Syllabus for
Master of Science (Statistics)
Academic Year  (2023)

Master of Science in Statistics at CHRIST (Deemed to be University) offers the students an amalgam of knowledge on theoretical and applied statistics on a broader spectrum. Further, it intends to impart awareness of the importance of the conceptual framework of statistics across diversified fields and provide practical training on statistical methods for carrying out data analysis using sophisticated programming languages and statistical software such as R, Python, SPSS, EXCEL, etc. The curriculum has been designed in such a way to cater for the needs of stakeholders to get placements in industries and institutions on successful completion of the course and to provide those ample skills and opportunities to meet the challenges at the national level competitive examinations like CSIR NET in Mathematical Science, SET, Indian Statistical Service (ISS), etc.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: To strengthen analytical and problem-solving skill through real-time applications.

PO3: To gain practical experience in computational techniques and programming tools used in the statistical arena.

PO4: To provide a strong foundation in the best practices of collating and disseminating information.

PO5: To imbibe quality research and develop solutions to social issues.

Evaluation pattern for full CIA courses offered:

The “Theory and Practical” Type of courses offered in all UG/PG programmes will be considered as Full CIA courses.

For this type of courses, there is no exclusive Mid Semester Examination and End Semester Examination; instead there shall be a continuous evaluation during the semester as,

CAC – Continuous Assessment Component

Assessment components such as Hard copy / Soft copy Assignment, Quiz, Presentation, Video Making, MOOC, Project, Demonstration, Service Learning, etc

CAT – Continuous Assessment Test

A written / Lab test would be conducted on any working day

The total marks for the full CIA courses would vary based on the number of hours allocated in a week for the respective course. Out of the maximum marks allotted to the respective course, 50% marks will be considered as CIA and remaining 50% as ESE based on the combinations of the evaluation components (CAC and CAT). 

Probability measures uncertainty and forms the foundation of statistical methods. This course makes students use measure-theoretic and analytical techniques for understanding probability concepts.

Sets – functions - Sigma field – Measurable space – Sample space – Measure – Probability as a measure - Inverse function - Measurable functions – Random variable - Induced probability space - Distribution function of a random variable: definition and properties.

Expectation and moments: Definition and properties – Probability generating function - Moment generating functions – quantile of order p - Moment inequalities: Markov’s, Chebychev’s, Holder, Jenson and basic inequalities - Characteristic function and properties (idea and statement only).

Random vectors – joint distribution function – joint moments - Conditional probabilities - Bayes’ theorem – conditional distributions – independence - Conditional expectation and its properties

Modes of convergence: Convergence in probability, in distribution, in rth mean, almost sure convergence and their inter-relationships - Convergence theorem for expectation

Law of large numbers - Covergence of series of independent random variables - 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.

Probability distributions are used in many real-life phenomena. This course makes students understand different probability distributions and model real-life problems using them.

Modified power series family and properties - Binomial - Geometric- Negative binomial, Logarithmic series - hypergeometric distribution and its properties.

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

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 distribution - Dirichlet distribution.

Quadratic forms in normal variables: distribution and roperties - Cochran’ theorem: applications.

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

Matrix operations - Linear equations - row reduced and echelon form - Homogenous system of equations - Linear dependence

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

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.

This course aims to impart the concepts of survey sampling theory and the analysis of complex surveys, including methods of sample selection, estimation, sampling variance, and standard error of estimation in a finite population, development of sampling theory for use in sample survey problems and sources of errors in surveys.

 Getting started with R - installing R and R studio - getting help - installing and loading packages - simple arithmetic calculations - data structure – expressions - conditional statements– functions – loops - R–markdown Practical Assignments: 1. R program to illustrate different data structures 2. Defining functions and making the report in markdown

Population - sample - sampling vs census - simple random 

sampling (SRS): with 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.

Practical Assignments:

3. Illustration of simple random sampling schemes

4. Illustration of stratified random sampling schemes

 Unbiased ratio type estimator - ratio estimator in stratified random sampling - Difference estimator and Regression estimator - comparison of regression estimator with mean per unit and ratio estimator - regression estimator in stratified random sampling. Practical Assignments: 5. Estimation using ratio estimator 6. Estimation using regression estimator 7. Ratio estimator and regression estimator in stratified sampling

Varying probability sampling designs With and without replacement sampling schemes: PPS and PPSWR schemes - Selection of samples - estimators: ordered and unordered estimators - Πps sampling schemes. Practical Assignments: 8. Exercise on the PPS scheme 9. Exercise on the PPSWR scheme 10. 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 - CO2, CO3, CO4 Department of Statistics and Data Science CHRIST (Deemed to be University) 21 determination of optimum cluster size - Sampling and nonsampling errors Practical Assignments: 11. Exercise on the systematic sampling scheme 12. Exercise on cluster sampling

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

3. de Micheaux, P. L., Drouilhet, R., & Liquet, B. (2013). The R software. Springer. New York.

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.

6. Lumley, T. (2011). Complex surveys: a guide to analysis using R. John Wiley & Sons

ESE 50%

This course equips students with programming skill in Python and associated statistical libraries and to apply in data analysis.

 Installing Python - basic syntax - interactive shell - editing, saving and running a script. The concept of data types - variables - assignments - mutable type - immutable types - arithmetic operators and expressions - comments in the program - understanding error messages - Control statements - operators.

Practical Assignments:

1. Lab exercise on data types

2. Lab exercise on arithmetic operators and expressions

3. Lab exercise on Control statements.

Introduction to functions - inbuilt and user defined functions - functions with arguments and return values - formal vs actual arguments - named arguments - Recursive functions - Lambda function - OOP Concepts - classes - objects - attributes and methods - defining classes - inheritance - polymorphism.

Practical Assignments:

4. Lab exercise on inbuilt and user-defined functions

5. Lab exercise on Recursive and Lambda function

6. Lab exercise on OOP Concepts

Introduction to Pandas - Pandas data series - Pandas data frames - data handling - grouping - Descriptive statistical analysis and Graphical representation

Practical Assignments:

7. Lab exercise on Pandas data series, frame, handling and grouping

8. Lab exercise on statistical analysis 

 Hypothesis testing - data modeling - linear regression models - logistic regression model.

Practical Assignments:

9. Lab exercise on Hypothesis testing

10. Lab exercise on regression modellin

Line graph - Bar chart - Pie chart - Heat map - Histogram - Density plot - Cumulative frequencies - Error bars - Scatter plot - 3D plot.

Practical Assignments:

11. Lab exercise on graphical and diagrammatic representation.

12. Lab exercise on the density plot

13. Lab exercise on scatter and 3D plo

2.An Introduction to Statistics with Python. Springer International Publishing.

