Syllabus for
Master of Science (Statistics)
Academic Year  (2020)

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: Identify the need and scope of Interdisciplinary research.

PO3: Enhance research culture and uphold scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO2: Demonstrate the execution of statistical experiments or investigations, analyse and interpret using appropriate statistical methods, including statistical software and report the findings of experiments or studies accurately.

PSO3: Demonstrate acquaintance with contemporary trends in industrial/research settings and innovate novel solutions to existing problems.

CIA - 50%

ESE - 50%

To make students to use measure-theoretic and analytical techniques for understanding probability concepts. 

CO2: Analyse probability concepts using measure-theoretic approach

CO3: Identify applications of different limit theorems in statistical problems

CO4: Apply Radon-Nikodym theorem in conditional probability 

Algebra of sets, Fields, Sigma fields, Inverse function, Measurable functions, Random variables, Lebesgue measure, Lebesgue-Stieltjes measure, Counting measure, Discrete probability space, General probability space as normed measure space, Induced probability space. Distribution function of a random variable, Distribution function of random vectors. Indepence of random variables

Intgegration with respect to measure (Introduction only), Expectation and moments: Definition and properties, Moment generating functions, Moment inequalities:Chebychev’s, Holder, Jenson and basic inequalities, Product spaces and Fubini’s theorem, Charecteristic function and properties (idea and statement only).

Modes of convergence: Convergence in probability, in distribution, in rth mean, almost sure convergence and their inter-relationships, Convergence theorem for expectation such as Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem.

Law of large numbers, Covergence of series of independent random variables, Kolmogorov’s inequality, Weak law of large numbers (Kninchine’s and Kolmogorov’s), Kolmogorov’s strong law of large numbers, Central limit theorems for i.i.d random variables, Lindberg-Levy and Liaponov’s CLT, Lindberg-Feller CLT.

Conditional expectation and its properties, Conditional probabilities, Randon-Nikodym Theorem (Statement only) and its applications, Bayes’ theorem, Martingales, Submartingales, Martingale convergence theorem, Decomposition of submaritingales.

2. Bhat, B.R, (2014), Modern Probability Theory, 4 th Ed., New Age International.

3. Rohatgi, V.K. and Salah, A.K.E, (2014), An Introduction to Probability and Statistics, 3 rd Ed., John Wiley & Sons. 

2. Feller W, (2008), An Introduction to Probability Theory and its Applications, Volume II,3rd Ed., Wiley Eastern.

3. Basu A.K, (2012), Measure Theory and Probability, 2 nd Ed., PHI.

4. Durrett R, (2010), Probability: Theory and Examples. 4th ed. Cambridge University Press, 2010.

To make students to understand different probability distributions and to model real-life problems using it.

CO2: Analyse well-known probability distributions as special case of different families of distribution.

CO3: To identify different distributions arising from sampling from normal distribution.

CO4: To apply probability distribution in various statistical problems.

Modified power series family and properties. Binomial, Negative binomial, Logarithmic series and Lagrangian distributions and their properties as special cases of the results from modified power series family, hypergeometric distribution and its properties. 

Pearsonian system of distributions, Beta, Gamma, Pareto and Normal as special cases of the Pearson family and their properties. Exponential family of distributions. 

Sampling distributions of the mean and variance from normal population, independence of mean and variance, Chi-square, students t and F distribution and their non-central forms. Order statistics and their distributions.

Bivariate Poisson, Multinomial distribution, Multivariate normal (definition only), bivariate exponential distribution of Gumbel, Marshall and Olkin and Block and Basu, Dirichlet distribution.

Quadratic forms in normal variables: distribution and properties, Cochran’ theorem: applications.

2. Arnold B.C, Balakrishnan N and Nagaraja H.N (2012) A first course in order statistics.

3. Galambos J, and Kotz’s (1978): Characterization of Probability distributions, Springer - Verlag.

4. Elderton, W. P., & Johnson, N. L. (2009). Systems of frequency curves, Cambridge University press. 

2. Johnson N.L, Kotz S and Balakrishnan N (2017) Continuous univariate distributions I & II, John Wiley.

3. Johnson N.L, Kotz S and Balakrishnan N (2000) Multivariate Distribution, 2 nd Ed., John Wiley. 

This course is offered to make students understand the critical aspects of matrix theory and linear models which are used in different areas of statistics such as regression analysis, multivariate analysis, design of experiments and stochastic processes. 

