Available courses

DESIGN AND ANALYSIS OF EXPERIMENTS

MST2C06: DESIGN AND ANALYSIS OF EXPERIMENTS (4 Credits)
Unit-I: Randomization, Replication and local control, One way and two way classifications with equal and unequal number of observations per cell with and without interaction, Fixed effects and Random effects model. Model adequacy checking, CRD, RBD and Latin Square designs, Analysis of co-variance for completely randomized and randomized block designs. Analysis of experiments with missing observations.
Unit-II: Incomplete Block Designs: Balanced Incomplete Block designs, Construction of BIB Designs, Analysis with recovery of inter-block information and intra-block information. Partially balanced incomplete block designs, Analysis of partially balanced incomplete block designs with two associate classes, Lattice designs.
Unit-III: 2n  Factorial experiments. Analysis of 2n factorial experiments. Total confounding of 2n  designs in 2n  blocks. Partial confounding in2n
 blocks. Fractional factorial designs, Resolution of a design, 3n
factorial designs. Concepts of Split plot design and strip plot design
Unit-IV: Response surface designs, Orthogonality, Rotatability blocking and analysis - Method of Steepest accent, Models properties and Analysis.

Text Books
1. Montgomery D C (2001). Design and Analysis of Experiments, John Wiley.
 2. Das M N and Giri N C (1979). Design and Analysis of Experiments, second edition, Wiley.
 3. Hinkleman and Kempthrone C (1994). Design and Analysis of Experiments Volume I, John Wiley.
Reference Books:
1. Joshi D.D. (1987). Linear Estimation and Design of Experiments, Wiley Eastern.
 2. Chakrabarti, M.C. (1964). Design of experiments, ISI, Calcutta. 

PROBABILITY THEORY

Unit-I: Sets and classes of events – Sequences of sets and their limits – Fields, Sigma fields, Borel field. Random variables, Sigma fields induced by random variables, Vector random variables, limits of sequence of random variables, Probability space, General Probability space, Induced probability space,
Concepts of other measures.
Unit-II: Distribution functions of random variables. Decomposition of distribution functions, Distribution function of vector random variables, Correspondence theorem, Expectation and moments,
Properties of expectations, Moments and inequalities, Characteristic functions, Properties, Inversion theorem, Characteristic functions and moments, Bochner’s theorem (No proof required), Independence
of classes of events; Independence of random variables; Kolmogorov 0-1 law; Borel 0-1 law.
Unit-III: Convergence of random variables: Convergence in probability, Convergence almost surely, Convergence in distribution, Convergence in rth mean – their inter-relations- examples and counterexamples. Convergence of distribution functions; Weak convergence, Helly-Bray Lemma and Helly –
Bray theorem, Levy continuity theorem.
Unit- IV: Law of Large Numbers – Kolmogorov inequality, Kolmogorov three series theorem; Weak law of large numbers (both IID and Non-IID cases). Strong Law of large numbers (Law of iterated logarithm not included), Central Limit Theorem(CLT),Lindeberg-Levy theorem, Liapounov form of
CLT. Lindeberg-Feller CLT (no proof required). Association between Liapounov’s condition and Lindeberg conditions; Simple applications of CLT