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

ESE 50%

To acquaint students with different methodologies in statistical research and to make them prepare 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: installing packages and editor, preparing title page - mathematical expressions - tables - importing graphics - bibliography - 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%

This course is designed to provide the strong conceptual foundations of testing of hypotheses, procedures for testing hypotheses, likelihood ratio tests, sequential tests and non-parametric tests.

Monotone Likelihood Ratio (MLR) property - Testing in one-parameter exponential families - Generalised Neyman-Pearson lemma (Statement only) - Unbiased and invariant tests - Locally most powerful tests

One-sided uniformly most powerful tests - Unbiased and Uniformly Most Powerful Unbiased tests for different two-sided hypothesis - Extension of these results to Pitman family when only upper or lower end depends on the parameters - UMP test from α-similar tests and α-similar tests with Neyman structure.

Likelihood ratio test (LRT) - asymptotic properties - LRT for the parameters of binomial and normal distributions - Generalized likelihood ratio tests - Chi-

Square tests - t-tests - F-tests

Non-parametric tests: Sign test - Chi-square tests - Kolmogorov-Smirnov one sample and two samples tests - Median test - Wilcoxon Signed Rank test - Mann- Whitney U-test - Test for Randomness - Runs up and runs down test - Wald–Wolfowitz run test for equality of distributions - Kruskal–Wallis one-way analysis of variance - Friedman’s two-way analysis of variance - Power and asymptotic relative efficiency.

Large sample confidence interval - shortest length confidence interval - Methods of finding confidence interval: Inversion of the test statistic, pivotal quantities,

pivoting CDF- evaluation of confidence interval: size and coverage probability.

2. Rajagopalan, M. and Dhanavanthan, P. (2012). Statistical Inference, PHI Learning Pvt Ltd, New Delhi.

3. Kendall, M.G. and Stuart, A. (1967). The Advanced Theory of Statistics, vol 2, 2nd edition. Mc-Millan, New York.

4. Kale, B. K. and Muralidharan, K. (2015). Parametric Inference: An Introduction. Alpha Science Int. Ltd.

5. Mukhopadhyay, P.(2015): Mathematical Statistics, Books and Allied (P) Ltd., Kolkata.

6. Gibbons J.K. (1971). Non-Parametric Statistical Inference, McGraw Hill.

7. Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 2/e, JohnWiley, NewYork.

ESE - 50%

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

A sequence of random variables - definition and classification of the stochastic process - autoregressive processes and Strict Sense and Wide Sense stationary processes.

Markov Chains: Definition, Examples - Transition probability matrix -  Chapman-Kolmogorv 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, inter-arrival time 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 - Generalisations and variations of renewal processes -  Brownian motion - Introduction to Markov renewal processes.

2.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.

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

ESE 50%

Regression models are mainly used to establish a relationship among variables and predict future values. It got applications in various domains such as finance, life science, management, psychology, etc. This course is designed to impart knowledge of statistical model building using regression techniques. 

 Simple and multiple - Least squares estimation - Properties of the estimators - Maximum likelihood estimation - Estimation with linear restrictions - Hypothesis testing - confidence intervals.

Practical Assignments:

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

2. Construct a confidence interval for the simple linear model

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

4. Construct a confidence interval for multiple linear

 Residual analysis - Departures from underlying assumptions - Effect of outliers - Collinearity - Nonconstant variance and serial correlation - Departures from normality - Diagnostics and remedies. Practical Assignments: 5. Carry out residual analysis and validate the model assumptions. 6. Construct residual plots for checking outliers, leverage, and influential points. 7. Checking the assumption of homoscedasticity and its remedial measures 8. Detecting multicollinearity and its remedial measures.

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

Practical Assignments:

9. Selecting the best model using step-wise regression.

10. Selecting the best model using Forward and backward 

Nonlinear regression Introduction to general non-linear regression - least-squares in non-linear case - estimating the parameters of a non-linear system - re-parameterization of the model - logistic regression- Non-linear growth models

Practical Assignments:

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

 Linear absolute deviation regression - M estimators: properties - Breakdown and Efficiency points - Bootstrapping in Regression- Jackknife techniques and least squares approach based on M-estimators.

Practical Assignments:

12. Illustrate resampling procedures in regression models.

13. Build a regression model robust regression procedure.

1. S.N Sivanandam, S.N Deepa (2018). Principles of soft computing. Wiley India.
2. S Lovelyn Rose, L Ashok Kumar, Karthika Renuka (2019). Deep Learning using Python. Wiley India.

3. Francois Chollet (2017). Deep Learning with Python. Manning Publishing.

4. Andreas C. Muller & Sarah Guido (2017). Introduction to Machine Learning with Python. O’Reilly Media, Inc.

5. Chatterjee, S., & Hadi, A. S. (2015). Regression analysis by example. John Wiley & Sons.

6. Draper, N. R., & Smith, H. (2014). Applied regression analysis. 3rd edition. John Wiley & Sons. 

ESE 50%

This course provides a strong foundation for data science and application area related to it and caters for the underlying core concepts and emerging technologies in data science.

Definition – Big Data and Data Science Hype – Why data science – Getting Past the Hype – The Current Landscape – Who is a Data Scientist? - Data Science Process Overview – Defining goals – Retrieving data – Data preparation – Data exploration – Data modeling – Presentation. Data science ethics – valuing different aspects of privacy – The five C’s of data.

MACHINE LEARNING Machine learning – Modeling Process – Training model – Validating model – Predicting new observations –Supervised learning algorithms – Unsupervised learning algorithms

Practical Assignments:

1. Implement any one supervised algorithm

2. Implement any one unsupervised algorithm

INTRODUCTION TO DATA WAREHOUSE Defining Features, Database and Data Warehouses, Architectural Types, Overview of the Components, Metadata in the Data warehouse, The Star Schema, Star Schema Keys, Advantages of the Star Schema, Star Schema: Examples, Snowflake Schema, Aggregate Fact Tables, ETL process, Reporting services.

Practical Assignments:

11. Analysis Services

12. ETL process

13. Reporting Service

[2] Mining of Massive Datasets, Jure Leskovec, Anand Rajaraman, Jeffrey David Ullman, Cambridge University Press, 2nd edition, 2014

[3] Sinan Ozdemir, Principles of Data Science learn the techniques and math you need to start making sense of your data. Birmingham Packt December, 2016. [4]D J Patil, Hilary Mason, Mike Loukides, Ethics and Data Science, O’ Reilly, 2018. 

This course provides an understanding of various statistical methods in describing and analyzing biological data. Students will be equipped with an idea about the applications of statistical hypothesis testing, related concepts and interpretation of biological data.

Presentation of data - graphical and numerical representations of data - Types of variables, measures of location - dispersion and correlation - inferential statistics - probability and distributions - Binomial, Poisson, Negative Binomial, Hyper geometric and normal distribution. Sensitivity, Specificity, Positive predictive value, Negative predictive value. ROC Curves.

