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.
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.
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,
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.
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 (two-way 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
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.
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.
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.
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.