The standard of papers in General English and General Studies will be such as may be expected of a graduate of an Indian University.

The standard of papers in the other subjects will be that of the Master’s degree examination of an Indian University in the relevant disciplines. The candidates will be expected to illustrate the theory by facts, and to analyze problems with the help of theory. They will be expected to be particularly conversant with Indian problems in the field(s) of Economics/Statistics.

GENERAL ENGLISH

Candidates will be required to write an essay in English. Other questions will be designed to test their understanding of English and workman-like use of words. Passages will usually be set for summary or precis.

GENERAL STUDIES

General knowledge including knowledge of current events and of such matters of everyday observation and experience in their scientific aspects as may be expected of an educated person who has not made a special study of any scientific subject. The paper will also include questions on Indian Polity including the political system and the Constitution of India, History of India, and Geography of nature which a candidate should be able to answer without special study.

STATISTICS-I (OBJECTIVE TYPE)

(i) Probability:

Classical and axiomatic definitions of Probability and consequences. Law of total probability, Conditional probability, Bayes' theorem, and applications. Discrete and continuous random variables. Distribution functions and their properties.

Standard discrete and continuous probability distributions - Bernoulli, Uniform, Binomial, Poisson, Geometric, Rectangular, Exponential, Normal, Cauchy, Hypergeometric, Multinomial, Laplace, Negative binomial, Beta, Gamma, Lognormal. Random vectors, Joint and marginal distributions, conditional distributions, Distributions of functions of random variables. Modes of convergences of sequences of random variables - in distribution, in probability, with probability one and in mean square. Mathematical expectation and conditional expectation. Characteristic function, moment and probability generating functions, Inversion, uniqueness, and continuity theorems. Borel 0-1 law, Kolmogorov's 0-1 law. Tchebycheff's and Kolmogorov's inequalities. Laws of large numbers and central limit theorems for independent variables.

(ii) Statistical Methods:

Collection, compilation, and presentation of data, charts, diagrams, and histogram. Frequency distribution. Measures of location, dispersion, skewness, and kurtosis. Bivariate and multivariate data. Association and contingency. Curve fitting and orthogonal polynomials. Bivariate normal distribution. Regression-linear, polynomial. Distribution of the correlation coefficient, Partial and multiple correlations, Intraclass correlation, Correlation ratio.

Standard errors and large sample tests. Sampling distributions of sample mean, sample variance, t, chi-square, and F; tests of significance based on them, Small sample tests.

Non-parametric tests-Goodness of fit, sign, median, run, Wilcoxon, Mann-Whitney, Wald- Wolfowitz and Kolmogorov-Smirnov. Order statistics-minimum, maximum, range, and median. Concept of Asymptotic relative efficiency.

(iii) Numerical Analysis:

Finite differences of different orders: E and D operators, factorial representation of a polynomial, separation of symbols, sub-division of intervals, differences of zero.

Numerical differentiation: Trapezoidal, Simpson’s one-third, and three-eight rule, and Waddle's rule.

Summation of Series: Whose general term (i) is the first difference of a function (ii) is in geometric progression.

Numerical solutions of differential equations: Euler's Method, Milne’s Method, Picard’s Method, and Runge-Kutta Method.

(iv) Computer application and Data Processing:

Basics of Computer: Operations of a computer, Different units of a computer system like central processing unit, memory unit, arithmetic and logical unit, an input unit, output unit, etc., Hardware including different types of input, output and peripheral devices, Software, system and application software, number systems, Operating systems, packages and utilities, Low and High-level languages, Compiler, Assembler, Memory – RAM, ROM, unit of computer memory (bits, bytes, etc.), Network – LAN, WAN, internet, intranet, basics of computer security, virus, antivirus, firewall, spyware, malware, etc.

Basics of Programming: Algorithm, Flowchart, Data, Information, Database, an overview of different programming languages, frontend and backend of a project, variables, control structures, arrays, and their usages, functions, modules, loops, conditional statements, exceptions, debugging and related concepts.

STATISTICS- II (OBJECTIVE TYPE)

(i) Linear Models:

Theory of linear estimation, Gauss-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, the 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-Keul-Duncan.

(ii) Statistical Inference and Hypothesis Testing:

Characteristics of a good estimator. Estimation methods of maximum likelihood, minimum chi-square, moments, and least squares. Optimal properties of maximum likelihood estimators. Minimum variance unbiased estimators. Minimum variance bound estimators. Cramer-Rao inequality. Bhattacharya bounds. Sufficient estimator. Factorization theorem. Complete statistics. Rao-Blackwell theorem. Confidence interval estimation. Optimum confidence bounds. Resampling, Bootstrap, and Jackknife.

