STATISTICS
PAPER - I
1. Probability:
Sample space and events, probability
measure and probability space, random
variable as a measurable function, distri-bution function of a random variable, dis-crete and continuous-type random vari-able, probability mass function, probability
density function, vector-valued random
variable, marginal and conditional distri-butions, 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 prob-ability, in p-th mean and almost every-where, their criteria and inter-relations,
Chebyshev’s inequality and Khintchine‘s
weak law of large numbers, strong law of
large numbers and Kolmogoroff’s theo-rems, probability generating function, mo-ment generating function, characteristic
function, inversion theorem, Linderberg
and Levy forms of central limit theorem,
standard discrete and continuous probabil-ity distributions.
2. Statistical Inference:
Consistency, unbiasedness, efficiency,
sufficiency, completeness, ancillary statis-tics, 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 likeli-hood, least squares, minimum chi-square
and modified minimum chi-square, prop-erties 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 ra-tio, 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 consis-tency, Kolmogorov-Smirnov two-sample
test, run test, Wilcoxon-Mann-Whitney test
and median test, their consistency and as-ymptotic normality.
Wald’s SPRT and its properties, OC and
ASN functions for tests regarding param-eters for Bernoulli, Poisson, normal and
exponential distributions. Wald’s funda-mental 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, re-gression analysis, linear regression, cur-vilinear regression and orthogonal poly-nomials, multiple regression, multiple and
partial correlations, estimation of variance
and covariance components, multivariate
normal distribution, Mahalanobis-D2and
Hotelling’s T2statistics and their applica-tions and properties, discriminant analy-sis, canonical correlations, principal com-ponent analysis.
4. Sampling Theory and Design of Ex-periments:
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 involv-ing one or more auxiliary variables, two-phase sampling, probability proportional
to size sampling with and without replace-ment, the Hansen-Hurwitz and the Horvitz-Thompson estimators, non-negative vari-ance estimation with reference to the
Horvitz-Thompson estimator, non-sam-pling 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 or-thogonality and balance, BIBD, missing plot
technique, factorial experiments and 2n
and 32, confounding in factorial experi-ments, 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 sequen-tial 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 vari-ables, Use of Dodge-Roming tables.
Concept of reliability, failure rate and reli-ability functions, reliability of series and
parallel systems and other simple configu-rations, renewal density and renewal func-tion, Failure models: exponential, Weibull,
normal, lognormal.
Problems in life testing, censored and trun-cated experiments for exponential models.
2. Optimization Techniques:
Different types of models in Operations Re-search, their construction and general meth-ods 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, sensi-tivity analysis, transportation and assign-ment problems, rectangular games, two-person zero-sum games, methods of solu-tion (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, clas-sification of states and ergodic theorems,
homogeneous continuous-time Markov
chains, Poisson process, elements of queu-ing theory, M/M/1, M/M/K, G/M/1 and M/G/1
queues.
Solution of statistical problems on comput-ers using well-known statistical software
packages like SPSS.
3. Quantitative Economics and Official
Statistics:
Determination of trend, seasonal and cy-clical components, Box-Jenkins method,
tests for stationary series, ARIMA models
and determination of orders of
autoregressive and moving average com-ponents, 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, in-dex number of wholesale prices, consumer
prices, agricultural production and indus-trial production, test for index numbers -proportionality, time-reversal, factor-rever-sal and circular .
General linear model, ordinary least
square and generalized least squares
methods of estimation, problem of
multicollinearity, consequences and solu-tions of multicollinearity, autocorrelation
and its consequences, heteroscedasticity
of disturbances and its testing, test for in-dependence 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, indus-trial production, trade and prices, methods
of collection of official statistics, their reli-ability 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, registra-tion, NSS other surveys, their limitations and
uses, definition, construction and uses of
vital rates and ratios, measures of fertility,
reproduction rates, morbidity rate, standard-ized 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 de-mographic parameters, standard classifica-tion by cause of death, health surveys and
use of hospital statistics.
Methods of standardisation of scales and
tests, Z-scores, standard scores, T-scores,
percentile scores, intelligence quotient and
its measurement and uses, validity and
reliability of test scores and its determina-tion, use of factor analysis and path analy-sis in psychometry.
