| 3 hrs. duration | 100 Marks |
Multivarate normal distribution and its properties. Marginal and conditional distribution, characteristics function distribution of Quadratic form.
Random sampling from a multivariate normal distribution. Maximum likelihood estimators of Parameters. Distribution of sample mean vector.
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.
Distribution of sample intra-class correlation - coefficient in a random sample from a symmetric multivariate normal distribution. Application in testing and interval estimation.
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.
Multivariate linear regression model- estimation of parameters, tests of linear hypotheses about regression coefficients. likelihood ratio test criterion. Multivariate Analysis of Variance (MANOVA) one and two-way classified data.
Classification and discrimination producers for discrimination between two multivariate normal populations - sample discriminant function, tests associated with discriminant funcions, probabilities of misclassification and their estimation, classification into more than two multivariate normal populations.
Principal components, Dimension reduction, Canonical variables and canonical correlation - definition use, estimation and computation.
Review of convergence in proability and convergence in distribution. Cramer and Slutsky's theorems.
Consistent Estimation o real and vector valued parameters. Invariance of Consistent estimator under continuous transformation. Consistency of estimators by method of moments and method of percentiles. Mean squared error criterion. Asymptotic relative efficiency. Error probabilities and their rates of convergence. Minimum sample size required to attain given level of accuracy.
Consistent Asymprotic Normal (CAN) estimators. Invariance of CAN estimator under differentiable transformation. CAN property of estimators obtained by moments and percentiles. CAN estimators obtained by moments and MLE method in one parameter exponential family. Extension to multiparameter exponential family. Examples of consistent but not asymptotically normal from Pitman family. Method of maximum likelihood. CAN estimators for one parameter Cramer family. Cramer- Huzurbazar theorem. Solution of likelihood equations. Method of scoring. Newton-Raphson and other iterative procedures. Fisher Lower Bound to asymptotic variance, extension to multiparameter case (without proof). Multinomial distribution with cell probabilities depending on a parameter
MLE in pitman family and double Exponent distribution. MLE in censored and truncated distributions. UMVUE of probability densities (univariate case only)
Likelihood ratio test (LRT). Asymptotic distribution of LRT statistic wald test. Rao's Score test. Pearon chi-square test for Goodness of fit. Bartlett's test for hamogeneity of variances, Large sample tests. Consistency of Large Sample Tests. Asymptotic power of large sample tests.
References:
| 3 hrs. duration | 100 Marks |
Linear estimation. Guass Mark off theorem. Testing of hypothesis (involving several linear functions. test of sub-hypothesis and test involving equality of some of the parameters). Introduction to one way random model and estimation of variance components.
General theory of analysis of experimental designs. Designs for two way elimination of heterogeneity. Desirable properties of a good Design: Orthogonality, Connectedness and Balance, Various Optimality Criteria and their interpretations. Relation Between blocks of incomplete block designs, duality, resolvability and affine resolvability. Theorems on bounds.
Group divisible, lattice and linked block designs- intrablock analysis. Latin square and Youden square designs. Combination of result in group of experiments
Construction of orthogonal latin squares- (i) for prime power numbers and (ii) by Mann- Mechneish theorem, simple methods of construction of BIB designs. Construction of symmetrical fractional factorial designs.
Quenouille's Technique of Bias Reduction and its application to ratio type estimators. Ratio-method of estimation under Midzune Scheme of Sampling When X is Known.
Multivariate Extension of Ratio and Regrression estimators.
Double Sampling for Ratio and Regression Methods of Estimation
Definition of T-classes. Varying probability sampling with and without replacement. Unbaised estimators of variance of Horvitz and Thompson's estimator. Rao-Hartley and Cochran Sampling Scheme and their estimation procedure.
The theory of Multi-stage Sampling with varying probabilities with or without replacement. Issues in small area estimation-Syntheitic and generalized regression estimatotrs.
Non Sampling errors and baised response randomized responses for variables, Errors in surveys, Modelling Observational errors, estimation of variance components, application to longitudinal studies (repetitive surveys).
Variance estimation, method of random groups, balanced half samples (IPNSS), Jack-Knife method.
Introduction to super population models.
Reference:| 3 hrs. duration | 100 Marks |
Statistical Quality Control: Meaning of Specification limits, item, quality, process and process control, Objectives of S.Q.C., Control Charts for measurable quality characteristic. Chance variation and assignable causes of variation. Mean X and R charts. Determination of control limits and central line in various situations.
