Structure of the examination:
Date of Advertisement: July 17th , 2012
Date of Examination: August 17th , 2012 (Friday) 2.30 p.m. – 4.30 p.m. at the Department of Statistics, University of Calcutta
Date of Interview: September 14th , 2012 (Friday) from 2 pm at the Department of Statistics, University of Calcutta
Publication of result: October 12th , 2012
Commencement of course work: November 20th , 2012
One Semester Course Work of 20 credits as follows
|1||Review of candidate’s research topic||:||5 credit points|
|2||Seminar presentation by the candidate||:||5 credit points|
|3||Seminar attendance||:||2 credit points|
|3||Research Methodology||:||5 credit points|
|5||Statistical Computing||:||3 credit points|
Detailed Syllabus for Entrance Examination:
Real Number System, Cluster Points of sets, Closed and open sets, Compact sets, Bolzano-Weierstrass Property, Heine-Borel Property.Sets of Real Vectors, Sequences and Series, Convergence. Real valued functions. Limit, Continuity and Uniform continuity. Differentiability of univariate and multivariate functions. Mean value theorems. Extrema of functions. Riemann integral. Improper integrals. Riemann-Stieltjes integral. Sequences and Series of functions, Uniform convergence, Power series.Term by term differentiation and integration, Differentiation and integration under the integral sign.
Classes of sets, Fields, Sigma-fields, Minimum sigma-field, Borel sigma-field in Rk, Sequence of sets, limsup and liminf of a sequence of sets. Measure, Probability Measure, Properties of a measure, Caratheodory extension theorem (statement only) Lebesgue and Lebesgue-Stieltjes measures on Rk . Measurable functions, Random variables, Expectation, Sequences of random variables, Almost sure convergence, Convergence in probability. Distribution function, Convergence in distribution. Integration of a measurable function with respect to a measure, Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem. Generating functions and Characteristic function, Inversion theorem and Continuity theorem (Statement only) ,Borel –Cantelli lemma, Independence,Weak law and Strong law of large numbers, Kolmogorov inequality. Central limit theorem for iid random variables, CLT for a sequence of independent random variables (Statement only).Radon-Nikodym theorem(Statement only) and its applications, Product measure and Fubini’s theorem (Statement only).
Linear Algebra and Linear Programming
Vectors and Matrices;Graphical Solution and Simplex Algorithm
Non-central x2, t & F distributions – definitions and properties.Distribution of quadratic forms – Cochran’s theorem. Multivariate normal distribution – independence of sample mean vector and variance-covariance matrix. Wishart distribution.Distributions of partial and multiple correlation coefficients and regression coefficients, distribution of intraclass correlation coefficient. Hotelling T2 and Mahalanobis’s D2 application in testing and confidence set construction.
Large Sample Theory
Asymptotic distributions of sample moments and functions of moments, Asymptotic distributions of Order Statistics and Quantiles. Consistency and Asymptotic Efficiency of Estimators, Large sample properties of Maximum Likelihood estimators. Asymptotic distributions and properties of Likelihood ratio tests, Rao’s test and Wald’s tests in the simple hypothesis case.
Sufficiency & completeness, Notions of minimal sufficiency,bounded completeness and ancillarity, Exponential family.Point estimation : Bhattacharya system of lower bounds to variance of estimators. Minimum variance unbiased estimators – Applications of Rao – Blackwell and Lehmann – Scheffe theorems.Testing of Hypothesis : nonrandomized and randomized tests, critical function, power function. MP tests – Neyman – Pearson Lemma. UMP tests. Monotone Likelihood Ratio families. Generalized Neyman – Pearson Lemma. UMPU tests for one parameter families. Locally best tests. Similar tests. Neyman structure. UMPU tests for composite hypotheses.
Confidence sets: relation with hypothesis testing. Optimum parametric confidence intervals. Sequential tests. Wald’s equation for ASN. SPRT and its properties – fundamental identity. O.C. and ASN. Optimality of SPRT (under usual approximation).
