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 p-th mean and almost everywhere, their
criteria and interrelations, Borel-Cantelli lemma, Chebyshevs and Khinchines
weak laws of large numbers, strong law of large numbers and Kolmogorovs
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 (UMV
U) 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, Bhattacharyyas bounds,
estimation by methods of moments, maximum likelihood, least squares, 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, VIP tests, Neyman-Pearson lemma, UMP tests, monotone likelihood ratio, generalised Neyman-Pearson lemma, similar and unbiased test. 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.
Kolmogorovs test for goodness of fit and its consistent), sign test and its optimality, Wilcoxori signed-ranks test and its consistency, Kolmogorov-Smirnove two-samples test, run test, Wilcoxon-Mann-Whitney test and median test, their consistency and asymptotic normality. Walds SPR I and its properties, OC and ASN functions, Walds 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 Hotellings T2
statistics and their applications and properties, discriminant analysis,
canonical correlations, one-way M ANOVA, 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 estimationinvolving 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, Warners 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.
PAPER II
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 producers and consumers risk, AQL, LTPD and AOQL, sampling plans
for variables, use of Dodge-Romig and Military Standard tables.
Concepts of reliability, maintainability and availability, reliability o( series and parallel systems and other simple configurations, renewal density and renewal function, survival models (exponential), Weibull, lognormal, Rayleigh and bath-tub), different types o( redundancy and use of redundancy in reliability improvement, problems in life-testing, censored and turncated 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 algebraic).
Replacement of failing or deteriorating items, group and individual replacement policies, concept of scientific inventor) 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 continuous-time Markov chains, Poisson process, elements of queueing theory, M/M/l, M/ M/K, G/M/l and M/G/l 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 stationary of series, ARIMA models and determination of
orders of autoregressive and moving average components, forecasting.
Commonly used index numbers Laspeyres, Paasches and Fishers 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 index numbers like proportionality test, time-reversal test, factor-reversal test, circular test and dimensional invariance test.
General linear model, ordinary least, squares and generalised least squares methods of estimation, problem of multicollinearity, consequences and solutions of multicollinearity, auto-correlation and its consequences, heteroscedasticity of disturbances and its testing, tests for independence of disturbances, Zellners seemingly unrelated regression equation model and its estimation, concept of structure and model for simultaneous equations, problem of identification -rank and order conditions of identifiability, two-stage least squares method of estimation.
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, fifting a logistic curve,
population projection, stable population 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.
