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
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, 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, Rao-Blackwell and Lehmann-Scheffe
theorems, Cramer-Rao inequality for single parameter. Estimation by methods of
moments, maximum likelihood, least squares, minimum chi-square and modified
minimum chi-square, properties of maximum likelihood and other estimators,
asymptotic efficiency, prior and posterior distributions, loss function, risk function,
and minimax estimator. Bayes estimators.
Non-randomised and randomised tests, critical function, MP tests, Neyman-Pearson
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 signed-ranks test and its consistency, Kolmogorov-Smirnov two-sample
test, run test, Wilcoxon-Mann-Whitney 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, 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, twoway
and three-way 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, Mahalanobis-D2 and Hotelling’s T2 statistics and their applications and
properties, discriminant analysis, canonical correlations, principal component analysis.
4. 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 , 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.
Fixed effects model (two-way 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 2n and 32, confounding in factorial experiments,
split-plot 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 Dodge-Roming 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 Monte-Carlo methods 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 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 continuous-time 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 well-known statistical software
packages like SPSS.
3. Quantitative Economics and Official Statistics:
Determination of trend, seasonal and cyclical components, Box-Jenkins method,
tests for stationary series, ARIMA models and determination of orders of autoregressive
and moving average components, forecasting.
Commonly used index numbers-Laspeyre’s, Paasche’s and Fisher’s ideal index
numbers, chain-base 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, time-reversal, factor-reversal 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 identification-rank and order
conditions of identifiability, two-stage 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, quasi-stable population, techniques in estimation of
demographic parameters, 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 and
reliability of test scores and its determination, use of factor analysis and path analysis
in psychometry.
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