CO2: Analyse system of linear equations using matrix theoretic approach

CO3: Identify applications of matrix theory in statistical problems

CO4: Apply matrix theory in linear models

Vectors, Operations on vector space, subspace, nullspace and column space, Linearly independent sets, spanning set, bases, dimension, rank, change of basis.

Matrix operations, Linear equations, row reduced and echelon forms, Homogenous system of equations, Linear dependence 

Algebra of linear transformations, Matrix representations, rank nullity theorem, determinants, eigenvalues and eigenvectors, Cayley-Hamilton theorem, Jordan canonical forms, orthogonalisation process, orthonormal basis.

Reduction and classification of quadratic forms, Special matrices: symmetric matrices, positive definite matrices, idempotent and projection matrices, stochastic matrices, Gramian matrices, dispersion matrices

Fitting the model, ordinary least squares, estimability of parametric functions, Gauss – Markov theorem, applications: regression model, analysis of variance. 

2. Gentle, J. E. (2017) Matrix algebra- Theory, Computations and Applications in Statistics. Springer texts in statistics, Springer, New York.

3. Strang, G. (2006) Linear Algebra and its Applications: Thomson Brooks. Cole, Belmont, CA, USA.

2. Rencher, A. C., & Schaalje, G. B. (2008) Linear models in statistics, 2nd Ed., John Wiley & Sons.

3. Christensen, R. (2011) Plane answers to complex questions: the theory of linear models. Springer Science & Business Media.

4. Khuri, A. I. (2003). Advanced calculus with applications in statistics, 2nd Ed., John Wiley & Sons. 

To acquint students with different methodologies in statistical research and to make them prepare scientific articles using LaTeX.

CO2: To identify suitable methodology for solving the research problem

CO3: To produce scientific articles using LaTeX

Objectives, Motivation, Utility. Concept of theory, empiricism, deductive and inductive theory. Characteristics of scientific method , Understanding the language of research , Concept, Construct, Definition, Variable. Research Process Problem Identification & Formulation , Research Question – Investigation Question , Logic & Importance

Principles of mathematical writing, LaTeX: writing a research paper, survey article, thesis writing, Beamer: preparing presentations

2. L. Lamport (2014), LaTeX, a Document Preparation System, 2nd ed, Addison-Wesley.

ESE - 50%

To impart the knowledge of different sample survey designs useful in the collection of scientific data

CO2: Analyse different sample survey designs and find estimators.

CO3: Identify the use of different sample survey designs.

CO4: Apply suitable sample survey design in real-life problems.

Sampling vs census, simple random sampling: with (SRS) and without replacement (SRSWOR) of units, estimators of mean, total and variance, determination of sample size, sampling for proportions, Stratified sampling scheme: estimation and allocation of sample size, comparison with simple random sampling schemes. 

Lab Exercises:

1. Drawing samples with SRSWR and SRSWOR and estimation of parameters

2. Estimation of parameters using a sample of proportions

3. Drawing stratified sample and estimation of parameters

Bias and mean square error, estimation of variance, confidence interval, comparison with mean per unit estimator, optimum property of ratio estimator, unbiased ratio type estimator, ratio estimator in stratified random sampling, Difference estimator and Regression estimator:- Difference estimator, regression estimator, comparison of regression estimator with mean per unit and ratio estimator, regression estimator in stratified random sampling.

Lab Exercises:

4. Estimation using ratio estimator

5. Estimation using regression estimator

6. Ratio estimator and regression estimator in stratified sampling

With and without replacement sampling schemes: PPS and PPSWR schemes, Selection of samples, estimators: ordered and unordered estimators. Πps sampling schemes.

Lab Exercises:

7. Exercise on the PPS scheme

8. Exercise on the PPSWR scheme

9. Exercise on Πps sampling scheme

Systematic sampling scheme: estimation of population mean and variance, comparison of systematic sampling with SRS and stratified random sampling, circular systematic sampling, Cluster sampling: estimation of population mean, estimation of efficiency by a cluster sample, variance function, determination of optimum cluster size, Multistage sampling: estimation population total with SRS sampling at both stages, multiphase sampling (outline only), quota sampling, network sampling; Adaptive sampling: introduction and estimators under adaptive sampling. Introduction to small area estimation.

