1

SARDAR PATEL UNIVERSITY

DEPARTMENT OF STATISTICS FACULTY OF SCIENCE

COURSE OF STUDY

RULES OF DEGREE OF THE MASTER OF SCIENCE (M.Sc.) STATISTICS

R.PG.Sc.1: A candidate who has obtained the degree of Bachelor of Science/Arts

(Statistics) of this University or of any other University recognized as

equivalent thereto, or a candidate who has obtained Bachelor of Science

(General) degree with statistics as one of subjects or a student who has

studied statistics at second year B.Sc of the University and obtained his

B.Sc in Mathematics or Computer Science is eligible for the M.Sc. course

in the Department.

R.PG.Sc.2: The M.Sc. course is divided in to four semesters. The teaching, evaluation

of the various theory papers and laboratory work will be conducted under

the semester system. For this purpose each academic year is divided into

two semesters.

R.PG.Sc. 3: Candidates will be examined in each Theory paper for 100 marks and

practicals for 100 marks wherever prescribed at the end of each semester.

Project work will be undertaken during IV semester. There shall be a viva-

voce examination of 50 marks at the end of each semester to be held by

the University.

For deciding result of M.Sc. examination at each semester the ratio

between the internal assessment and external assessment will be 30:70.For

the purpose of internal assessment, the Department concerned will conduct

at least one test in each semester. The Department will also arrange Quiz,

Seminar etc. for internal assessment in theory course work. The

distribution of marks will be as under: -

1. Structure for each theory paper:

a) Quiz .. .. .. 5 marks

b) Seminar .. .. .. 5 marks

c) Test .. .. .. 20 marks

-----------

Total 30 marks

2

2. Structure for each practical

a) Regularity, records and results … 10 marks

b) Test … … … 20 marks

-------------

Total 30 marks

R.PG.Sc. 4: Every student is supposed to under take a project work under the guidance

of a teacher assigned by the Head of the Department. He/She may be

allowed to have external guide from the place he/she will be doing the

project. At the end of the project work students should submit, dully

certified by the guide(s), two typed and bound volumes of the project

report. The guide(s) evaluate the thesis for 50 marks. There will be a

common board of examiners with at least an external examiner for the

conduct of Project Viva. The students will be assessed for 50 marks by

this board. In the final mark sheet the sum total of marks given by the

guide(s) and the marks obtained in Project Viva will be entered against the

Head “ S-707 Project”. A student will be declared pass if he/she scores at

least 40 marks.

R.PG.Sc.5: Candidate shall be required to attend at least 75% of total theory lectures,

practicals organized under each of the courses during the semesters. In

project work student should work up to the satisfaction of the assigned

project guide(s).

R.PG.Sc.6: (i) The Head of the Department in consultation with other teachers of the

department will prepare in the beginning of the year a detailed scheme of

seminars, home work, quizzes, etc, and the Programme for the test

examinations and the same will be announced to the candidates.

(ii) The records of the test examinations as well as seminars, homework,

quizzes etc. will be maintained by the concerned faculty.

(iii)Every candidate shall maintain a regular record of his/her practical work

that shall be duly certified by his/her teacher(s) from time to time.

R.PG.Sc.7: Candidates will be required to obtain at least 33% marks in the internal

evaluation separately in each head of passing. A candidate who fails to

obtain 33% marks in not more than two heads of passing, may be allowed

to appear at the University examination by the head of the department

concerned on the recommendation of the committee appointed by him to

assess the candidate's overall performance.

(Note: A Head of passing will mean a course in theory or practical or

project work).

R.PG.Sc.8: A candidate desirous of appearing at each semester examination may

forward his application in the prescribed form to the Registrar through the

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Head of the University Post-graduate Department concerned on or before

the date prescribed.

R.PG.Sc.9: The final result for the award of the degree will be declared on the basis of

the grand total of all the Theory papers, Practicals, Project work and Viva-

Voce Examinations prescribed for all the four semesters.

R.PG.Sc.10: Only those students who fail in not more than two heads of passing at each

semester examination be allowed to keep terms at the next semester. No

candidate will be allowed to reappear for the course in which he/she has

already passed.

R.PGSc.11: Standard of passing:

The standard of passing at the M.Sc. degree examination will be as under:

(a) To pass any semester for the M.Sc. degree, a candidate must obtain at least 40%

marks at the University Examination and 40% marks in the aggregate of

University and Internal examination in each course of Theory, Practical and

Project work and 40% marks in Viva-Voce Examination.

(b) Award of Classes:

(i) The successful candidates will be placed in Second class if they obtained at least

50% or more marks but less than 60% in the aggregate of all semesters

examination taken together.

(ii) The successful candidates will be placed in First Class if they obtain at least 60%

or more but less than 70% of the total marks in the aggregate of the all the

semesters examinations taken together.