Hypothesis testing: Simple and composite hypotheses. Two kinds of error. Critical region. Different types of critical regions and similar regions. Power function. Most powerful and uniformly most powerful tests. Neyman-Pearson fundamental lemma. Unbiased test. Randomized test. Likelihood ratio test. Wald's SPRT, OC, and ASN functions. Elements of decision theory.

(iii) Official Statistics:

National and International official statistical system

Official Statistics: (a) Need, Uses, Users, Reliability, Relevance, Limitations, Transparency, its visibility (b) Compilation, Collection, Processing, Analysis and Dissemination, Agencies Involved, Methods

National Statistical Organization: Vision and Mission, NSSO and CSO; roles and responsibilities; Important activities, Publications, etc.

National Statistical Commission: Need, Constitution, its role, functions, etc.; Legal Acts/ Provisions/ Support for Official Statistics; Important Acts

Index Numbers: Different Types, Need, Data Collection Mechanism, Periodicity, Agencies Involved, Uses

Sector Wise Statistics: Agriculture, Health, Education, Women, and Child, etc. Important

Surveys & Census, Indicators, Agencies, and Usages, etc.

National Accounts: Definition, Basic Concepts; issues; the Strategy, Collection of Data and Release.

Population Census: Need, Data Collected, Periodicity, Methods of data collection, dissemination, Agencies involved.

Misc.: Socio-Economic Indicators, Gender Awareness/Statistics, Important Surveys, and Censuses.

STATISTICS- III (DESCRIPTIVE TYPE)

(i) Sampling Techniques:

Concept of population and sample, need for sampling, complete enumeration versus sampling, basic concepts in sampling, sampling and Non-sampling error, Methodologies in sample surveys (questionnaires, sampling design, and methods followed in field investigation) by NSSO.

Subjective or purposive sampling, probability sampling or random sampling, simple random sampling with and without replacement, estimation of population mean, population proportions, and their standard errors. Stratified random sampling, proportional and optimum allocation, comparison with simple random sampling for fixed sample size. Covariance and Variance Function.

ratio, product and regression methods of estimation, estimation of the population mean, evaluation of Bias and Variance to the first order of approximation, comparison with simple random sampling. Systematic sampling (when population size (N) is an integer multiple of sampling size (n)). Estimation of population mean and standard error of this estimate, comparison with simple random sampling. Sampling with probability proportional to size (with and without replacement method), Des Raj and Das estimators for n=2, Horvitz- Thomson’s estimator Equal size cluster sampling: estimators of population mean and total and their standard errors, comparison of cluster sampling with SRS in terms of an intra-class correlation coefficient. Concept of multistage sampling and its application, two-stage sampling with an equal number of second stage units, estimation of population mean and total. Double sampling in ratio and regression methods of estimation. Concept of interpenetrating sub-sampling.

(ii) Econometrics:

Nature of econometrics, the general linear model (GLM) and its extensions, ordinary least squares (OLS) estimation and prediction, generalized least squares (GLS) estimation and prediction, heteroscedastic disturbances, pure and mixed estimation.

Autocorrelation, its consequences, and tests. Theil BLUS procedure, estimation, and prediction, multi-collinearity problem, its implications and tools for handling the problem, ridge regression.

Linear regression and stochastic regression, instrumental variable estimation, errors in variables, autoregressive linear regression, lagged variables, distributed lag models, estimation of lags by OLS method, Koyck’s geometric lag model.

Simultaneous linear equations model and its generalization, identification problem, restrictions on structural parameters, rank and order conditions.

Estimation in simultaneous equations model, recursive systems, 2 SLS estimators, limited information estimators, k-class estimators, 3 SLS estimator, full information maximum likelihood method, prediction, and simultaneous confidence intervals.

(iii) Applied Statistics:

Index Numbers: Price relatives and quantity or volume relatives, Link and chain relatives

composition of index numbers; Laspeyre's, Paasches’, Marshal Edgeworth and Fisher index numbers; chain base index number, tests for index number, Construction of index numbers of wholesale and consumer prices, Income distribution-Pareto, and Engel curves, Concentration curve, Methods of estimating national income, Inter-sectoral flows, Interindustry table, Role of CSO. Demand Analysis

Time Series Analysis: Economic time series, different components, illustration, additive and multiplicative models, determination of trend, seasonal and cyclical fluctuations.

