Wednesday, December 18, 2024
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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
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.
Paper-II
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
producer’s and consumer’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.
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.
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 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.
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.
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