ANALYTICAL TOOLS FOR STATISTICS – II

Unit-I- Basics of linear algebra
Definition of vector space, sub spaces, linear dependence and independence, basis and dimensions, direct sum and compliment of a subspace, quotient space, Inner product and orthogonality.
Unit-II- Algebra of Matrices
Linear transformations and matrices, operations on matrices, properties of matrix operations, Matrices with special structures-triangular matrix, idempotent matrix, Nilpotent matrix, symmetric, Hermitian and skew Hermitian matrices, unitary matrix. Row and column space of matrix, inverse of a matrix. Rank of product of matrix, rank factorization of a matrix, rank of a sum and projections, Inverse of a partitioned matrix, Rank of real and complex matrix.
Unit-III- Eigen values, spectral representation and singular value decomposition Cayley-Hamilton theorem, minimal polynomial, eigen values, eigen vectors and eigen spaces, spectral representation of a semi simple matrix, algebraic and geometric multiplicities, Jordan canonical form,
spectral representation of a real symmetric, concepts of Hermitian and normal matrices, singular value decomposition.
Unit- IV- Linear equations generalized inverses and quadratic forms
Homogenous system, general system, Rank Nullity Theorem (statement only), generalized inverse, properties of g-inverse, Moore-Penrose inverse, properties, computation of g-inverse, definition of quadratic forms, classification of quadratic forms, rank and signature, positive definite and non-negative definite matrices, extreme of quadratic forms, simultaneous diagonalisation of matrices.

ANALYTICAL TOOLS FOR STATISTICS – I

Unit-I-Multivariable Functions
Limits and continuity of multivariable functions. Derivatives, directional derivatives and continuity. Total derivative in terms of partial derivatives, Taylor’s theorem. Inverse and implicit functions. Optima of multivariable functions. Method of Lagrangian multipliers, Riemann integral of a multivariable function.
Unit-II-Analytic functions and complex integration
Analytical functions, Harmonic functions, Necessary condition for a function to be analytic, Sufficient condition for a function to be analytic, Polar form of Cauchy- Riemann equation, Construction of analytic function. Complex integral, Cauchy’s theorem , Cauchy’s integral formula and its generalized form. Poisson integral formula, Morera’s theorem.Cauchy’s inequality, Liouville's theorem, Taylor’s theorem, Laurent’s theorem.
Unit-III- Singularities and calculus of residues
Zeros of a function, singular point, different types of singularities. Residue at a pole, residue at infinity, Cauchy’s residue theorem, Jordan’s lemma, Integration around a unit circle. Poles on the real axis, Integration involving many valued function.
Unit-IV- Laplace transform and Fourier Transform
Laplace transform, Inverse Laplace transform. Applications to differential equations, Infinite Fourier transform, Fourier integral theorem. Different forms of Fourier integral formula, Fourier series.

DISTRIBUTION THEORY

Unit-1: Discrete distributions: Random variables, Moments and Moment generating functions, Probability generating functions, Discrete uniform, Binomial, Poisson, Geometric, Negative binomial, Hyper geometric and Multinomial distributions, Power series distributions.
Unit-2: Continuous distributions: Uniform, Normal, Exponential, Weibull, Pareto, Beta, Gamma, Laplace, Cauchy and Log-normal distributions. Pearsonian system of distributions, location and scale
families.
Unit-3: Functions of random variables: Joint and marginal distributions, Conditional distributions and independence, Bivariate transformations, Covariance and Correlations, Bivariate normal distributions,
Hierarchical models and Mixture distributions, Multivariate distributions, Inequalities and Identities. Order statistics.
Unit-4: Sampling distributions: Basic concept of random sampling, Sampling from normal distributions, Properties of sample mean and variance. Chi-square distribution and its applications, tdistribution and its applications. F-distribution- properties and applications. Non-central Chi-square, t,
and F- distributions.

Biostatistics

Unit-I: Biostatistics-Example on statistical problems in Biomedical Research-Types of Biological dataPrinciples of Biostatistical design of medical studies- Functions of survival time, survival distributions and their applications viz. exponential, gamma, Weibull, Rayleigh, lognormal, distribution having bathtub shape hazard function. Parametric methods for comparing two survival distributions ( L.R test and Cox’s F-test).
Unit-II: Type I, Type II and progressive or random censoring with biological examples, Estimation of mean survival time and variance of the estimator for type I and type II censored data with numerical examples. Non-parametric methods for estimating survival function and variance of the estimator viz.
Acturial and Kaplan –Meier methods.
Unit-III: Categorical data analysis (logistic regression) - Competing risk theory, Indices for measurement of probability of death under competing risks and their inter-relations. Estimation of probabilities of death under competing risks by ML method. Stochastic epidemic models: Simple and
general epidemic models.
Unit-IV: Basic biological concepts in genetics, Mendel’s law, Hardy- Weinberg equilibrium, random mating, natural selection, mutation, genetic drift, detection and estimation of linkage in heredity.  Planning and design of clinical trials, Phase I, II, and III trials. Sample size determination in fixed sample designs. Planning of sequential, randomized clinical trials, designs for comparative trials; randomization techniques and associated distribution theory and permutation tests (basic ideas only); ethics behind
randomized studies involving human subjects; randomized dose-response studies(concept only).

