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{\bf The list of courses taken at the Indian
Statistical Institute(Calcutta) in the B.Stat (Hons..) and M.Stat.
programmes by Mr/Ms XYZ}
\end{center}
\vspace{.1in}
{\bf List of Courses taken in Bachelor of Statistics (Honors) Programme :}
{\bf B1 FIRST YEAR}
{\bf B1.1 Calculus I (Dr. Ashoke K. Roy) :}
Real numbers . Functions . Sequences . Limits . Limsup and Liminf
. Series . Tests of convergence for sequences and series, Absolute
convergence, rearrangement of terms . Cauchy sequences .Continuity .
Differentiation . Chain rule . Rolle's theorem . Mean value theorem .
Higher order derivatives. Leibnitz's formula . Taylor series expansion
L'Hospital's rule . Maxima and minima of functions . Integral
Calculus : Riemann integration .Riemann integrable functions on closed
intervals . Fundamental theorems of calculus . Computation of definite
integrals .
{\bf B1.2 Probability Theory and Its Applications I (Dr. B. V. Rao) :}
Elementary concepts : experiments, outcomes, sample spaces,
events . Coin tossing experiments . Discrete sample space .
Combinatorial probability, Composite experiments, Conditional
probability, Bayes theorem, independence, urn models . Random
variables, p.m.f and c.d.f for discrete random variables . Examples of
standard discrete random variables : binomial, geometric, Poisson,
negative binomial, hypergeometric etc.Stirling's formula.
Expectation,mean,variance. Moments and moment generating Functions.
Probability generating functions . Median and quartiles . Joint
distribution of discrete random variables . Functions of discrete
random variables . Symmetric random walk in one dimension .
{\bf B1.3 Vectors and Matrices I (Dr. P. S. S. N. V. P. Rao) :}
Groups, rings and fields . Vector space over an arbitrary field .
Real vector space : Subspaces, linear independence, basis, dimension,
sum and intersection of subspaces . Linear transformations :
representation in terms of matrices . Sum and product of matrices,
partitioned matrices . Rank, trace, elementary operations, canonical
reductions, inverse of nonsingular matrices, sweep out method . Linear
systems of equations : homogenous and nonhomogeneous systems, solution
space, consistency and general solution, numerical examples . Singular
matrices . generalized inverse of matrices : properties, applications
{\bf B1.4 Statistical Methods I (Dr. Probal Chaudhuri) :}
Types of investigation and collection of data . Types of
observations . Classification and tabulation of data . Summarization
of univariate data . Simple linear regression : introduction.
Logistic regression . Mean absolute deviation regression ( Gauss'
method ) . Weighted least square . Iteratively reweighted least
squares . BLUE : examples in linear regression and standard univariate
cases . Bivariate data; summarization and quantitative measures.
Bivariate medians : Spatial medians,Oja's median, Liu's median.
Tukey's concept of depth and related concept of median . Examples of
uses of statistics : a) Linear discrimination in Fisher's Irish data,
b) Clinical trials .
{\bf B1.5 Computational Techniques and Programming I (Dr. S. K. Pal) :}
Introduction to algorithms . Computers : structure and
characteristics, storage, logic and program. Basic computer operations.
Algorithms and flow charts . High level languages
Syntax of FORTRAN . Programming in FORTRAN . Input-output facilities,
Computational and control instructions, labels and jump statements,
loops. Data structures . Segmentation of programs : functions and
subroutines . Local and global variables . Program structure and
debugging . Files as auxiliary storage medium . File handling . Use of
subroutine libraries like NAG .
{\bf B1.6 Calculus II (Dr. Ashoke K. Roy) :}
Improper integrals . Sequences and series of functions . Double
sequence. Introduction to point set topology : Metric
spaces,definition of limits of sequences and functions . Continuity.
Compactness, Heine-Borel theorem . Cantor intersection theorem .
Properties of complete metric spaces. Uniform continuity : related
results .
Pointwise and uniform convergence . Term by term differentiation
and integration . Stone-Weierstrass approximation theorem .
Calculus of several variables . Continuity . Concept of
differentiability : directional derivatives . Partial derivatives. .
Frechet differentiability.
Chain rule . Jacobian .