Time-series as discrete parameter stochastic process, autocovariance and autocorrelation functions, and their properties.

Exploratory time Series analysis, tests for trend and seasonality, exponential and moving average smoothing. Holt and Winters smoothing, forecasting based on smoothing.

A detailed study of the stationary processes: (1) moving average (MA), (2) autoregressive (AR), (3) ARMA, and (4) AR integrated MA (ARIMA) models. Box-Jenkins models, choice of AR, and MA periods.

Discussion (without proof) of estimation of mean, autocovariance, and autocorrelation functions under large sample theory, estimation of ARIMA model parameters.

STATISTICS-IV (DESCRIPTIVE TYPE)

(A) Operations Research and Reliability:

Definition and Scope of Operations Research: phases in Operation Research, models and their solutions, decision-making under uncertainty and risk, use of different criteria, sensitivity analysis.

Transportation and assignment problems. Bellman’s principle of optimality, general formulation, computational methods, and application of dynamic programming to LPP.

Decision-making in the face of competition, two-person games, pure and mixed strategies, the existence of solution and uniqueness of value in zero-sum games, finding solutions in 2x2, 2xm, and mxn games.

Analytical structure of inventory problems, EOQ formula of Harris, its sensitivity 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 constant and random lead times.

Queuing models – specification 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 Pollazcek-Khinchine result.

Sequencing and scheduling problems. 2-machine n-job and 3-machine n-job problems with identical machine sequence for all jobs

Branch and Bound method for solving traveling salesman problem.

Replacement problems – Block and age replacement policies.

PERT and CPM – basic concepts. Probability of project completion.

Reliability concepts and measures, components and systems, coherent systems, reliability of coherent systems. Life-distributions, reliability function, hazard rate, common univariate life distributions – exponential, Weibull, gamma, etc. Bivariate exponential distributions.

Estimation of parameters and tests in these models.

Notions of aging – IFR, IFRA, NBU, DMRL, and NBUE classes and their duals. Loss of memory property of the exponential distribution.

Reliability estimation based on failure times in variously censored life-tests and in tests with the replacement of failed items. Stress-strength reliability and its estimation.

(B) Demography and Vital Statistics:

Sources of demographic data, census, registration, ad-hoc surveys, Hospital records, Demographic profiles of the Indian Census.

Complete life table and its main features, Uses of life table. Markham's and Gompertz's curves. National life tables. UN model life tables. Abridged life tables. Stable and stationary populations.

Measurement of Fertility: Crude birth rate, General fertility rate, Age-specific birth rate, Total fertility rate, Gross reproduction rate, Net reproduction rate.

Measurement of Mortality: Crude death rate, Standardized death rates, Age-specific death rates, Infant Mortality rate, the Death rate by cause.

Internal migration and its measurement, migration models, the concept of international migration.

Net migration. International and postcensal estimates. Projection method including logistic curve fitting. Decennial population census in India.

(C) Survival Analysis and Clinical Trial:

Concept of time, order and random censoring, likelihood in the distributions – exponential, gamma, Weibull, lognormal, Pareto, Linear failure rate, inference for these distributions.

Life tables, failure rate, mean residual life, and their elementary classes and their properties. Estimation of survival function – actuarial estimator, Kaplan – Meier estimator, estimation under the assumption of IFR/DFR, tests of exponentiality against non-parametric classes, total time on the test.

Two sample problems – Gehan test, log-rank test.

Semi-parametric regression for failure rate – Cox’s proportional hazards model with one and several covariates, rank test for the regression coefficient.

Competing risk model, parametric and non-parametric inference for this model.

Introduction to clinical trials: the need and ethics of clinical trials, bias and random error in clinical studies, the conduct of clinical trials, an overview of Phase I – IV trials, multicenter trials.

Data management: data definitions, case report forms, database design, data collection systems for good clinical practice.

Design of clinical trials: parallel vs. cross-over designs, cross-sectional vs. longitudinal designs, review of factorial designs, objectives, and endpoints of clinical trials, design of Phase I trials, design of single-stage and multi-stage Phase II trials, design and monitoring of phase III trials with sequential stopping,

Reporting and analysis: analysis of categorical outcomes from Phase I – III trials, analysis of survival data from clinical trials.

