| PART I |
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Paper - IA: Real Analysis
| Real Number System, Cluster Points of sets, Closed and open sets, Compact sets, Bolzano-Weierstrass Property, Heine-Borel Property(Statement Only).Sets of Real Vectors. |
(15) |
| Sequences and Series, Convergence. Real valued functions. Limit, Continuity and Uniform continuity. Differentiability of univariate and multivariate functions. Mean value theorems. Extrema of functions. |
(17) |
| Riemann integral. Improper integrals. Riemann-Stieltjes integral. Sequences and Series of functions, Uniform convergence, Power series. |
(12) |
| Term by term differentiation and integration, Differentiation and integration under the integral sign. |
(6) |
| References : |
| T.M.Apostol |
: |
Mathematical Analysis |
| W.Rudin |
: |
Principles of Mathematical Analysis |
| R.R.Goldberg |
: |
Methods of Real Analysis |
| J.C.Burkill |
: |
First Course of Mathematical Analysis |
| J.C.Burkill & H.Burkill |
: |
Second Course of Mathematical Analysis |
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Paper - IB : Probability
| 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. |
(12) |
| Measurable functions, Random variables, Expectation, Sequences of random variables, Almost sure convergence, Convergence in probability. Distribution function, Convergence in distribution. |
(11) |
| Integration of a measurable function with respect to a measure, Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem. |
(7) |
| Generating functions and Characteristic function, Inversion theorem and Continuity theorem (Statement only) |
(7) |
| Borel -Cantelli lemma, Independence,Weak law and Strong law of large numbers, Kolmogorov inequality. |
(6) |
| Central limit theorem for iid random variables, CLT for a sequence of independent random variables (Statement only). |
(3) |
| Radon-Nikodym theorem(Statement only) and its applications, Product measure and Fubini’s theorem (Statement only). |
(4) |
| References : |
| A.K.Basu |
: |
Measure theory and Probability |
| B.R.Bhat |
: |
Modern Probability Theory |
| P.Billingsley |
: |
Probability and Measure |
| J.F.C. Kingman & S.J.Taylor |
: |
Introduction to Measure and Probability |
| R.G.Laha & V.K.Rohatgi |
: |
Probability theory |
| R.Ash |
: |
Real Analysis and Probability |
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Paper - IIA : Large Sample Theory and Sampling Distributions
| Sampling Distributions (25) |
| Prerequisites of matrix algebra. |
(2) |
| Non-central Χ2 , t & F distributions - definitions and properties. |
(3) |
| Distribution of quadratic forms - Cochran’s theorem. |
(7) |
| Multivariate normal distribution - independence of sample mean vector and variance-covariance matrix. Wishart distribution. |
(6) |
| Distributions of partial and multiple correlation coefficients and regression coefficients, distribution of intraclass correlation coefficient. |
(4) |
| Hotelling T2 and Mahalanobis’s D2 application in testing and confidence set construction. |
(3) |
| References : |
| C.R.Rao |
: |
Linear Statistical Inference and its Applications |
| T.W.Anderson |
: |
Introduction to Multivariate Analysis |
| A.M.Khirsagar |
: |
Multivariate Analysis |
| S.S.Wilks |
: |
Mathematical Statistics |
| M.S.Srivastava & C.G.Khatri |
: |
Introduction to Multivariate Statistics |
| R.J.Muirhead |
: |
Aspects of Multivariate Statistical Theory |
| Large Sample Theory (25) |
| Asymptotic distributions of sample moments and functions of moments, Asymptotic distributions of Order Statistics and Quantiles. |
(7) |
| Consistency and Asymptotic Efficiency of Estimators, Large sample properties of Maximum Likelihood estimators. |
(10) |
| Asymptotic distributions and properties of Likelihood ratio tests, Rao’s test and Wald’s tests in the simple hypothesis case. |
(8) |
| References : |
| R.J.Serfling |
: |
Approximation Theorems of Mathematical Statistics |
| E.L.Lehmann |
: |
Large Sample Theory |
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Paper - IIB : Statistical Inference I
| Review of notions of sufficiency & completeness,Notions of minimal sufficiency,bounded completeness and ancillarity, Exponential family. |
(8) |
| Point estimation : Bhattacharya system of lower bounds to variance of estimators. Minimum variance unbiased estimators - Applications of Rao - Blackwell and Lehmann - Scheffe theorems. |
(7) |
| 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. |
(24) |
| Confidence sets: relation with hypothesis testing. Optimum parametric confidence intervals. |
(3) |
| Sequential tests. Wald’s equation for ASN. SPRT and its properties - fundamental identity. O.C. and ASN. Optimality of SPRT (under usual approximation). |
(8) |
| References : |
| E.L.Lehmann |
: |
Testing Statistical Hypotheses |
| S.Zacks |
: |
The Theory of Statistical Inference |
| C.R.Rao |
: |
Linear Statistical Inference and its Applications |
| E.L.Lehmann |
: |
Theory of Point Estimation |
| T.S.Ferguson |
: |
Mathematical Statistics |
| B.K.Ghosh |
: |
Sequential Tests of Statistical Hypotheses |
| D.A.S.Fraser |
: |
Nonparametric methods in Statistics |
| J.O.Berger |
: |
Statistical Decision Theory and Bayesian Analysis |
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Paper - IIIA : Linear Models and Regression Analysis
| Linear Models (25) |
| Pre-requisites of matrix algebra. |
(4) |