Curves and surfaces . Arc length : rectifiable curves .
Reparametrization of curves . Tangents to curves, Velocity . Concept
of curvature . Tangent planes to surfaces. Fundamental vector product.
Points of
extrema and their calculation . Lagrange multiplier technique.
Taylor series expansion for several variables . Inverse function
theorem and implicit function theorem .
Line Integrals
. Reparametrization . Line integral with respect
to arc length .
{\bf B1.7 Probability theory and its applications II (Dr. B. V. Rao) :}
Univariate continuous distributions : uniform, beta, gamma
exponential, Cauchy, normal, lognormal . Normal distribution
properties. Expectation of a continuous random variable, mean
variance, moments, m.g.f., moments of standard distributions.
Functions of a random variable . Chi-squared distribution .
Bivariate continuous distribution : properties . Conditional and
marginal distributions . Expectation and conditional expectation.
Regression, correlation . Example : bivariate normal distribution.
Introduction to multivariate continuous distributions. Dirichlet
distribution .
{\bf B1.8 Vectors and matrices II ( Dr . P S S N V P Rao) :}
Determinant of matrices : definition and properties . Trace of a
matrix . Quadratic forms : simultaneous reduction . Fisher-Cochran
theorem for matrices . Inner product and norms . Projection operators
. Solution of characteristic equation : characteristic roots and
vectors . Cayley- Hamilton theorem . Spectral decomposition of
symmetric matrices . Singular value decomposition . Jacobi's method
for spectral decomposition . Special forms of g-inverses . Jordan
canonical form. Applications to statistics . Computational Aspects .
{\bf B1.9 Statistical Methods II (Dr. T. Krishnan) :}
Gauss' law of error . Introduction to concept of likelihood .
Methods of estimation : Method of moment, maximum likelihood estimator
. Criteria of estimation : Unbiasedness, UMVUE, consistency .
Cramer-Rao lower bound . Testing of hypotheses . Null and alternative
hypotheses. Two kinds of error. Size and power of a test . Simple and
composite hypotheses . Neyman-Pearson lemma .Most powerful and
uniformly most powerful test : simple examples. Confidence intervals
.Various sampling distributions . Derivation of t-test for normal
equality of mean case . Simulation of probability sampling
distributions. Multivariate data : partial and multiple correlation,
regression. Introduction to Bayes decision rules and Bayes estimation.
{\bf B1.10 Computational Techniques and Programming II (Dr S K Pal) :}
Finite computational processes . Propagation of errors .
Approximation of infinite series . Solution of nonlinear equations :
Newton-Raphson, Gauss- Siedel and Gauss-Newton techniques.
Computational linear algebra : solution of linear equations,
calculation of inverse and spectral decomposition of matrices.
Householder transformation and QR decomposition of matrices.
Interpolation : finite difference, divided difference, Lagrangian
techniques . Inverse interpolation . Numerical differentiation .
integration . Difference equations : stable and unstable techniques.
{\bf B2 SECOND YEAR }
{\bf B2.1 Calculus III (Dr. Ashoke K. Roy) :}
Multiple integrals . Change of variable formula . Surface
integral . Definition of curl, divergence and wedge product .Green's
theorem for simply connected and multiply connected regions . Stokes'
theorem . Gauss' divergence theorem .
Infinite products . Artin's characterization of gamma function
Introduction to Fourier transformation and Fourier series
Fourier coefficients . Fejer's kernel . Fejer's theorem, Abel's
theorem .
{\bf B2.2 Probability theory and its applications III (Dr. B. V. Rao) :}
Properties of Multivariate normal : conditional and marginal
distributions, expectation and conditional expectation, regression.
Multivariate distributions . Distribution of order statistic
Definition of expectation in the general setup. DCT, Fatou's lemma
(assuming MCT ). Development of the theory of conditional expectation
Examples. DeMoivre-Laplace Central Limit Theorem.
Various modes of convergence : convergence in probability, law,
almost everywhere, in pth moment. Slutsky's theorem. Polya's theorem,
Borel-Cantelli lemma. WLLN. SLLN( assuming existence of fourth moment.)
Characteristic functions. Inversion formula. Helly's selection
theorem. Levy's continuity theorem. Levy's CLT. Multivariate CLT.