(D) Quality Control:

Statistical process and product control: Quality of a product, need for quality control, the basic concept of process control, process capability, and product control, general theory of control charts, causes of variation in quality, control limits, sub grouping summary of out of control criteria, charts for attributes p chart, np chart, c-chart, V chart, charts for variables: R, (X, R), (X, σ) charts.

Basic concepts of process monitoring and control; process capability and process optimization.

General theory and review of control charts for attribute and variable data; O.C. and A.R.L. of control charts; control by gauging; moving average and exponentially weighted moving average charts; Cu-Sum charts using V-masks and decision intervals; Economic design of X-bar chart.

Acceptance sampling plans for attributes inspection; single and double sampling plans and their properties; plans for inspection by variables for the one-sided and two-sided specification.

(E) Multivariate Analysis:

Multivariate normal distribution and its properties. Random sampling from a multivariate normal distribution. Maximum likelihood estimators of parameters, distribution of the sample mean vector.

Wishart matrix – its distribution and properties, distribution of sample generalized variance, null and non-null distribution of multiple correlation coefficients.

Hotelling’s T2 and its sampling distribution, application in test on mean vector for one and more multivariate normal population and also on equality of components of a mean vector in the multivariate normal population.

Classification problem: Standards of good classification, the procedure of classification based on multivariate normal distributions.

Principal components, dimension reduction, canonical variates, and canonical correlation — definition, use, estimation, and computation.

(F) Design and Analysis of Experiments:

Analysis of variance for one-way and two-way classifications Need for the design of experiments, the basic principle of experimental design (randomization, replication, and local control), complete analysis and layout of the completely randomized design, randomized block design, and Latin square design, Missing plot technique. Split Plot Design and Strip Plot Design.

Factorial experiments and confounding in 2n and 3n experiments. Analysis of covariance. Analysis of non-orthogonal data. Analysis of missing data.

(G) Computing with C and R:

Basics of C: Components of C language, the structure of a C program, Data type, basic data types, Enumerated data types, Derived data types, variable declaration, Local, Global, Parametric variables, Assignment of Variables, Numeric, Character, Real and String constants, Arithmetic, Relation and Logical operators, Assignment operators, Increment and decrement operators, conditional operators, Bitwise operators, Type modifiers and expressions, writing and interpreting expressions, using expressions in statements. Basic input/output.

Control statements: conditional statements, if-else, nesting of if-else, else if ladder, switch statements, loops in c, for, while do-while loops, break, continue, exit ( ), goto, and label declarations, One dimensional two dimensional and multidimensional arrays. Storage classes: Automatic variables, External variables, Static variables, Scope, and a lifetime of declarations.

Functions: classification of functions, functions definition and declaration, assessing a function, return statement, parameter passing in functions. Pointers (concept only).

Structure: Definition and declaration; structure (initialization) comparison of structure variable; Array of structures: array within structures, structures within structures, passing structures to functions; Unions accessing a union member, the union of structure, initialization of a union variable, uses of the union. Introduction to the linked list, linearly linked list, insertion of a node in the list, removal of a node from the list.

Files in C: Defining and opening a file, input-output operation on a file, creating a file, reading a file. Statistics Methods and techniques in R.

_______________________________________________________

RBI GRADE B DSIM

1. Theory of Probability and Probability Distributions Classical and axiomatic approach of probability and its properties, Bayes theorem and its application, strong and weak laws of large numbers, characteristic functions, central limit theorem, probability inequalities. Standard probability distributions – Binomial, Poison, Geometric, Negative binomial, Uniform, Normal, exponential, Logistic, Log-normal, Beta, Gamma, Weibull, Bivariate normal etc. Exact Sampling distributions - Chi-square, student’s t, F and Z distributions and their applications. Asymptotic sampling distributions and large sample tests, association and analysis of contingency tables. Sampling Theory: Standard sampling methods such simple random sampling, Stratified random sampling, Systematic sampling, Cluster sampling, Two stage sampling, Probability proportional to size etc. Ratio estimation, Regression estimation, non-sampling errors and problem of non-response, and Correspondence and categorical data analysis.

2. Linear Models and Economic Statistics Simple linear regression - assumptions, estimation, and inference diagnostic checks; polynomial regression, transformations on Y or X (Box-Cox, square root, log etc.), method of weighted least squares, inverse regression. Multiple regression - Standard Gauss Markov setup, least squares estimation and related properties, regression analysis with correlated observations. Simultaneous estimation of linear parametric functions, Testing of hypotheses; Confidence intervals and regions; Multicollinearity and ridge regression, LASSO. Definition and construction of index numbers, Standard index numbers; Conversion of chain base index to fixed base and vice-versa; base shifting, splicing and deflating of index numbers; Measurement of economic inequality: Gini's coefficient, Lorenz curves etc.

