**STANDARD AND SYLLABI**

The standard of papers in General English and General Studies will be such as maybe

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 analyse problems with the help of theory. They

will be expected to be particularly conversant with Indian problems in the field(s) of

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

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 correlation, Intraclass correlation,

Correlation ratio.

Standard errors and large sample test. 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, WaldWolfowitz 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.

Concept of interpolation and extrapolation: Newton Gregory’s forward and backward

interpolation formulae for equal intervals, divided differences and their properties, Newton’s

formula for divided difference, Lagrange’s formula for unequal intervals, central difference

formula due to Gauss, Sterling and Bessel, the concept of error terms in interpolation formula.

Inverse interpolation: Different methods of inverse interpolation.

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

Waddles 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

the central processing unit, memory unit, arithmetic and logical unit, an input unit, an 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

14

variance, estimation with correlated observations, properties of least square estimators,

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 the field of

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 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 intra-class correlation

coefficient.

Concept of multistage sampling and its application, two-stage sampling with 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.

15

**(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, Paasche’s, 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.

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.

Spectral analysis of the weakly stationary process, periodogram and correlogram analyses,

computations based on Fourier transform.

**STATISTICS-IV (DESCRIPTIVE TYPE)**

(Equal number of questions i.e. 50% weightage from all the subsections below

and

candidates have to choose any two subsections and answer)**(i) 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,

existence of solution and uniqueness of value in zero-sum games, finding solutions in 2×2,

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 travelling 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.

16

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.

**(ii) 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

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.

**(iii) 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 test.

Two sample problem – 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, 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.

**(iv) Quality Control:**

Statistical process and product control: Quality of a product, need for quality control, 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 a one-sided and two-sided specification.

**(v) Multivariate Analysis:**

Multivariate normal distribution and its properties. Random sampling from 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 multiple correlation coefficients.

17

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 a 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.

(vi) Design and Analysis of Experiments:

Analysis of variance for one way and two-way classifications, Need for 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.

(vii) Computing with C and R :

Basics of C: Components of C language, 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 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, union of structure,

initialization of a union variable, uses of union. Introduction to linked list, linear linked list,

insertion of a node in list, removal of a node from 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.