PAPER - I
1. Probability:
Sample space and events, probability
measure and probability space, random
variable as a measurable function, distri-bution function of a random variable, dis-crete and continuous-type random vari-able, probability mass function, probability
density function, vector-valued random
variable, marginal and conditional distri-butions, 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 prob-ability, in p-th mean and almost every-where, their criteria and inter-relations,
Chebyshev’s inequality and Khintchine‘s
weak law of large numbers, strong law of
large numbers and Kolmogoroff’s theo-rems, probability generating function, mo-ment generating function, characteristic
function, inversion theorem, Linderberg
and Levy forms of central limit theorem,
standard discrete and continuous probabil-ity distributions.
2. Statistical Inference:
Consistency, unbiasedness, efficiency,
sufficiency, completeness, ancillary statis-tics, 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 likeli-hood, least squares, minimum chi-square
and modified minimum chi-square, prop-erties 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 ra-tio, 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 consis-tency, Kolmogorov-Smirnov two-sample
test, run test, Wilcoxon-Mann-Whitney test
and median test, their consistency and as-ymptotic normality.
Wald’s SPRT and its properties, OC and
ASN functions for tests regarding param-eters for Bernoulli, Poisson, normal and
exponential distributions. Wald’s funda-mental 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, re-gression analysis, linear regression, cur-vilinear regression and orthogonal poly-nomials, multiple regression, multiple and
partial correlations, estimation of variance
and covariance components, multivariate
normal distribution, Mahalanobis-D2and
Hotelling’s T2statistics and their applica-tions and properties, discriminant analy-sis, canonical correlations, principal com-ponent analysis.
4. Sampling Theory and Design of Ex-periments:
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 involv-ing one or more auxiliary variables, two-phase sampling, probability proportional
to size sampling with and without replace-ment, the Hansen-Hurwitz and the Horvitz-Thompson estimators, non-negative vari-ance estimation with reference to the
Horvitz-Thompson estimator, non-sam-pling 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 or-thogonality and balance, BIBD, missing plot
technique, factorial experiments and 2n
and 32, confounding in factorial experi-ments, 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 sequen-tial 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 vari-ables, Use of Dodge-Roming tables.
Concept of reliability, failure rate and reli-ability functions, reliability of series and
parallel systems and other simple configu-rations, renewal density and renewal func-tion, Failure models: exponential, Weibull,
normal, lognormal.
Problems in life testing, censored and trun-cated experiments for exponential models.
2. Optimization Techniques:
Different types of models in Operations Re-search, their construction and general meth-ods 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, sensi-tivity analysis, transportation and assign-ment problems, rectangular games, two-person zero-sum games, methods of solu-tion (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, clas-sification of states and ergodic theorems,
homogeneous continuous-time Markov
chains, Poisson process, elements of queu-ing theory, M/M/1, M/M/K, G/M/1 and M/G/1
queues.
Solution of statistical problems on comput-ers using well-known statistical software
packages like SPSS.
3. Quantitative Economics and Official
Statistics:
Determination of trend, seasonal and cy-clical components, Box-Jenkins method,
tests for stationary series, ARIMA models
and determination of orders of
autoregressive and moving average com-ponents, 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, in-dex number of wholesale prices, consumer
prices, agricultural production and indus-trial production, test for index numbers -proportionality, time-reversal, factor-rever-sal and circular .
General linear model, ordinary least
square and generalized least squares
methods of estimation, problem of
multicollinearity, consequences and solu-tions of multicollinearity, autocorrelation
and its consequences, heteroscedasticity
of disturbances and its testing, test for in-dependence 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, indus-trial production, trade and prices, methods
of collection of official statistics, their reli-ability 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, registra-tion, NSS other surveys, their limitations and
uses, definition, construction and uses of
vital rates and ratios, measures of fertility,
reproduction rates, morbidity rate, standard-ized 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 de-mographic parameters, standard classifica-tion by cause of death, health surveys and
use of hospital statistics.
Methods of standardisation of scales and
tests, Z-scores, standard scores, T-scores,
percentile scores, intelligence quotient and
its measurement and uses, validity and
reliability of test scores and its determina-tion, use of factor analysis and path analy-sis in psychometry.
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