POST GRADUATE TEACHING DEPARTMENT OF STATISTICS,

RASHTRASANT TUKADOJI MAHARAJ NAGPUR

UNIVERSITY, NAGPUR

M. A. / M.Sc. STATISTICS

SEMESTER PATTERN SYLLABUS (CBCS)

As per G.R of higher & Technical Education, Maharashtra Govt. dated 16th May 2023

TO BE IMPLEMENTED

FROM

2023 – 24

&

ONWARDS PHASE WISE

 Choice Based Credit System for Post Graduate Courses

Scheme of Examination for

M. Sc. STATISTICS

Semester I

M.Sc. Statistics Semester I

Teaching scheme
Examination Scheme
(Hours/Week)
Minimum
Theory/Practical

Max.Marks
Passing

Credits

Duration in hrs.
Code Marks

Total Marks
Practical

Total
Theory

Practical
External

Internal

Theory
Marks

Ass
Paper1:Probability & - -
MST1T0 4 4 4 3 60 40 100 50
Distribution Theory
1
Paper 2: Statistical
MST1T0 Inference 4 - 4 4 3 60 40 100 50 -
2
Paper 3:
Electives
Industrial
Process & - -
MST1T0 4 4 4 3 60 40 100 50
Quality Control –
3
A
OR
Survival
Analysis – B
OR
Bioassay - C

Paper4:Research - -
MST1T0 4 4 4 3 60 40 100 50
Methodology
4
Practical 1 - 4 -
MST1P0 6 6 3 50 50 100 50
1
Practical 2 - 4 -
MST1P0 6 6 3 50 50 100 50
2
TOTAL 16 12 28 22 -- 340 260 600 200 100
Semester II

M.Sc. Statistics Semester II

Teaching
scheme Examination Scheme
(Hours/Week)
Theory/Practical

Max.Marks MinimumP

Duration in hrs.
Credits
Code assingMarks

TotalMarks
Practical
Theory

Total

Practical
External

Internal

Theory
Marks

Ass
MST2T01 Paper5:Sampling & 4 - 4 4 3 60 40 100 50 -
design of Experiment
Paper 6: 4 - 4 4 3 60 40 100 50 -
MST2T02
Stochastic
Processes &
Time Series
Paper7: Electives 4 - 4 4 3 60 40 100 50 -
Industrial Statistics
–A
OR
MST2T03 Reliability Theory –
B OR
Statistical Genetics
– C1/ Statistical
Ecology – C2
MST2T04 Paper8: - 8 8 4 3 60 40 100 50 -
On Job Training
MST2P03 Practical 3 - 6 6 3 4 50 50 100 - 50
MST2P04 Practical 4 - 6 6 3 4 50 50 100 - 50
TOTAL 12 20 32 22 - 340 260 600 200 100
Semester III

M.Sc. Statistics Semester III

Teaching
scheme(Hours/ Examination Scheme
Week)
Theory/Practical

Max. Minimum

Credits

Duration in hrs.
Passing
Code Marks

TotalMarks
Marks
Practical
Theory

Total

Practical
External

Internal

Theory
Marks

Ass
MST Paper 9:Decision - -
3T01 4 4 4 3 60 40 100 50
Theory & Non
Parametric
Methods
MST Paper10:Linear - -
3T02 4 4 4 3 60 40 100 50
&Non linear
Modelling
Paper11: Electives
MST3T
Data Mining – A - -
03 OR 4 4 4 3 60 40 100 50
Mathematical
Programming – B
OR
Demography - C
Practical 5

MST3
P05 - 6 6 3 4 50 50 100 - 50

MST3 Practical 6 - -
6 6 3 4 50 50 100 50
P06
MST3R Research - -
8 8 4 4 60 40 100 50
P1 Project(RP)
Minor Work
TOTAL 16 12 28 22 - 340 260 600 150 150
Semester IV

M.Sc. Statistics Semester IV

Teaching
scheme Examination Scheme
(Hours/Week)
Theory/Practical

Max.Marks MinimumP

Duration in hrs.
Credits
Code assingMarks

TotalMarks
Practical
Theory

Total

Practical
External

Internal

Theory
Marks

Ass
Paper 4 - 4 4 3 60 40 100 50 -
MST4T01
12Multivariate
Analysis
Paper13:Computat 4 - 4 4 3 60 40 100 50 -
MST4T02
ional Statistics
Paper14:Electives 4 - 4 4 3 60 40 100 50 -
Big Data
Analysis
MST4T03 –A
Or
Operations
Research – B
Or
Actuarial
Statistics - C
Practical 7

MST4
P07 - 4 4 2 4 50 50 100 - 50
Practical 8

MST4
P08 - 4 4 2 4 50 50 100 - 50

Research Project(RP) - 12 12 6 4 60 40 100 - 100

MST4RP2
Major Work
TOTAL 12 20 32 22 - 340 260 600 150 200

General Rules and Regulations regarding pattern of question paper:

A] Pattern of Question Papers

1. There will be four units in each paper.

2. Question paper will consist of five questions.
3. Four questions will be on four units with no internal choice (one question on each
unit)
4. Fifth question will be compulsory with questions from each of the four units having
equal weight age and there will be no internal choice.
5. Maximum marks of each paper will be 60.
6. Each paper will be of three hours duration
7. Projects shall be evaluated by both Internal and External examiners.
8. Practical / laboratory examination of 100 marks. Distribution of marks shall be 50
 internal and 50 external.
9 Minimum passing marks in each head (theory, practical & project) will be 50%.
10. Internal marks will be given on the basis of Regularity, Midterm / Test and
Assignment.

B] Duration of practical examination will be of four hours.

Distribution of marks of practical examination of 100 marks will be as follows:

Practical Viva Voce Practical Record Total

performance

50 25 25 100

C] Project Work:
 Project work will carry 100 marks and distribution of these 100 marks will be as
follows.
 For written project work: 40 Marks- Evaluated jointly by External and Internal
Examiner
 Presentation: 20 Marks- Evaluated jointly by External and Internal Examiner
 Internal Assessment: 20 Marks- Evaluated by Internal Examiner
 Viva Voce: 20 Marks- Evaluated by External Examiner

List of Electives

Subject Semester – I Subject code Semester - II

code
MST1T03- Industrial Process and MST2T03- A Industrial Statistics
A Quality Control
MST1T03- B Survival Analysis MST2T03- B Reliability Theory
MST1T03- Bioassay MST2T03- C1 Statistical Genetics
C MST2T03- C2 Statistical Ecology

List of Electives

Subject Semester – III Subject code Semester - IV

code
MST3T04- Data Mining MST4T04- A Big Data Analytics
A
MST3T04- B Mathematical MST4T04- B Operations Research
Programming
MST3T04- Demography MST4T04- C Actuarial Statistics
C

Note: The exit option at the end of one year of the Master’s
degree program will commence from AY 2024- 25.
 Students who have joined a two-year Master’s degree
program may opt for exit at the end of the first year and
earn a PG Diploma.

POST GRADUATE TEACHING DEPARTMENT OF STATISTICS,

RASHTRASANT TUKADOJI MAHARAJ NAGPUR UNIVERSITY, NAGPUR
M. A. / M.Sc. STATISTICS
SEMESTER PATTERN SYLLABUS (CBCS)

SEMESTER I
PAPER I – MST1T01
Probability & Distributions Theory

Objectives ● A majority of topics in Statistics depend upon a strong foundation of

Probability theory. Another basic concept is that of a random variable , its distribution
and associated properties. Sequences of random variables is also a part of this course
so that asymptotic or long run behavior of the sequence can be studied. The entire
analysis in statistics is based on probability distribution of the random variable under
consideration. This course includes a study of some basic standard discrete and
continuous distributions along with their properties. Besides, important concepts like
truncated and mixture distributions are also introduced. The concepts and distributional
properties of order statistic are also introduced in this course.