{\bf B2.3 Statistical methods III (Dr. S. Bendre) :}
Criteria and methods of estimation . Principle of data reduction.
. Concept of sufficiency, minimal sufficiency. Neyman factorization
theorem . Exponential family of distributions.
Natural parameter
space . MLE for exponential family . Complete sufficient statistics
Cramer-Rao lower bound and Rao-Blackwell theorems . Computation of
UMVUE . Ancillary statistics, Basu's theorem .
Testing of Hypotheses . Neyman-Pearson lemma and Likelihood ratio
test . Two sample and paired t-tests . Behrens-Fisher problem .
Conditional tests . Combination of tests . Confidence intervals,
criteria of goodness .
Large sample tests and confidence intervals . Associated sampling
distributions, central and non-central t,central and non-central F.
Logit and Probit analysis . Fisher scoring method .
{\bf B2.4 Introduction to anthropology and Human Genetics
(Dr. P. DasGupta and Dr. P. Bharati)}
Anthropology : Definition, scope. Bio-cultural evolution of man.
Man as a social animal. Human variation and adaptation to environment;
causes of variation; climatic, biotic and socio-cultural environments.
Human somatic and germ cells; human chromosomes; Mendelian
genetics; basic concepts in genetics; inheritance; population
genetics.
{\bf B2.5 Economics I (Dr. S. R. Chakraborty and Dr. A. Sarkar) :}
Microeconomics : Demand and supply. Theory of the consumer.
Theory of production. Theory of market. Topics in general equilibrium.
{\bf B2.6 Elements of Algebraic Structures (Dr. K. S. Vijayan) :}
Elementary concepts : Sets and related notations, Functions,
Relations, Equivalence Relations,Integers and Fundamental Theorem of
Arithmetic.
Group : Binary operations, semi-groups, monoid. Definition of
Group, Subgroup, Cosets, Lagrange's Theorem, Fermat's Little Theorem,
Euler's Theorem. Group Homomorphism, Kernel, Normal Subgroup,
Isomorphism and Isomorphism Theorems, Quotient Group. Action of a group
on a set, Cayley's Theorem,Cauchy's Theorem, Class Equation, Conjugate
Classes, Sylow Theorems. Symmetric Groups, Alternating Groups,
Permutation Groups. Structure Theorem for finite Abelian Group.
Ring : Definition, subrings, ideals, quotient rings. Concept of
Zero Divisors, Domain, Integral Domain. Concept of divisor,g.c.d.,
prime element, unit element. Principal Ideal Domain, Euclidian Domain,
prime factorization theorem for Euclidian Domain, Ascending Chain
Condition, prime factorization for the PIDs, Unique Factorization
Domains. Maximal Ideals and Prime Ideals, fields. Ring of Gaussian
Integers. Polynomial Rings over a field, irreducible
polynomials,Eisenstein's Criterion, derivative of a polynomial.
Field : Definition, sub and quotient fields. Extension of a
field, Isomorphism of fields, root of a polynomial over a field,
Algebraic Extension, degree of extension,Algebraic Numbers,Splitting
Fields, Normal extensions. Introduction to Galois Theory.
Vector Spaces : Definitions, subspaces, quotient spaces, linear
transformations, direct sum and its universal property. Linearly
independent set, spanning set, basis. Algebraic dual of a vector
space.
Module : Definition, sub and quotient modules, homomorphism,
isomorphism theorems, irreducible modules, decomposition of a module
over a Euclidian Ring.
{\bf B2.7 Statistical Methods IV (Dr. S. Bendre) :}
Bivariate distributions : Properties; measures of association :
Spearman's rank correlation coefficient,Kendall's tau Inference on
parameters . Multivariate discrete distributions . Analysis of
contingency table . Contingency tables : Fisher's exact test . Large
sample test : Pearson's chi-square and contingency chi-square .
Multivariate normal distribution : Distribution of linear and
quadratic forms. Independence of sample mean and covariance matrix .
Estimation of parameters . Testing for mean, Hotelling's $T^2$ statistic .
Wishart distribution, definition and properties . Tests on product
moment, partial and multiple correlation.
Large sample distribution of sample correlation .
