University of Lucknow Courses

M.Sc. (Statistics)

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Duration: 2 Years

Eligibility: B.Sc.

Course Structure

Semester - I

• 101 : Real Analysis
• 102 : Linear Algebra
• 103 : Measure Theory and Probability
• 104 : Sample Surveys
• 105 : Statistics Practical
  

Semester - II

• 201 : Complex Analysis, Transforms and Special Functions.
• 202 : Theory of Probability and Limit Theorems.
• 203 : Linear Models & Regression Analysis
• 204 : Multivariate Analysis
• 205 : Statistics Practical
  

Semester - III

• 301 : Sequential Analysis and Reliability Theory
• 302 : Inference
• 303 : Block Designs and their Analysis
• 304 : Econometrics
• 305 : Statistics Practical
  

Semester - IV

• 401 : Asymptotic Inference
• 402 : Stochastic Processes
• 403 : Decision Theory and Bayesian Analysis
• 404 : Factorial Experiments and Response surfaces
• 405 : Statistics Practical & Project

 

Course Detail

101 : Real Analysis

Unit - I

Recap of elements of set theory; Introduction to real number, introduction to ndimensional Euclidian space, open and closed intervals (rectangles), compact sets, Bolzano-Weirstrass theorem,closed, open and compact sets and their properties, Heine - Borel theorem.

Unit - II

Sequences and their limits. series and their convergence, Tests of convergence, Differentiation : Mean value theorem, maxima & minima of functions. Functions of several variables: constrained maxima & minima of functions.

Unit - III

Real valued functions, continuous functions, Uniform continuity, sequences of functions, uniform convergence, Power series and radius of convergence, Riemann Integration, Mean value theorems of integral calculus.

Unit - IV

Improper integrals and their convergence, Convex functions and their properties, Jensen's, Minkowski's and Holder's inequalities, Multiple integrals and their  evaluation by repeated integration, Change of variables in multiple integration, Uniform convergence in improper integrals, differentiation under the sign of integral, Leibnitz rule.

 

102 : Linear Algebra

Unit - I

Fields, vector spaces, subspaces, linear dependence and independence, basis and dimension of a vector space. Finite dimensional vector spaces, completion theorem, examples of vector spaces over real and complex fields. Vector space with an inner product, Gram-Schmidt orthogonalization process, orthonormal basis and orthogonal projection of a vector.

Unit - II

Linear transformation, algebra of matrices, row and column spaces of a matrix, elementary matrices, determinants, rank and inverse of a matrix, null space and nullity, partitioned matrices, Kronecker product. Hermite canonical form. Important results on g-inverse. Idempotent matrices. Solutions of matrix equations.

Unit - III

Real quadratic forms, reduction and classification of quadratic forms, index and signature, triangular reduction of a positive definite matrix.Characteristic roots and vectors, Caley-Hamilton theorem, minimal polynomial, similar matrices, algebraic and geometric multiplicity of a characteristic root, spectral decomposition of a real symmetric matrix.

Unit - IV

Reduction of a pair of symmetric matrices, Hermitian matrices. Singular values and singular value decomposition, Jordan decomposition. Extrema of quadratic forms. Vector and matrix differentiation.

 

103 : Measure Theory and Probability

Unit - I

Sets and sequences of sets. Fields, sigma field, minimal sigma field. Borel field in Rk, Monotone classes, Set function, Measure, Probability measure, Properties of measure, Caratheodory extension theorem (without proof). Lebseque measure, Lebesgue-Stieljes measure, Measurable functions and properties.

Unit - II

Sequence of random variables, convergence in probability, convergence in r-th mean, almost sure convergence, convergence in distribution, Interrelationship among different modes of convergences.

Unit - III

Integral with respect to a measure and dominated convergence theorems, Product spaces, Fubini Theorem (without proof), Signed measure, Absolute continuity.

Unit - IV

Radon-Nikodym Theorem (without proof). Lebesgue decomposition theorem. Helly-  Bray theorem, Expectation of random variables, Conditional expectation, Martingales and simple properties, Jensen, Holder, Schwartz Minkowski's Liapounov's inequalities.

 

104 : Sample Survey

Unit - I

Unequal probability sampling: pps wr and wor methods (including Lahiri's scheme) and related estimators of a finite population mean (Hansen-Hurwitz and Desraj estimators for general sample size and Murthy's estimator for a sample of size 2).

Unit - II

Horvitz-Thompson estimator, its variance and unbiased estimator of variance, IPPS schemes of sampling due to Midzuno-Sen, Rao-Hartley-Cochran and Samphord.

