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CSIR Syllabus

Mathematics Part II Continued

24. Large sample statistical methods :  Various modes of convergence, Op and op, CLT Sheffe’s theorem, Polya’s theorem and Stutsky’s theorem.  Transformation and variance stabilizing formula.  Asymptotic distribution of function of sample moments.  Sample quantities.  Order statistics and their functions.  Tests on correlations, coefficients of variation, skewness and kurtosis.  Pearson Chi-square, contingency.  Chi – square and likelihood ratio statistics.  U – statistics. Consistency of Tests.  Asymptotic relative efficiency.

25. Multivariate Statistical Analysis  :  Singular and non – singular multivariate distributions.  Characteristics functions.  Multivariate normal distribution ; marginal and conditional distribution, distribution of linear forms, and quadratic forms, Cochran’s theorem.

Inference on parameters of multivariate normal distributions :  one – population and two – population cases.  Wishart distribution. Hotellings T2, Mahalanobis D2, Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.

26. Linear Models and Regression  :  Standard Gauss – Markov models ;  Estimability of parameters ; best linear unbiased estimates ( BLUE ); Method of least squares and Gauss – Markov theorem ; Variance – covariance matrix of BLUES.

Tests of linear hypothesis ;  One – way and two – way classifications.  Fixed, random and mixed effects models ( two – way classifications only ); variance components, Bivariate and multiple linear regression; Polynomial regression ; use of orthogonal polynomials.  Analysis of covariance.  Linear and nonlinear regression.  Outliers.

27. Sample Surveys  :  Sampling with varying probability of selection, Hurwitz – Thompson estimator ;  PPS sampling ;  Double sampling,  Cluster sampling.  Non-sampling errors ; interpenetrating samples.  Multiphase sampling.  Ratio and regression methods of estimation.

28. Design of Experiments  :  Factorial experiments, confounding and fractional replication.  Split and strip plot designs ;  Quesi – Latin square designs ;  Youden square.  Design for study of response surfaces ; first and second order designs.  Incomplete block designs ;  Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information, PBIBD with 2 associate classes.  Analysis of sense of experiments, estimation of residual effects.  Construction of orthogonal – Latin squares, BIB designs, and confounded factorial designs.  Optimality criteria for experimental designs.
29. Time – Series Analysis  :  Discrete – parameter stochastic processes ; strong and weak stationarity ; autocovariance and autocorrelation, Moving average, autoregressive, autoregressive moving average and autoregressive integrated moving average processes.  Box – Jenkins models.  Estimation of the parameters in ARIMA models, forecasting.  Perfodogram analysis.

30. Stochastic Processes  :  Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution ; branching processes ; Random walk ; Gambler’s ruin.  Markov processes in continuous time ;  Poisson processes, birth and death processes, Wiener process.

31. Demography and Vital Statistics  :  Measures of fertility and mortality, period and Cohort measures.
Life tables and its applications ;  Methods of construction of abridged life tables.  Application of stable population theory to estimate vital rates.  Population projections,  Stochastic models of fertility and reproduction.

32. Industrial Statistics  :  Control charts for variables and attributes ;  Acceptance sampling by attributes ; single, double and sequential sampling plans ;  OC and ASN functions, AOQL and ATI ;  Acceptance sampling by varieties.  Tolerance limits.  Reliability analysis :  Hazard function, distribution with DFR and IFR ;  Series and parallel systems.  Life testing experiments.

33. Inventory and Queueing theory  :   Inventory ( S, s ) policy, periodic review models with stochastic demand.  Dynamic inventory models.  Probabilistic re-order point, lot size inventory system with and without lead time.  Distribution free analysis.  Solution of inventory problem with unknown density function.  Warehousing problem.  Queues ;  Imbedded Markov Chain method to obtain steady state solution of M/G/1, G/M/1 AND M/D/C, Network models.  Machine maintenance models.  Design and control of queueing systems.

34. Dynamic Programming and Marketing  :  Nature of dynamic programming, Deterministic processes, Non-sequential discrete optimization – allocation problems, assortment problems.  Sequential discrete optimization long – term planning problems, multi stage production processes.  Functional approximations.  Marketing systems, application of dynamic programming to marketing problems.  Introduction of new product, objective in setting market price and its policies, purchasing under fluctuating prices, Advertising and promotional decisions, Brands switching analysis, Distribution decisions.