Delta method . Variance stabilizing transformations and their use
Introduction to nonparametric density estimation and
nonparametric regression based on kernels . Resampling techniques :
Jackknife and Bootstrap . Method of crossvalidation . Simulation
studies . Introduction to robust statistic .
{\bf B2.8 Economic Statistics and Official Statistics (Dr. A. Majumder) :}
Index numbers. Elements of time series analysis. Analysis of
income and allied size distributions. Demand analysis. Official
statistics.
{\bf B2.9 (a) Demography, (Dr. P. K. Majumder) :}
Need and uses of demography. Sources of demographic data. Census;
Indian census. Rates and ratios; birth, fertility, mortality. Measures
of fertility and reproduction. Direct and indirect standardization of
vital rates. Elements of life table construction and usage. Population
estimates and forecasts. Logistic law of population growth.
{\bf B2.9 (b) Statistical Quality Control and Operation Research
(Dr. A. R. Mukherjee) :}
Introduction to SQC. Control charts. Acceptance Sampling.
Illustrations and application. Introduction to OR. Introduction to
queueing theory.
{\bf 3 THIRD YEAR :}
{\bf 3.1 Linear Statistical Models (Dr. T. Krishnan) :}
Linear statistical models; illustrations. Linear estimation. Test
of linear hypotheses. Gauss-Markov theorem, Fundamental Theorems of
least square . Interaction, Tukey's 1 degree of freedom test for
nonadditivity . Repeated measures . Random effect model, Nested model
.Multiple comparisons. Linear regression. ANOVA. Analysis of
covariance. Log-linear models.
{\bf 3.2 Sample Surveys and Applications (Dr. T. P. Tripathi) :}
Scientific basis of sample surveys. Principal steps of a sample
survey; illustrations. Methods of drawing random samples. SRSWR,SRSWOR
and VPSWR; estimation of population mean,total and covariance. Sample
size determination. Stratified sampling; estimation, allocation,
illustrations. Systematic sampling, linear and circular. Variance
estimation. PPS sampling;selection and estimation. Two-stage sampling.
Cluster sampling. Nonsampling errors. Product, Ratio and Regression
methods, delta method. VPSWOR : Horvitz-Thompson estimator. Double
sampling. Use of auxiliary information in sample surveys.
Post-stratified sampling .
{\bf B3.3 Theory of Statistical Inference I (Dr. R. DasGupta) :}
Formulation of problems. Reduction of Data. Sufficiency,
Factorization Theorem (proof in continuous case), minimal sufficiency,
Ancillarity, Basu's theorem. Lehmann-Scheffe method . MLR, Exponential
and location-scale family of distributions.
Point Estimation : Criteria of goodness, m.s.e., unbiasedness,
relative efficiency, Chapman-Robbins and Cramer-Rao inequality,
Bhattacharya bounds, UMVUE, Rao-Blackwell theorem, completeness,
methods of estimation and their simple properties, Method of least
squares, weighted least squares, maximum likelihood methods
Consistency; illustrations. BAN estimators.
Test of Hypotheses : Statistical Hypothesis, simple and composite
tests with critical regions, randomized tests, generalized
Neyman-Pearson lemma, error probabilities, level, size and power of a
test; MP, UMP, UMPU, LMP, LMPU tests, illustrations, Likelihood Ratio
tests. Consistency.
Confidence Intervals : Criteria of goodness,
illustrations,relationship with tests of hypotheses.
{\bf B3.4 Differential Equations (Dr. Ashoke K. Roy) :}
First and Second Order linear differential equations with
constant and variable coefficients, homogenous equations, Bernoulli's
equation, Ricati's equation. Linear differential equation with
constant coefficient, exponential shift,method of undetermined
coefficient. Power series solutions and special functions, Legendre,
Bessel, Hypergeometric equations . System of linear differential
equations : Calculus of matrices. Existence of solution of $dx/dt=f(x,t).$
Picard's method. Peano's theorem. Calculus of variation. Euler's
differential equations.
{\bf B3.5 Introduction of Stochastic Processes (Dr. B. V. Rao) :}
Renewal Theory, Random Walk; Applications. Discrete Markov chains
with countable state space. Stationary distribution. Limit theorems.