Lab Exercises:

9. Exercise on the systematic sampling scheme

10. Exercise on cluster sampling

11. Exercise on multi-stage sampling

12. Exercise on small area estimation

Sampling and non-sampling errors, the effect of unit nonresponse in the estimate, procedures for unit nonresponse

Lab Exercises:

13. Exercise on the sensitivity of efficiency due to sampling errors

14. Procedures for non-response

2. Singh, D. and Chaudharay, F.S. (2018) Theory and Analysis of Sample Survey Designs, New Age International.

2. Singh, S. (2003). Advanced Sampling: Theory and Practice. Kluwer.

3. Des Raj and Chandhok, P. (2013) Sampling Theory, McGraw Hill. 4. Mukhopadhay, P (2009) Theory and methods of survey sampling, Second edition, PHI Learning Pvt Ltd., New Delhi.

5. Sampath, S. (2005) Sampling theory and methods, Alpha Science International Ltd., India.

ESE - 50%

To equip students with knowledge of R programming to develop statistical models for real world problems.

CO2: To perform graphical representation of data using R

CO3: To demonstrate the usage of R for data analysis

Variables, Functions, Vectors, Expressions and assignments, Logical expressions, Matrices, The workspace, R markdown.

Lab Exercises:

1. Demonstrate variables and functions in R

2. Creating vectors and matrices and associated operations in R

3. Logical and arithmetic operations in R

Loops: if, for, while , Program flow , Basic debugging ,Good programming habits, Input and outputs: Input from a file , Output to a file –Plotting. 

Lab Exercises:

4. Illustration of control structures: if, else, for

5. Illustration of control structures: while, repeat, break, next and ifelse

6. Getting data in and out of the R environment: reading tables, reading CSV files, readLines(), opening url, user inputs, writing files.

Functions, Optional arguments and default values, Vector-based programming using functions, Recursive programming, Debugging functions, Sophisticated data structures - Factors - Dataframes - Lists - The apply family.

Lab Exercises:

7. Creating user-defined functions and doing vector-based programming

8. Creating lists and data frames and associated operations

9. Demonstration of recursive functions, apply functions in R

10. Demonstration of factors and arrays in R

Visualizing data, Graphical summaries of data-Bar chart, Pie chart, Histogram, Box-plot, Stem and leaf plot, Frequency table, Plotting of probability distributions and sampling distributions, PP plot, Q-Q Plot , ggplot2, lattice – 3D plots, Graphics parameters, par -Graphical augmentation.

Lab Exercises:

11. Visualization of numerical variables in R using ‘base R’, ‘ggplot2’ and ‘lattice 3D’ packages

12. Contingency tables and visualization of categorical variables using ‘base R’, ‘ggplot2’ and ‘lattice 3D’  packages

13. Construction  of probability plots and quantile plots in R

Numerical methods- Root-finding algorithms, Simulating iid uniform samples, Congruential generators, Seeding, Simulating discrete random variables, Inversion method for continuous random variables, Rejection method, generation of normal variates: Rejection with exponential envelope, Box-Muller algorithm.

Lab Exercises:

14. Root finding algorithms for the non-linear system of equations.

15. Simulation of discrete variables in R

16. Simulation of continuous variables- inversion method, rejection method

2. Matloff, N. (2016). The art of R programming: A tour of statistical software design. No Starch Press.

2. Chambers, J. M. (2008). Software for Data Analysis-Programming with R. SpringerVerlag, New York.

ESE - 50%

To provide a strong mathematical and conceptual foundation in the methods of parametric estimation and their properties.

CO2: To identify the suitable estimation method.

CO3: To analyse likelihood function and apply different root solving methods to find estimators

CO4: To construct confidence intervals for parameters involved in the model.

Sufficiency: factorization theorem, minimal sufficiency, exponential family and completeness. Ancillary statistics and Basu's theorem.

UMVUE: Fisher Information and Cramer-Rao inequality, Chapman-Robbin’s and Bhattacharya bounds, Rao-Blackwell theorem, Lehman-Scheffe theorem. Unbiased estimation.