Practical Assignments:

1. Exercise on the representation of data

2. Exercise on reporting data by descriptive statistics

Parametric methods - one sample t-test - independent sample t-test - paired sample t-test - one-way analysis of variance - two-way analysis of variance - analysis of covariance - repeated measures of analysis of variance - Pearson correlation coefficient - Non-parametric methods: Chi-square test of independence and goodness of fit - Mann Whitney U test - Wilcoxon signed-rank test - Kruskal Wallis test - Friedman’s test - Spearman’s correlation test.

Practical Assignments:

1. Exercise on various parametric methods of analysis

2. Exercise on various non-parametric methods of analysis

Review of simple and multiple linear regression - introduction to generalized linear models - parameter estimation of generalized linear models - models with different link functions - binary (logistic) regression - estimation and model fitting - Poisson regression for count data - mixed effect models and hierarchical models with practical examples.

Practical Assignments:

1. Exercise on simple linear and multiple linear regression

2. Exercise on logistic regression

3. Exercise on Poisson regression

Introduction to epidemiology, measures of epidemiology, observational study designs: case report, case series correlational studies, cross-sectional studies, retrospective and prospective studies, analytical epidemiological studies-case control study and cohort study, odds ratio, relative risk, the bias in epidemiological studies.

Practical Assignments:

1. Exercise on analysis of observational study data

2. Exercise on analysis of cross-sectional study data

3. Exercise on analysis of case-control study data

4. Exercise on analysis of cohort study data

2. Dobson, A. J., & Barnett, A. G. (2018). An introduction to generalized linear models. CRC press.

3. Gordis, L. (2013). Epidemiology e-book. Elsevier Health Sciences.

2. David Moore S. and George McCabe P., (2017) Introduction to practice of statistics, 9th Edition, W. H. Freeman.

3. Sundar Rao and Richard J., (2012) Introduction to Biostatistics and research methods, PHI Learning Private limited, New Delhi.

4. Abhaya Indrayan and Rajeev Kumar M., (2018) Medical Biostatistics, 4th Edition,Chapman and Hall/CRC Press.

5. Gordis Leon (2018), Epidemiology, 6th Edition, Elsevier, Philadelphia

6. Park K., (2019), Park's Text Book of Preventive and Social Medicine, Banarsidas Bhanot, Jabalpur.

ESE -50%

This course is designed to train the students to develop their modelling skills in mathematics through various methods of optimization. The course helps the students to understand the theory of optimization methods and algorithms developed for solving various types of optimization problems.

Introduction to optimization – convex set and convex functions –simplex method: iterative nature of simplex method – additional simplex method: duality concept - dual simplex method - generalized simplex algorithm - revised simplex method: revised simplex algorithm – development of the optimality and feasibility Conditions.

Practical Assignments:

1. Formulate the LPP.

2. Solve the LPP using simplex method.

3. Solve the LPP using revised simplex method.

Branch and bound algorithm – cutting plane algorithm – transportation problem: north-west method, least-cost method, vogel’s approximation and method of multipliers – assignment problem: mathematical statement, Hungarian method, variation of assignment problems.

Practical Assignments:

4. Solve integer LPP by cutting plane method.

5. Formulate and solve transportation problems.

6. Formulate and solve assignment problems.

Introduction – unimodal function – one-dimensional optimization: Fibonacci method – golden Section Method – quadratic interpolation methods - cubic interpolation methods – direct root method: newton method and quasi newton method – Multidimensional unconstrained optimization: univariate method– Hooks and Jeeves method – Fletcher – Reeves method - Newton’s method and quasi newton’s method.

Practical Assignments:

7. Solve a non LPP problem.

8. Solve an unconstrained optimization problem by a univariate method

Single variable optimization – multivariable optimization with no constraints: semi-definite case and saddle point – multivariable optimization with equality constraints: direct substitution –method of constrained variation – method of Lagrange multipliers - Kuhn-Tucker conditions - constraint qualification – convex programming problem.

Practical Assignments:

9. Solve a single variable optimization problem.

10. Solve multivariable optimization problems with equality constraints.

11. Solve a convex optimization problem.

Unconstrained minimization problem – solution of an unconstrained geometric programming problem using arithmetic- geometric inequality method – primal dual relationship -constrained minimization - dynamic programming: Dynamic programming algorithm – solution of linear programming problem by dynamic programming.

Practical Assignments:

12. Formulate and solve a dynamic programming problem.

13. Solve LPP through dynamic programming problems.

14. Solve a geometric programming problem.

2. S. S. Rao (2019), Engineering Optimization, 5th Edition, New Age International Pvt. Ltd., Publishers, Delhi.

2. Hadley, G. (2002), Linear Programming, Addison Wesley.

3. G. Srinivasan (2007), Operations Research: Principles & Applications, Prentice Hall of India, New Delhi, India.

ESE-50%

This course is designed to provide the strong conceptual foundations of testing of hypothesis, procedures of testing hypothesis, likelihood ratio tests, sequential tests and non-parametric tests

Monotone Likelihood Ratio (MLR) property - Testing in one-parameter exponential families - Generalised Neyman-Pearson lemma (Statement only) - Unbiased and invariant tests - Locally most powerful tests

One-sided uniformly most powerful tests - Unbiased and Uniformly Most Powerful Unbiased tests for differenttwo-sided hypothesis - Extension of these results to Pitman family when only upper or lower end depends onthe parameters - UMP test from α-similar tests and α-similar tests with Neyman structure.

Likelihood ratio test (LRT) - asymptotic properties -LRT for the parameters of binomial and normal distributions - Generalized likelihood ratio tests - Chi-Square tests - t-tests - F-tests

Non-parametric tests: Sign test - Chi-square tests - Kolmogorov-Smirnov one sample and two samples tests -Median test - Wilcoxon Signed Rank test - Mann-Whitney U-test - Test for Randomness - Runs up and runs down test - Wald–Wolfowitz run test for equality of distributions - Kruskal–Wallis one-way analysis of variance - Friedman’s two-way analysis of variance -Power and asymptotic relative efficiency.

 Large sample confidence interval - shortest length 

confidence interval - Methods of finding confidence 

interval: Inversion of the test statistic, pivotal quantities, 

pivoting CDF- evaluation

2nd edition. Mc-Millan, New York.

4. Kale, B. K. and Muralidharan, K.  (2015). Parametric Inference: An Introduction. Alpha Science Int. Ltd.
5. Mukhopadhyay, P.(2015): Mathematical Statistics, Books and Allied (P) Ltd., Kolkata.
6. Gibbons J.K. (1971).  Non-Parametric Statistical Inference, McGraw Hill.Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 2/e, JohnWiley, NewYork.