Unit - III

The Jackknife and Bootstrap : estimate of bias, estimate of variance. Ratio Estimation in reference to Jackrite and bootstraps, Relationship between the jackknife and the bootstrap. Interpenetrating sub sampling.

Unit - IV

Non-sampling errors. Randomized Response techniques (Warner's method : related and unrelated questionnaire methods).

 

105 : Statistical Computing/Statistics Practical

• Microsoft Excel:
  • Spread sheet
  • Descriptive Statistics (Univariate)
  • Regression
  • Different kinds of charts including histogram, pie charts & bar charts.
  • Frequency curves.
• Calculation of bases
• Gram-Smidt orthonormalization
• Inverse of matrix
• Solution of a set of non-homogeneous equations
• g-inverse of matrix
• Characteristics roots and vectors
• Reduction and classification of quadratic forms
• Practical on pps.
  • to draw samples by cumulative total method/Lahiri method
  • to estimate population mean/population total of the characteristics under study using ordered and unordered samples : Desraj, Murthy and H-T estimators.
• Other practical based on the topics of papers – 101, 102, 103 & 104.

 

Semester - II

201 : Complex Analysis, Transforms and Special Functions

Unit - I

Functions of a complex variable, limit, Continuity, differentiation, Candy-Riemann equations, Power series, Analytic functions.

Unit - II

Cauchy's theorem and integral formula, Taylor's and Lauret's series, Residue theorem, Evaluation of standard integrals by contour integration.

Unit - III

Laplace transform and its properties, Laplace transforms of important functions, Inverse Laplace transforms, Convolution theorem, Solution of ordinary differential equations.

Unit - IV

Gamma, Hypergeornetric Legendre's and Bessel's functions, Elementary properties of these functions. 

 

202 : Theory of Probability and Limit Theorms.

Unit - I

Weak and strong law of large numbers for independent random variables, Kolmogorov's inequality and theorm, Hazek-Renyi inequality, Levy's inequality and theorem, Uniform integrability.

Unit - II

Central limit theorems , Lindberg-Levy theorem, Liapounoff theorem, Lindberg-Feller theorem (without proof), Glivenko-Cantelli Theorem.

Unit - III

Distribution function, Stieltjes integrals and Riemann Integral, Characteristic function and moments, Inversion theorem, continuity theorem and its applications (CLT for iid random variables and Khintechine's weak law etc.).

Unit - IV

Infinitely divisible distributions, Convergence of infinitely divisible distributions, Borel-Cantellilemma, borel-zero one law.

 

203 : Linear Models & Regression Analysis

Unit - I

Generalized inverse, Moore-Penrose generalized inverse. Important results on g-inverse, Use of generalized inverse of matrices, Distribution of quadratic forms for multi-variate normal random vector, Cochran Theorem.

Unit - II

Linear models of full rank and not of full rank, Normal equations and least squares estimates, BLUE, Gauss-Markov Theorem, Error and estimation spaces, variance and covariances of least squares estimates, estimation of error variance.

Unit - III

Models containing function of the predictors, including polynominal models, Use of orthogonal models, Hypotheses for one and more than one linear parametric functions, Confidence regions, Analysis of Variance, Power of F-test. Multiple comparison tests due to Tukey and Scheffe, Simultaneous confidence intervals.

Unit - IV

Selecting the best regression equation : Stepwise regression, backward elimination. Criteria for evaluating equations, residual mean square, Cp and its use, Residuals and their plots. Tests for departure from assumptions of linear models such as normality, homogeneity of variances, Detection of outliers & its remedies, Transformation: Box - Cox transformation. Introduction to non-linear models.

 

204 : Multivariate Analysis

Unit - I

Wishart matrix - its distribution and properties, Distribution of sample generalized variance, Null and non-null distribution of simple correlation coefficient, Null distribution of partial and multiple correlation coefficient, Distribution of sample regression coefficients, Application in testing and interval estimation.

Unit - II

Null distribution of Hotelling's T2 statistic, Application in tests on mean vector for one and more multivariate normal populations and also on equality of the components of a mean vector in a multivariate normal population.

Unit - III

Classification and discrimination procedures for discrimination between two multivariate normal populations-sample discriminant function, test associated with discriminant functions, probabilities of misclassification and their estimation, classification into more than two multivariate normal populations, Fisher Behren Problem.

Unit - IV

Multivariate Analysis of variance (MANOVA) for one way classified data only, Principal components, dimension reduction, Canonical variables and canonical correlations: definition, use, estimation and  computation.