Consistency, Weak and strong consistency, Marginal and joint consistent estimators, CAN estimators, equivariance, Pitman estimators

Method of moments,  Minimum chi-square and its modification, Least square estimation, Maximum likelihood, Properties of maximum likelihood estimators, Cramer-Huzurbazar Theorem, Likelihood equation - multiple roots, Iterative methods, EM Algorithm.

Large sample confidence interval, shortest length confidence interval. Methods of finding confidence interval: Inversion of test statistic, pivotal quantities, pivoting CDF, evaluation of confidence interval: size and coverage probability, loss function and test function optimality.

2.      Srivastava, A. K. , Khan, A. H. and Srivastava, N. (2014). Statistical Inference: Theory of Estimation. PHI Learning Pvt. Ltd, New Delhi.

3.      Lehmann, E. L., & Casella, G. (2006). Theory of point estimation, 2^nd Ed. Springer.

4.      Robert, C., & Casella, G. (2013). Monte Carlo statistical methods. Springer.

2.      Silvey, S. D. (2017). Statistical inference. Routledge.

3.      Trosset, M. W. (2009). An introduction to statistical inference and its applications with R.

Chapman and Hall/CRC.

4.      Dixit, U. J. (2016). Examples in parametric inference with R, Springer.

ESE - 50%

Total - 100%

To equip the students with theoretical and practical knowledge of stochastic models which are used in economics, life sciences, engineering etc.

CO2: To identify ergodic Markov chains.

CO3: To analyse queuing models using continuous-time Markov chains.

CO4: To apply Browning motion in finance problems.

sequence of random variables, definition and classification of a stochastic process, autoregressive processes and stationary processes.

Markov Chains: Definition, Examples, Transition probability matrix, Chapman-Kolmogorov equation, classification of states, limiting and stationary distributions, ergodicity, discrete renewal equation and basic limit theorem, Absorption probabilities, Criteria for recurrence. Generic application: hidden Markov models

Transition probability function, Kolmogorov differential equations, Poisson process: homogenous process, interarrival distribution, compound process, Birth and death process. Service applications: Queuing models- Markovian models.

Galton-Watson branching processes, Generating function, Extinction probabilities, Continuous-time branching processes, Extinction probabilities, Branching processes with general variable lifetime.

Renewal equation, Renewal theorem, Applications, Generalizations and variations of renewal processes, Applications of renewal theory, Brownian motion, Introduction to Markov renewal processes.

2.      Cinlar, E. (2013). Introduction to stochastic processes. Courier Corporation.

3.      S. M. Ross (2014). Introduction to Probability Models. Elsevier.

2.      J. Medhi (2009) Stochastic Processes, 3rd Edition, New Age International.

3.      Dobrow, R.P. (2016), Introduction to Stochastic Processes with R, Wiley Eastern.

ESE - 50%

Total - 100%

To equip the students with the theory and methods to analyse and categorical responses.

CO2: to identify test for contingency tables. 

CO3: To apply regression models for count data. 

CO4: To analyse contingency tables using log-linear models. 

Categorical response data, Probability distributions for categorical data, Statistical inference for discrete data

Probability structure for contingency tables, Comparing proportions with 2x2 tables, The odds ratio, Tests for independence, Exact inference, Extension to three-way and larger tables

Components of a generalized linear model, GLM for binary and count data, Statistical inference and model checking, Fitting GLMs

Interpreting the logistic regression model, Inference for logistic regression, Logistic regression with categorical predictors, Multiple logistic regression, Summarizing effects, Building and applying logistic regression models, Multicategory logit models

Loglinear models for two-way and three-way tables,  Inference for Loglinear models, the log linear-logistic connection, Independence graphs and collapsibility, Models for matched pairs: Comparing dependent proportions,  Logistic regression for matched pairs, Comparing margins of square contingency tables, symmetry issues 

2.       Agresti, A. (2010). Analysis of ordinal categorical data. John Wiley & Sons.

2.      Stokes, M. E., Davis, C. S., & Koch, G. G. (2012). Categorical data analysis using SAS. SAS Institute.

3.      Agresti, A. (2018). An introduction to categorical data analysis. John Wiley & Sons.

4.      Bilder, C. R., & Loughin, T. M. (2014). Analysis of categorical data with R. Chapman and Hall/CRC.

ESE - 50%

Total - 100%

To impart the knowledge statistical model building using regression technique. 