Unit – I: Probability measure on a sigma field. Probability space, Properties of

Probability measure, Continuity, mixture of Probability measures. Axiomatic definition of
Probability. Independence of events, mutual independence, sequence of independent
events, independent classes of events, Borel-Cantelli lemma. Random variables,
Expectation of random variables.Distribution function and its properties.

Unit – II: Convergence of a sequence of rvs, convergence a.s, convergence in

probability, convergence in distribution, convergence in rth mean, Weak and Strong law
of large numbers;Chebyshev Weak Law of large numbers, Khinchins weak law of large
numbers, Kolmogorov strong law of large numbers (statement only). Characteristic
function, Central limit theorem – De-Moivre Laplace, Lindeberg Levy, Lindeberg- Feller
(Sufficiency only)

Unit – III: Joint, marginal and conditional pmfs and pdfs, Basic discrete and continuous
distributions - Binomial, Poisson, negative binomial, geometric, uniform, multinomial,
hyper geometric distribution, Normal, bivariate normal, exponential. Joint distribution of
sample and induced sampling distribution of a statistic. Beta, Gamma, Cauchy, Log-
normal, Weibull, Laplace distributions.

Unit – IV: z, t and F distributions and their properties and applications. Chi-square
distribution and its properties. Compound, truncated and mixture distributions.
Distributions of quadratic forms under normality and related distribution theory. Order
statistics, their distribution and their properties, joint and marginal distribution of order
statistics. Extreme values and their asymptotic distributions (statement only) with
applications.

Outcomes :

● Students will learn the concept of probability as a measure and its various properties .
Besides this, they learn the concept of random variable, its types, probability distribution,
cumulative distribution function, and its various properties .

● Types of convergences for sequences of random variables and their inter-

relationships. Weak and strong law of large numbers based on the above types of
convergence

● Concept and importance of Central limit Theorem which helps in understanding the
limiting behavior and the concept of characteristic function and its properties.

●This course gives a strong base for advanced theoretical as well as advanced applied
probability.

● Students will get knowledge of some standard discrete and continuous distributions
along with their properties.

● Students also learn the applications of the random variables studied in real life
situations. They will be able to test whether the given random variable follows any of
the standard distributions based on which further analysis can be done.

● Students will get knowledge about some sampling distributions with their applications
and some useful inequalities.

● Students learn the concept of truncation and mixture distributions and also an
important concept of order statistics. These concepts are useful in many areas like
Acturial Science, Telecommunication and in non-parametric density estimation

References :

1) B. R. Bhat : Modern Probability theory

2) A. K. Basu : Measure and Probability Theory
3) M Fisz : Probability theory and Mathematical Statistics.
4) V. K. Rohatgi : Introduction to Probability theory and its application.
5) Johnson S and Kotz : Distributions in statistics Vol I, II and III.

PAPER II – MST1T02

Statistical Inference

Objectives ● Estimation theory is an important part of statistical Inference. The

objective of this course is to train the students in obtaining point and interval estimate
of the parameter based on given data.

There are some standard statistical methods useful in testing various types of
hypotheses. These methods are widely used in almost all disciplines. Sometimes we
need to construct a test procedure if the situation is not as required in the standard
methods. The course includes basic lemma useful in construction of the test. The test
procedure also changes according to the nature of null hypothesis, alternative
hypothesis, the distribution of a random variable under consideration etc. The course
helps in constructing most powerful or uniformly most powerful tests for different types
of hypotheses for any given distribution. Some situations require sequential test
procedure.

Unit I

Problem of point estimation, Properties of good estimators, Unbiasedness, Consistency,

Efficiency, Sufficiency , Factorization Theorem ,Minimal sufficient statistics for
exponential family and Completeness. Cramer-Rao lower bounds. Minimum variance
unbiased estimators, Rao-Blackwell Theorem. Lehman-Scheffe theorem .

Unit II

Method of Estimation, Method of Maximum Likelihood, Method of Moments, Method of

Chi-Square, properties of M.L.E.Interval estimation : Confidence level , construction of
confidence intervals using Pivots.

Unit III

Test of hypothesis, concept of critical regions, test functions, two kinds of errors, size
function, power function, level, MP and UMP test in the class of size  tests. N.P lemma,
MP test for simple against simple alternative hypothesis. UMP tests for simple null
hypothesis against one sided alternative and for one sided null against one sided
alternative in one parameter exponential family.

Unit IV

Likelihood ratio test. Asymptotic distribution of LRT statistics (without proof). Wald test,
Rao’s score test,Sequential testing. Sequential probability ratio test. Relation among
parameters Application of SPRT to Binomial, Poisson, Normal Distribution. unbiased
test, UMPUT and their existence in case of exponential family similar tests and tests
with Neyman structure.

Outcome : At the end of the course , students become well versed with

 The basic problem of estimation, various methods of estimation and properties

that are to be satisfied by a good point estimator.

 Methods to improve an unbiased estimator to MVUE.

 Construction of confidence interval of the parameters of various distributions.

 A good understanding of the concept of hypotheses helps the students to

differentiate between types of hypotheses like, null and alternative, simple and
composite, one sided and two sided etc.

 They learn about the two types of errors committed while constructing any test
and the probabilities associated with them.

References :
1) E. L. Lehman : Theory of Point estimation
2) B. K. Kale : First course on Parametric inference
3) C.R. Rao : Linear statistical inference and its applications
4) Ferguson T. S. : Mathematical statistics.
5) Zacks S. : Theory of statistical inference.

PAPER III
Elective Paper MST1T03-A
Industrial Process and Quality Control
Objective: In industry, it is important to maintain quality of the product. The quality of
the product can be checked with the help of quality control techniques available in
statistics. The course includes various methods for monitoring the quality of the
product. The control charts can be used to monitor quality of product, process, and
services in various disciplines.

Unit – I:Basic concept of process monitoring General theory and review of Shewhart
control charts for measurements and attributes (p, d = np, C, X and R chart) O.C. and
ARL for X control chart. General ideas on economic designing of control chart.
Assumptions and costs. Duncan’s model for the economic design of X chart. Moving
average and exponentially weighted moving average charts. Cu-sum charts using v
masks and decision intervals.

Unit – II: Classification of nonconformities and their weighting modification of the c

chart for Quality scores and Demerit classifications ,Q chart for no. of nonconformities
per (u chart) Multivariate Quality control. Hotelling’s T2.
Unit – III: Concept of six sigma. Evolution of six sigma Quality approach practical
approach to six sigma quality Basic steps involved in application of six sigma Define-
measure-Analyzeimprove and control approach, Barriers in implementation of six sigma
in Indian manufacturing industries( small and medium enterprises).Impact of six sigma
in a developing economy.
Unit – IV: Principle of acceptance sampling problem of lot acceptance . Acceptance
sampling plans for attributes. Single double and sequential sampling plans and their
properties Dodge Romig sampling plans for attributes (AOQL and LTPD),MIL std plans,
continuous sampling plans, Dodge type CSPI, CSPII and CSPIII .
Outcome:

Students develop the ability to apply various types of control charts as given below in
different industries.

 Various control charts for measurements and attributes useful in industries.

 EWMA charts, multivariate control charts.

 Concept of six sigma and its importance.

 Principle of acceptance sampling problem ,various acceptance sampling plans.

References:
1] Montgomery D.C. (1985) Introduction to statistical quality control. Wiley.
2) Montgomery D.C. (1985) Design and Analysis of Experiments Wiley
3) Grant E. L. & Leaver worth R. S. statistical Quality control McGraHill publication.

