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# (IAS Planner) Optional Subjects Syllabus: (Statistics)

## Syllabus of Examination

## Statistics (Optional Subjects)

## Paper-I

**1. Probability: **Sample space and events, probability
measure and probability space, random variable as a measurable function,
distribution function of a random variable, discrete and continuous-type random
variable, probability mass function, probability density function, vector-valued
random variable, marginal and conditional distributions, stochastic independence
of events and of random variables, expectation and moments of a random variable,
conditional expectation, convergence of a sequence of random variable in
distribution, in probability, in p-th mean and almost everywhere, their criteria
and inter-relations, Chebyshev’s inequality and Khintchine‘s weak law of large
numbers, strong law of large numbers and Kolmogoroff’s theorems, probability
generating function, moment generating function, characteristic function,
inversion theorem, Linderberg and Levy forms of central limit theorem, standard
discrete and continuous probability distributions.

**2. Statistical Inference: **Consistency, unbiasedness,
efficiency, sufficiency, completeness, ancillary statistics, factorization
theorem, exponential family of distribution and its properties, uniformly
minimum variance unbiased (UMVU) estimation, Rao-Blackwell and Lehmann-Scheffe
theorems, Cramer-Rao inequality for single parameter. Estimation by methods of
moments, maximum likelihood, least squares, minimum chi-square and modified
minimum chi-square, properties of maximum likelihood and other estimators,
asymptotic efficiency, prior and posterior distributions, loss function, risk
function, and minimax estimator. Bayes estimators.

Non-randomised and randomised tests, critical function, MP tests, Neyman-Pearson
lemma, UMP tests, monotone likelihood ratio, similar and unbiased tests, UMPU
tests for single parameter likelihood ratio test and its asymptotic
distribution. Confidence bounds and its relation with tests.

Kolmogoroff’s test for goodness of fit and its consistency, sign test and its
optimality. Wilcoxon signed-ranks test and its consistency, Kolmogorov-Smirnov
two-sample test, run test, Wilcoxon-Mann-Whitney test and median test, their
consistency and asymptotic normality.

Wald’s SPRT and its properties, OC and ASN functions for tests regarding
parameters for Bernoulli, Poisson, normal and exponential distributions. Wald’s
fundamental identity.

**3. Linear Inference and Multivariate Analysis: **Linear
statistical models’, theory of least squares and analysis of variance, Gauss-Markoff
theory, normal equations, least squares estimates and their precision, test of
significance and interval estimates based on least squares theory in one-way,
two-way and three-way classified data, regression analysis, linear regression,
curvilinear regression and orthogonal polynomials, multiple regression, multiple
and partial correlations, estimation of variance and covariance components,
multivariate normal distribution, Mahalanobis-D2 and Hotelling’s T2 statistics
and their applications and properties, discriminant analysis, canonical
correlations, principal component analysis.

**4. Sampling Theory and Design of Experiments: **An
outline of fixed-population and super-population approaches, distinctive
features of finite population sampling, probability sampling designs, simple
random sampling with and without replacement, stratified random sampling,
systematic sampling and its efficacy , cluster sampling, two-stage and
multi-stage sampling, ratio and regression methods of estimation involving one
or more auxiliary variables, two-phase sampling, probability proportional to
size sampling with and without replacement, the Hansen-Hurwitz and the
Horvitz-Thompson estimators, non-negative variance estimation with reference to
the Horvitz-Thompson estimator, non-sampling errors.

Fixed effects model (two-way classification) random and mixed effects models
(two-way classification with equal observation per cell), CRD, RBD, LSD and
their analyses, incomplete block designs, concepts of orthogonality and balance,
BIBD, missing plot technique, factorial experiments and 2n and 32, confounding
in factorial experiments, split-plot and simple lattice designs, transformation
of data Duncan’s multiple range test.