Ratio-Limit theorems; Markov pure jump processes. Poisson Process.
Birth and Death processes Applications.
{\bf B3.6 Design of Experiments (Dr. G. M. Saha) :}
Need for designs, fundamental principles, blocks and plots;
illustrations. Uniformity trials. Basic design; Completely Randomized,
Randomized Block, Complete Latin Square, Repeated Latin Square
Graeco-Latin Square; description, appropriateness,advantages
analysis, illustrations.Multiple comparison methods. Euler's
conjecture and results of Bose, Parker and Srikhande . Orthogonality
of classification of two and higher order layouts .ANOVA and ANCOVA
in the context of design. Missing plot techniques. Elements of
Split-Plot and Strip-Plot designs. Factorial designs; main effects
interactions, confounding; $2^n, 3^2,3^3$ and 3 series. $(2^n,2^k)$
confounded
designs, confounding in 3 series,Asymmetric designs .
{\bf B3.7 Theory of Statistical Inference II
(Dr. Jayanta K. Ghosh and Dr. S. Ghoshal) :}
Introduction to Nonparametric Methods and Inference : Formulation
of the problems. Order statistics and their distributions. Tests and
confidence intervals for population quantiles. Sign test. Test for
symmetry. Wilcoxon-Mann-Whitney test. Run test. Tests for
independence. Concepts of relative asymptotic efficiency. Estimation
of location and scale parameters. U-statistics,V-Statistics,
L-Statistics,R-Statistics.Nonparametric density estimation :
Histogram,Kernel estimator,nearest neighbour,splines. Nonparametric
regression . Resampling techniques : Bootstrap and Jackknife.
Elements of Sequential Analysis : Need for sequential tests.
Stein's test. Stop time,Wald's equations,Wald's fundamental
identity.Wald's SPRT, ASN, OC function. Optimality of
SPRT.Illustration for binomial and normal distributions. Elements of
sequential estimation. Bayesian sequential analysis . Stopping time
principle and dynamic programming : secretary problem.
{\bf B3.8 Statistics Comprehensive (Dr. T. Krishnan and Dr. S. Bendre) :}
Brief review of historical development of statistical theory and
methods. Problem sessions on statistical theory, methods and
applications . Robust statistics : influence function, L and M
estimation . Distance sampling . EM algorithm . Tests for normality :
Skewness-Kurtosis test, Shapiro-Wilks test,chi-squared test . Power
series distribution . Distribution of extremal values : Gumbel
distribution. Independence of quadratic forms of normal random
variables . Generalized linear models . Data analysis.
{\bf B3.9 Physics I (Dr. D. Bannerjee and Dr. G. Goswami) :}
Thermodynamics. Modern physics : atomic structure, basic nuclear
physics. Classical mechanics : Lagrangian and Hamiltonian formulation.
Relativistic mechanics.
\begin{center}
{\bf List of Courses taken in Master of Statistics Programme}
\end{center}
{\bf M1 FIRST YEAR :}
{\bf M1.1 Regression Techniques (Dr. T. Krishnan) :}
Regression and outliers. Regression and collinearity. Ridge
regression. Subset selection in regression. Graphical techninques in
regression. Missing values in regression. Non-parametric regression.
Non-linear regression. Generalized Linear Models. Robust Regression.
Computational techniques in regression. Bootstrap methods in
regression. Growth curves. Quantile regression.
{\bf M1.2 Multivariate Statistical Analysis (Dr. Ayanendranath Basu) :}
Review of multivariate distributions; correlations and
regression. Multivariate normal distribution; properties; distribution
of linear and quadratic forms; distribution of sample mean vector and
covariance matrix; Wishart distribution; inference on mean vector,
product-moment correlation, partial correlation, multiple correlation,
and distribution of related test statistics along with associated
confidence regions; two-sample problem and discriminant analysis;
MANOVA.Use of S-PLUS software .
{\bf M1.3 Probability Theory II (Dr. T. Chandra) :}
Fields and $\sigma$-fields. Monotone class theorem.
Dynkin's $ \pi - \lambda $ theorem. Measures. Caratheodory extension and
outer
measures. Integration. Change of variable formula. Borel fields and
probabilities on Borel fields. Probability spaces. Random variables.
