University
of Calcutta
Admission to the Ph.D. Programme in Statistics: 2014
Procedure:
1. The conditions for eligibility will be guided
overall by the rules specified in the notification titled University of
Calcutta (Regulations for the Degree of Doctor of Philosophy, Ph.D.),
Regulations 2009
2. Eligibility: Candidates with an M.Sc. or
equivalent degree in Statistics and allied subjects from any UGC recognized
University/Institute are eligible to apply for admission in the Ph.D.
programme.
3. The admission procedure consists of a written
test followed by an interview for candidates successful in the same. Those who
have qualified NET / SET (Mathematical Sciences) / GATE (Mathematics) or hold a
UGC Teachers’ Fellowship or already obtained M.Phil. in Statistics and allied
subjects or M. Tech. (QR&OR) degree of ISI prior to the application
deadline will be exempted from the written examination but will have to appear
in the interview.
4. The written test is common to the Ph.D. and
M.Phil. programmes for the year 2014. A candidate who qualifies for the Ph.D.
programme is automatically eligible to join the M. Phil. programme instead, if
he/she chooses. However, for exercising this choice, such a candidate must have
applied also for the M. Phil. programme by the relevant deadline.
The two programmes cannot be pursued simultaneously.
5. Number of seats in the Ph.D. programme: 58
6. Reservations will be applicable as per existing
rules.
Date of Advertisement
:
Last date of submission of application form
: September 26, 2014
Date of common written test :
October 14, 2014 (12 noon ‐ 2 p.m.)
Result of common written
test : October
21, 2014
Date of Interview
: November 05, 2014 (from 12 noon)
Date of publication of selection list
: November 17, 2014
Please note that candidates who are eligible for
waiver of the written test are also required to complete and submit the
application form by the above deadline.
Course
Work (PhD Programme):
One Semester Course Work of 20 credits as follows:
1. Literature Review and seminar : 4 credits
2. Seminar Presentation : 4 credits
3. Research Methodology : 4 credits
4. Evolution of Statistics : 4 credits
5. Statistical Computing : 4 credits
Structure of the written examination:
1. There will be 25 multiple choice questions each
carrying 2.5 marks out of which one has to answer 20 questions. If a candidate
answers more than 20, only the first 20 answered will be evaluated.
2. There will be 15 short answer type questions of
5 marks each out of which one has to answer 10 questions. If a candidate
attempts more than 10, only the first 10 attempted will be evaluated.
3. Pass marks for the test will be announced in due
course.
4. Candidates successful in the written examination
would have to compete with other eligible candidates who have already cleared
NET / SET / GATE / M. Phil / M. Tech. (QR&OR) at the interview stage. The
list of finally selected candidates would be posted in the University website
and Departmental Notice Board.
5. Candidates selected for the final interview will
be required to specify his/her areas of interest by a specified date before the
interview. However, the selection committee may, at its discretion, require a
candidate to opt for a topic/area other than his/her initial choice before
admitting him/her into the Ph.D. programme. The final date for submitting the brief
will be announced along with the intimation for the interview.
Detailed
Syllabus for common M.Phil-PhD Entrance Examination:
Real Analysis
Real number system, cluster points of sets, closed,
open and compact sets, Bolzano-Weierstrass property,
Heine-Borel property. Sets of real vectors.
Sequences and series, convergence. Real valued functions.
Limit, continuity and uniform continuity.
Differentiability of univariate functions. Mean value theorems.
Extrema of functions. Riemann integrals. Improper
integrals. Sequences and series of functions, uniform
convergence. Power series. term by term
differentiation and integration,
Probability
Fields, sigma-fields and generators, semifields,
Borel sigma-field on R and R^k. Monotone classes,
Measurable functions and properties, compositions;
product sigma-fields, Borel sigma-field on Euclidean
spaces.
Measures, finite, sigma-finite measures.
Probability measures, properties. Independence of events, Borel Cantelli
lemmas.
Extensions of measures, Lebesgue measure on R and
R^k.
induced measures. Random variables, Distribution
functions, measures in R and R^k. Probability distributions. Discrete and
absolutely continuous distributions. probability densities.
Convergence in probability and almost sure.
Integration: simple, nonnegative, general
measurable functions, integrability, MCT, DCT, Fatou's lemma.