Meaning of Statistical Control and its relation with specification limits. Modified control limits, warning limits and tolerances limits. Methods of estimation inherent variability, Rational sub-Grouping, sucessive estimate; control charts for defectives.'P' and 'nP'control charts for number of defects per unit, 'C' chart.
Advantages of S.Q.C. Comparision of Mean X and R chart with P chart when both can be used for same situation
Acceptance sampling by attributes. Need for sampling inspection methods for acceptance. Lot quality and lot by lot acceptances. A.O.Q.L., producer's risk customer's risk, recification O.C. Function, A.S.N. and Average to inspection of an acceptance procedure. Single and Double sampling plans and their mathematical analysis.
Knowledge of Standard Sampling inspectiontables. Dodge and Roming table and Mil Std. 105A. Sampling inspectionplans for continuous production process where lots can be formed.
Sampling inspection plans by variables - one sided specification standard (known and unknown). Two sided specifications (known and unknown). Use of design in experiments in SPC. factorial experiments.fractional factorial designs.
Definition and Scope of Operational Research; Phases in operations Research; Models and their solutions; decision-making under uncertainity and risk, use of different criteria; sensitivity analysis
Review of L.P.problems: duality theorem; transportation and assignment problems; Non-linear programming - Kuhn Tucker cvonditions. Wolfe's and Beale's algorithms for solving quadratic programming problems. Bellman's principle of optimality general formulation. Computational methods and application of dynamic programming.
Analytical structure of inventory problems; EOQ formula of Harris. its sesitivity analysis and extensions allowing quantity discounts and shortages, Multi-item inventory subject to constraints. Models with random demand, the static risk model. P and Q-systems with constsant and random lead times.
Queuing models- specifications and effectiveness measures. steady-state solutions of M/M/1 and M/M/C models with associated distributions of Queue-length and waiting time. M/G/1 queue and Pollazeek Khinchine result. Steady-state solutions of M/EK/1 and EK/M/1 queues. Machine interference problem.
Sequencing and scheduling problems. 2 machine n-job and 3-machine n-job problems with identical machine sequence for all jobs, 2-job n-machine problem with different routings. Branch and bound method for solving travelling salesman problem.
Reference:| 4 hrs. duration | 100 Marks |
| 4 hrs. duration | 100 Marks |
Any one of the following papers with the permission of the Institution concerned.
| 3 hrs. duration | 100 Marks |
Components of time series. Methods of their determination Variate difference method. Yule-slut-Sky effect. Correlogram, Autoregressivemodels of first and second order. Periodogram analysis.
Index numbers of prices and quantities and their relative merits, construction of index numbers of wholesale and consumer prices.
Income distribution Pareta and Engel curves, concentration curve, methods of estimating national income. Intersectoral flows, inter-industry table.
Census and Vital Statistical data. Vital Rates and ratios, standardisation of rates, trends and differentials in mortality and fertility.
Stationary population construction of life table, gross and net reproduction rates: stable population theory, population estimation and projection.
Demographic trends in India. Labour force analysis, Birth and Death stochastic processes.
Stochastic population models. logistic models, bivariate growth models, migration models, fertility analysis models. mortality analysis models.
Reference| 4 hrs. duration | 100 Marks |
| 3 hrs. duration | 100 Marks |
STOCHASTIC PROCESSES: 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 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; Poission process, birth and death process; Applications to queues and storage problems. 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; moving average and auto regressive processes; Galton-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.
ECONOMETRICS: Nature of ecnometrics. The general linear model (GLM) and its extensions. Ordinary least squares (OLS)estimation and prediction. Generalized least squares(GLS) estimation and prediction. Heterosecdastic disturbances. Pure and mixed estimation. Grouping of observations and equations.
Auto Correlation its consequences and test. Theil BLUS procedure Estimation and prediction. Multicollinearity problem. its implications and tools for handling the problem. Ridge regression.
Linear regression with Stochastic regressors. Instrumental variable estimation. Errors in Variables. Autoregressive linear regression. Distributed lag models. Use of principal components, canonicalcorrelations and discriminant analyses in econometrics.
Simultaneous linear equations model. Examples. Identification problem. Restrictions on structural parameters - rank and order conditions Restrictions on variances and covariances.
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, Mote Carlo studies and simulation.
