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# BHU MA/MSc (Statistics) 1st &2nd Semester Syllabus

 Posted Date: 28-Feb-2011 Last Updated: 28-Feb-2011 Category: General Author: md jerajuddin Member Level: Silver Points: 2

BHU Varanasi

SEMESTER - I

Course Code Title Credit

STM – 101 Real Analysis, Complex Analysis and Linear Algebra (3 credits)
STM – 102 Distribution Theory (4 credits)
STM – 103 Statistical Inference -I (4 credits)
STM – 104 Statistical Computing (4 credits)
STM – 105 Practical Paper based on the contents of Papers
[STM – 103 and STM - 104] (3 credits)
STM106M - Statistical Methods( Minor Elective for students of other programmes*
(3 credits)
Descriptive Statistics : Measures of central tendency, dispersion, skewness and kurtosis for the study of nature of data.
Idea of correlation and regression for two and three variables; correlation coefficient, correlation ratio, multiple and partial correlations.
Important statistical distributions : Binomial, Poisson, hypergeometric, negative binomial, normal, uniform, exponential and multinomial distributions.
Statistical Inference : concepts of point estimation, interval estimation and testing.

Total: 21 credits

STM – 101 :
REAL ANALYSIS, COMPLEX ANALYSIS AND LINEAR ALGEBRA: 3 credits

Recap of elements of set theory, introductions to real numbers, open and closed intervals (rectangles), compact sets, Bolzano-Weirstrass theorem.
Sequence and series, their convergence, real valued function, continuous functions, Uniform continuity, sequences of functions, Uniform convergence.

Differentiation, maxima-minima of functions, functions of several variables, multiple integrals, change of variables in multiple integration, uniform convergence in improper integral.
Complex numbers, Analytic function, Cauchy fundamental theorem, Cauchy integral theorem, Contour integrations.
Vector spaces, sub-spaces, linear dependence and independence, orthogonalization process, orthonormal basis

STM – 102 : DISTRIBUTION THEORY: 4 credits

Brief review of basic distribution theory, joint, marginal conditional p.m.f.'s and p.d.f's, standard discrete and continuous distributions, bivariate normal, bivariate exponential, multivariate normal and multinomial distributions, functions of random variables and their distributions using Jacobian of transformation and other tools.
Compound, truncated and mixture distributions, conditional expectation, multiple and partial correlations, linear and multiple regressions. Markov, Holder, Jensen, Liapunov inequalities.
Sampling distributions, non-central chi-square, t and F distributions and their properties. distributions of quadratic forms under normality and related distribution theory.
Order statistics, their distributions and properties, joint and marginal distributions of order statistics, extreme values and their asymptotic distributions (statement only) with applications. approximating distributions Delta method and its applications, approximating distributions of sample moments, transformations of statistics

STM – 103 : STATISTICAL INFERENCE –I: 4credits
Extension of Cramer-Rao inequality for multi-parameter case, Bhattacharya bounds, information in data aboutthe parameters as variation in likelihood function.
Ideas of sufficient and minimal complete-sufficient statistics, sufficiency when the range of variate depends on parameter, minimum variance unbiased estimators, Rao-Blackwell and Lehman-Scheffe theorems, examples based on some standard distributions.
Asymptotic properties of maximum likelihood estimators, solution of likelihood equations, method of scoring, Newton-Raphson method.
General decision problems, loss function, risk function, estimation and testing viewed as general decision problems, minimax decision, Bayes decision, least favourable prior, Bayes estimation under squared error loss, some simple illustrations based on binomial, Poisson, and normal distributions, procedure for obtaining minimax estimators from Bayes estimators.

STM – 104 : STATISTICAL COMPUTING: 4 Credits
Programming in a high level such as C (preferred) or FORTRAN. The purpose of this unit is to introduce programming with the eventual aim of developing skills required to write statistical software. Topics should include simple syntax, loops, pointers and arrays, functions, input/output, and linking to databases.
Numerical analysis and statistical applications. The purpose of this unit is to apply programming skills in methods and alogrithms useful in probability, statistics, and data analysis. Topics should include numerical integration, root extraction, random number generation, Monte Carlo integration, and matrix computations.

STM– 105 :
Practical paper based on the contents of Papers Statistical Inference –I: 3(credits)
based on (STM - 103) and Statistical Computing ( STM – 104)
Practical paper will be of 100 marks out of which 30 marks will be assigned on sessionals / tutorials / class tests/seminars in class / group discussions and 70 marks will be assigned on the end semester examination out of which 50 marks will be on the performance in practical examination and 10 marks will be assigned each on practical record book and viva – voice . The duration of the paper shall be FOUR HOURS ).

