STATISTICS - Chanakya Mandal Pariwar

STATISTICS (Medium –English)

PAPER–I (subject code 1056)

1. Probability :

Sample space and events, probability measure and probability space, random variable as a measurable
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 path 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 Kolmogoroffs 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
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.
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 paramet likelihood
ratio test and its asymptotic distribution. Confidence bounds and its relation with tests.
Kolmogorov’s test for goodness of fit and its consistency, sign test and its optimality. Wilcoxon
signedranks 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 oneway, 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’s 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, propability sampling designs, simple random sampling with and without
replacement, stratified random sampling, systematic sampling and its efficacy, cluster sampling,
twostage 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 HorvitzThompson 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 24 and 32, confounding in factorial experiments, split-plot and simple lattice designs,
transformation of data Duncan’s multiple range test.

PAPER II (subject code 1057)

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 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-Romin
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 metbod and the M-technique
with artificial variables, the duality theory of LP and its economic interpretation, sensitivity analysis,
transpotation and assignment problems, rectangular games, two-person 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 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/MI, M/M/K, G/M/l 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, Box-Jenkins method, tests for stationary
series, ARIMA models and determination of orders of autoregressive and moving average
components, fore-casting.
Commonly used index numbers – Laspeyre’s, Paasche’s and Fisher’s ideal index numbers, cham-base
index number, uses and limitations of index numbers, index number of wholesale prices, consumer
price, agricultural production and industrial production, test fot index numbers -proportionality, timereversal,
factor-reversal and circular.
General linear model, ordinary least square and generalized least squares methods of estimation,
problem of multi-collinearity, consequences and solutions of multi-collinearity, 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, two-stage least sauare 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.