CO2: To identify the correct regression model for the given problem 

CO3: To apply non-linear regression in real-life problems. 

CO4: To analyse the robustness of the regression model. 

Linear Regression Model: Simple and multiple, Least squares estimation, Properties of the estimators,  Maximum likelihood estimation, Estimation with linear restrictions, Hypothesis testing, confidence intervals.

Lab Exercises:

1.     Build a simple linear model and interpret the data.

2.     Construct confidence interval for simple linear model

3.     Build a multiple linear model and estimate its parameters.

4.     Construct confidence interval for multiple linear model

Residual analysis, Departures from underlying assumptions, Effect of outliers, Collinearity, Nonconstant variance and serial correlation, Departures from normality, Diagnostics and remedies. 

Lab Exercises:

1.     Carry out residual analysis and validate the model assumptions.

2.     Construct residul plots for checking outliers and non constant variance.

selection of input variables and model selection Methods of obtaining the best fit - stepwise regression Forward selection and backward elimination

Lab Exercises:

1.     Selecting best model using step wise regression.

2.     Selecting best model using Forward and backward selection procedure.

Introduction to general non-linear regression, least-squares in non-linear case, estimating the parameters of a non-linear system, reparametrisation of the model Non-linear growth models 

Lab Exercise:

1.Estimate parameters in non-linear models using least square procedure

Linear absolute deviation regression, M estimators, robust regression with rank residuals, resampling procedures for regression models methods and its properties (without proof) - Jackknife techniques and least squares approach based on M-estimators.

Lab Exercises:

1.     Illustrate resampling procedures in regression models.

2.     Build a regression model robust regression procedures.

2.Draper, N. R., & Smith, H. (1998). Applied regression analysis. John Wiley & Sons. 

3.Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis, John Wiley & Sons. 

2.      Keith, T. Z. (2014). Multiple regression and beyond: An introduction to multiple regression and structural equation modelling. Routledge.

3.      Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.

4.      Fox, J., & Weisberg, S. (2018). An R companion to applied regression. Sage publications.

ESE - 50%

Total - 100%

To equip the students with programming skill in python and to apply in data analysis.

CO2: To understand  functions and data modelling 

CO3: To analyze statistical datasets and visualize it. 

installing Python; basic syntax, interactive shell, editing, saving, and running a script, The concept of data types; variables, assignments; immutable variables; numerical types; arithmetic operators and expressions; comments in the program; understanding error messages; Conditions, boolean logic, logical operators; ranges; Control statements: if-else, loops

hiding redundancy, complexity; arguments and return values; formal vs actual arguments, named arguments. Program structure and design. Recursive functions. Classes and OOP: classes, objects, attributes and methods; defining classes; design with classes, data modelling

Pandas, Statsmodels, Seaborn, displaying statistical data, distributions and hypothesis testing, linear regression models. 

2.      Haslwanter, T. (2016). An Introduction to Statistics with Python. Springer International Publishing:.

2.Anthony, F. (2015). Mastering pandas. Packt Publishing Ltd. 

ESE - 50%

Total - 100%

To provide a strong foundation for data science and application area related to it and understand the underlying core concepts and emerging technologies in data science.

CO2: Understand data analysis techniques for applications handling large data 

CO3: Demonstrate various databases and Compose effective queries 

Introduction – Big Data and Data Science  – Data science Hype – Getting Past the Hype – The Current Landscape – Role of Data Scientist – Exploratory Data Analysis –  Data Science Process Overview – Defining goals – Retrieving data – Data preparation – Data exploration – Data modeling – Presentation. Problems in handling large data – General techniques for handling large data – Big Data and its importance, Four Vs, Drivers for Big data, Big data analytics, Big data applications, Algorithms using map reduce, Matrix-Vector Multiplication by Map Reduce. Steps in big data – Distributing data storage and processing with Frameworks – Data science ethics – valuing different aspects of privacy – The five C’s of data.

1.      1. Lab exercise for feature engineering

2.     2. Lab exercise for big data processing

Machine learning – Modeling Process – Training model – Validating model – predicting new observations – Supervised learning algorithms – Unsupervised learning algorithms. Introduction to deep learning – Deep Feed Forward networks – Regularization – Optimization of deep learning – Convolutional networks – Recurrent and recursive nets – applications of deep learning.

