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, Borel-Cantelli lemma, Chebyshevâ€™s and Khinchineâ€˜s weak
laws of large numbers, strong law of large numbers and kolmogorovâ€™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.
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, Bhattacharyyaâ€™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.
Kolmogorovâ€™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. Waldâ€™s SPRT and its properties, OC and
ASN functions, Waldâ€™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;â€™ D2 and Hotellingâ€™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, Warnerâ€™s randomised response technique for sensitive
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.
I. Industrial Statistics
Process and product control, general theory of control charts, different types
of control charts for variables and attributes, X, R, s, p, np and c charts,
cumulative sum chart, V-mask, single, double, multiple and sequential sampling
plans for attributes, OC, ASN, AOQ and ATI curves, concepts of producerâ€™s
and consumerâ€™s risks, AQL, LTPD and AOQL, sampling plans for variables, use
of Dodge-Romig and Military Standard tables.
Concepts of reliability, maintainability and availability, reliability of
series and parallel systems and other simple configurations, renewal density
and renewal function, survival models (exponential), Weibull, lognormal,
Rayleigh, and bath-tub), different types of redundancy and use of redundancy
in reliability improvement, problems in life-testing, censored and truncated
experiments for exponential models.
II. Optimization Techniques
Different, types of models in Operational Research, their construction and
general methods of solution, simulation and Monte-Carlo methods, the structure
and formulation of linear programming (LP) problem, simple LP model and its
graphical solution, the simplex procedure, the two-phase method and the
M-technique with artificial variables, the duality theory of LP and its
economic interpretation, sensitivity analysis, transportation and assignment
problems, rectangular games, two-person zero-sum games, methods of solution
(graphical and algerbraic).
Replacement of failing or deteriorating items, group and individual
replacement policies, concept of scientific inventory management and
analytical structure of inventory problems, simple models with deterministic
and stochastic demand with and without lead time, storage models with
particular reference to dam type.
Homogeneous discrete-time Markov chains, transition probability matrix,
classification of states and ergodic theorems, homogeneous continous-time
Markov chains, Poisson process, elements of queueing theory, M/M/1, M/M/K,
G/M/1 and M/G/1 queues.
Solution of statistical problems on computers using well known statistical
software packages like SPSS.
III. Quantitative Economics and Official Statistics
Determination of trend, seasonal and cyclical components, Box-Jenkins method,
tests for stationery of series, ARIMA models and determination of orders of
autoregressive and moving average components, forecasting.
Commonly used index numbers-Laspeyre's, Paashe's and Fisher's ideal index
numbers, chain-base index number uses and limitations of index numbers, index
number of wholesale prices, consumer price index number, index numbers of
agricultural and industrial production, tests, for mdex numbers lve
proportonality test, time-reversal test, factor-reversal test, circular test
and dimensional invariance test.
General linear model, ordinary least squares and generalised least squires
methods of estimation, problem of multicollineaity, consequences and solutions
of multicollinearity, autocorrelation and its consequeces, heteroscedasticity
of disturbances and its testing, test for independe of disturbances, Zellner's
seemingly unrelated regression equation model and its estimation, concept of
structure and model for simulaneous equations, problem of identification-rank
and order conditions of identifiability, two-stage least squares method of
Present official statistical system in India relating to population,
agriculture, industrial production, trade and prices, methods of collection of
official statistics, their reliability and limitation and the principal
publications containing such statistics, various official agencies responsible
for data collection and their main functions.
IV. Demography and Psychometry
Demographic data from census, registration, NSS and other surveys, and
their limitation and uses, definition, construction and uses of vital rates
and ratios, measures of fertility, reproduction rates, morbidity rate,
standardized death rate, complete and abridged life tables, construction of
life tables from vital statistics and census returns, uses of life tables,
logistic and other population growth curves, fitting a logistic curve,
population projection, stable population theory, uses of stable population and
quasi-stable population techniques in estimation of demographic parameters,
morbidity and its measurement, standard classification by cause of death,
health surveys and use of hospital statistics.
Methods of standardisation of scales and tests, Z-scores, standard scores,
T-scores, percentile scores, intelligence quotient and its measurement and
uses, validity of test scores and its determination, use of factor analysis
and path analysis in psychometry.