Semester II

SEMESTER - II
STM – 201 Survey Sampling 3 credits
STM – 202 Linear Models and Regression Analysis 3 credits
STM – 203 Statistical Inference –II 3 credits
STM – 204 Stochastic Processes 3 credits
STM – 205 Probability 3 credits
STM – 206 Practical Paper based on the contents of Papers STM – 201 and
STM - 203 2credits
STM – 207 Practical Paper based on the contents of Papers STM – 202 and
STM - 204 2 credits
STM 208M Sampling Theory, Design of Experiments and Data Analysis Using Softwares
(Minor Elective for students of other programmes)* 3 credits

Total: 22 credits

STM – 201 : SURVEY SAMPLING: 3 Credits
Fixed population and super-population approaches. Distinct featues of finite population sampling, Probability sampling design and estimators along with basic statistical properties. Review of some important results in SRSWOR and SRSWR. Estimation of population mean/Total in stratified population, Allocation problem in stratified random sampling in case of fixed cost and also for specified precision. Expression for variance of stratified sample mean in case of fixed cost, Post stratification, Double sampling with post stratification, Deep stratification, Controlled sampling.
Unequal probability sampling: PPSWR/WOR methods (including Lahiri’s scheme) and Des Raj estimator, Murthy estimator (n=2). Horvitz Thompson Estimator of a finite population total/mean, Expression for V(HTE) and its unbiased estimator, Issue in non-negative variance estimation.
IIPS Scheme of sampling due to Midzuno-Sen, Double sampling scheme, Some double sampling estimators for mean using auxiliary character (Ratio, regression and product) method of estimation, Some unbiased ratio type estimators for population mean, Concept of cluster sampling, two stage sampling, interpenetrating subsampling, Two phase sampling.Non-sampling error with special reference to non-response problems, Some well-known randomised response techniques for sensitive characteristics.

STM – 202 : LINEAR MODELS AND REGRESSION ANALYSIS: 3 Credits

Grauss-Markov linear models, estimable functions, error and estimation space, normal equations and least square estimators, estimation of error variance, estimation with correlated observations, properties of least square estimators, generalized inverse of a matrix and solution of normal equations, variances and covariances of least square estimators.
One way and two-way classifications, fixed, random and mixed effects models. Analysis of variance (two-way classification only), Multiple comparison tests due to Tukey, Scheffe and Student-Newmann-Karl.
Simple linear regression, multiple, regression, fit of polynomials and use of orthogonal polynomials. Residuals and their plots as tests for departure from assumptions such as fitness of the model, normality, homogeneity of
variances and detection of outliers. Remedies.
Multi co-linearity, ridge regression, sub-set selection of explanatory variables, Mallows Cp Statistics

STM – 203 : STATISTICAL INFERENCE –II: 3 Credits

Consistent Asymptotic normal estimators and their properties, CAN estimators obtained by ML method in one parameter exponential case, Invariant estimators, location and scale invariant estimators, Pitman’s method for obtaining location and scale invariant estimators.
Interval estimation by confidence sets, Neyman theory, general method for constructing confidence intervals, shortest confidence intervals, uniformly most accurate intervals, Bayes intervals, example based on normal distribution.
Neyman-Pearson lemma, generalized Neyman-Pearson lemma, monotone likelihood ratio families, UMP tests for one and two sided alternatives, admissibility and unbiasedness of tests, type A and type A1 tests, similar tests, tests having Neyman structure, likelihood ratio test (LRT) asymptotic distribution of LRT statistic.
Wald’s sequential probability ratio test and its properties, OC and ASN function, derivation of OC and ASN
functions.

TM – 204 : STOCHASTIC PROCESSES: 3 Credits

Introduction to stochastic processes (SPs): Classification of SPs according to state space and time domain.
Countable state Markov chains (MC’s), Chapman-Kolmogorov equations; calculation of n-step transition probability and its limit. Stationary distribution, classification of states; transient MC; random walk and gambler’s ruin problem; Applications from social, biological and physical sciences.
Discrete state space continuous time MC: Kolmogorov – Feller differential equations; Poisson process, birth and death process; Wiener process as a limit of random walk; first-passage time and other problems.
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 processes;
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 MC and Markov processes.

STM – 205 : PROBABILITY: 3 Credits

Classes of sets, fields, sigma fields, minimal sigma field, Borel sigma field, sequence of sets, lim sup and lim inf of a sequence of sets, measure, probability measure, properties of measure, Caratheodory extension theorem
(statement only), Lebesgue and Lebesgue - Steiltzes measures.
Measurable functions, random variables, sequence of random variables, almost sure convergence, convergence in probability (and in measure). Integration of a measurable function with respect to a measure, monotone convergence theorem, Fatou’s lemma, dominated convergence theorem.
Borel-Cantelli lemma, independence, weak law and strong law of large numbers for independently and identically distributed sequences.
Convergence in distribution, characteristic function, uniqueness theorem, Levy’s continuity theorem (statement only). CLT for a sequence of independent random variables under Lindeberg’s condition, CLT for independently and identically distributed random variables.

STM – 206 : Practical paper based on the contents of Papers Survey Sampling
(STM–201)and Statistical Inferencs–II(STM–203): 2 Credits
STM – 207 : Practical paper based on the contents of Papers Linear Models and
Regression Analysis ( STM - 202 ) and Stochastic Processes
(STM-204): 2 Credits

STM208M : SAMPLING THEORY, DESIGN OF EXPERIMENTS AND DATA ANALYSIS USING
SOFTWARES: 3 Credits
(The course will involve only the concepts and uses of theories
rather than rigorous derivations of theresults. )

Basic concepts of sampling from a finite population; sampling versus complete enumeration; simple random sampling; sample size determination; stratified random sampling; systematic sampling; cluster sampling and multi – stage sampling ( all sampling schemes without proof of expressions ).
Analysis of variance techniques : One way and two way classified data.
Design of experiments : Randomization, replication, local control; completely randomized design; randomized block design and Latin square design; factorial experiments.
Data analysis : The students will be trained to use SPSS and SYSTAT softwares for data analysis. The main focus of the training will also include the use of parametric and non – parametric tests and the interpretation of the results.

It does not matter how slowly you walk,
matter is that walk as long as you don't stop.

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