1.     1.  Lab exercise on Linear and Logistic discrimination

2.      2. Lab exercise on K means clustering and Hierarchical clustering

Concept and Overview of DBMS, Data Models, Database Languages, Database Administrator, Database Users, Three Schema architecture of DBMS. Basic concepts, Design Issues, Mapping Constraints, Keys, Entity-Relationship Diagram, Weak Entity Sets, Functional Dependency, Different anomalies in designing a Database, Normalization: using functional dependencies, 1NF, 2NF, 3NF and Boyce-Codd Normal Form

1. Lab Exercise on Database Design

Top-Down Approach

Bottom-up Approach 

SQL Basic Structure - DDL, DML, DCL-Integrity Constraints - Domain Constraints, Entity Constraints - Referential Integrity Constraints, Concept of Set operations, Joins, Aggregate Functions, Null Values, , assertions, views, Nested Sub queries – procedural extensions – stored procedures – functions- cursors – Intelligent databases – ECA rule – Data Integration – ETL Process

1.         Lab Exercise on SQL

2.         Lab Exercise on PL/SQL

3.         Lab Exercise on ETL

Data Warehousing - Defining Feature – Data warehouses and data marts –Metadata in the data warehouse – Data design and Data preparation - Dimensional Modeling - Principles of dimensional modeling – The star schema – star schema keys – Advantages of the star schema – Updates to the dimension tables – The snowflake schema – Aggregate fact tables – Families Oo Stars – MDX queries – Reporting services.

1. Lab Exercise on Analysis Services

2. Lab Exercise on Reporting Services

2. Thomas Cannolly and Carolyn Begg, (2007), Database Systems, A Practical Approach to Design, Implementation and Management”, 3rd Edition, Pearson Education. 

2. D J Patil, Hilary Mason, Mike Loukides, (2018), Ethics and Data Science, O’ Reilly.

3. LiorRokach and OdedMaimon, (2010), Data Mining and Knowledge Discovery Handbook. 

ESE - 50%

Total - 100%

This course will provide an introduction to the principles and methods for the analysis of time-to-event data. This type of data occurs extensively in both observational and experimental biomedical and public health studies. 

CO2: Understand Non-Parametric Survival techniques for applications lifetime data 

CO3: Demonstrate various Competing Risks and their effects 

The hazard and survival functions. Mean residual life function, competing risk, right, left and interval censoring, truncation, likelihood for censored and truncated data. Parametric and non-parametric estimation in truncated and censored cases.

Lab Exercises:

1.Lab exercise on the parametric estimation of left and right-censored data

2.Lab exercise on the parametric estimation of truncated data

3.Lab exercise on the non-parametric estimation of censored and truncated data

Parametric forms and the distribution of log time. The exponential, Weibull, Gompertz, Gamma, Generalized Gamma, Coale-McNeil, and generalized F distributions. The U.S. life table. Approaches to modelling the effects of covariates. Parametric families. Proportional hazards models (PH). Accelerated failure time models (AFT). The intersection of PH and AFT. Proportional odds models (PO). The intersection of PO and AFT. Recidivism in the U.S. 

Lab Exercises:

1.Lab exercise on parametric modelling pf survival data

2.Lab exercise on the proportional hazard model

3.Lab exercise on AFT models

One-sample estimation with censored data. The Kaplan-Meier estimator. Greenwood's formula. The Nelson-Aalen estimator. The expectation of life. Comparison of several groups: Mantel- Haenszel and the log-rank test. 

Regression: Cox's model and partial likelihood. The score and information. The problem of ties. Tests of hypotheses. Time-varying covariates. Estimating the baseline survival. Martingale residuals.

Lab Exercises:

7.Lab exercise on Kaplan-Meier estimator and Nelson-Aalen estimator

8.Lab exercise on Mantel- Haenszel and the log-rank test

9.Lab exercise on the Cox model with time-varying covariate

Cox's discrete logistic model and logistic regression. Modelling grouped continuous data and the complementary log-log transformation. Piece-wise constant hazards and Poisson regression. Current status data versus retrospective data. Open intervals and time since the last event. Backward recurrence times. Interval censoring. 