Paper III – MST1T03-B

Survival Analysis
Objective : Life time is an important random variable observed in epidemiology, and
various industries. The analysis of this variable is studied in survival analysis.
Depending upon the distribution of life time, there are separate statistical methods for
this kind of analysis . In survival analysis the data is many times a censored data.

Unit – I:Concepts of Time, order and Random Censoring.

Life distributions Exponntial, Gamma, Weiball, Lognormal, Pareto, Linear failure rate,
parameteric inference, point estimation, confidence intervals, scores, tests based on LR,
MLE.

Unit – II:Life tables, Failure rate, mean residual life and their elementary
properticsAgeing classes IFR ,IFRA, NBU, NBUE, HNBUE and their duals, Bathtub Failure
rate.

Unit – III:Estimation of survival function Actuarial estimator, Kaplan Meier Estimator,

Estimation under the assumption of IFR/DFR.
Tests of exponntiality against nonparametric classes Total time on test, Deshpande test,
Two sample Problem, Gehan test, Long rank test, Mantel Haenszd Test, Tarone Ware
tests.

Unit – IV:Semi Parametric regression for failure rate Cox’s Proportional hazards model
with one and several covariates.

Outcome: Students will get knowledge of

 Types of censoring and estimation of parameters of the distribution using

censored data.

 Life tables and related rates and properties.

 Survival function and its estimation using parametric and nonparametric

methods .

 Semi parametric regression for failure rate, Cox’s proportional hazard models.
References:
1. Cox, D.R. and Oakes, D.(1984) Analysis of Survival Data, Chapman and Hall,
Newyork.

2. Gross A. J. and Clark VA (1975) survival Distrubutions : Reliability Applications in

Biomedical Sciences, John Wiley & Sons.

3. Elandt Johnson, R.E. Johnso9n NL (1980) Survival models and Data Analysis,
John Wiley and sons.

4. Miller, R.G. (1981) Survival Analysis

5. Zacks, S. Reliability.

Paper III – MST1T03- C

Bioassay

Objective: Bioassay is an analytical method to determine concentration of a substance

by its effect on living cells, tissues, insects, etc. There are various types of Bioassays
like qualitative or quantitative, direct or indirect. These analytical methods are useful in
environmental science, microbiology etc. The method of dose and response
relationship in this analysis is used in pharmaceutical sciences. Objective of this course
is to train students in analytical methods used in these fields

Unit I: Types of biological assays, Direct assays, ratio estimators, asymptotic

distributions, Fieller’s theorem. Regression approaches to estimating dose-response
relationships, Logit and Probit approaches when dose-response curve for standard
preparation is unknown.

Unit II:.Methods of estimation of parameters, estimation of extreme quantiles., dose

allocation schemes.Quantal Responses, Polychotomousquantal responses. estimation
of points on the quantal response function.

Unit III: Sequential procedures, estimation of safe doses.

Unit IV: ANOVA and Bayesian approach to Bioassay.
Outcome: Students get knowledge about the application of statistics in biological
sciences through topics like

●Types of biological assays and methods for estimating dose response relationship.
●Logit and probit approach for estimating dose-response relationship.

●Methods of estimation of parameters and dose allocation schemes.

●Sequential procedures, estimation of safe dose, ANOVA and Bayesian approach to

Bioassays.

References:
1.GovindRajulu,Z.(2000):Statistical Techniques in Bioassay,S.Karger.
2. Finney, D.J.(1971):Statistical Methods in Bioassay, Griffin.
3.Finney, D.J.(1971): Probit Analysis 3rdED.), Griffin.
4.Weatherile, G.B.(1966):Sequential Methods in Statistics, Methuen.

Paper IV – MST1T04

Research Methodology

Course objectives:
o To understand the role of research methodology in Engineering/Science/Pharmacy
o To understand literature review process and formulation of a research problem
o To understand methods and basic instrumentation
o To learn technical writing and communication skills required for research
o To learn about intellectual property rights and patents

Unit I: Introduction to Research

Definition of research, Characteristics of research, Types of research- Descriptive vs.
Analytical,
Applied vs. Fundamental, Quantitative vs. Qualitative, Conceptual vs. Empirical,
Overview of
research methodology in various areas, Introduction to problem solving, basic research
terminology such as proof, hypothesis, lemma etc., Role of Information and
Communication Technology(ICT) in research.

Unit II: Research Problem Formulation and Methods

Literature review, sources of literature, various referencing procedures, maintain
literature data
using Endnote2, Identifying the research areas from the literature review and research
database,
Problem Formulation, Identifying variables to be studied, determining the scope,
objectives,
limitations and or assumptions of the identified research problem, Justify basis for
assumption,
Formulate time plan for achieving targeted problem solution.
Important steps in research methods: Observation and Facts, Laws and Theories,
Development of
Models, Developing a research plan: Exploration, Description, Diagnosis and
Experimentation.

Unit III: Research reports and Thesis writing

Introduction: Structure and components of scientific reports, types of report, developing
research
proposal. Thesis writing: different steps and software tools in the design and
preparation of thesis,
layout, structure and language of typical reports, Illustrations and tables, bibliography,
referencing
and footnotes, Oral presentation: planning, software tools, creating and making
effective
presentation, use of visual aids, importance of effective communication.

Unit IV: Research Ethics, IPR and Publishing

Ethics: Ethical issues. IPR: intellectual property rights and patent law, techniques of
writing a Patent, filing procedure, technology transfer, copy right, royalty, trade related
aspects of intellectual property rights Publishing: design of research paper, citation and
acknowledgement, plagiarism tools, reproducibility and accountability.

Reference Books:
1. Ranjit Kumar, "Research Methodology: A Step by Step Guide for Beginners",
SAGE Publications Ltd., 2011.
2. Wayne Goddard, Stuart Melville, "Research Methodology: An Introduction"
JUTA and Company Ltd, 2004.
3. C.R. Kothari ,"Research Methodology: Methods and Trends",
New Age International,2004
4. S.D. Sharma , "Operational Research", Kedar Nath Ram Nath & Co.,1972
5. B.L. Wadehra,"Law Relating to Patents,Trademarks, Copyright Designs and
Geographical Indications", Universal Law Publishing, 2014.
6. Donald Cooper, Pamela Schindler, "Business Research Methods",
McGraw-Hill publication, 2005.

Practical 1 : It Will be Based on Theory Paper.

Practical 2 : It Will be Based on Theory Paper.

SEMESTER II

Paper I – MST2T01
Sampling & Designs of Experiments

Objectives ● Statisticians draw conclusion on the basis of samples. The conclusions

are close to reality only when they are based on a sample which is a proper
representative on the population. This course is designed to train the students to obtain
an appropriate sample in any given situation. Design of Experiments is a systematic
method to determine the relationship between factors affecting a process and its
output. It is useful in understanding cause and effect relationships. This is useful in
managing inputs that will be reflected as variations in outputs. It is used in areas like
agriculture, Psychology, medicine, engineering, biochemistry etc.

Unit I:-

Basic method of sample selection: -Simple random sampling with replacement, Simple
random sampling without replacement. Unequal probability sampling: PPS WR/WORand
related estimators of finite population mean (Des-Raj estimators for general sample
size.) Horvitz Thompson’s estimator

Stratified random sampling: Estimation of population mean, total and variance,

Construction of strata and number of strata,

Unit – II: Use of supplementary information for estimation: - Ratio and Regression
method of estimation based on SRSWOR. Double sampling for estimating strata sizes
in ratio and regression method of estimation. Cluster sampling, equal and unequal
sizes.

Unit – III:- Analysis of variance, elementary concepts (one and 2 way classified data )
Review of elementary design (CRD, RBD, LSD) Missing plot technique in RBD and LSD
with one and two missing values (only estimation of missing values)
BIBD : Elementary parametric relations, Analysis. Definitions and parametric relations of
SBIBD, RBIBD, ARBIBD, PBIBD.

