Horvitz-Thompson estimator, 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 per
cell), CRD, RBD, LSD and their analyses, incomplete block designs, concepts of
orthogonality and balance, BIBD, missing plot technique, factorial designs : 2n,
32 and 33, confounding in factorial experiments, split-plot and simple lattice
designs.
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