| 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. |
(10) |
| 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. |
(7) |
| Random and Mixed effects models (balanced case), estimation of variance components. |
(4) |
| References : |
| H.Scheffe |
: |
The Analysis of Variance |
| S.R.Searle |
: |
Linear Models |
| G.A.F.Seber |
: |
Linear Regression Analysis |
| N.Giri |
: |
Analysis of Variance |
| D.D.Joshi |
: |
Linear Estimation & Design of Experiments |
| Regression Analysis (25) |
| Residuals and their plots. Tests of fit of a model. Q-Q plots. Transformations. |
(4) |
| Detection of outliers. |
(3) |
| Departures from the usual assumptions : heteroscedasticity, autocorrelation, multicollinearity, non-normality - detection and remedies. |
(14) |
| Variable selection problems. |
(4) |
| References : |
| N.R.Draper & H.Smith |
: |
Applied Regression Analysis |
| D.A.Belsley, Kuh & Welsch |
: |
Regression Diagnostics-
identifying influential data & sources of collinearity |
| R.D.Cook & S.Weisberg |
: |
Residual and its Influence in Regression |
| C.R.Rao |
: |
Linear Statistical Inference and its Applications |
| J.Johnston |
: |
Econometric Methods |
| Judge, Griffith, et.al. |
: |
The Theory and Practice of Econometrics |
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Paper - IIIB : 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. |
(18) |
| Construction of complete classes of mutual orthogonal Latin squares (MOLS); construction of BIBD - through MOLS and Bose’s fundamental method of differences. |
(10) |
| Factorial experiments, confounding and balancing in symmetric factorials. |
(16) |
| Response Surface Experiments - first order designs. |
(6) |
| References : |
| M.C.Chakraborty |
: |
Mathematics of Design and Analysis of Experiments |
| A.Dey |
: |
Theory of Block Designs |
| D.Raghavarao |
: |
Constructions & Combinatorial Problems in Design of
Experiments |
| R.C.Bose |
: |
Mathematical Theory of Symmetric Factorial Design
(Sankhya - Vol. 8) |
| R.C.Bose |
: |
On the Construction of Balanced Incomplete Block Design
(Annals Eugenics - Vol. 9) |
| D.C.Montogomery |
: |
Design and Analysis of Experiments |
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Paper - IV : Sample Surveys and Statistics for National Development
| Sample Surveys (25) |
| Probability sampling from a finite population - Notions of sampling design, sampling scheme, inclusion probabilites, Horvitz-Thompson estimator of a population total. |
(3) |
| 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. |
(8) |
| Stratified sampling - Allocation problem and construction of strata. |
(3) |
| Ratio, Product, Difference and Regression estimators. Unbiased Ratio estimators - Probability proportional to aggregate size sampling, Hartley - Ross estimator in simple random sampling. |
(4) |
| 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. |
(4) |
| Double sampling for stratification. Double sampling ratio and regression estimators. Sampling on successive occasions. |
(3) |
| References : |
| W.G.Cochran |
: |
Sampling Techniques, 3rd edition, Wiley |
Des Raj
& Chandak |
: |
Sampling Theory, Narosa |
A.S.Hedayat
& B.K.Sinha |
: |
Design & inference in finite population sampling, Wiley |
| P.Mukhopadhyay |
: |
Theory & Methods of Survey Sampling, Prentice-Hall, India |
| M.N.Murthy |
: |
Sampling Theory and Methods. Statistical Publishing
Society, Calcutta |
| Statistics for National Development (25) |
| Concept of economic development - role of statistics. Development indices. |
(3) |
| National and international statistical systems. |
(3) |
| National accounts - estimation of national and state incomes and their components. |
(4) |
| Projection of populations. |
(2) |
| Projections of labour force and manpower, measurement of unemployment. |
(3) |
| Distribution of income and consumption - measurement of inequality and poverty. |
(6) |
| Other indicators of development : agriculture - crop-forecasting and estimation, crop insurance, procurement and buffer stock management ; water resources management ; industrial growth ; foreign trade and balance of payments ; planning and allocation of resources ; evaluation of family welfare programmes. |
(4) |
| (for all the topics statistical aspects only are to be emphasized ) |
| References : |
| CSO (1980) |
: |
National Accounts Statistics |
| N.Keyfitz |
: |
Applied Mathematical Demography |
| UNESCO |
: |
Principles of Vital Statistics System - Series M-12 |
| A.Sen |
: |
Poverty and Inequality |
| Chaubey Y. P. |
: |
Poverty Measurements: issues, approaches and indices |
| UNO |
: |
Yearly Human Development Reports |
| World Bank |
: |
Yearly Reports |
|
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Statistical Computing
| Overview of C language: Simple Syntax, loops, pointers, arrays, functions, files. |
(5) |
| C-programming for solving problems related to: Statistical Inference, Sorting and Searching, Generation of Random Numbers and selection of samples, Simulation, Monte Carlo techniques. |
(15) |
| R- programming for solving problems related to: Reading data, Basic operations, Graphical display, Descriptive statistics, Linear models, Correlated data, Creating user defined functions, Treatment of missing values, Interface with C, Numerical optimization, Simulations. |
(20) |
| Statistical Package : SPSS/ Matlab. |
(10) |
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