Unit – IV: Analysis of covariance of one-way and two-way classified data., split plot
design: construction. factorial experiments, factorial effects, best estimates and testing
3
the significance of factorial effects, study of 2 factorial experiments in RBD.
Confounding in factorial experiments, complete and partial confounding.

Outcome :

 Students will get knowledge about various methods of sampling like SRSWOR,
SRSWR, Stratified random sampling, Cluster sampling, systematic sampling etc.

 Students will get knowledge about the efficiency of one method of sampling with
respect to the other so that in any given situation, they will be able to apply an
appropriate method of sampling.

 They also learn about ratio method of estimation and regression method of
estimation when the data on the associated variable is also available along with
a study variable.

 Students also get knowledge about Warner’s model of randomized response

technique and two stage sampling.

 Students acquire knowledge about how to design an experiment for comparing

various effects on the output of an experiment through following designs.

 They learn basic design like CRD, RBD, LSD, techniques of estimating missing
values (if any) in the data. Use of BIBD and its various types such as RBIBD,
ARBIBD, SBIBD.

 They learn use of factorial experiments as per the requirement of the situation
where the effects of treatments are studied at two levels and their interactions.
Use of confounding in factorial experiments to maintain homogeneity of the
block.

References:

1) Sukhatme : Sampling theory of surveys with applications.

2) Singh D and Chaudhary F. S. : Theory and analysis of sample survey designs.

3) Murthy M. N. : Sampling theory and methods.

4) Des Raj and Chandak : Sampling theory.

5) Das M. N. and Giri N (1997) : Design and Analysis of experiments . Wiley Eastern.
6) Joshi D. D. (1987) : Linear estimation and design of experiments. Wiley
Eastern.
7) Montgomery. C. D. (1976) : Design and analysis of experiments. Wiley, New York.

Paper II – MST2T02
Stochastic Processes & Time Series
Objectives :

Almost all processes that we come across in real life are stochastic in nature. The
course includes some basic standard stochastic processes which help in understanding
the real life situations. Many processes are observed to be Markovian in nature. Study
of such processes is also included in this course. A time series is a series of data points
indexed in time order. When this data is plotted, one can observe trends and other
variations. Time series analysis is used to study the patterns of variations in data with
time. These methods are useful in finance, weather forecasting, agriculture, etc. where
the primary goal is forecasting. Objective here is to introduce the concept of time series
,different methods of analyzing and modeling the time series data and their use in
forecasting.

Unit – I: Definition of Stochastic Process, Classification of Stochastic processes

according to state space and time domain. Examples of various Stochastic Processes.
Definition of Markov Chain, Examples of Markov Chain Formulation of Markov Chain
models, initial distribution, Stationary transition Probability Matrix, Chapman-
Kolmogorov equation, calculation of n-step transition probabilities.
Classification of states, closed and irreducible classes, transient, recurrent, and null
states, Periodic States, Criteria for the various types of states, Ergodic theorem.
Unit – II: Random walk and Gambler’s Ruin problem Absorbing and reflecting barriers.
First Passage Probability. Expected duration of game. Random walk in 2 and 3
dimensions. Discrete state space continuous time Markov Chain, Poisson Process,
Continuous state space continuous time Markov chain : Kolmogrov’s equation Wiener
process as a limit of random walk model, properties of Wiener process. Covariance
stationary processes.
Branching process: Galton-Watson branching process for the size of nth generation, the
relation between the generating function, probability of ultimate extinction, distribution
of population size.
Unit III :Exploratory time series analysis, tests for trend and seasonality. Exponential
and Moving average smoothing. Holt -Winters smoothing. Forecasting based on
smoothing, adaptive smoothing. Time - series as a discrete parameter stochastic
process. Auto covariance and autocorrelation functions and their properties.

Unit IV : Stationary processes: General linear processes, moving average (MA), auto
regressive
(AR), and autoregressive moving average (ARMA). Auto regressive integrated moving
average
(ARIMA) models, Box –Jenkins models Stationarity and inevitability conditions.
Nonstationary and seasonal time series models: Seasonal ARIMA (SARIMA) models,
Transfer function models (Time series regression). Estimation of ARIMA model
parameters, maximum likelihood method, large sample theory (without proofs). Choice
of AR and MA periods, FPE, AIC, BIC, residual analysis and diagnostic checking.

Outcome : Students become well be acquainted with

 The definition of stochastic processes and its classification. Markov chain and
its applications in different areas, Classification of states and various results
associated with Markov Chain.

 Random walk model, Poisson processes & branching processes and will be able
to apply them in real life situations.

 Different methods of data smoothing to remove random variations.

 Stationary and Non-stationary processes. Estimation of the parameters of the

time series model and using the models for forecasting .

References :
1. J. Medhi : Stochastic Processes.
2. S. Karlin and H Taylor : First course in stochastic processes.
3. W. Feller : Introduction to probability theory and its applications Vol.
1.
4. Brockwell, P.J. and Davis, R. A. (2003). Introduction to Time Series Analysis,
Springer
5. Chatfield, C. (2001). Time Series Forecasting, Chapmann &Hall, London
6. Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nd Ed. Wiley.
7. Hamilton N. Y. (1994). Time Series Analysis. Princeton University press.
8. Box, G.E.P & Jenkins G.M (1976): Time Series Analysis – Forecasting & Coontrol ,
Holden-Day, San Francisco .
9. Montgomery, D.C & Johnson, L.A ( 1977) : Forecasting and Time Series Analysis,
McGraw Hill

Elective Paper

PAPER III MST2T03 - A

Industrial Statistics
Objective: Objective here is to introduce to the students a branch of statistics that helps
in maintaining quality in the industry. Course contains various methods for quality
maintenance.

Unit – I: Quality Systems : ISO 9000 standards. QS 9000 standards.

Total quality management (TQM) : Different definitions and dimensions of quality,basic
concept , Total Quality Management Models, Quality Management Tools, Six Sigma and
Quality Management, What is Kaizen ? - Five S of Kaizen, Role of Managers in TQM, Role of
Customers in Total Quality Management, Comparison of Six Sigma and TQM, Reasons for
failure of TQM, Deming’s 14 point program, Continuous quality improvement, PDSA cycle ,
Juran triology, Quality Gurus
Unit –II: Use of Design of experiments in SPC factorial experiments, fractional
factorial design. Half fraction of the 23 factorial design Basic ideas of response surface
methodology. Specification limits, Natural tolerance limits and control limits. Process
capability analysis (PCA) : Process capability analysis using Histogram, and using
control chart.
Unit – III: Probability plotting capability indices Cp, Cpk and Cpm comparison of
capability indices. Estimation confidence intervals and tests of hypothesis relating to
capability indices for Normally distributed characteristics. Index Cpc for non normal
data.
Unit – IV: Quality at Design stage. Quality function deployment failure mode and effect
analysis. Taguchi philosophy system parameter and tolerance designs. Loss functions.
Determination of manufacturing Tolerances. Signal to noise ratio and performance
measures critique of S/N ratios.
Outcome: Students get theoretical knowledge and industrial application of the following
topics

●Working of quality systems ISO 9000 and QS 9000.

●Use of design of experiments for quality improvement and process Capability Analysis,
various capability indices.

●Taguchi philosophy, 6 sigma.

●Failure model, effect and criticality analysis.

References :

1) Montgomery D.C. :(1985) Introduction to statistical quality control. Wiley.

