📉 OPTIONAL — STATISTICS

Papers: VI & VII · 500 marks. GS overlap: LOW (helps with data interpretation and economy). Best for: statistics/mathematics graduates. Highly technical — not for beginners.

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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, 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.
2. 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 ratio tests, asymptotic properties of likelihood ratio tests. Basic ideas of sequential probability ratio test and its properties. Confidence bounds and its relation with tests. 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.
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, MANOVA, 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 Horvitz-Thompson estimators, non-negative variance estimation with reference to the Yates-Grundy 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 designs 2^n and 3^2, confounding in factorial experiments, split-plot and simple lattice designs.

PAPER II

(Candidates choose any TWO of the four sections)

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, 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.
2. 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 under mixed strategies, equivalence between games and LP; 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 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.
3. Quantitative Economics and Official Statistics: determination of trend, seasonal and cyclical components, Box-Jenkins method, tests for stationarity of 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 price index number, index numbers of agricultural and industrial production, test for index numbers — proportionality test, time-reversal, factor-reversal and circular tests; theory and analysis of consumer demand — specification and estimation of demand functions, demand elasticities, theory of production, supply functions and elasticities, input demand functions, estimation of parameters of single-equation linear regression model, ordinary least squares method, generalized least squares method, heteroscedasticity, serial correlation, multicollinearity, errors in variable models, simultaneous equations models — identification, rank and order conditions, indirect least squares and two-stage least squares; 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 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, standard classical and prospective methods of studying population growth. 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.

BOOKLIST

STRATEGY

• Derivations plus numericals — practise both; show complete working.
• Choose your Paper II sections early and prepare them thoroughly rather than sampling all four.
• Time management is the main failure point — practise full timed papers.
• Solve 15 years of PYQs.
• Only choose with a genuine statistics/maths background — the measure-theoretic probability alone deters most candidates.

Pros: highly objective; very small competition pool; static syllabus. Cons: extremely technical; almost no GS overlap; limited guidance material available.