University Grants Commission

  NET Bureau

  NET Syllabus  

Subject: Statistics                      Code:107                                

 Total: 10 Units  

Unit I: Probability and Distributions    

Basic concepts of probability, conditional probability, Bayes theorem, independent events.   Random variables and distribution functions, expectation and moments, moment generating  function. Standard discrete and continuous univariate distributions. Jointly distributed random  variables, marginal and conditional distributions. Chebyshev inequality. Sampling distributions,  transformation of random variables. Characteristic function and its properties. Modes of  convergence of random variables, weak and strong laws of large numbers, central limit theorems  (i.i.d. case).  

Unit II: Real Analysis and Matrix Algebra  Real Analysis: Finite, countable and uncountable sets; sequences of real numbers, convergence  of sequences, bounded sequences, monotonic sequences, Cauchy criterion for convergence;  Series of real numbers, convergence, tests of convergence, alternating series, absolute and  conditional convergence; Power series and radius of convergence; Functions of a real variable:  Limit, continuity, monotone functions, uniform continuity, differentiability, Rolle’s theorem,  mean value theorems, Taylor’s theorem, L  Hospital’s rule, Riemann integration and its  properties, improper integrals.   Functions of two real variables: Limit, continuity, partial derivatives, total derivative, maxima  and minima, saddle point, method of Lagrange multipliers, double and triple integrals and their  applications.   Matrix Algebra: Vector spaces, subspaces, span, linear independence, basis and dimension, row  space and column space of a matrix, rank and nullity, row reduced echelon form, trace and  determinant, inverse of a matrix, systems of linear equations; Gram-Schmidt orthogonalization;  Characteristic roots and characteristic vectors, characteristic polynomial, Cayley-Hamilton  theorem, symmetric matrices, skew-symmetric matrices, orthogonal matrices and their  characteristic roots, positive definite and positive semi-definite matrices and their properties,  quadratic forms.  

Unit III: Sampling Methods and Design of Experiments  Sampling Methods: Simple random sampling, stratified random sampling, systematic sampling.  Ratio and regression methods of estimation, cluster sampling for equal and unequal clusters,  double sampling, sampling with varying probabilities with and without replacement.  Nonnegative variance estimation, ordered and unordered estimators.  Design of Experiments: Analysis of variance in one-way and two-way classification (with and  without interaction) in fixed effects model, principles of design of experiments, completely  randomized design, randomized block design, Latin square design, missing plot techniques.  Factorial experiments-       ,  confounding in factorial experiments, Incomplete block designs  and its intra-block and inter-block analysis, connectedness and orthogonality of block designs,  balanced incomplete block design (BIBD), inter-block analysis and recovery of intra-block  information of BIBD.  

Unit IV: Estimation Theory   

Point Estimation: Unbiasedness, consistency, method of moments and maximum likelihood  estimators, efficiency, uniformly minimum variance unbiased estimators, Rao-Cramer lower  bound, sufficiency, factorization theorem, minimal sufficiency, ancillary statistic, completeness,  Rao-Blackwell theorem, Lehmann-Scheffe theorem, Basu’s theorem.  

Interval estimation: method of pivoting, confidence intervals for parameters in one sample and  two sample normal populations. confidence intervals based on large samples.   

Nonparametric Inference: Distributions of order statistics, empirical distribution function and  its properties. Rank correlation coefficients of Spearman and Kendall.  

Unit V: Testing of Hypotheses   

Basic concepts, construction of tests: Neyman-Pearson lemma, families with monotone  likelihood ratio. Uniformly most powerful, uniformly most powerful unbiased and uniformly  most powerful invariant tests, likelihood ratio tests: applications to one sample and two sample  problems. Wald’s sequential probability ratio test, operating characteristic and average sample  number.  Chi-square tests (goodness of fit, independence of attributes, homogeneity in contingency  tables), sign test, Wilcoxon signed rank test, Mann-Whitney U-test, linear rank tests for location  and scale problems, Kruskal-Wallis test.  

Unit VI: Linear Estimation, Regression Analysis and Econometrics  Simple and multiple linear regression model, Gauss-Markov model, least squares and maximum  likelihood estimation, testing of hypothesis related to regression parameters, Analysis of variance  for linear model,    ,  squares  estimation,  adjusted    ,  tests of linear hypothesis, generalized and weighted least  indicator/dummy variables, multicollinearity, heteroscedasticity,  autocorrelation, Durbin-Watson test, logistic regression models.  Restricted regression estimation under exact, stochastic and mixed restrictions. Model with  stochastic regressors and errors in variable model, instrumental variable estimator, simultaneous  equations model, identification problem, two-stage least squares estimation, k-class estimator.  

 Unit VII: Time Series  Time series data, descriptive measures, autocovariance, autocorrelation functions (ACVF, ACF),  and partial autocorrelation function (PACF), correlogram. Strong and weak stationarity,  ergodicity. General linear process and Wold decomposition. Moving Average (MA),  Autoregressive (AR) and mixed ARMA processes, stationarity and invertibility conditions.  Yule–Walker equations. Identification, estimation and order selection of AR, MA and ARMA  models, forecasting with stationary and invertible processes.  Non-stationary time series: random walk, ARIMA (p, d, q) models and parameter estimation.  Frequency domain analysis: Spectral representation of time series, spectral density of AR, MA  and ARMA processes, periodogram analysis and estimation of spectral density.  

Unit VIII: Multivariate Analysis  Multivariate normal distribution and its properties, estimation of mean vector and covariance  matrix in multivariate normal distribution, distribution of sample mean vector, Wishart  distribution and its properties, distribution of simple, partial and multiple correlation coefficients  and related tests, inference for parameters. Test of hypothesis related to mean vector and  generalized     statistic, discriminant analysis,   correlation analysis.   

Unit IX: Stochastic Processes   principal component analysis, canonical  Markov chains with finite and countable state space, classification of states, Chapman Kolmogorov equations, limiting behaviour of n-step transition probabilities, stationary  distribution, Gambler’s ruin problem, simple random walk.  Poisson process, inter-arrival and  waiting time distributions, Birth and death processes, M/M/1 queues.  

Unit X: Indian Statistical System and Research Methodology  Indian Statistical System: Ministry of Statistics and Programme Implementation and its  different wings, National Statistical Commission, National Statistics Office, census and large  sample surveys. Contributions of P C Mahalanobis, P V Sukhatme, R C Bose, S N Roy, C R  Rao, and other prominent Indian Statisticians.  Research Methodology: ‘R’ software: R as a calculator, functions and matrix operations, built  in functions, missing data and logical operators. Conditional executions and loops; data  management with sequences, repeats, sorting, ordering and strings; lists, factors, display and  formatting. Data frames, data input and output, graphics and plots. Basics of programming,  scripts and functions.        and other word processing software.