Expectation. MCT. Fatou's lemma. DCT. Scheffe's theorem. Essential
supremum. Radon-Nikodym theorem. Jordan, Hahn and Lebesgue
decomposition. Independence. Product spaces. Fubini's theorem.
Discrete and continuous distributions on R. Uniform integrability.
Various modes of convergences : in law, in probability, almost
everywhere, in p-th mean and their relations.Polya's theorem,
Slutsky's theorem, Extended Scheffe's theorem, Riesz-Fischer theorem .
Skorohod representation of R , Borel-Cantelli lemmas,Erdos-Renyi
theorem,Renyi-Lamperti lemma .Subgrobabilty measures,Vague
convergence,Helly-Bray lemma,characteristic functions. Inversion
formula.Levy's continuity theorem . Helly's selection theorem .
WLLN : Bernstein,Khintchin,Markov,Kolmogorov-Feller.
SLLN : Kolmogorov,Borel,Rajchman.
CLT : Levy, Liapunov, Lindeberg-Feller.
Glivenko-Cantelli theorem.Kolmogorov 3-series theorem,Liapounov
inequality, $C_r$-inequality of Loeve,Kolmogorov and Hajek-Renyi
inequality . Tail $\sigma$-fields, Kolmogorov O-1 law.
{\bf M1.4 Elements of Statistical Decision Theory and Statistical Inference
(Dr. Probal Chaudhuri) :}
Formulation of a statistical decision problem with illustrations.
Bayes rules, admissible rules, minimax rules, complete class, minimal
complete class. Detailed analysis when nature's action space is
finite; geometric interpretation,separating hyperplane theorem;
minimax theorem; complete class theorem. Illustration of Bayes and
minimax rules. Convex loss function; Rao-Blackwell theorem. Unbiased
tests and similar region, Neyman structure. Elements of invariant
rules. Invariant and maximal invariant statistics. Invariant tests.
UMPI tests. Reduction of data using sufficiency and invariance.
Equivariant estimates. Pitman estimates. EM algorithm. Kernel density
estimation. Kernel regression estimates. Curse of dimensionality.
Generalized additive model.
{\bf M1.5 Complex Analysis (Dr. Ashoke K. Roy) :}
Limits and infinite series with complex terms. Power series,
domain of convergence. Continuity. Complex derivatives.Analytic
functions. Cauchy-Riemann equations. Complex line integrals. Total
variation. Cauchy's theorem. Cauchy's integral theorem. Zeros of
analytic function.Entire Functions.Liouville's Theorem. Proof of
Fundamental Theorem of Algebra. The Maximum Modulus Principle.Laurent
expansion of an analytic function . Open mapping theorem for analytic
function. Poles and meromorphic functions. Classification of
singularities .Calculus of residues. Contour integration;
illustrations. Rouche's theorem . Uniform Convergence on Compacta.
Normal functions. Montel's Theorem . Extended Complex Plane, Concept
of pole at infinity. Conformal Mappings. Mobius Transformations,
Schwartz's Lemma. Harmonic, Sub-Harmonic and Super-Harmonic functions,
Maximum Principle, Poisson Kernel,Poisson Summation Formula,Solution
of Dirichlet Problem . Riemann Mapping Theorem. Simple notion of
patching and extensions of analytic functions.
{\bf M1.6 Applied Stochastic Processes (Dr. Anup Dewanji) :}
Markov chains, Markov processes in continuous time, birth and
death processes, population growth processes, migration,
multidimensional processes, mutation in bacteria, epidemic processes,
simple diffusion processes, queueing processes.
{\bf M1.7 Large Sample Statistical Methods
(Dr. Jayanta K. Ghosh and Dr. S. Ghoshal) :}
Review of various models of convergence of random variables. CLT,
Scheffe's theorem. Polya's theorem and Slutsky's theorem. Portmanteau
theorem. Edgeworth expansion. Asymptotic distribution of function of
sample moments, sample quantiles. Order statistics and their
functions. Bahadur's theorem. Tests on correlation coefficients.