This   course   considers   statistical   techniques   to    evaluate    processes    occurringthrough   time.   It   introduces   students   to    time    series    methods    and    theapplicationsofthesemethodstodifferenttypesof   data   in   various   fields.   Timeseriesmodelling   techniques   including   AR,   MA,   ARMA,   ARIMA   and   SARIMAwillbeconsidered   with   reference   to   their   use   in   forecasting.   The   objective   ofthiscourseistoequipstudentswith   various   forecasting   techniques   and   tofamiliarize themselves

Stochastic Process - Time series as a discrete parameter stochastic process Auto – Covariance - Autocorrelation and their properties - Exploratory time series analysis graphical analysis - classical decomposition model - concepts of trend, seasonality and cycle - Estimation of trend and seasonal components-Elimination of trend. Seasonality - Method of differencing - Moving average smoothing, Method of seasonal differencing

Practical Assignments:

1.Graphical representation of time series, plots of ACF and

PACF and their interpretation

2.Examples of trend, seasonal and cyclical time series and estimation of trend and

seasonal components

3.Exercise on Moving average smoothing to eliminate trend and illustration on the

method of differencing to eliminate trend and seasonality. 4.Exercise on least-square fitting to estimate and eliminate the trend component

Stationary time series models - Concepts of weak and strongstationarity – General linear Process - Auto-Regressive(AR),MovingAverage(MA)andAuto-RegressiveMovingAverage(ARMA)processes – theirproperties -conditionsforstationarity and invertibility -model identification based onACF and PACF- Maximum likelihood estimation – YuleWalkerEstimation-orderselection(AICandBIC)-Residual Analysis- - Box Jenkins methodology to theidentificationofstationarytime

seriesmodels

PracticalAssignments:

5.ExerciseonfittingARmodel

6.  ExerciseonfittingMAmodel

7.  Exerciseonfitting ARMAmodel

8.Model-identificationusing ACFandPACF,ModelselectionusingAIC andBIC

9.ResidualanalysisanddiagnosischeckforAR, MAandARMA models

Concept of non-stationarity - Spurious trends and regressions- unit root tests : Dickey-Fuller (DF) test - Augmented Dickey- Fuller(ADF) test – Auto-Regressive Integrated Moving

Average(ARIMA(p,d,q)) models - Difference equation form

of ARIMA- Random shock form of ARIMA - An inverted form of ARIMA

Practical Assignments:

10.Exercise on the identification of non-stationary series from various plots.

11.Exercise on testing non-stationarity using ADF test,

Exercise on fitting ARIMA models.

12.Residual analysis and diagnosis check for ARIMA model.

Analysis of seasonal models - parsimonious models for seasonal time series - Seasonal unit root test (HEGY test) - General multiplicative seasonal models - Seasonal ARIMA models - estimation - Residual analysis for seasonal time series.

 Practical Assignments:

 13.  Exercise on the identification of additive and Multiplicative time series

 14.Exercise on testing the presence of seasonality and on fitting Seasonal ARIMA

  models

 15.  Residual analysis and diagnosis check for Seasonal ARIMA model

In sample and out of sample forecast - Simple exponential and

moving average smoothing - Holt Exponential Smoothing - Winter exponential smoothing - Forecasting trend and

seasonality in Box Jenkins model:- Method of minimum mean squared error (MMSE) forecast - their properties – forecast- error

 Practical Assignments:

16.Exercise on Simple exponential smoothing and Holt Exponential Smoothing

17.Exercise on Winters exponential smoothing. 18.Exercise on forecasting using ARIMA models. 19.Exercise on forecasting using seasonal ARIMA models.

2.  Chatfield, C., & Xing, H. (2019). The analysis of time series: an introduction with R.CRCPress.

2.  Brockwell, P. J., & Davis, R. A. (2016). Introduction to time series and forecasting.springer.

ESE 50%

Machine learning has a wide array of applications that belongs to different fields, such as biomedical research, reliability of large structures, space research, digital marketing, etc. This course will equip students with a wide variety of models and algorithms for machine learning and prepare students for research or industry application of machine learning techniques. 

Statistical learning: definition-prediction accuracy and model interpretability-supervised and unsupervised learning-assessing model accuracy- important problems in data mining: classification, regression, clustering, ranking, density estimation- Concepts: training and testing, cross-validation, overfitting, bias/variance tradeoff, regularized learning equation- simple and multiple linear regression algorithms

 Practical Assignments:

1.     Lab exercise on data preparation and using simple linear regression

2.     Lab exercise on model assessment simple linear regression

3.     Lab exercise on data preparation with multiple linear regression

Logistic model- training and testing the model-linear discriminant analysis-quadratic discriminant analysis- Use of Bayes’ theorem-k- nearest neighbours - Naive Bayes’- Adaboost

 Practical Assignments:

4.     Lab exercise on the logistic model

5.     Lab exercise on discriminant analysis

6.     Lab exercise on  Naïve Bayes’ and k-NN classifiers

7.     Lab exercise on  Adaboost

Optimal model-shrinkage methods: ridge and lasso regression-Dimension reduction methods: principal component (PC) regression and partial least square (PLS) regression: Non-linear models: regression splines-polynomial – Generalized additive models

 Practical Assignments:

8.     Lab exercise on ridge regression

9.     Lab exercise on Lasso regression

10.     Lab exercise on PC regression

11.     Lab exercise on PLS regression

Decision tree-regression trees - bagging - random forests - boosting - classification trees-boosting-tree vs linear models

 Practical Assignments:

12.     Lab exercise on decision trees

13.     Lab exercise on regression trees

14.     Lab exercise on random forests

15.     Lab exercise on classification trees

Maximal classifier-support vector classifiers-support - rank boost (ranking algorithm) - hierarchical Bayesian modelling for density - resampling techniques-bootstrap- clustering algorithms: K-means algorithm. 

Practical Assignments:

16.     Lab exercise on SVM classifier

17.     Lab exercise on rank boost algorithm

18.     Lab exercise on kernel density estimation

19.     Lab exercise on k-means clustering

2. Müller, A. C., & Guido, S. (2016). Introduction to machine learning with Python: a guide for data scientists. “O’Reilly Media, Inc.".
3. Murphy, K. P. (2012). Machine learning: a probabilistic perspective. MIT press 

Java programming is essential tool for data science and statistical computation. This course enables students to create programs in Java which can be used for statistical analysis

Introduction Installing java Development kit - Java JVM, JRE and JDK - Working with the Terminal –First Program – primitive data Types:– Variables and constants - Operators - Input and Output - Expressions & Blocks – Comment, Conditional Statements.