Change of variables. Holder's and Minkowski's
inequalities. Expectations, moments. Jensen's inequality.
Product measures. Fubini's theorem.
Independence of random variables. Sums, variances,
covariances. Second Borel-Cantelli lemma.
Kolmogorov's 0-1 law. Weak and strong laws of large
numbers. Kolmogorov's inequality.
Convergence in distribution. Integration of
complex-valued functions, characteristic functions. Inversion
and Continuity theorems. Central Limit Theorems.
Lp-convergence of random variables. Connections
between various modes of convergence (in distribution, in probability, L_p,
almost sure).
Absolute continuity and singularity of measures.
Radon-Nikodym theorem (statement).
Linear Algebra and Linear Programming
Vectors and Matrices: Vector spaces and subspaces,
Linear dependence and independence, span, basis,
orthogonality and orthonormality,
Matrix algebra
Linear programming: Graphical Solution and Simplex
Algorithm
Sampling Distributions
Non-central χ2
, t & F distributions – definitions and
properties.Distribution of quadratic forms Cochran’s theorem.
Large Sample Theory
Scheffe's theorem, Slutsky's theorem. Asymptotic
normality, multivariate CLTs, delta method. Glivenko Cantelli Lemma
Asymptotic distributions of sample moments and
functions of moments, Asymptotic distributions of Order Statistics and
Quantiles. Consistency and Asymptotic Efficiency of Estimators, Large sample properties
of Maximum Likelihood estimators. Asymptotic distributions and properties of
Likelihood ratio tests, Rao’s test and Wald’s tests in the simple hypothesis
case.
Statistical Inference
Sufficiency & completeness, Notions of minimal
sufficiency,bounded completeness and ancillarity, Exponential family.Point
estimation : Bhattacharya system of lower bounds to variance of estimators.
Minimum variance unbiased estimators – Applications
of Rao – Blackwell and Lehmann – Scheffe theorems.Testing of Hypothesis :
nonrandomized and randomized tests, critical function, power function.
MP tests – Neyman – Pearson Lemma. UMP tests.
Monotone Likelihood Ratio families. Generalized Neyman – Pearson Lemma. UMPU
tests for one parameter families. Locally best tests. Similar tests. Neyman
structure. UMPU tests for composite hypotheses.
Confidence sets: relation with hypothesis testing.
Optimum parametric confidence intervals. Sequential
tests. Wald’s equation for ASN. SPRT and its
properties – fundamental identity. O.C. and ASN.
Optimality of SPRT (under usual approximation).
Linear Models
Gauss Markov Model: Estimable function, error
function, BLUE, Gauss Markov theorem. Correlated set up, least squares estimate
with restriction on parameters.
Linear Set, General linear hypothesis –related
sampling distributions, Multiple comparison techniques
due to Scheffe and Tukey.
Analysis of variance: Balanced classification,
Fixed Effects Model, Random Effects Model and Mixed Effects Model; Inference on
Variance components.
Regression analysis, Analysis of covariance.
Regression Analysis
Building a regression model: Transformations –
Box-Cox model, Stepwise regression, Model selection (adjusted R2, cross
validation and Mallow's Cp criteria, AIC and BIC), Multicollinearity.
Detection of outliers and influential observations:
residuals and leverages, DFBETA, DFFIT and Cook’s Distance.
Checking for normality: Q-Q plots, Normal
Probability plot, Shapiro-Wilks test.
Departures from the Gauss-Markov set-up:
Heteroscedasticity and Autocorrelation – detection and remedies.
Longitudinal Data Analysis – introduction with
motivation. Exploring longitudinal data. Linear models for longitudinal data
–introduction, mean models, covariance models, mixed effects models.
Predictions.
Types of data. Two-way classified data –
Contingency Tables and associated distributions, Types of studies, Relative
Risk and Odds Ratio and their properties. More-than-two-way classified data –
partial
associations, marginal and conditional odds.
Generalized Linear Models: Introduction, Components
of a GLM, Goodness of fit – deviance, Residuals,
Maximum likelihood estimation.
Binary data and Count data: ungrouped and grouped.
Polytomous data.
Over dispersion, Quasi-likelihood.
Models with constant coefficient of variation,
joint modeling of mean and variance, Generalized additive
models.
Discrete longitudinal data - generalized linear
marginal models, GEE for marginal models, Generalized linear subject specific
models and transition models.
