STATISTICS
PAPER – I
1. 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 p-th mean and
almost everywhere, their criteria and inter-relations, Chebyshev’s inequality
and Khintchine‘s weak law of large numbers, strong law of large numbers and
Kolmogoroff’s theorems, probability generating function, moment generating
function, characteristic function, inversion theorem, Linderberg and Levy forms
of central limit theorem, standard discrete and continuous probability
distributions.
2. Statistical
Inference:
Consistency,
unbiasedness, efficiency, sufficiency, completeness, ancillary statistics,
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 parameter.
Estimation by methods of
moments, maximum likelihood, least squares, minimum chi-square and modified minimum
chi-square, properties of maximum likelihood and other estimators, asymptotic
efficiency, prior and posterior distributions, loss function, risk funct n, and
minimax estimator. Bayes estimators. Non-randomised and randomised tests,
critical function, MP tests, Neyman-Pearson lemma, UMP tests, monotone
likelihood ratio, similar and unbiased tests, UMPU tests for single parameter
likelihood ratio test and its asymptotic distribution. Confidence bounds and
its relation with tests.
Kolmogoroff’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-Whitney test and median test, their consistency and asymptotic
normality.
Wald’s SPRT and its
properties, OC and ASN functions for tests regarding parameters for Bernoulli,
Poisson, normal and exponential distributions. Wald’s fundamental identity.
3. 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, estimation of variance and covariance components,
multivariate normal distribution, Mahalanobis-D2 and Hotelling’s T2 statistics
and their applications and properties, discriminant analysis, canonical
correlations, principal component analysis.
4. 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 , cluster sampling, twostage 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.
Fixed effects model
(two-way classification) random and mixed effects models (two-way
classification with equal observation per cell), CRD, RBD, LSD and their
analyses, incomplete block designs, concepts of orthogonality and balance,
BIBD, missing plot technique, factorial experiments and 2n and 32, confounding
in factorial experiments, split-plot and
simple lattice designs, transformation of data Duncan’s multiple range test.
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