References:| 3 hrs. duration | 100 Marks |
RELIABILITY: Reliabilityconcepts and measures; components and systems; coherent systems; reliability of coherent systems; cuts and paths; modular decomposition; bounds on system reliability; structural and reliabilityimportance of compositions.
Life distribution; reliability function; hazard rate; common life distributions - exponential, Weibull, gamma etc. Estimation of parameters and tests in these models
Notions of ageing: IFR, IFRA, NBU, DMRL and NBUE classes and their duals; loss of memory property of the exponential distribution; closures of these classes under formation of coherent systems, convolutions and mixtures.
Univariate shock models and life distributions arising out of them; bivariate shock models; common bivariate exponential distributions and their properties.
Reliability estimation based on failure times in variously censored life tests and in tests with replacement of failed items; stress-strength reliability and its estimation.
Maintenance and replacement policies; avail;ability of repairable systems; modelling of a repairable system by a non-homogeneous Poission process.
Reliability growth models; probability plotting techniques; Hollander-Proschan and Deshpandey tests for exponentially; tests for HPP vs. NHPP with repairable systems.
Basic ideas of accelerated life testing
SURVIVAL ANALYSIS: Concepts of time, order and random Consoring, likelihood in these cases. Life distributions-Exponential Gamma. Weibull, Lgonormal. Pareto, Linear failure rate, Parametric inference Point estimation. Confidence Intervals, Scores, LR, MLE tests (Rao-Willks-Wald) for these distributions.
Life tables, failure rate, mean residual life and their elementary properties. Ageing classes and their properties, Bathtub Failure rate.
Estimation of survival function - Acturial Estimator, Kaplan-Meier Estimator, Estimation under the assumption of IFR/DFR. Tests of Exponentiality against non-parametric classes. Total time on test, Deshpandey test.
Two sample problem- Gehan test, Log rank test, Mantel-Haenszel test, Taron-Ware tests.
Semi Parametric regression for failure rate- Cox's proportional hazards model with one and several convariates. Rank test for the regression coefficients
Competing risks model, parametric and non-parametric inference for this model.
Multiple decrement life table.
References:| 3 hrs. duration | 100 Marks |
STATISTICAL DECISION THEORY: Decision problem and 2-person game, utility theory, loss functions, expcted loss, decision rules (non-randomized and randomized), decision principles (conditional Bayes frequentist), inference problems as decision problems, optimal decision rules.
Concepts of Admissiblity and completeness, Bayes rules, admissibility of Bayes rules.
Supporting and separating hyperplane theorems, minimax theorem of for finite parameter space, minimax estomators of Normal and Poission means, admissibility of minimax rules.
Invariant decision rules- location parameter problems, invariance and and essentially complete classes in simple estimation and testing situations, estimation of a distribution function.
Multivariate normal distribution, exponential family of distributions, sufficient statistics, essentially complete classes of rules based on sufficient statistics, complete sufficient statistics.
Sequential decision rules, Bayes and minimax sequential decision rules, invariant sequential decision problems, sequential tests of a simple hypothesis against a simple alternative ASPRT and stopping rule principle.
BAYESIAN INFERENCE:Subjective interpretation of probability in terms of fair odds. Evaluation of (i) subjective probability of an event using a subjectively unbaised 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. conjugate families for (i)exponential family models (ii) models admitting sufficient statistics of fixed dimensions. Enlarging the netural conjugate family by (i)enlarging hyper parameter space (ii)mixtures from conjugate family. choosing an appropriate member of conjugate prior family. Non informative. Improper and invariant priors. jeffrey's invariant prior.
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. Generalization to convex loss functions. Evaluation of the estimated in terms of the posterior risk.
Bayesian interval estimation: Credible intervals. Highest posterior density regions. Interpretation of the confidence coefficient for classical confidence interval.
Bayesain testing of Hypothesis: Specification of the appropriate form of the prior distribution for a Bayesain testing of hypothesis problem. Prior odds, Posterior odds, Bayes factors for varioustypes of testing hypothesis problems depending upon whether the null hypothesis and the alternative hypothesis are simple or composite. Specification of the Bayes tests in the above cases. Discussion of Lindley's paradox for testing a point hypothesis for normal mean against the two sided alternative hypothesis.
Bayesain prediction problem
Large sample approximations for the posterior distribution. Bayesain calculatis for non conjugate priors: (i) importance sampling (ii) Obtaining a large sample of parameter values from the posterior distribution using acceptance - Rejection methods markov Chain Monte Carlo Methods and .
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