Practical Assignments:

1. Lab exercise on data types using various operators in Java

2. Lab exercise on control structures

 Primitive data structure and Non-primitive data structure - Array Processing - Multi-dimensional Arrays - String Methods - String Manipulation – Regular Expressions

Practical Assignments:

3. Lab exercise on Arrays

4. Lab exercise on Strings 

Objects - Encapsulation – Classes – Inheritance – Inheritance Hierarchies - Polymorphism – Abstraction – Protected Classes – Exceptions –Assertions –Loggin – Generic Programming- Operations in Dictionaries Practical Assignments: 5. Lab exercise on classes and objects 6. Lab exercise on exception handling and assertions 7. Lab exercise on dictionaries

8. Lab exercise on strings and regular expressions

9. Lab exercise on dictionaries

Data Science Libraries- data processing library, Math and Stats libraries, machine learning and data mining libraries. Standard Java Library -Collections-Input-Output,Acessing Data-CSV,JSON,DataFrames.

Practical Assignments:

8. Lab exercise on handling CSV files

9. Lab exercise on handling JSON.

Statistical Data Analysis and Data Visualization Working with mean median mode, hypothesis testing, Regression Analysis. Understanding plots and graphs

Practical Assignments:

10. Lab exercise on data visualization

This course is designed to train the students in the design and conduct of clinical trials and provide knowledge about the methods of statistical data analysis of clinical trials.

Historical background of clinical trials - the need for clinical trials - ethics and planning of clinical trials - main features of study protocol - the selection of study subjects -treatment schedule - evaluation of patient response - follow-up studies -GCP/ICH guidelines

Different phases of clinical trials: phase I, phase II, phase III, phase IV - Basic study designs - randomized controlled trials - non-randomized concurrent controlled trials -historical controls - cross over design - withdrawal design - hybrid designs – group allocation designs and studies of equivalency.

Fixed allocation randomization - stratified randomization - adaptive randomization - unequal randomization - Intervention and placebos - blinding in clinical trials: unblended trials - single-blind trials - double- blind trials and triple-blind trials.

Various methods for determining sample size for clinical trials: method for dichotomous response variable - continuous response variable - repeated measures – cluster randomization and equivalency of intervention - Multicenter trials.

Design of case report form - data collection - intention to treat analysis and per-protocol analysis - interim analysis - reporting adverse events - issues in data analysis - non- adherence - poor quality and missing data

2. Meinert Curtis L., (2012), Clinical Trials-Design, conduct and Analysis, 2 nd Edition Oxford University Press, New York.

2. Tom Brody (2016), Clinical Trials – Study Design, End points, Biomarkers, Drug safety, FDA and ICH guidelines.

3. Jo Ann Pfeiffer and Cris Wells (2017), A practical Guide to managing Clinical Trials, CRC Press, Taylor and Francis Group

This course will provide knowledge in different probability models in the reliability evaluation of the system and its components. Reliability engineering is applied in the industry to reduce failures, ensure effective maintenance and optimize repair time.

Reliability of a system - failure rate - mean, variance and percentile residual life: identities connecting them - notions of ageing - IFR, IFRA, NBU, NBUE, DMRL, HNBUE, NBUC, etc. and their mutual implications - TTT transforms and characterization of ageing classes.

Practical Assignments:

1. Exercise on failure rate function and mrl function

2. Exercise on comparison ageing classes 

Non-monotonic failure rates and mean residual life functions - study of lifetime models: exponential, Weibull, lognormal, generalized Pareto, gamma with reference to basic concepts and ageing characteristics - bathtub and upside-down bathtub failure rate distributions

Practical Assignments:

3. Exercise on exponential lifetime model

4. Exercise on Weibull lifetime model

5. Exercise on bathtub shaped lifetime model 

Reliability systems with dependents components: Parallel and series systems, k out of n Systems - ageing properties with dependent and independents components - concepts and measures of dependence on reliability - RCSI, LCSD, PF2, WPQD.

Practical Assignments:

6. Exercise on reliability evaluation of series system

7. Exercise on reliability evaluation of a parallel system

8. Exercise on reliability evaluation of k out of n system

9. Exercise on reliability evaluation of dependent component system 

Reliability estimation using MLE: exponential, Weibull and gamma distributions based on censored and non-censored samples - UMVU estimation of reliability function - Bayesian reliability estimation of exponential and Weibull models

Practical Assignments:

10. Exercise on ML estimation under non-censored samples.

11. Exercise on ML estimation under censored samples.

12. Exercise on Bayesian estimation of reliability. 

Life testing: basics – modelling lifetime – Accelerated Life Time (ALT) models- cumulative exposure models (CEM) - exponential CEM – stress-strength reliability – exponential stress-strength model.

Practical Assignments:

13. Exercise on basic life testing procedure.

14. Exercise on exponential CEM model.

15. Exercise on stress-strength reliability.

2.     Bain, L. (2017). Statistical analysis of reliability and life-testing models: theory and methods. Routledge.

ESE 50%

This will equip the student to apply statistical methods they have studied in various courses

and present their work through research articles.

1.Research Problem Identification 

2.Formulation

https://kp.christuniversity.in/KnowledgePro/images/Regulations/CRCE.pdf

Research Articles from academic databases.

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.

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.

Practical Assignments:

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. 

Practical Assignments:

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.

Practical Assignments:

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. 

Practical Assignments:

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.

Practical Assignments:

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.

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

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

ESE - 50%

This course will provide students with a mathematical background of various basic designs involving one-way and two-way elimination of heterogeneity and characterization properties. To prepare the students in deriving the expressions for analysis of experimental data and selection of appropriate designs in planning a scientific experimentation

Basic principles of design of experiments - Randomization - Replication and Local control - Uniformity trials - Size and Shape of plots and blocks - Elements of linear estimation - Analysis of variance - Completely Randomized Design (CRD) - Randomized Complete Block Design (RCBD) and Latin Square Design (LSD) -  Missing plot techniques.

Analysis of covariance - Ancillary/Concomitant variable and study variable - Linear model for ANCOVA - Adjustment of treatment sum of squares in ANCOVA - One - Way and two-way classification with a single concomitant variable in CRD and RCBD designs.

Factorial experiments - Simple experiment (single factor) vs Factorial experiments - Mixed and Fixed factor experiments - Treatment combination in a factorial experiment - Simple effect - Main effect and Interaction effect in a factorial experiment - Yates method of computing factorial effects totals - Complete and partial confounding in symmetrical factorial experiments (22, 23, 33, 2nand 3n series) - Gain in the factorial experiments.

Split - Plot, Split - Split plot and Strip - Plot (Split Block) design - Situation for the usage of the design - Layout and analysis of the designs - Difference in the error components in the designs - Selection of factor for allocation in plots (main/sub) - Combined experiments -  Cross - Over designs.

Balanced Incomplete Block (BIB) designs - General properties and Analysis with and without recovery of information - Construction of BIB designs - Parameter relationship - Intra and inter-block Analysis - Partially Balanced Incomplete Block Design (PBIBD) - Youden square designs - Lattice designs.