 

205 : Statistics Practical

• Experiments based on multivariate analysis.
• Hotelling T2/D2 (Discriminant analysis)
  • To test Ho :μ = μo from N(μ, Σ ), Σ unknown.
  • To test Ho : μ1
    • (1) = μ2
    • (2) inN ( ( ), ),N ( (2), )p1p μ Σ μ Σ− −, Σ unknown.
  • Discriminant Analysis
  • Problem of Misclassification
• Multivariate Analysis of variance (One way classified data only).
• Principal components
• Canonical correlations
• Factor Analysis
• Other practical based on the topics of papers – 201, 202, 203 & 204.

 

Semester - III

301 : Sequential Analysis And Reliability Theory

Unit - I

Need for sequential procedures, SPRT and its properties. Wald's equation and identity, OC and ASN functions optimality of SPRT.

Unit - II

Sequential estimation, Stein's two stage procedure. Anscombe Theorem. Chow-Robbin's procedure, Asymptotic consistency and risk efficiency, Estimation of normal mean.

Unit - III

Reliability concepts and measures, components and systems, coherent systems, reliability of coherent systems cuts & paths, Bounds on system reliability. Life distributions, reliability functions, hazard rate, Common life distributions : Exponential, gamma and Weibull, estimation of parameters and tests in these models.

Unit - IV

Notion of aging IFR, IFRA, NBU, DMRL and NBUE classes. Different types of redundancy and use of redundancy in reliability improvement, Problem of life testing, censored and truncated experiments for exponential models.

 

302 : Inference

Unit - I

Likelihood function, Sufficiency, Factorization Theorem, Minimal sufficient statistics,  Completeness Exponential families of distributions and their properties. Distribution admitting sufficient Statistics, Extension of results to multiparameter case.

Unit - II

Cramer-Rao bounds, Bhattacharya bounds. Minimum variance unbiased estimators, Rao-Blackwell Theorem. Lehman-Scheffe theorem and their applications.

Unit - III

Non-randomized and randomized tests. Size, power functions. unbiasedness. NP-Lemma and its applications in construction of MP tests for simple null hypotheses. MLR families.

Unit - IV

UMP tests for one sided null hypotheses against one-sided composite alternative. Generalized NP lemma, Locally best test, UMPU tests, Similar tests, Neyman structure, UMPU tests against one-sided and two-sided alternatives, Confidence set estimation, Relation with hypothesis testing, optimum parameteric confidence sets.

 

303 : Block Designs and Analysis

Unit - I

Fixed, mixed and random effects models; Variance components estimation : study of various methods, Tests for variance components.

Unit - II

General block design and its information matrix (C), criteria for connectedness, balance design and orthogonality: Intrablock analysis (estimability, best point estimates/Interval estimates of estimablelinear parametric functions and testing of linear hypotheses).

Unit - III

BIBD - recovery of interblock information, Youden design - intrablock analysis, Analysis of covariance in a general Gauss-Markov model and its applications to standard designs, Missing plot technique - general theory and applications,

Unit - IV

Finite group and finite field, Finite geometry: projective and Euclidean, Construction of complete set of mutually orthogonal latin square (mols), Construction of BIBD's using mols and finite geometries,Symmetrically repeated differences, Steiner Triples and their use in construction of BIBD, Lattice Design, Split plot design.

 

304 : Econometrics

Unit - I

Natute of econometrics, The general linear model (GLM) and its extensions, Use of dummy varibles and seasonal adjustment, Generalized least squares (GLS) estimation and prediction, Heteroscedastic disturbances, Pure and mixed estimation, Grouping of observations and of equations.

Unit - II

Auto correlation, its consequences and tests, Theil BLUS procedure: estimation and prediction, Multicollinearity problem, its implications and tools for handling the problem, Ridge regression.

Unit - III

Linear regression with stochastic regressors, Instrumental variable estimation, Errors in variables, Autoregressive linear regression, Distributed lag models, Simultaneous linear equations model, Examples, Identification problem, Restrictions on structural parameters - rank and order conditions, Restrictions on variances and covariances.

Unit - IV

Estimation in simultaneous equations model, Recursive systems, 2 SLS Estimators. Limited information estimators, k - class estimators. 3 SLS estimation, Full information maximum likelihood method, Prediction and simultaneous confidence intervals, Monte Carlo studies and simulation.