3. Statistical Inference: Estimation, Testing of Hypothesis and Non-Parametric Test Estimation: Concepts of estimation, unbiasedness, sufficiency, consistency and efficiency. Factorization theorem. Complete statistic, Minimum variance unbiased estimator (MVUE), Rao Blackwell and Lehmann-Scheffe theorems and their applications. Cramer-Rao inequality. Methods of Estimation: Method of moments, method of maximum likelihood estimation, method of least square, method of minimum Chi-square, basic idea of Bayes estimators. Principles of Test of Significance: Type-I and Type-II errors, critical region, level of significance, size and power, best critical region, most powerful test, uniformly most powerful test, Neyman Pearson theory of testing of hypothesis. Likelihood ratio tests, Tests of goodness of fit. Bartlett's test for homogeneity of variances. Non-Parametric Test: The Kolmogorov-Smirnov test, Sign test, Wilcoxon Signed-rank test, Wilcoxon Rank-Sum test, Mann Whitney U-test, Kruskal-Walls one way ANOVA test, Friedman's test, Kendall's Tau coefficient, Spearman's coefficient of rank correlation.

4. Stochastic Processes Poisson Processes: Arrival, interarrival and conditional arrival distributions. Non-homogeneous Processes. Law of Rare Events and Poisson Process. Compound Poisson Processes. Markov Chains: Transition probability matrix, Chapman- Kolmogorov equations, Regular chains and Stationary distributions, Periodicity, Limit theorems. Patterns for recurrent events. Brownian Motion - Limit of Random Walk, its defining characteristics and peculiarities; Martingales.

5. Multivariate Analysis Multivariate normal distribution and its properties and characterization; Wishart matrix, its distribution and properties, Hotelling’s T2 statistic, its distribution and properties, and its applications in tests on mean vector, Mahalanobis’ D2 statistics; Canonical correlation analysis, Principal components analysis, Factor analysis and cluster analysis.

6. Econometrics and Time Series General linear model and its extensions, ordinary least squares and generalized least squares estimation and prediction, heteroscedastic disturbances, pure and mixed estimation. Auto correlation, its consequences and related tests; Theil BLUS procedure, estimation and prediction; issue of multi-collinearity, its implications and tools for handling it; Ridge regression. Linear regression and stochastic regression, instrumental variable regression, autoregressive linear regression, distributed lag models, estimation of lags by OLS method. Simultaneous linear equations model and its generalization, identification problem, restrictions on structural parameters, rank and order conditions; different estimation methods for simultaneous equations model, prediction and simultaneous confidence intervals. Exploratory analysis of time series; Concepts of weak and strong stationarity; AR, MA and ARMA processes and their properties; model identification based on ACF and PACF; model estimation and diagnostic tests; Box-Jenkins models; ARCH/GARCH models. Inference with Non-Stationary Models: ARIMA model, determination of the order of integration, trend stationarity and difference stationary processes, tests of non-stationarity.

7. Statistical Computing Simulation techniques for various probability models, and resampling methods jack-knife, bootstrap and cross-validation; techniques for robust linear regression, nonlinear and generalized linear regression problem, tree-structured regression and classification; Analysis of incomplete data – EM algorithm, single and multiple imputation; Markov Chain Monte Carlo and annealing techniques, Gibbs sampling, Metropolis-Hastings algorithm; Neural Networks, Association Rules and learning algorithms.

8. Data Science, Artificial Intelligence and Machine Learning Techniques Introduction to supervised and unsupervised pattern classification; unsupervised and reinforcement learning, basics of optimization, model accuracy measures. Supervised Algorithms: Linear Regression, Logistic Regression, Penalized Regression, Naïve Bayes, Nearest Neighbour, Decision Tree, Support Vector Machine, Kernel density estimation and kernel discriminant analysis; Classification under a regression framework, neural network, kernel regression and tree and random forests. Unsupervised Classification: Hierarchical and non-hierarchical methods: k-means, k-medoids and linkage methods, Cluster validation indices: Dunn index, Gap statistics. Bagging (Random Forest) and Boosting (Adaptive Boosting, Gradient Boosting) techniques; Recurrent Neural Network (RNN); Convolutional Neural Network; Natural Language Processing.

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