Montgomery, D.C. (2019). Design and Analysis of Experiments. John Wiley and Sons, Inc. New York

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

CO1: List different stochastic models.

CO2: Identify ergodic Markov chains.

CO3: Analyse queuing models using continuous-time Markov chains.

CO4: Apply Brownian motion in finance problems.

A sequence of random variables - definition and classification of the stochastic process - autoregressive processes and Strict Sense and Wide Sense stationary processes.

Markov Chains: Definition, Examples - Transition probability matrix -  Chapman-Kolmogorv 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, inter-arrival time 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 - Generalisations and variations of renewal processes -  Brownian motion - Introduction to Markov renewal processes.

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

1. Feller, W. (2008) An Introduction to Probability Theory and its Applications, Volume I&II , 3^rd Ed., Wiley Eastern.
2. J. Medhi (2009) Stochastic Processes, 3rd Edition, New Age International.
3. Dobrow, R.P. (2016), Introduction to Stochastic Processes with R, Wiley Eastern.
4. Cinlar, E. (2013). Introduction to stochastic processes. Courier Corporation.

The objective of this course is to provide fundamental knowledge of neural networks and deep learning. This course gives a brief idea of the basics of neural networks, shallow and deep neural networks and other methods to build various research projects. 

Fundamental concepts of Artificial Neural Networks (ANN) - Biological neural networks - Comparison between biological neuron and artificial neuron - Evolution of neural networks - Scope and limitations of ANN - Basic models of ANN - Learning methods - Activation functions - Important terminologies of ANN: Weights - Bias - Threshold - Learning Rate - Momentum factor - Vigilance parameters.

Concept of supervised learning algorithms - Perceptron networks - Adaptive linear neuron (Adaline) - Multiple adaptive linear neuron - Back-Propagation network: Learning factors - Initial weights - Learning rate ɑ - Momentum factor - Generalization - Training and testing of the data.

Concept of unsupervised learning algorithms - Fixed weight competitive net: Maxnet - Mexican Hat net - Hamming networks - Kohonen self-organizing feature maps - Learning vector quantization.

Introduction - Components of Convolution Neural Networks (CNN) architecture: Padding - Strides - Rectified linear unit layer - Exponential linear unit - Pooling - Fully connected layers  - Local response normalization - Hierarchical feature engineering -  Training CNN using Backpropagation through convolutions - Case studies: AlexNet - GoogLeNet.

Stateless algorithms: Naive algorithms - Upper bounding methods - Simple reinforcement learning for Tic-Tac-Toe - Straw-Man algorithms - Bootstrapping for value function learning - One policy versus off policy methods: SARSA - Policy gradient methods: Finite difference method - Likelihood ratio method - Monte Carlo tree search. 

ESE-50%

To enable the students to understand and apply different concepts of statistical genetics in large populations with selection, mutation and migration. The students would be exposed to the physical basis of inheritance, detection and estimation of linkage, estimation of genetic parameters and development of selection indices.

Physical basis of inheritance - Analysis of segregation - Detection and Estimation of linkage for qualitative characters - Amount of information about linkage - Combined estimation - Disturbed segregation.

Practical Assignments:

1.      Analysis of segregation, detection and estimation of linkage.

2.      Estimation of Amount of information about linkage.

3.      Calculation of combined estimationof linkage.

Gene and genotypic frequencies - Random mating and Hardy-Weinberg law - Application and extension of the equilibrium law - Fisher’s fundamental theorem of natural selection - Disequilibrium due to linkage for two pairs of genes - Sex - Linked genes.

Practical Assignments:

4. Estimation of disequilibrium due to linkage for two pairs of genes.

5. Estimation of path coefficients.

6. Estimation of equilibrium between forces in large populations.

Forces affecting gene frequency: Selection - Mutation and Migration - Equilibrium between forces in large populations - Polymorphism.

Practical Assignments:

7. Estimation of changes in gene frequency due to systematic forces.

8. Estimation of the Inbreeding coefficient.

Polygenic system for quantitative characters - Concepts of breeding value and Dominance deviation - Genetic variance and its partitioning.

Practical Assignments:

 9. Analysis of genetic components of variation.

10. Estimation of breeding values.

Correlation between relatives – Heritability - Repeatability and Genetic correlation - Response due to selection - Selection index and its applications in plants and animals improvement Programme - Correlated response to selection - Restricted selection index - Inbreeding and crossbreeding - Changes in mean and variance.

Practical Assignments:

11. Estimation of Heritability and repeatability coefficient,

12. Estimation of the genetic correlation coefficient.

2. Balding DJ, Bishop, M. and Cannings, C. (2001). Handbook of Statistical Genetics. John Wiley.

3. Shizhong Xu.(2013). Principles of Statistical Genomics. Springer.

4.Falconer, D.S. (2009). Introduction to Quantitative Genetics. English Language Book Society. Longman. Essex.

ESE- 50 %

This course is designed to equip students with the knowledge of actuarial models and their applications

CO2: Identify various actuarial models.

CO3: Illustrate survival models and life tables.

CO4: Interpret the real-life data based on exploratory data analysis.

CO5: Apply actuarial models to real-life data.

Utility theory-introduction - insurance and utility theory - models for individual claims and their sums - curtate future lifetime - the force of mortality - assumptions for fractional ages - some analytical laws of mortality - multiple life functions - joint life and last survivor status - insurance and annuity benefits through multiple life functions - evaluation for special mortality laws.

 Practical Assignments:

               1.      Problems based on multiple life functions

               2.   Illustrate discrete and continuous annuity benefits

Introduction to survival analysis - life table and its relation with survival function - examples - assumptions for fractional ages - estimate empirical survival and loss distribution using Kaplan-Meier estimator - Nelson Aalen estimator - Cox proportional hazards and Kernel density estimators.

Practical Assignments:

             3. Apply survival models to simple problems in long-term insurance, pensions and banking.

             4. Preparation of life tables based on the real life data.

             5. Estimation of survival distribution using Kaplan-Meier estimator, Nelson Aalen estimator, Cox proportional hazards and Kernel density estimators

Principles of actuarial modelling - stochastic and deterministic models - their advantages and disadvantages - frequency models: distributions suitable for modelling frequency of losses (Poisson, Binomial, negative binomial and geometric distributions) - fundamentals of aggregate models - computation of aggregate claims distributions and calculation of loss probabilities - evaluate the effect of coverage modifications (deductibles, limits and coinsurance) - inflation on aggregate models.