Stochastic Processes

Unit-I: Concept of Stochastic processes, examples, Specifications; Markov chains- Chapman Kolmogorov equations – classification of states – limiting probabilities; Gamblers ruin problem and Random Walk – Mean time spent in transient states – Branching processes (discrete time), Hidden
Markov chains.
Unit-II: Exponential distribution – counting process – inter arrival time and waiting time distributions. Properties of Poisson processes – Conditional distribution of arrival times. Generalization of Poisson processes – non-homogenous Poisson process, compound Poisson process, conditional mixed Poisson process. Continuous time Markov Chains – Birth and death processes – transition probability functionlimiting probabilities.
Unit-III: Renewal processes-limit theorems and their applications. Renewal reward process. Regenerative processes, Semi-Markov process. The inspection paradox, Insurers ruin problem.
Unit-IV: Basic characteristics of queues – Markovian models – network of queues. The M/G/I system. The G/M/I model, Multi server queues. Brownian motion Process – hitting time – Maximum variable – variations on Brownian motion – Pricing stock options – Gaussian processes – stationary and weakly
stationary processes.

Applied Regression Analysis

Unit-I: Linear Regression Model, Least squares estimation, Gauss Markov Theorem, Properties of the estimates, Distribution Theory, Maximum likelihood estimation, Estimation with linear restrictions,
Generalised least squares; Hypothesis testing - likelihood ratio test, F-test; Confidence intervals.
Unit-II: Residual analysis, Departures from underlying assumptions, Effect of outliers, Collinearity, Non-constant variance and serial correlation, Departures from normality, Diagnostics and remedies.
Unit-III: Polynomial regression in one and several variables, Orthogonal polynomials, Indicator variables, Subset selection of explanatory variables, stepwise regression and Mallows Cp -statistics, Introduction to non-parametric regression.
Unit-IV: Introduction to nonlinear regression, Least squares in the nonlinear case and estimation of parameters, Models for binary response variables, estimation and diagnosis methods for logistic and Poisson regressions. Prediction and residual analysis, Generalized Linear Models – estimation and
diagnostics.

Time Series Analysis

Unit-I: Motivation, Time series as a discrete parameter stochastic process, Auto – Covariance, AutoCorrelation and spectral density and their properties. Exploratory time series analysis, Test for trend and seasonality, Exponential and moving average smoothing, Holt – Winter smoothing, forecasting based
on smoothing, Adaptive smoothing.
Unit-II: Detailed study of the stationary process: Autoregressive, Moving Average, Autoregressive Moving Average and Autoregressive Integrated Moving Average Models. Choice of AR / MA periods.
Unit-III: Estimation of ARMA models: Yule – Walker estimation for AR Processes, Maximum likelihood and least squares estimation for ARMA Processes, Discussion (without proof) of estimation of mean, Auto-covariance and auto-correlation function under large samples theory, Residual analysis and diagnostic checking. Forecasting using ARIMA models, Use of computer packages like SPSS.
Unit-IV: Spectral analysis of weakly stationary process. Herglotzic Theorem. Periodogram and correlogram analysis. Introduction to non-linear time Series: ARCH and GARCH models.

