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Optional Statistics

 

STATISTICS

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.

 

 

 

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, 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

 

4. Sampling Theory and Design of Experiments:

 

 

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

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 numbersLaspeyre’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

multi-collinearity, consequences and solutions of mult-icollinearity, auto-correlation 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 standardization 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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