2) Montgomery D.C. : (1985) Design and Analysis of Experiments Wiley
3) Grant E. L. & Leaver worth R. S. :Statistical Quality control, McGraHill publications.
4) Amitava Mitra :Fundamentals of quality control and improvement
5) Oakland J. S. : (1989) Total quality management, Butterworth Heinemaah 14
6) K. Shridhara Bhat : Total quality management, Himalaya Publishing House
7) C. B. Michna: D. H. Besterfield Total quality management, Pearson Education.
8) Phadke M. S. (1989) Quality Engineering through Robust design. Practice Hall.
9) Logothelis N. (1992) Managing total quality, Prentice Hall of India.
10) Oakland J. S.: Statistical Process control Heinemach Professional publishing.
 Paper III – MST2T03 - B
Reliability Theory
Objective: Man made systems suffer from imperfections for several reasons. Often
these imperfections lead to improper functioning resulting in failure of the system It
may be the result of defect in the system while producing it or may be because of
natural component deterioration on some interacting factors. Probability of non failure
is termed as reliability. Reliability models can be developed for predicting the reliability
of a component or of system prior to its implementation.
Unit I:.Reliability concepts and measures ,components and systems, coherent systems,
reliability of coherent systems, cuts and paths, modular compositions, bounds on
system reliability, structural and reliability importance of components. Life distributions,
reliability functions, hazard rate, common life distributions, exponential, Gamma ,
Weibull, Lognormal etc. Estimation of parameters, confidence intervals, LR and MLE
tests for these distributions.
Unit II:. Notions of ageing: IFR, IFRA, NBU, DMRL and NBUE classes and their duals, loss
of memory property of the exponential distribution, closures of these classes under
formation of coherent systems, convolutions and mixtures. Univariate shock models
and life distributions arising out of them, bivariate shock model, common bivariate
exponential distributions and their properties.

Unit III:. Reliability estimation based on failure times in variously censored life tests and
in tests with replacement of failed items, stress and strength reliability and its
estimation. aintenance and replacement policies, availability of repairable systems,
modeling of repairable system by a non-homogeneous Poisson process.

Unit IV: Reliability growth models, probability plotting techniques, Hollander- Proschan
and Deshpande tests for exponentially, tests for HPP vs. NHPP with repairable systems.
Outcome: Students learn about the following concepts in reliability and their
applications

● Failure time distribution, reliability function, hazard function etc.

●Increasing failure rate as an effect of ageing, shock models.

●Reliability estimation in various cases.

●Reliability growth models.

REFERENCES:

1. Barlow R E and Proschan F (1985), Statistical Theory of Reliability and Life Testing .
2. Lawless J.F. (1982) Statistical Models and Methods of Life Time Data.
3. Bain L. J Engelhardt (1991), Statistical Analysis of Reliability and Life Testing Model.
4. Zacks S, Reliability Theory.
5. D C Montgomery-Design and Analysis of Experiments.
6. R H Myers and D C Montgomery –Response Surface Methodology.
7. J Fox: Quality through Design
8. J A Nelder and P McCullasn Generalized Linear Models.

Paper III – MST2T03- C1

Statistical Genetics
Objective: The course includes basic results in genetics from their probability
distribution point of view, population equilibrium, dominating alleles and estimation of
their probability. Objective is to make the students aware about the concept of natural
selection, inbreeding ,linkage, segregation etc which are probability distribution based
concepts in genetics. This probability distribution can be used to estimate the frequency
of the gene in coming generations. This is very useful in some rare diseases.

Unit I: Basic biological concepts in genetics. Mendel’s law. Hardy Weinberge equilibrium.
Matrix theory of random mating. Mating tables. Estimation of allel frequency for
dominant and co dominant cases. Approach to equilibrium for X-linked gene.
Unit II:. Non random mating. Inbreeding. Coefficients of inbreeding. Inbreeding in
randomly mating populations of finite size. Phenotypic assortative mating.
Unit III: Natural selection, mutation, genetic drift. Equilibrium when both natural
selection and mutation are operative. Statistical problems in human genetics, Blood
group analysis.
Unit IV: Analysis of family data : (a) Relative pair data, I ,T,O matrices, identity by
descent. (b) Family data- estimation ofsegregation ratio under ascertainment bias. (c)
Pedigree data –Elston- Stewart algorithm for calculation of likelihoods, linkage,
Detection and estimation of linkage, estimation of recombination fraction, inheritance of
quantitative trials models and estimation of parameters.
Outcome: Students learn about the basic concepts of genetics and their development
using statistical tools

●Concept of random mating, Hardy Weinberg equilibrium, calculating probabilities for

various gene combination and study of X-linked genes.

●Nonrandom mating, inbreeding, various coefficients of inbreeding.

●Concept of Natural selection, its effect on equilibrium .

●Analysis of family data, linkage detection , and estimation of linkage.

References:
1. Li,C.C.(1976):First Course on Population genetics. Boxwood Press, California.
2. Ewens, W.J.(1979):Mathematical Population genetics ,Springer Verlag.
3.Nagylaki, T.(1992): Introduction to theoretical population genetics. Springer
Verlag
4. Elandt – Johnson Probability Models and Statistical Methods in Genetics. John
Wiley

Paper III MST2T03

Statistical Ecology
Objective: Ecology is study of interaction of organisms that include biotic and abiotic
components and their environment. Ecologists can explain life processes, interactions,
adoptions, movement of materials, distribution of organisms, biodiversity etc. by using
various statistical methods. The course gives knowledge of these methods and models
in this particular branch.

Unit – I:Population Dynamics One species exponential, logistic and Gompertz models,.
Two species competition, coexistence, predator prey oscillation, Lotka-Volterra
Equations, isoclines, Lestie matrix model for age structured populations. Survivorship
curves constant hazard rate, monotone hazard rate and bath tub shaped hazard rates.
Unit – II:Population density estimation: Capture recapture modesls, nearest neighbor
models, Line transect sampling, Ecological Diversin, Simpsons index, Diversity as
average rarity.
Unit – III:Optimal Harvesting of Natural Resources, Maximum Sustainble field, tragedy
of the commons Game theory in ecology, concepts of Evolutionarily stable strategy, its
Properties, simple cases such as Hawk-Dove geme.
Unit – IV: Foraging Theory : Diet choice Problem, patch choice problem meanvariance
trade off.

Outcome: Students develop the ability to analyze life processes, biodiversity etc using
statistical methods in ecology which includes
  Various models of population dynamics for one species, two species, their co-
existence.

 Population density estimation, ecological diversion.

 Concept of evolutionary stable strategy and its properties.

 Optimal harvesting of natural resources.

References:
1. Gore, A.P. and Paranjpe S.A. (2000) A course on Mathematical and Statistical
Ecology, Kluwer Academic Publishers.
2. Pielou, E.C. (1977) An Introduction to Mathematical Ecology (Wiley)
3. Seber, G.A.F (1982) The Estimation of animal abundance and related parameters
(2nd Ed) (Grittin)
4. Clark, C.W. (1976) Mathematical bio-economics : the optimal management of
renewable resources (John wiley)
5. Maynard Smith J. (1982) Evolution and the theory of games (Cambridge
University Press)
6. Stephenes, D.W. & Krebs JR (1986) Foraging Theory (Princeton University Press)
PAPER IV

PGST3RP2 Field Work or On Job Training

Practical 3 : It will be based on Theory Paper .

Practical 4 : It will be based on Theory Paper .

SEMESTER III
Paper I – MST3T01
Decision Theory & Non Parametric methods
Objective : Decision making is very important in all walks of life. The statistical aspect
of decision making is based on the risk involved in any decision. There are methods
which come out with the decision having minimum risk. The objective here is to
introduce the topic to the students giving only basic knowledge of it. There are many
real life situations where the assumptions required for the application of parametric
statistical methods are not satisfied. Nonparametric inference is a branch of statistics
which has solutions in such situations. The course includes various non-parametric
methods.