Lab Exercises:

10.Lab exercise on the discrete logistic model for survival data

11.Lab exercise on Poisson regression for survival data

12.Lab exercise on Piece-wise regression for survival data

Modelling multiple causes of failure. Research questions of interest. Cause-specific hazards. Overall survival. Cause-specific densities. Estimation: one-sample and the generalized Kaplan- Meier and Nelson-Aalen estimators. The Incidence function. Regression models. Weibull regression. Cox regression and partial likelihood. Piece-wise exponential survival and multinomial logits. The identification problem. Multivariate and marginal survival. The Fine-Gray model.

Lab Exercises:

13.Lab exercise on non-parametric modelling of competing risk data 

14.Lab exercise on parametric modelling of competing risk data

15.Lab exercise on multivariate survival data

2. Cleves, M.; W. G. Gould, and J. Marchenko (2016). An Introduction to Survival Analysis using Stata. Revised 3rd Ed. College Station, Texas: Stata Press. 

3. Kalbfleisch, J. D., & Prentice, R. L. (2011). The statistical analysis of failure time data,2nd Ed. John Wiley & Sons. 

4. Moore, D. F. (2016). Applied survival analysis using R. Switzerland: Springer. 

2. Therneau, T. M. and P. M. Grambsch (2000). Modelling Survival Data: Extending the Cox Model, Springer, NY

3. Collett, D. (2015). Modelling survival data in medical research. Chapman and Hall/CRC. 

ESE - 50%

Total - 100%

This course provides an introduction to the application of statistical tools on the industrial environment to study, analyze and control the quality of products. 

CO2: Apply the idea of Sampling Plans to control the quality of industrial outputs. 

Meaning and scope of statistical quality control - Causes of quality variation - Control charts for variables and attributes - Rational subgroups - Construction and operation of, σ, R, np, p, c and u charts - Operating characteristic curves of control charts. Process capability analysis using histogram, probability plotting and control chart - Process capability ratios and their interpretations. 

1.      Lab exercise on control charts for variables

2.      Lab exercise on control charts for attributes

3. Lab exercise on operating characteristic curve

Specification limits and tolerance limits - Modified control charts - Basic principles and design of cumulative-sum control charts – Concept of V-mask procedure – Tabular CUSUM charts. Construction of Moving range, moving-average and geometric moving-average control charts.

4.      Lab exercise on CUSUM charts

5.      Lab exercise on moving average charts

6. Lab exercise on geometric moving average charts

Acceptance sampling: Sampling inspection by attributes – single, double and multiple sampling plans – Rectifying Inspection. Measures of performance: OC, ASN, ATI and AOQ functions. Concepts of AQL, LTPD and IQL. Dodge – Romig and MIL-STD-105D tables 

7.      Lab exercise on single sampling scheme

8.      Lab exercise on double sampling scheme

9.      Lab exercise on Dodge-Romig sampling scheme

Sampling inspection by variables - known and unknown sigma variables sampling plan - Merits and limitations of variables sampling plan - Derivation of OC curve – determination of plan parameters. 

10.      Lab exercise on variable sampling scheme with known variance

11.      Lab exercise on variable sampling scheme with unknown variance

12. Lab exercise on OC curves

Continuous sampling plans by attributes - CSP-1 and its modifications - concept of AOQL in CSPs - Multi-level continuous sampling plans - Operation of multi-level CSP of Lieberman and Solomon – Wald - Wolfowitz continuous sampling plans. Sequential Sampling Plans by attributes – Decision Lines - OC and ASN functions. 

13.      Lab exercise CSP-1

14.      Lab exercise on multi-level CSP

15. Lab exercise on sequential sampling plan

2. Duncan, A. J. (2003.). Quality Control and Industrial Statistics, Irwin-Illinois, US. 

2. Schilling, E. G., and Nuebauer, D.V. (2009). Acceptance Sampling in Quality Control, Second Edition, CRC Press, New York. 

3. Ross, S. M. (2009). Introduction to Probability Models, Tenth Edition, Academic Press, MA, US. 

ESE - 50%

Total - 100%

The course will be inculcating research culture which will enhance the employability skills to the students. 

Students will do the following,

1. Identify a domain for the research project

2. Literature survey 

3. Identifying the existing methodology and models

4. Writing a problem statement 

5. Project presentation at the end of the process

ESE - 50%