Practical Assignments:

6. Compute relevant moments, probabilities and other distributional quantities for collective risk models.

7. Compute aggregate claims, distributions and use them to calculate loss probabilities.

8. Evaluate the effect of coverage modifications and inflation on aggregate models.

Principles of compound interest -Nominal and effective rates of interest and discount - the force of interest and discount - compound interest - accumulation factor - continuous - compounding - life insurance - life annuities - net premiums - net premium reserves - some practical considerations - premiums that include expenses - general expenses - types of expenses - per policy expenses - claim amount distributions - approximating the individual model - stop-loss insurance.

Practical Assignments:

9.      Illustrate discrete and continuous insurance benefits

10.      Illustrate discrete and continuous annuity benefits

Data as a resource for problem-solving - exploratory data analysis: single and multiple linear regression - principal component analysis and survival analysis - statistical learning: difference between supervised and unsupervised learning - professional and risk management issues - ethical and regulatory issues involved in using personal data and extremely large data sets - visualizing data and reporting.

Practical Assignments:

11.      Apply principal component analysis to reduce the dimensionality of a complex data set.

12.      Fit a simple and multiple linear models to a data set and interpret the results.

13.      Fit a survival model to a data set and interpret the output.

2.         Grimmett Geoffery and David Stizaker (2001), Probability and random processes. Oxford University Press.

3.         J. Medhi, Stochastic Processes (2009), John Wiley.

ESE 50%

This will equip the student to apply statistical methods they have studied in various courses and present their work through research articles.

CO1: Apply statistical techniques to a real-life problem.

CO3: Interpret and conclude the statistical analysis scientifically.

CO4: Present the work done through presentation and research article. 

1. Apply various statistical methods in solving a real-life problem.

2. Comparison with the existing models or results.

ESE 50%

This course is to enhance the verbal and written presentation skills of students and to develop analytical skills as students learn new areas and ideas in Statistics. 

CO1: Demonstrate presentation and writing skills.

1. Prepare a report on a relevant topic.

2. Present it well before the class and panel members.

ESE 50%

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

CO2: Apply the idea of sampling distributions of difference statistics in testing of hypotheses.

CO3: Infer the concept of nonparametric tests for single sample and two samples.

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.

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.

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

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

 Continuous Sampling Plans (CSP): CSP-1- CSP-2 - CSP-3 - Skip-Lot Sampling Plans (SkSP): SkSP-1 - SkSP-2  with SSP as reference plan - Chain Sampling Plans (ChSP - 1) with SSP as reference plan - Tighten-Normal-Tighted (TNT) sampling plan with SSP as reference plan– Decision Lines.

[2]. Stochastic Processes, S.M Ross, Wiley India Pvt. Ltd, 2008.

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

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

ESE - 50%

The exposure provided to the multivariate data structure, multinomial and multivariate normal distribution, estimation and testing of parameters, various data reduction methods would help the students in having a better understanding of research data, its presentation and analysis. This course helps to understand multivariate data analysis methods and their applications in various research areas.

Basic concepts on multivariate variables - Multivariate normal distribution - Marginal and conditional distribution - Concept of random vector - Its expectation and Variance - Covariance matrix. Marginal and joint distributions - Conditional distributions and Independence of random vectors - Multinomial distribution - Characteristic functions in higher dimensions - Multiple regressions and multiple correlations - Partial regression and Partial correlation (illustrative examples).

Multivariate analysis of variance (MANOVA) and Covariance (MANCOVA) of one and two-way classified data with their interactions - Univariate and Multivariate Two-Way Fixed-effects Model with Interaction.

Wishart distribution (definition, properties) - Construction of tests - Union - Intersection and likelihood ratio principles - Inference on mean vector - Hotelling's T2- Comparing Mean Vectors from Two Populations - Bartlett’s Test. 

Concepts of discriminant analysis - Computation of linear discriminant function (LDF) - Classification between k multivariate normal populations based on LDF - Fisher’s Method for discriminating two or several populations - Evaluating Classification Functions - Probabilities of misclassification and their estimation - Mahalanobis D2.

Factor analysis: - Orthogonal factor model - Factor loadings - Estimation of factor loadings - Factor scores and Its applications.

Cluster Analysis: - Distances and similarity measures - Hierarchical clustering methods - K- Means method. 

Wiley. New York.

2. Johnson, R.A. and Wichern, D.W. (2018). Applied Multivariate Statistical Analysis.

6th edn. Prentice- Hall. London.

Mathematical Statistics. 2nd edn. John Wiley & Sons. New York.

2. Srivastava, M.S. and Khatri, C.G. (1979). An Introduction to Multivariate Statistics.

North Holland.

3. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley. New

York.

CIA-50%

ESE-50%

This course has been designed to train the students in handling different types of Big data sets and provide knowledge about the methods of handling these types of data sets.  

Concepts of Data Analytics: Descriptive, Diagnostic, Predictive, Prescriptive analytics -Big Data characteristics: Volume, Velocity, Variety, Veracity of data - Types of data: Structured, Unstructured, Semi-Structured, Metadata - Big data sources: Human-Human communication, Human-Machine Communication, Machine-Machine Communication - Data Ownership - Data Privacy.

Practical Assignments:

1. Setting up infrastructure and Automation environment

2. Case study for identifying Data Characteristics

Standard Big data architecture - Big data application - Hadoop framework - HDFS Design goal - Master-Slave architecture - Block System - Read-write Process for data - Installing HDFS - Executing in HDFS: Reading and writing Local files and Data streams into HDFS - Types of files in HDFS - Strengths and alternatives of HDFS - Concept of YARN.

Practical Assignments:

3.  Exercise on Installing HDFS

4.  Exercise on Reading and Writing Local files into HDFS

5.  Exercise on Reading and Writing Data streams into HDFS

Introduction to MapReduce - Sample MapReduce application: Wordcount - MapReduce Data types and Formats - Writing MapReduce Programming - Testing MapReduce Programs - MapReduce Job Execution - Shuffle and Sort - Managing Failures - Progress and Status Updates.

Practical Assignments:

6. Exercise on MapReduce applications

7. Exercise on writing and testing MapReduce Programs

8. Exercise on Shuffle and Sort

9. Exercise on Managing Failures

Stream processing Models and Tools - Apache Spark - Spark Architecture: Resilient Distributed Datasets, Directed Acyclic Graph - Spark Ecosystem - Spark for Big Data Processing: MLlib, Spark GraphX, SparkR, SparkSQL, Spark Streaming - Spark versus Hadoop

Practical Assignments:

10. Exercise on installing Spark

11. Exercise on Directed Acyclic Graph

12. Exercise on Spark using MLlib, Spark GraphX

13. Exercise on Spark using SparkR, Spark Streaming

Hive Architecture - Components - Data Definition - Partitioning - Data Manipulation - Joins, Views and Indexes - Hive Execution - Pig Architecture - Pig Latin Data Model - Latin Operators - Loading Data - Diagnostic Operators - Group Operators - Pig Joins - Row Level Operators - Pig Built-in function - User-defined functions - Pig Scripts.