PROBABILITY DISTRIBUTIONS AND SAMPLING THEORY

Module 1: Standard distributions: Discrete type-Bernoulli, Binomial, Poisson, Geometric, Negative Binomial (definition only), Uniform (mean, variance and mgf). Continuous type-Uniform, exponential and Normal (definition, properties and applications); Gamma (mean, variance, mgf); Lognormal, Beta, Pareto and Cauchy (Definition only)
Module 2: Limit theorems: Chebyshev’s inequality, Sequence of random variables, parameter and Statistic, Sample mean and variance, Convergence in probability (definition and example only), weak law of large numbers (iid case), Bernoulli law of large numbers, Convergence in distribution (definition and examples only), Central limit theorem (Lindberg levy-iid case)
Module 3: Sampling methods: Simple random sampling with and without replacement, systematic sampling (Concept only), stratified sampling (Concept only), Cluster sampling(Concept only)
Module 4: Sampling distributions: Statistic, Standard error, Sampling from normal distribution, distribution of sample mean, sample variance, chi-square distribution, t-distribution, and F distribution (definition, derivations and relationships only).

PROBABILITY DISTRIBUTIONS AND SAMPLING THEORY

Module 1: Standard distributions: Discrete type-Bernoulli, Binomial, Poisson, Geometric, Negative Binomial (definition only), Uniform (mean, variance and mgf). Continuous type-Uniform, exponential and Normal (definition, properties and applications);Gamma (mean, variance, mgf); Lognormal, Beta, Pareto and Cauchy (Definition only)
Module 2: Limit theorems: Chebyshev’s inequality, Sequence of random variables, parameter and Statistic, Sample mean and variance, Convergence in probability (definition and example only), weak law of large numbers (iid case), Bernoulli law of large numbers, Convergence indistribution (definition and examples only), Central limit theorem (Lindberg levy-iid case)
Module 3: Sampling methods: Simple random sampling with and without replacement, systematic sampling (Concept only), stratified sampling (Concept only), Cluster sampling(Concept only)
Module 4: Sampling distributions: Statistic, Standard error, Sampling from normal distribution, distribution of sample mean, sample variance, chi-square distribution, tdistribution, and F distribution (definition, derivations and relationships only).

INTRODUCTORY STATISTICS

Module 1: Official statistics: The Statistical system in India: The Central and State Government organizations, functions of the Central Statistical Office (CSO), National Sample Survey Organization (NSSO) and the Department of Economics and Statistics.
Module 2: Introduction to Statistics: Nature of Statistics, Uses of Statistics, Statistics in relation to other disciplines, Abuses of Statistics. Concept of primary and secondary data. Designing a questionnaire and a schedule. Concepts of statistical population and sample from a population, quantitative and qualitative data, Nominal, ordinal and time series data, discrete and continuous data. Presentation of data by table and by diagrams, frequency distributions by histogram and frequency polygon, cumulative frequency distributions (inclusive and exclusive methods) and ogives. Measures of central tendency (mean, median, mode, geometric mean and harmonic mean) with simple applications. Absolute and relative
measures of dispersion (range, quartile deviation, mean deviation and standard deviation) with simple applications. Co-efficient of variation, Box Plot. Importance of moments, central and non-central moments, and their interrelationships. Measures of skewness based on quartiles and moments; kurtosis based on moments.
Module 3: Correlation and Regression: Scatter Plot, Simple correlation, Simple regression, two regression lines, regression coefficients. Fitting of straight line, parabola, exponential, polynomial (least square method).
Module 4:Time series: Introduction and examples of time series from various fields, Components of times series, Additive and Multiplicative models. Trend: Estimation of trend by free hand curve method, method of semi averages, method of moving averages and fitting various mathematical curves. Seasonal Component: Estimation of seasonal component by
Method of simple averages, Ratio to Trend. Index numbers: Definition, construction of index numbers and problems thereof for weighted
and unweighted index numbers including Laspeyre’s, Paasche’s, Edgeworth-Marshall and Fisher’s.

QUANTITATIVE METHODS FOR ECONOMIC ANALYSIS I

QUANTITATIVE METHODS FOR ECONOMIC ANALYSIS I