Design of Experiments
Block Designs: Connectedness, Orthogonality,
Balance and Efficiency; Resolvable designs; Properties of
BIB designs, Designs derived from BIB designs.
Intrablock
analysis of BIB, Lattice and PBIB designs, Row column designs, Youden Square
designs;
Recovery of inter-block information in BIB designs;
Missing plot technique.
Construction of mutually orthogonal Latin Squares
(MOLS); Construction of BIB designs through MOLS
and Bose’s fundamental method of differences.
Factorial designs: Analysis, confounding and
balancing in symmetric factorials.
Sample Surveys
Probability sampling from a finite population –
notions of sampling design, sampling scheme, inclusion probabilites, Horvitz-Thompson
estimator of a population total. Basic sampling schemes – simple random
sampling with and without replacement, unequal
probability sampling with and without replacement, systematic sampling. Related
estimators of population total/mean, their variances and variance estimators
– mean per distinct unit in simple random with
replacement sampling, Hansen-Hurwitz estimator in unequal probability sampling
with replacement, Des Raj and Murthy’s estimator (for sample of size two) in
unequal probability sampling without replacement.Stratified sampling –
Allocation problem and construction of strata. Ratio, product, difference and
regression estimators. Unbiased ratio estimators probability proportional to
aggregate size sampling, Hartley – Ross estimator in simple random sampling.
sampling and sub-sampling of clusters. Two-stage
sampling with equal/unequal number of second stage
units and simple random sampling without
replacement / unequal probability sampling with replacement
at first stage, ratio estimation in two-stage
sampling. Double sampling for stratification. Double sampling
ratio and regression estimators. Sampling on
successive occasions.
Bayesian Analysis
Different priors and related posteriors
Estimation, testing and prediction for univariate
normal distribution with known/unknown mean and/or
variance.
Hierarchical and Empirical Bayes under normal
setup.
Prior and posterior analysis in generalized linear
models
Decision Theory
Risk function, Admissibility of decision rules,
Complete, essentially complete, minimal complete and minimal essentially
complete classes. Essential completeness and completeness of class of rules
based on
sufficient statistic and the class of nonrandomized
rules for convex loss
Resampling Techniques
Empirical distribution function and its properties
Jackknife and Bootstrap procedures for estimating
bias and standard error.
Consistency of the Jackknife variance estimate in
an iid setup.
Bootstrap confidence intervals.
Stochastic Processes
Poisson process. Renewal Theory: renewal processes,
renewal function, elementary renewal theorem,
applications, Blackwell's theorem and key renewal
theorem (statements), applications, alternating renewal processes, applications
to limiting excess and age.
Markov chains: time-homogeneity, one-step &
multi-step transition probabilities, Chapman-Kolmogorov
equations, Markov times, strong Markov property,
classification of states, stationary distributions,
periodicity, ergodicity, convergence, convergence
rate. Examples: birth-and-death processes, branching
processes.
Jump-Markov processes: conservativeness, transition
probabilities, holding times, embedded Markov
chain, Chapman-Kolmogorov equations, Kolmogorov backward
and forward equations, stationary
distributions. Examples: pure birth,
birth-and-death chains, Markovian queues.
Time Series Analysis
Stationary time series. Autocorrelation and partial
autocorrelation functions. Correlogram. Box-Jenkins
Models – identification, estimation and diagnostic
checking.
Volatility – ARCH, GARCH models.
Multivariate Analysis:
Multivariate normal distribution and its
properties- marginal and conditional distributions. Random
sampling from a multivariate normal distribution-
UMVUE and MLE of parameters, joint distribution of
sample mean vector and SS-SP matrix; Wishart
distribution and its properties. Distribution of sample
correlation coefficients, partial and multiple
correlation coefficients, partial regression coefficient and
intraclass correlation coefficient. Distributions
of Hotelling’s T 2 and Mahalanobis’ D 2 statistics- their
applications in testing and confidence set
construction. Multivariate linear model, MANOVA for one-way
and two-way classified data.
Applied Multivariate Analysis
Clustering: Hierarchical clustering for continuous
and categorical data- different choices of proximity
measures, Agglomerative and divisive
algorithms.K-means clustering- optimum choice of the number of clusters.
Classification and discrimination procedures:
Discrimination between two known populations