Practical Assignments:

14. Exercise on Hive Architecture

15. Exercise on Pig Architecture

        3.Seema Acharya, Subhasini Chellappan (2019), Big Data and Analytics. 2nd Edition, Wiley India Pvt Ltd.

ESE - 50%

This course has been designed to train students in the applications of statistics in demographic studies

Definition and scope of demography -Importance of demographic analysis - Sources of demographic data - census, registration, ad-hoc surveys - Hospital records - Demographic profiles of the Indian Census

Practical Assignments:

1. Case study for demographic data

Complete life table and its main features - Uses of life table - Makehams and Gompertz curves - National life tables - UN model life tables - Abridged life tables - Stable and stationary populations.

Practical Assignments:

2.Exercise on lifetables

3.Exercise on Makehams and Gompertz curves

4.Exercise on abridged life tables

Crude birth rate - General fertility rate - Age-specific birth rate - Total fertility rate - Gross reproduction rate - Net reproduction rate.

Crude death rate - Standardized death rates - Age-specific death rates - Infant Mortality rate - Death rate by cause.

Practical Assignments:

1. Exercise on standardised death rates
2. Exercise on death rate by cause

Internal migration and its measurement - migration models - concept of international migration - Net migration - International and postcensal estimates - Projection method including logistic curve fitting - Decennial population census in India.

Practical Assignments:

1. Exercise on International and postcensal estimates

Exercise on Projection method

ESE50%

This course will equip students with a wide variety of statistical methods for modelling risk.

Distinguishing Characteristics Of Risk Analysis - Traditional Health Risk Analysis - Defining Risks: Source, Target, Effect, Mechanism - Basic Quantitative Risk Models - Risk as Probability of a Binary Event - Hazard Rate Models

Practical Assignments:

1. Lab exercise on the quantitative risk model.

2.. Lab exercise on the hazard rate model

Conditional Probability Framework for Risk Calculations - Population Risks Modeled by Conditional Probabilities - Trees, Risks and Martingales - Compartmental Flow Simulation Models - Monte Carlo Uncertainty Analysis - Introduction to Exposure Assessment - Uncertainty Analysis

Practical Assignments:

3.  Lab exercise on risk calculations.

4.  Lab exercise on risk modelling by conditional probabilities.

5. Lab exercise on  compartment flow simulation model.

6. Lab exercise on  Monte Carlo uncertainty analysis

Statistical Dose-Response Modeling - Exposure and Response Variables - Risk, Confidence Limits, and Model Fit - Model Uncertainty and Variable Selection - Dealing with Missing Data

Practical Assignments:

7.  Lab exercise on Dose-Response modelling.

8. Lab exercise on the estimation of risk and confidence limit.

9. Lab exercise on variable selection procedures.

10. Lab exercise on missing data algorithms

Statistical vs Causal Risk Modeling - Criteria for Causation - Epidemiological Criteria for Causation - Criteria for Inferring Probable Causation - Causal Graph Models and Knowledge Representation - Testing Hypothesized Causal Graph Structures - Causal Graphs in Risk Analysis - Probabilistic Inferences in DAG Models - Using DAG Models to Make Predictions

Practical Assignments:

11.  Lab exercise on causal risk models.

12. Lab exercise on causal graph models.

13. Lab exercise on Testing Hypothesized Causal Graph Structures.

14. Lab exercise on DAG models

Value Functions and Risk Profiles - Rational Individual Risk-Management via Expected Utility - EU Decision-Modeling Basics - Decision-Making Algorithms and Technologies - Axioms for EU Theories - Cognitive Heuristics and Biases Violate Reduction - Subjective Probability and Subjective Expected Utility (SEU)

Practical Assignments:

15. Lab exercise on EU decision modelling.

16. Lab exercise on optimization of decision-making algorithms

ESE - 50%

Students who complete this course will gain a solid foundation in how to apply and understand Bayesian statistics and how to understand Bayesian methods vs frequentist methods.  Topics covered include: an introduction to Bayesian concepts; Bayesian inference for binomial proportions, Poisson means, and normal means; modelling

Basics of minimaxity - subjective and frequentist probability - Bayesian inference -  prior distributions - posterior distributions -  loss function - the principle of minimum expected posterior loss - quadratic and other common loss functions - advantages of being Bayesian - Improper priors - common problems of Bayesian inference - Point estimators - Bayesian confidence intervals, testing - credible intervals                                                                                       

Practical Assignments:                                                                 

1. Construction of prior, conditional and posterior probabilities for the chosen data set
2. Computation minimum expected posterior loss.
3. Computation of Bayesian confidence intervals.

Two Equivalent Ways of Using Bayes' Theorem - Bayes' Theorem for Binomial with Discrete Prior - Important Consequences of Bayes' Theorem - and Bayes' Theorem for Poisson with Discrete prior.                                                        

Practical Assignments:                                                     

1. Bayes Classification
2. Examples on Binomial distribution with discrete prior.
3. Examples of Poisson distribution with discrete prior.

Using a Uniform Prior - Using a Beta Prior - Choosing Your Prior - Summarizing the Posterior Distribution - Estimating the Proportion - Bayesian Credible Interval

Comparing Bayesian and Frequentist Inferences for Proportion :

Frequentist Interpretation of Probability and Parameters - Point Estimation -  Comparing Estimators for Proportion - Interval Estimation - Hypothesis Testing - Testing a One-Sided Hypothesis - Testing a Two-Sided Hypothesis.                Bayesian Inference for Poisson: Some Prior Distributions for Poisson - Inference for Poisson Parameter.                       

Practical Assignments:                                                                 

1. Estimation of  binomial proportion using uniform prior distribution.
2. Estimation of binomial proportion using beta prior  distribution.
3. Estimation of Poisson parameter using some prior distributions

Bayes' Theorem for Normal Mean with a Discrete Prior - Bayes' Theorem for Normal Mean with a Continuous Prior - Normal Prior, Bayesian Credible Interval for Normal Mean - Predictive Density for Next Observation.        

Practical Assignments:     

1. Bayes estimator for Normal Mean with a Discrete Prior.
2. Bayes estimator for Normal Mean with a Continuous Prior.
3. Bayes Credible interval for the normal mean.
   

Analytic approximation - E-M Algorithm - Monte Carlo sampling - Markov Chain Monte Carlo Methods - Metropolis-Hastings Algorithm - Gibbs sampling: examples and convergence issues.                                     

Practical Assignments:     

1. E-M algorithm.
2. Monte-Carlo sampling.
3. Gibbs Sampling.
4. Markov Chain Monte-Carlo application.

2.  Jim, A. (2009). Bayesian Computation with R, 2nd Edition, Springer.