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Syllabus:Statistics(Main) |
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-Whitney 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,
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
distribution,
Mahalanobis;’ D2 and Hotelling’s T2 statistics and their
applications and
properties, discrimi nant analysis, canonical correlatons, 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
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, 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.
II. 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.
III. 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, test for index numbers like
proportionality 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 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
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.
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, fitting a logistic curve, population
projection,
stable population 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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