Pre-requisites of matrix algebra.Gauss-Markov set-up, estimable function, BLUE and Gauss-Markov Theorem, estimation and error spaces, estimation with correlated observations, least squares estimates with restriction on parameters.
Tests of linear hypotheses and associated confidence sets – related sampling distributions, Multiple comparison techniques due to Scheffe and Tukey; Applications of general linear hypothesis to regression, analysis of variance and covariance.Random and Mixed effects models (balanced case), estimation of variance components.
Residuals and their plots. Tests of fit of a model. Q-Q plots. Transformations.Detection of outliers.Departures from the usual assumptions : heteroscedasticity, autocorrelation, multicollinearity, non-normality – detection and remedies. Variable selection problems.
Design of Experiments
Block designs – concepts of connectedness, orthogonality and balance; intrablock analysis – BIB and PBIB designs; extension to row-column designs – Latin Square design and Youden Square design, Recovery of interblock information in BIB designs.Construction of complete classes of mutual orthogonal Latin squares (MOLS); construction of BIBD – through MOLS and Bose’s fundamental method of differences.Factorial experiments, confounding and balancing in symmetric factorials.Response Surface Experiments – first order designs.
Probability sampling from a finite population – Notions of sampling design, sampling scheme, inclusion probabilites, Horvitz-Thompson estimator of a population total. Basic sampling schemes – Simple random sampling with and without replacement, Unequal probability sampling with and without replacement, Systematic sampling. Related estimators of population total/mean, their variances and variance estimators – Mean per distinct unit in simple random with replacement sampling, Hansen-Hurwitz estimator in unequal probability sampling with replacement, Des Raj and Murthy’s estimator (for sample of size two) in unequal probability sampling without replacement.Stratified sampling – Allocation problem and construction of strata.Ratio, Product, Difference and Regression estimators. Unbiased Ratio estimators – Probability proportional to aggregate size sampling, Hartley – Ross estimator in simple random sampling. Sampling and sub-sampling of clusters. Two-stage sampling with equal/unequal number of second stage units and simple random sampling without replacement / unequal probability sampling with replacement at first stage, Ratio estimation in two-stage sampling.Double sampling for stratification. Double sampling ratio and regression estimators. Sampling on successive occasions.
Decision problem and two-person game. Nonrandomized and randomized rules. Risk function.
Introduction to Jackknife and Bootstrap-methods for estimating bias, standard error and distribution function based on iid random variables. Standard examples. Consistency of the Jackknife estimate of the variance in the iid set up.Jackknife and Bootstrap estimates of regression parameters and the error variance.Bootstrap confidence intervals.
Introduction to Stochastic Process, Markov Chains with finite and countable state space, Chapman-Kolmogorov equation, Classification of States, Calculation of n-step transition probability and its limit, Stationary distribution, Discrete state space continuous time Markov Chains, Poisson Process, Birth-Death Process, Renewal theory
Applied Multivariate Analysis
Hierarchical and non-hierarchical clustering methods.Classification and discrimination procedures for discrimination between two known populations – Bayes, Minimax and Likelihood Ratio procedures. Discrimination between two multivariate normal populations.Population and sample principal components and their uses. Large sample inferences.The orthogonal factor model, Estimation of factor loading, Factor rotation, Estimation of Factor scores, Interpretation of Factor Analysis.
Generalized Linear Models and Data Analytic Techniques
Types of data. Two-way classified data – Contingency Tables and associated distributions, Types of studies, Relative Risk and Odds Ratio and their properties. More-than-two-way classified data – partial associations, marginal and conditional odds. Generalized Linear Models – introduction, components of a generalized linear model, measuring the goodness of fit, deviance, residuals, maximum likelihood estimation.Applications to binary, count and polytomous data. Overdispersion. Ideas of marginal, conditional and quasi likelihoods. Longitudinal Data Analysis – introduction with motivation. Exploring longitudinal data.Linear models for longitudinal data –introduction, mean models, covariance models, mixed effects models. Predictions.Discrete longitudinal data- generalized linear marginal models, GEE for marginal models, Generalized linear subject specific models and transition models.