 

305 : Statistics Practical

• Experiments based on BIBD
• Experiment based on Lattice
• Analysis of Covariance
• Missing plot techniques
• Split plot designs
• Experiment based on system of Reliability
• ASN & OC functions for SPRT
• OLS estimation and prediction in GLM.
• Use of dummy variables (dummy variable trap) and seasonal adjustment.
• GLS estimation and prediction.
• Tests for heteroscdasticity ; pure and mixed estimation.
• Tests for autocorrelation. BLUS procedure.
• Ridge regression.
• Industrumental variable estimation.
• Estimation with lagged dependent variables.
• Identification problems - checking rank and order conditions.
• Estimation in recursive systems.
• Two SLS estimation.
• Simulation studies to compare OLS, 2SLS, LISE and FIML methods.
• Other practical based on the topics of papers – 301, 302, 303 & 304

 

Semester - IV

401 : symptotic Inference

Unit - I

Consistency (mean squared and weak), invariance of consistency under continuous transformation, consistency for several parameters, generating consistent estimators using weak law of large numbers, CAN estimators (single as well as multi-parameter cases), invariance of CAN estimators under differentiable transformations, generation of CAN estimators using central limit theorem.

Unit - II

Consistency of estimators by method of moments and method of percentiles, Minimum Chi square estimators and their modification and their asymptotically equivalence to maximum likelihood estimators.

Unit - III

Method of maximum likelihood: special cases as k-parameters exponential family of distribution and multinomial distributions, Computational routines : Newton – Raphson method, method of scoring, Consistency and inconsistency, Cramer Huzurbazar Theorem, Asymptotic efficiency of ML estimators, Best Asymptotically normal estimators. Concept of super efficiency.

Unit - IV

Large Sample tests : Likelihood ratio (LR) test, asymptotic distribution of LR statistic, Tests based on ML estimators, Wald Test, Score Test. Pearson’s chi-square test for goodness of fit and its relation to LR Test, Test consistency, Asymptotic power of test, Generalised likelihood ratio test, special cases such as multinomial distribution and Bartlett’s test for homogeneity of variances.

 

402 Stochastic Processes

Unit - I

Introduction to stochastic processes (sp's) : Classification of sp's according to state space and time domain, Countable state. Markov chains (MC's), Chapman-Kolmogorov equations, calculation of nstep transition probability and its limit, Stationary distribution, classification of states, transient MC, random walk and gambler's ruin problem.

Unit - II

Discrete state space continuous time, Markov Chains: Kolmogorov-Feller differential equations. Poisson process, birth and death process, application to queues and storage problems, Wiener process as a limit of random walk, first-passage time and other problems.

Unit - III

Renewal theory: Elementary renewal theorem and applications, Statement and uses of key renewal theorem, study of residual life time process, Stationary process, weakly stationary and strongly stationary process, Moving average and auto regressive processes.

Unit - IV

Branching process: Galton-Watson branching process, probability of ultimate extinction, distribution of population size, Martingale in discrete time, inequality, convergence and smoothing properties. Statistical inference in Markov Chains and Markov processes.

 

403 : Decision Theory and Bayesian Analysis

Unit - I

Decision problem and two person game, Utility theory, loss functions, Randomized and nonrandomized  decision rules, Optimal decision rules – unbiasedness, invariance, Bayes Rule, Minimax rule, concept of admissibility and completeness Bayes rules, Admissibility of Bayes and minimax rules.

Unit - II

Supporting and separating hyper plane theorems, complete class theorem. Minimax estimators of Normal and Poisson means.

Unit - III

Subjective interpretation of probability in terms of fair odds, Evaluation of (i) subjective probability of an event using a subjectively unbiased coin (ii) subjective prior distribution of a parameter, Bayes theorem and computation of the posterior distribution, Natural Conjugate family of priors for a model, Hyper parameters of a prior from conjugate family, Bayesian point estimation as a prediction problem from posterior distribution, Bayes estimators for (i) absolute error loss (ii) squared error loss (iii) 0 -1 loss.

Unit - IV

Bayesian interval estimation : credible intervals, Highest posterior density regions, Interpretation of the confidence coefficient of an interval and its comparison with the interpretation of the confidence coefficient for a classical confidence interval, Bayesian testing Hypothesis : Specification of the appropriate from of the prior distribution for a Bayesian testing of hypothesis problem, Prior odds, Posterior odds, Bayes factor.

 

404 : Factorial Experiments and Response Surfaces

Unit - I

General factorial experiments, factorial effects, symmetric factorial experiments, best estimates and testing the significance of factorial effects; analysis of 2n .

Unit - II

3n factorial experiments in randomized blocks, Complete and partial confounding, Fractional replication for symmetric factorials.

Unit - III

Response surface experiments, first order designs and orthogonal designs.

Unit - IV

Clinical trials, longitudinal data, treatment- control designs, Model validation and use of transformation, Tukey's test for additivity.

 

405 Statistics Practical and Project

• (a). Practical based on Factorial Experiment, Response surfaces & Bayesian Inference. Other practical based on the topics of papers – 401, 402, 403 & 404.
• (b). Project.