Unit – I: Decision problem, loss function, expected loss, decision rules (nonrandomized
and randomized), decision principles (conditional Bayes, frequentist) inference and
estimation problems as decision problems, criterion of optimal decision rules. Concepts
of admissibility and completeness, Bayes rules, minimax rules, admissibility of Bayes
rules. Existence of Bayes decision rules.
Unit – II: Definition of non-parametric test, Advantages and disadvantages of Non-
parametric tests. Single sample problems :
a) test of randomness
b) test of goodness of fit : Empirical distribution function.
Kolmogorov – Smirnov test, 2 test, Comparison of 2 test & KS test
c) One sample problem of location : sign. Test, Wilcoxon’s signed rank
test,
Wilcoxons paired sample signed rank test
Unit –III: Two sample problems : different types of alternatives, sign test, Wilcoxans
two sample rank sum test, Wald-Wolfowitz run test, Mann-Whitney-Wilcoxons test,
Median test, KS-two sample test. Klotz Normal score test.
One sample U-statistic, Kernel and symmetric Kernel Variance of U-Statistic, two-
sample U-statistic, Linear rank statistics and their distributional properties under null
hypothesis.
Unit – IV: Concept of time order and random censoring, likelihood in these cases,
survival function, hazard function Non-parametric Estimation of Survival function, Cox’s
proportional hazards model, the actuarial estimator, Kaplan – Meier Estimator.

Outcome: Students develop ability

●To formulate decision making problem, methods to solve the problem by defining
decision functions and risks involved and the methods of minimizing the risk.

● How and when to apply various non-parametric methods to different types of data for
testing various types of hypotheses.

●Various nonparametric tests for one sample, paired samples and two independent
samples problem.

●Concept of censoring the data, its need and its types. Parametric and nonparametric
methods of analyzing the censored data.

References:
1) Ferguson T. S. : Mathematical Statistics – A decision theoretic approach
2) Berger J. O. : statistical decision theory and Bayesian analysis
3) Gibbons J.D. : Non parametric Statistical inference
4) Randles and Wolfe : Introduction to the theory of non parametric statistics.

Paper II – MST3T02

Linear and Non-Linear Modeling

Objective : Regression analysis is the most common statistical modelling approach

used in data analysis and it is the basis for advanced statistical modelling.

The objective of this course is to impart knowledge about the use different useful tools
used in regression analysis. The relationship between variables can be of different
types like linear, nonlinear etc. The relationship is represented in terms of a model. The
adequacy of any model can be checked using residual plots and residual analysis.
Appropriate statistical tools are required to check for the violations of model
assumptions and for dealing with problems of multicollinearity etc.

Unit – I: MultipleLinear regression : Model assumptions and checking for the violations
of model assumption., Residual analysis – definition of residuals, standardized
residuals, residual plots, statistical tests on residuals, Press statistics. Transformation
of variables, Box-Cox power transformation.

Outliers : Detection and remedial measures, Influential observations : leverage,

measures of influence, Cook’s D, DFITS AND DFBETAS.
Unit –II: Multicolinearity : Concept and definition of M.C., sources of M.C. consequences
of M.C. identification of M.C. using the correlation matrix, VIF remedial measures
(collecting additional data, model respecification,), concept of ridge regression. Auto
correlation: consequences, Durbin-Watson test, Estimation of parameters in the
presence of autocorrelation.

Unit – III: Variable selection : Problem of variable selection, criteria for evaluation
subset regression models (choosing subsets), coefficient of multiple determination,
residual mean square, Mallow’s Cp Statistics. Computational Techniques for variable
selection-Forward selection, Backward elimination, stepwise regression.

Non-linear regression: Difference between Linear and Non-Linear Regression Models,

transformation to a linear model, Intrinsically linear and non-linear models. Parameter
estimation using the Newton-Gauss method, Hypothesis testing.

Unit – IV: Generalized linear models : Exponential families, Definition of GLM, Link
function, Estimation of parameters and inference in GLM.
Logistic regression model : Link function, logit, probit, complementary log-log,
estimation of parameters, odds ratio, hypothesis testing using model deviance.
Outcome: Students will get knowledge about

● Linear and Multiple regression.

● To interpret different types of plots such as residual plots, normal probability plots
etc. To check for the violations of model assumptions using residual analysis and other
statistical tests.

● To differentiate between linear and nonlinear regression under given situation.

● Generalized Linear Models including logistic regression.

References :
1) Draper N. R. and Smith H. :Applied Regression analysis
2) Montgomery D. C. : Linear regression analysis.

Elective Paper
Paper III MST3T03 - A
Data Mining

Objective: All over the globe, a huge amount of data is getting generated at a very high
rate. This huge data needs to be analyzed everywhere around us . Data mining is an
interdisciplinary subfield of computer science and statistics. The techniques are useful
in discovering patterns in large data sets. Objective here is to introduce the students to
this branch of statistics and impart knowledge in data processing, data management,
analysis of large data, model and inference consideration and online updating.

Unit – I:Review of classification methods from mullivariate analysis, classification and

decision trees, clustering methods from both statistical and data mining viewpoints,
vector quantization.
Unit – II:Unsupervised learning from univariate and multivariate data, Dimension
reduction and feature selections.
Unit – III:Supervised learning from moderate to high dimensional input. Spaces, artificial
neural networks and extensions of regression models, regression trees. Introduction to
data bases, including simple relational databases, data ware houses and introduction to
online analytical data processing.
Unit – IV:Association rules and prediction, data attributes, applications to electronic
commerce.
Outcome: Students get knowledge in data processing, data management analysis of
large data through the following topics

●Use of machine learning and statistical models to uncover the hidden patterns in large
volume of data.

●Clustering methods from statistical and data mining point of view.Unsupervised and
supervised learning of data in different cases.

●Dimension reduction, feature selection.Artificial neural network, extension of

regression models.

●Online analytical data processing.Association rules, predictions and applications.

References:
1. Berson, A and Smith, S.J. (1997) Data Ware housing, Data mining and OLAP
(McGraw-Hill)
2. Brieman, L. Friedman, J.H. Olshen, RA, and Stone, C.J. (1984) Classification and
regression Trees
3. Han, J and Kamber, M (2000) Data Mining, Concepts and Techniques (Morgan
Kaufmann)
4. Mitchell, T.M. (1997) Machine Learning (McGraw Hill)
5. Ripley, B.D. (1996) Pattern Recognition and Neural Networks (Cambridge
University Press)

Paper III MST3T03–B

Mathematical Programming
Objective: Optimization techniques have application in almost all disciplines. To get on
optimum solution to the problem under given constraints is always challenging. To get
the best solution to such problems, there are different methods depending on the
problem and constraints. Various such problems and methods to solve them are part of
this course.

Unit – I: L. P. : Simplex method, variants of simplex method, duality in L. P. duality

theorem, complementary slackness theorem, dual simplex method, transportation &
assignment problems, method of solving transportation & assignment problems.
Dynamic Programming : Dynamic programming approach for solving optimization
problems, forward & backward recursion formula, minimum path problem, single
additive constraint & additively separable return, single multiplicative constraint &
additively separable return, single additive constraint & multiplicatively separable return,
Goal Programming.
Unit – II:Sensitivity analysis of L. P. : Changes in R. H. S. constraint bi, changes in cost
coefficient cj, changes in coefficient of constraints aij, addition of new variables,
addition of new constraints. I.L.P.P. : Pure & mixed I.L.P.P. , methods for solving pure &
mixed I.L.P.P. Gomory’s cutting plane method, Branch & Bound technique.
Unit – III: N.L.P.P. : General N.L.P.P., convex & concave functions, text for concavity &
convexity, local optimum, global optimum, basic results for local optimum & global
optimum, Lagrange’s methods for optimality, KT conditions, Q.P.P. Wolfe’s & Beale’s
method for solving Q.P.P.
Unit – IV:Game theory : 2 person zero sum game, pure & mixed strategies, saddle point
of a matrix game, matrix game without saddle point, methods for solving matrix game
without saddle
point, 2X2 , mxn, mx2, 2Xn matrix games, dominance principle, use of dominance
principle in game theory, solving game problems by simplex method.
Outcome: Students will get knowledge

● To formulate and solve linear programming problem (LPP). They also learn various
methods to solve LPP.Application of LPP in industry, management, transportation,
assignment etc.