Properties of maximum likelihood estimators. Results of Bahadur and Le
Cam. Pearson chi-square, contingency chi-square, likelihood ratio,
Rao's statistics, Wald's statistics. Large sample non-parametric
inference. Asymptotic efficiency. Asymptotic distribution of posterior
distribution, Bayesian CLT.
{\bf M1.8 Time Series Analysis (Dr. T. Krishnan) :}
Discrete parameter stochastic processes, Kolmogorov's consistency
theorem; strong and weak stationarity; autocovariance and
autocorrelation. Moving average, autoregressive, autoregressive moving
average and ARIMA processes. Box-Jenkin's models. Estimation of the
parameters in ARIMA models; forecasting. Residuals and diagnostic
checking. Use of computer packages. Spectral analysis of weakly
stationary processes. Periodogram and correlogram analysis. Fast
Fourier transforms. State-space modelling. Kalman filter.
{\bf M1.9 Abstract Algebra (Dr. K. S. Vijayan) :}
Modules : left and right modules, sub and quotient modules,
module homomorphism, isomorphism theorems,free modules, free abelian
group, external and internal direct sum of modules and their universal
properties, Noetherian and Aritinian rings and modules, Hilbert Basis
Theorem, Torsion and Torsion free modules, Modules over PID,
Decomposition of Finitely Generated module over PID, Structure Theorem
for finite Abelian Groups. Irreducible and Completely reducible
modules. Nakayama's Lemma. Tensor Product of modules over commutative
rings, universal property, tensor product of free modules and vector
spaces.
Field : Algebraic Extension; Algebraic closure, existence and
uniqueness, separability, degree of separability, separable fields.
Splitting fields, Normal extensions. Galois Fields and Galois
Extensions. Galois Theory. Proof of Fundamental Theorem of Algebra.
Semisimple Rings : Concept of radical of a ring,semisimplicity,
Wedderburn Theorem for semisimple rings.
Concept of Group-Rings and simple concepts
of representations of groups.
Wedderburn Theorem on finite division rings. Jordan-Holder Theorem
for groups; introduction to lattices and Jordan- Holder-Dedekind
Theorem for lattices. Solvability of groups and solvability by
radicals.
{\bf M1.10 Optimization Techniques I (Dr. H. Sarbadhikari) :}
Lagrange method of multipliers. Maxima and minima of functions
of several variables. Elements of linear programming. Notion of
duality and related results. Applications. Max-flow-min-cut theorem.
Allocation problem. Matching problem. Application in Latin square.
{\bf M2 SECOND YEAR}
{\bf M2.1 Advanced Probability I (Dr. B. V. Rao) :}
Radon-Nikodym theorem. Hahn, Jordan and Lebesgue's decomposition
theorems. Conditional expectation. Regular conditional probability.
Bahadur, Pfanzagal, Burkholder's characterization of conditional
probabilities. Measure theoretic development of the concept of
sufficiency. Counterexamples due to Burkholder and others.
Finite and infinite products. Kolmogorov's consistency theorem.
Ionescu-Tulcea theorem.
Discrete parameter martingales. Martingale Convergence theorem,
Doob-Meyer decomposition. Various applications including Hewitt-Savage
O-1 law, Law of Large Numbers of U-statistics, DeFinneti's
Representation Theorem, conditional Borel-Cantelli's lemma.
Introduction to continuous parameter martingales and their path
properties. Martingale transforms. Applications to square functions;
Burkholder's inequalities.
{\bf M2.2 Functional Analysis (Dr. Ashoke K. Roy) :}
Topological vector spaces (tvs). Seminorms,Minkowski
functional.Normability of locally convex tvs . $l_p ,L_p$ spaces and their
duals,complex measures,Riesz representation theorem. Complete
metrizable tvs .Three principles : Hahn-Banach, Equicontinuity and
open mapping theorem . Closed graph theorem .
Locally convex spaces . Separation theorems.Dual systems and weak
topologies. Polar calculus.Banach-Alaoglu theorem .Mackey topology .
Strong topology. Adjoints,topological supplements,Extremal structure
of compact convex sets .
Frechet and Banach spaces. Reflexive spaces . Weak and weak-*
topology of Banach space . Bounded weak-* topology on Frechet spaces.
Krein-Smulyan theorem. Closed range theorem.Weak,strong and uniform
operator topologies .Compact operators on Banach spaces .
