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 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, 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, RaoBlackwell
and LehmannScheffe theorems, CramerRao inequality for
single parameter. Estimation by methods of moments, maximum
likelihood, least squares, minimum chisquare and modified
minimum chisquare, properties of maximum likelihood and
other estimators, asymptotic efficiency, prior and posterior
distributions, loss function, risk function, and minimax
estimator. Bayes estimators.
Nonrandomised and randomised tests, critical function, MP
tests, NeymanPearson 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 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 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, GaussMarkoff theory, normal equations, least
squares estimates and their precision, test of significance
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, estimation of variance and covariance
components, multivariate normal distribution, MahalanobisD^{2}
and Hotelling’s T^{2} statistics and their
applications and properties, discriminant analysis,
canonical correlations, principal component analysis.
4.
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 , 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.
Fixed
effects model (twoway classification) random and mixed
effects models (twoway 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 2^{n }and 3^{2}, confounding in
factorial experiments, splitplot and simple lattice
designs, transformation of data Duncan’s multiple range
test.
PAPER  II
1.
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. 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 DodgeRoming tables.
Concept of
reliability, failure rate and reliability functions,
reliability of series and parallel systems and other simple
configurations, renewal density and renewal function,
Failure models: exponential, Weibull, normal , lognormal.
Problems in
life testing, censored and truncated experiments for
exponential models.
2.
Optimization Techniques:
Different types of models in Operations Research, their construction and
general methods of solution, simulation and MonteCarlo
methods 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 algebraic).
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
continuoustime Markov chains, Poisson process, elements of
queuing theory, M/M/1, M/M/K, G/M/1 and M/G/1 queues.
Solution of
statistical problems on computers using wellknown
statistical software packages like SPSS.
3.
Quantitative Economics and Official Statistics:
Determination of trend, seasonal and cyclical components,
BoxJenkins method, tests for stationary series, ARIMA
models and determination of orders of autoregressive and
moving average components, forecasting.
Commonly
used index numbersLaspeyre's, Paasche's and Fisher's ideal
index numbers, chainbase index number, uses and limitations
of index numbers, index number of wholesale prices, consumer
prices, agricultural production and industrial production,
test for index numbers  proportionality, timereversal,
factorreversal and circular .
General
linear model, ordinary least square and generalized least
squares methods of estimation, problem of multicollinearity,
consequences and solutions of multicollinearity,
autocorrelation and its consequences, heteroscedasticity of
disturbances and its testing, test for independence of
disturbances, concept of structure and model for
simultaneous equations, problem of identificationrank and
order conditions of identifiability, twostage least square
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 limitations, principal publications
containing such statistics, various official agencies
responsible for data collection and their main functions.
4.
Demography and Psychometry:
Demographic data from census, registration, NSS other
surveys, their limitations 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, quasistable population, techniques in
estimation of demographic parameters, standard
classification by cause of death, health surveys and use of
hospital statistics.
Methods
of standardization of scales and tests, Zscores, standard
scores, Tscores, percentile scores, intelligence quotient
and its measurement and uses, validity and reliability of
test scores and its determination, use of factor analysis
and path analysis in psychometry.