●Sensitivity analysis of LPP by studying the effect of changes in coefficients of

constraints on the solutions to the problem. They also learn the effect of any other
changes in the constraints, addition of new constraint on the solution to the problem.

●Pure and mixed integer linear programming problem and formulation of non linear
programming problem and different methods to solve them.

●The problem and different methods of solving two person zero sum game.
References:

1) Gass : Linear programming

2) Taha H. A. : Operations Research
3) Philips, Ravindran and Solberg : Operations research – Principles and practice

Paper III MST3T03 -C

Demography
Objective: Demography is statistical analysis of human population which can be studied
with the help of different models. It offers a most appropriate approach to study human
development across the world. Demographic analysis focuses not only on the changing
composition of population by age and gender but also on other observable and
measurable characteristics like education, health etc. . The objective of the course is to
impart knowledge in various methods of analyzing population data.

Unit – I: Definition and scope : Development of demography as a interdisciplinary

discipline, Basic demographic concept and components of population dynamics
coverage and content errors in demographic data, use of balancing equations and
Chandras Kharan Deming formula to check completeness of registration data.
Adjustment of age data Use of whipple, myer and UN indices.Population composition,
dependency ratio.
Unit – II:Measure of Fertility : Stochastic models for reproduction, distribution of time to
first birth, inter live birth intervals and of number of births (for both homogeneous and
non homogeneous groups of women) estimation of Parameters estimation of parity
preregression ratios from open birth interval data.
Unit – III:Measure of Mortality : Various measures of mortality, infant mortality rate,
cause specific death rate and standardized death rates. Construction of a bridge life
table Distribution of life table functions and their estimation.
Migration : Migration Rates and Ratios : Indirect measures of net-internet migration
National growth rate method stochastic models for migration and for social and
occupational mo0bility based on Markov Chains estimation of Measures of Mobility.
Unit – IV:Measurement of population change : Linear, Geometric exponential, Gompretz,
Logistic population growth models, Methods of population projection, Use of Leslie
matrix. Stable and Quasi stable populations, intrinsic growth rate, Models for population
growth and their fitting to population data. Stochastic models for population growth.
Outcome: Students get knowledge in various methods of analyzing population data by
age, gender, education and health etc through

●Basic demographic concepts and different features of population data.

●Various measures of fertility and mortality.

●Migration, its rates and ratio.

●Various measures of population changes, linear and non-linear population growth
models.

References :

1. Benjamin, B (1969) Demographic analysis. (Georage, Akllen & Unwin).

2. Cox, P.R. (1970) Demography, Cambridge University Press.

3. Keyfitz, N. (1977) : Applied Mathematical Demographic analysis, Springer-Verlag .

4. Spiegelman M (1969) : Introduction to Demographic analysis (Harward University

Press)

Bartholomew , D.J. (1982) Stochastic models for social processes, John-Wiley

PAPER IV
Research Project (Minor)

Practical 5 : It will be based on Theory Paper.

Practical 6 : It will be based on Theory Paper .

SEMESTER IV

Paper I – MST4T01
Multivariate Analysis
Objective: Multivariate Analysis is a branch of statistics where data on two or more
variables are analyzed simultaneously. Most of the statistical methods in univariate
analysis can be extended to the case of two or more variables. There are many
situations where we need to study effect of two or more independent variables on one
variable. This is studied as an extension of simple correlation and regression in case of
one independent variable. This course consists of all such distributions, statistical
methods etc. which are multivariate analogues of corresponding univariate analysis.
Many results that are derived only for multivariate cases are also included in the course.

Unit – I: Correlation : multiple and partial correlation. Linear and multiple regression co-
efficient of determination and its uses. Tests of significance of multiple and partial
correlation coefficient. Multivariate normal distribution, singular and nonsingular
normal distribution, characteristic function, moments, marginal and conditional
distributions, maximum likelihood estimators of the parameters of multivariate normal
distribution .
Unit – II: Wishart matrix-its distribution without proof and properties.Distribution of
MLEs of parameters of Multivariate Normal distribution ,Distribution of sample
generalized variance, Applications in testing and interval estimation, Wilks λ
[Introduction, definition, distribution (statement only)].

Unit – III: Hotelling’s T2 statistic and its null distribution. Application in tests on mean
vector for one and more multivariate normal populations and also on the equality of
the components of a mean vector in a multivariate normal population. Application of
T2 statistic and its relationship with Mahalanobis’ D2 statistic. Confidence region for the
mean vector. Applications of D2 statistics.

Unit – IV: Classification and discrimination : procedures for discrimination between two
multivariate normal populations. Fisher’s discriminant function, tests associated with
discriminant function, Sample discriminant function. Probabilities of misclassification
and their estimation. Classification into more than two multivariate populations.
Principal components. Dimension reduction. Canonical variables and anonical
correlation, definition, uses, estimation and computation.
Outcome: Students develop the ability of handling multivariate data and to draw
inferences from such data using following methods.

●Concept of multiple and partial correlation and multiple regression.

●Multivariate normal distribution, estimation of parameters of the distribution and its

properties.Wishart distribution and its properties.

●Hotellings T2, its null distribution and its applications for testing hypotheses
associated with mean vector (vectors).

●Problem of classification, procedure of classification using Fisher’s discriminant

function.Principal component analysis useful in dimension reduction and analysis using
canonical correlation.

References :

1) Anderson T. W. : An introduction to multivariate statistical analysis.

2) Kshirsagar A. M. : Multivariate analysis
3) Rao C. R. : Linear statistical inference and its applications

Paper II- MST4T02

Computational Statistics
Objective: In many disciplines, results are established with the help of the data by fitting
a suitable model . Analyzing the data plays an important role in such cases. Advanced
statistical methods and different types of models can be applied to these data, even
very big data. The course deals with different computational methods and algorithms
necessary for analysis of the data. The course includes different methods that are
particularly useful in simulating data from various distributions and analyzing them with
the help of computers.

Unit – I: Exploratory data analysis: Components of EDA, transforming data, Clustering :

Similarity measures, similarity coefficients, Heirarchical clustering methods : single,
complete and average linkage methods, dendrograms. Graphical Methods: Quintile
plots, Box Plots, Histogram, Stem & leaf diagram, Q-Q plots, P-P plots,
Unit –II: Stochastic simulation: generating random variables from discrete and
continuous distributions, simulation bivariate/multivariate distributions, simulating
stochastic processes such as simple queues. Variance reduction technique: Importance
sampling for integration, control variates, antithetic variables.
MCMC methods : Essence of MCMC methods, Time reversible MC, Law of large
numbers for MC. Metropolis-Hastings algorithm, Gibbs sampling for bivariate/
multivariate simulation.
Simulated annealing for optimization, simulated annealing for M.C. Simulation based
testing : simulating test statistics and power functions, permutation/randomization
tests.
Unit –III: Resampling paradigms: Jack knife and Bootstrap : Delete one J-K, pseudo
values, Bias and S.E. Efron’s bootstrap, Bootstrap C.I. Bootstrap-t C.I, Bootstrap C.I.
(percentile method), Bootstrap in regression, Bootstrap C.I. for linear regression
parameters.
Unit – IV: EM algorithm: Application to missing and incomplete data problems. Mixture
models. Smoothing with Kernels: Density estimation, kernel density estimator for
univariate data, Bandwidth selection and cross validation, Max likelihood L CV, Least
square CV.
Outcome: Students get knowledge about
Visualization of data and exploratory data analysis.