## Paper-II

**1. Industrial Statistics: **Process and product control,
general theory of control charts, different types of control charts for
variables and attributes, X, R, s, p, np and c charts, cumulative sum chart.
Single, double, multiple and sequential sampling plans for attributes, OC, ASN,
AOQ and ATI curves, concepts of producer’s and consumer’s risks, AQL, LTPD and
AOQL, Sampling plans for variables, Use of Dodge-Roming tables.

Concept of reliability, failure rate and reliability functions, reliability of
series and parallel systems and other simple configurations, renewal density and
renewal function, Failure models: exponential, Weibull, normal , lognormal.

Problems in life testing, censored and truncated experiments for exponential
models.

**2. Optimization Techniques: **Different types of models
in Operations Research, their construction and general methods of solution,
simulation and Monte-Carlo methods formulation of linear programming (LP)
problem, simple LP model and its graphical solution, the simplex procedure, the
two-phase method and the M-technique with artificial variables, the duality
theory of LP and its economic interpretation, sensitivity analysis,
transportation and assignment problems, rectangular games, two-person zero-sum
games, methods of solution (graphical and algebraic).

Replacement of failing or deteriorating items, group and individual replacement
policies, concept of scientific inventory management and analytical structure of
inventory problems, simple models with deterministic and stochastic demand with
and without lead time, storage models with particular reference to dam type.

Homogeneous discrete-time Markov chains, transition probability matrix,
classification of states and ergodic theorems, homogeneous continuous-time
Markov chains, Poisson process, elements of queuing theory, M/M/1, M/M/K, G/M/1
and M/G/1 queues.

Solution of statistical problems on computers using well-known statistical
software packages like SPSS.

**3. Quantitative Economics and Official Statistics: **
Determination of trend, seasonal and cyclical components, Box-Jenkins method,
tests for stationary series, ARIMA models and determination of orders of
autoregressive and moving average components, forecasting.

Commonly used index numbers-Laspeyre's, Paasche's and Fisher's ideal index
numbers, chain-base index number, uses and limitations of index numbers, index
number of wholesale prices, consumer prices, agricultural production and
industrial production, test for index numbers - proportionality, time-reversal,
factor-reversal and circular .

General linear model, ordinary least square and generalized least squares
methods of estimation, problem of multicollinearity, consequences and solutions
of multicollinearity, autocorrelation and its consequences, heteroscedasticity
of disturbances and its testing, test for independence of disturbances, concept
of structure and model for simultaneous equations, problem of
identification-rank and order conditions of identifiability, two-stage least
square method of estimation.Present official statistical system in India
relating to population, agriculture, industrial production, trade and prices,
methods of collection of official statistics, their reliability and limitations,
principal publications containing such statistics, various official agencies
responsible for data collection and their main functions.

**4. Demography and Psychometry: **Demographic data from
census, registration, NSS other surveys, their limitations and uses, definition,
construction and uses of vital rates and ratios, measures of fertility,
reproduction rates, morbidity rate, standardized death rate, complete and
abridged life tables, construction of life tables from vital statistics and
census returns, uses of life tables, logistic and other population growth
curves, fitting a logistic curve, population projection, stable population,
quasi-stable population, techniques in estimation of demographic parameters,
standard classification by cause of death, health surveys and use of hospital
statistics.

## Suggested Reading:

**Introductory Probability and Statistical Applications - Paul Meyer**• An Introduction to Probability Theory & Mathematical Statistics -V K
Rohtagi

• Fundamentals of Statistics (2 Vol.)- A M Goon, M K Gupta and B Dass Gupta

• An Outline of Statistical Theory (2 Vol.) -A M Goon, M K Gupta and B .Dass Gupta

• Fundamentals of Mathematical Statistics-A C Gupta and V K Kapoor

• Fundamentals of Applied Statistics-S C Gupta and V K Kapoor

• Sampling Techniques-William G. Cochran

• Sampling Theory of Surveys with applications - B. V Sukhatme & B V Sukhatme.