Hilbert space . Cauchy-Schwartz inequality,parallelogram law,
Polarization identity. Projection theorem . Self adjoint,
normal,unitary operators . Orthonormal basis . Spectral theorem for
finite dimensional spaces . Spectral measure. Spectral theorem for
self adjoins and normal operator .
{\bf M2.3 Theory of Games and Statistical Decisions (Dr. Arup Bose) :}
Introduction to theory of two-person zero-sum games.
Sequential Decision Theory: Invariant SPRT and their termination
probabilities. Sufficiency and invariance in sequential analysis.
Stopping time of invariant SPRT. Stopping time. Principle of backward
induction, monotone case, extended stopping time.
Sufficiency : sufficient -fields,results of Halmos-Savage and
Bahadur . minimal sufficiency, complete sufficiency,ancillarity.
Invariance : Invariant decision rules,equivariant
estimation, invariant tests.Hunt-Stein theorem.Results of
Hall-Weisman-Ghosh.
{\bf M2.4 Pattern Recognition (Dr. T. Krishnan) :}
Decision-theoretic analysis. Study of risk function, and
conditional risk function given the training sample.
Multivariate normal distributions : classification problem and
optimal rules, Fisher's linear and quadratic discrimination. Study of
error rates. Asymptotic results.
Logistic discrimination. Estimation of parameters. Mixture
distribution. Hidden Markov models. Nonparametric rules. Nearest
neighbour rules. Adaptive rules.Classification trees . Image
processing. Markov random field. Cluster analysis. Dendogram. K-means.
Projection pursuit.
{\bf M2.5 Topology and Set Theory (Dr. Amartya Dutta) :}
Topological spaces, continuity, countability axioms. Subspaces,
products, quotients. Separation properties including Urysohn and
Tietze's extension theorem. Compactness, local compactness,
Tychonoff's theorem; one-point and Stone-Cech compactification.
Connectedness, local connectedness, path connectedness, local path
connectedness. Covering axioms. Metrization theorem of Urysohn.
Completeness. Definition and examples of manifolds. Introduction to
topological groups: Definition, examples and basic properties.
{\bf M2.6 Stochastic Processes I (Dr. B. V. Rao) :}
Weak convergence of probability measures on polish spaces
including C[0,1] Brownian motion : Construction, simple properties of
paths. Poisson processes. Stationary processes. Markov processes and
generators.
{\bf M2.7 Advanced Statistical Inference
(Dr. Jayanta K. Ghosh and Dr. Arup Bose) :}
Advanced topics from classical inference.
Foundation of Statistics : Coherence, Bayesian analysis and
Likelihood principle.
{\bf M2.8 Advanced Algebra
(Dr. Amartya K. Dutta) :}
Topics from Commutative Algebra :
Ideals, prime ideals, maximal ideals, nil-radical and Jacobson
radical, operations of ideals, extension and contraction. Rings and
modules of fractions, local properties, extended and contracted ideals
in rings of fractions. Primary decomposition . Chain conditions,
Noetherian rings, Artin rings, Discrete valuation rings and Dedekind
domains.
{\bf M2.9 Bayesian Inference (Dr. Jayanta K. Ghosh) :}
Bayesian nonparametrics
{\bf M2.10 Graph Theory and Combinatorics (Dr. S. B. Rao
and Dr. P. Sarkar) :}
Definitions. data structures. pigeon hole principle. Applications.
Ramsey numbers. Partitioning integers. Generating functions.
Euler's theorem. Eulerian
and hamiltonian graphs. Tournaments. Subgraphs and restrictions.
Trees. Minimal spanning tree. Kruskal's algorithm and its proof.
Prim's algorithm and its proof. Application to computer science.
Fleury's algorithm. Graph isomorphism. Degree sequences. Havel-Hakimi
theorem. Erd\"{o}s-Gallai theorem. Dirac's theorem. Closure of a graph.
Travelling salesman problem. Matching. Tutte's theorem. Applications :
restricted personnel assignment problem, system of distinct
representatives. Graph coloring. 5-color theorem. Chromatic polynomial.
Edge-coloring number. Vertex-coloring number. Total-coloring number.
Max-flow-min-cut theorem.
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