●Stochastic simulation techniques like MCMC.

●Some important methods of handling missing data and incomplete data problems like
EM algorithm etc.

● Jack knife, Bootstarp and nonparametric density estimation using kernels.

References :

1. Jun S. Liu : Monte Carlo Strategies in Scientific Computing, Springer

series
in statistics, 2001.
2. Efron B. and Tibsirani J. R. : An Introduction to Bootstrap
3. Ross S. M. : Applied Probability models

Elective Paper

Paper III- MST4T03- A

Big Data Analytics
Objective: The course includes basic exploratory data analysis, Hadoop and statistical
methodologies for big data analytics which is very useful in analyzing any type of data
using computers. The main purpose of this paper is to make aware of implementing the
statistical concepts using latest technology and programming in R.

Unit I: Review of Exploratory Data Analysis univariate, bivariate, multivariate

visualization, clustering, k-means, dimension reduction techniques, definition of big
data,4V characteristics - volume, variety, velocity & veracity, advantages of big data
analytics, techniques used in big data analysis ensemble, association, high dimensional
analysis, deep and precision analysis.

Unit II: Structured &Semi-structured data, relational and non relational data bases, real
time analytics using Hadoop systems, Hadoop ecosystem, Hadoop distributed file
system, stream processing engines, introducing &understanding text mining processes.

Unit III: Statistical methodologies for big data, re-sampling based methods, bag of little
bootstrap (BLB), leveraging, mean log likelihood, MCMC methods, divide & conquer
methods for linear regression (univariate & multivariate) models and GLM, online
updating method, implementation of factor analysis, cohort analysis and Time series
analysis.

Unit IV: Open source R and R packages, command level R, learning programming
concepts of R, breaking memory barriers,data management, numerical calculation,
sentiment analysis, R packages bigmemory & ff.

Outcome :

 Students will learn application of statistical concepts to very large data sets.

 Students will get knowledge of programming concepts in R.

  Students will learn data base systems and relational data base systems in details.

 Students will learn application of univariate & multivariate data representation

and analysis of their models using big data analytic techniques using R.

References:

1. BigData (covers Hadoop 2, MapReduce, Hive, YARN, Pig, Rand Data Visualization)
Black Book, DTE ditorial Services, Dreamtech Press.

2. Data Science & BigData Analytics Discovering, Analyzing, Visualizing and Presenting
data EMC Education Services, Wiley Publication.

3. Beginner’s guide for data analysis using R : Jeeva, Jone, Khanna Publication.

4. Practical Statistics for Data Scientists: By Peter Bruce and Andrew Bruce

5. Big Data analytics by Radha Shankarmani & M. Vijayalakshmi , second edition.

Paper III- MST4T03 - B

Operations Research
Objective: Operations research deals with the application of advanced analytical
methods which helps in taking better decisions. The course includes advanced
techniques that are useful in business, management, industry, project planning etc.

Unit – I:Inventory problems : Structure of inventory problem, EOQ formula, EOQ model
with uniform rate of demand & having no shortages, EOQ model with different rate of
demand in different cycles having no shortages, EOQ model with uniform rate of
demand & finite rate of replenishment having no shortages, EOQ model with uniform
rate of demand & finite rate of replenishment having shortages, EOQ model with
uniform rate of demand, infinite rate of replenishment having shortages, EOQ model
with single & double price breaks.
Unit – II:Single period probabilistic inventory models with
i) instantaneous demand & discrete units
ii) instantaneous demand& continuous units
iii) Continuous demand & discrete units
iv) Continuous demand & continuous units

Unit – III: Sequencing Problems :

Processing n jobs through two machines,
Processing n jobs through three machines,
Processing 2 jobs through m machines,
Processing n jobs through m machines,
Traveling salesman problem
Queuing Models : M/M/1 : FCFS/ /  / & its generalization
M/M/1 : FCFs/N/ ,
M/M/C/ : FCFS/  / ,
M/Ek/1 : FCFS / / ,
Unit – IV: Networking : Basic steps in PERT & CPM, methods of solving PERT problem,
crashing the network, updating (PERT & CPM) max. flow min. cut theorem, problems
based on max. flow min. cut theorem.
Outcome: Students get theoretical knowledge and the applications of the advanced
techniques in O.R. in business, management, industry, project planning etc. through the
topics like

 Concept of inventory problem, need of inventory and types of inventory models

 and types of probabilistic inventory models.

 Sequencing problems and methods to solve sequencing problems in different

situations

 Concept of queues, different types of queues and their analysis .

 Concept of Networking, CPM, PERT and methods of obtaining optimum

solutions to the problems.

References:
1) Taha H. A. : Operations Research
2) Hiller & Liberman ; Introduction to Operations research.
3) Kantiswaroop Gupta and Singh : Operations research.
4) Gross D and Harris C. M. : Fundamentals of queueing theory.

Paper III – MST4T03 - C

Actuarial Statistics

Objective: Actuarial science includes statistical methods to assess risk mainly in

insurance and finance. The course includes these statistical methods based on
probability theory and stochastic models. Objective here is to make the students aware
about this important branch of statistics.

Unit I: Life table and its relation with survival function, assumptions for fractional ages,
some analytical laws of mortality, select and ultimate tables. Multiple life functions, joint
and last survivor status, insurance and annuity benefits through multiple life functions.
Multiple decrement models, deterministic and random survivor groups, associated
single decrement tables, central rates of multiple decrement, net single premiums and
their numerical evaluations.
Unit II:.Principals of compound interest: Nominal and effective rates of interest and
discount, force of interest and discount, compound interest, accumulation factor,
continuous compounding.
Life insurance : Insurance payable at the moment of death and at the end of the year of
death-level benefit insurance, endowment insurance, diferred insurance and varying
benefit insurance, recursion, commutation functions.
Unit III: Life annuities : Single payment, continuous life annuities, discrete life annuities,
life annuities with monthly payments, commutation functions, varying annuities,
recursion, complete annuities- immediate and apportionable annuities-due. Net
premiums : Continuous and discrete premiums, true monthly payments premiums,
apportionable premiums, commutation functions, accumulation type benefits.
Unit IV: Net premium reserves : Continuous and discrete net premium reserves on a
semi continuous basis, reserves based on true monthly premiums, reserves on an
apportionable or discounted continuous basis, reserves at fractional duration,
allocations of loss to policy years, recursive formulas and differential equations for
reserves, commutation functions. 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.
Outcome: The course will make the students aware about an important branch of
Statistics called Actuarial Statistics by studying the following topics

●Life tables, its relation with survival function and application to life insurance.

●Calculation of premiums under different conditions.

●Net premium reserves, some practical considerations.

●Claim amount distribution.

References:
1. Bowers, N.L.; Gerber, H.U.; Hickman,J.C.; Jones D.A. and Nesbitt, C.J.(1986) :
Actuarial Mathematics. Society of Actuarials, Ithaca, Illiois, U.S.A. Second Ed
(1977).
2. Deshmukh S.R (2009): An introduction to Actuarial Statistics using R, Uni.Press
3. Spurgeon E.T (1972): Life Contingencies, Cambridge University.
Paper IV

MST4RP2 Project (Major)

Practical 7 : It will be based on Theory Paper.

Practical 8 : It will be based on Theory Paper.