PaperI
Probability : Sample space and events, probability measure and probability space, random variable as a measurable function, distribution function of a random variable, discrete and continuoustype random variable probability mass function, probability density function, vectorvalued 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 interrelations, BorelCantelli lemma, Chebyshev’s and Khinchine‘s weak laws of large numbers, strong law of large numbers and kolmogorov’s theorems, GlivenkoCantelli 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, RaoBlackwell and LehmannScheffe theorems, CramerRao inequality for single and severalparameter family of distributions, minimum variance bound estimator and its properties, modifications and extensions of CramerRao inequality, ChapmanRobbins inequality, Bhattacharyya’s bounds, estimation by methods of moments, maximum likelihood, least squares, minimum chisquare and modified minimum chisquare, properties of maximum likelihood and other estimators, idea of asymptotic efficiency, idea of prior and posterior distributions, Bayes estimators.
Nonrandomised and randomised tests, critical function, MP tests, NeymanPearson lemma, UMP tests, monotone likelihood ratio, generalised NeymanPearson lemma, similar and unbiased tests, UMPU tests for single and severalparameter families of distributions, likelihood rotates and its large sample properties, chisquare 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 signedranks test and its consistency, KolmogorovSmirnov twosample test, run test, WilcoxonMannWhitney 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, GaussMarkoff theory, normal equations, least squares estimates and their precision, test of signficance and interval estimates based on least squares theory in oneway, twoway and threeway 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, discrimi nant analysis, canonical correlatons, oneway MANOVA, principal component analysis, elements of factor analysis.
Sampling Theory and Design of Experiments
An outline of fixedpopulation 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, twostage and multistage sampling, ratio and regression, methods of estimation involving one or more auxiliary variables, twophase sampling, probability proportional to size sampling with and without replacement, the HansenHurwitz and the HorvitzThompson estimators, nonnegative variance estimation with reference to the HorvitzThompson estimator, nonsampling errors, Warner’s randomised response technique for sensitive characteristics.
Fixed effects model (twoway classification) random and mixed effects models (twoway 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, splitplot and simple lattice designs.
PaperII
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, Vmask, single, double, multiple and sequential sampling plans for attributes, OC, ASN, AOQ and ATI curves, concepts of producer’s and consumer’s risks, AQL, LTPD and AOQL, sampling plans for variables, use of DodgeRomig 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 bathtub), different types of redundancy and use of redundancy in reliability improvement,
problems in lifetesting, 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 MonteCarlo methods, the structure and formulation of linear programming (LP) problem, simple LP model and its graphical solution, the simplex procedure, the twophase method and the Mtechnique with artificial variables, the duality theory of LP and its economic interpretation, sensitivity analysis, transportation and assignment problems, rectangular games, twoperson zerosum 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 discretetime Markov chains, transition probability matrix, classification of states and ergodic theorems, homogeneous continoustime 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, BoxJenkins method, tests for stationery of series, ARIMA models and determination of orders of autoregressive and moving average components, forecasting.
Commonly used index numbersLaspeyre's, Paashe's and Fisher's ideal index numbers, chainbase index number uses and limitations of index numbers, index number of wholesale prices, consumer price index number, index numbers of agricultural and industrial production, test for index numbers like proportionality test, timereversal test, factorreversal test, circular test and dimensional invariance test.
General linear model, ordinary least squares and generalised least squires methods of estimation, problem of multicollinearlity, 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 identificationrank and order conditions of identifiability, twostage 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 quasistable 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, Zscores, standard scores, Tscores, 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.