(iii) The successful candidates in First Class who obtain at least 70% or more marks in

the aggregate all the semesters examinations taken together will be declared to

have passed the examination in First Class with Distinction.

R.PG.SC.12:(i) A candidate who fails in more than two courses (any two of the total

heads of passing in the particular semester) in a particular semester

will not be admitted for further study at a subsequent semester and will

be required to repeat the courses in which he/she has failed by joining

the department as a regular student in the semester in which these

courses are again offered. A candidate failing in not more than two

courses at any semester examination will be promoted to the

subsequent semester according to the following scheme:

(ii) A candidate failing in the First Semester will be permitted to prosecute

his/her study up to the Third Semester but will not be permitted to go

to the Fourth Semester until he/she has cleared all the courses of the

First Semester even though he/she may have passed in the Second

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and/or Third semester. A candidate failing in the Second Semester;

will be permitted to prosecute his studies up to the Fourth Semester.

R.PG.Sc. 13: The following will be the course structure of M.Sc Programme in Statistics.

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M.Sc. Statistics

I SEMESTER

S 401 Probability Theory I

S 402 Matrix Algebra

S 403 Distribution Theory

S 404 Statistical Inference I

S 405 Statistical Computing

S 406 Practicals

S407 Viva-voce

II SEMESTER

S 501 Probability Theory II

S 502 Linear Models and Regression Analysis

S 503 Statistical Inference II

S 504 Theory of Sample Surveys

S 505 Operations Research

S 506 Practicals

S 507 Viva-voce

Semester III Semester IV

S 601 Deign of Experiments S 701 Computer Oriented Statistical Methods

S 602 Multivariate Analysis S 702 Statistical Quality Control Techniques

Optional Courses S 603 Reliability and Life Testing

Financial Statistics Biostatistics

S 604 Stochastic Processes SF 703(A) Econometrics

OR

SF 703(B) Time Series

SB 703 Bioassays

S 605 Practicals

S 606 Practicals.

SF 704 Actuarial Statistics SB 704 Clinical Trials

S 607 Viva Voce S 705 Practicals

S 706 Project

S 707 Viva Voce

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R.PG.Sc. 14:The following will be the marking scheme for the above-mentioned course.

MARKING SCHEME IN DETAIL

MAXIMUM PASSING MARKS SEMESTER SUBJECT

INTERNAL EXTERNAL TOTAL

S 401 Probability Theory I 30 70/28 100/40

S 402 Matrix Algebra 30 70/28 100/40

S 403 Distribution Teory 30 70/28 100/40

S 404 Statistical Inference I 30 70/28 100/40

S 405 Statistical Computing 30 70/28 100/40

S 406 Practicals 30 70/28 100/40

I

S 407 Viva-Voce 50/20 50/20

S 501 Probability Theory II 30 70/28 100/40

S 502 Linear Models and

Regression Analysis 30

70/28 100/40

S 503 Statistical Inference II 30 70/28 100/40

S 504 Theory of Sample

Surveys 30

70/28 100/40

S 505 Operations Research 30 70/28 100/40

S 506 Practicals 30 70/28 100/40

II

S 507 Viva-Voce 50/20 50/20

S 601 Deign of Experiments 30 70/28 100/40

S 602 Multivariate Analysis 30 70/28 100/40

S 603 Reliability and Life

Testing 30

70/28 100/40

S 604 Stochastic Processes 30 70/28 100/40

S 605 Practicals 30 70/28 100/40

S 606 Practicals. 30 70/28 100/40

III

S 607 Practicals 50/20 50/20

S 701 Computer Oriented

Statistical Methods 30 70/28 100/40

S 702 Statistical Quality

Control Techniques 30 70/28 100/40

Financial Statistics

SF 703(A) Econometrics

OR

SF 703(B) Time Series

30 70/28 100/40

SF 704 Actuarial Statistics 30 70/28 100/40

Biostatistics

SB 703 SB 703 Bioassays 30 70/28 100/40

SB 704 SB 704 Clinical Trials 30 70/28 100/40

S 705 Practicals 30 70/28 100/40

S 706 Project 60 40 100/40

IV

S 707 Viva-Voce -- 50/20 50/20

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S 401 PROBABILITY THEORY- I

Convergence of sequence of sets, fields and sigma fields, monotone

classes, Borel sets in R and Rn.

Additive set function, measures, probability measures, 9-finite

measures, properties of measures, Caratheodory extension theorem

(statement only) , its application for the construction of Lebesgue and

Lebeasgue-Steiltjes measures.

Measurable functions, Borel measurable functions, convergence almost

everywhere, convergence in measure.

6L

6L

7L

Integration of measurable functions with respect to a measure,

properties of integral.

Monotone convergence theorem. Fatou’s lemma, the dominated

convergence theorem, its application to differentiation under integral

sign.

Absolute continuity and singularity of measure, Lebeasgue

decomposition theorem and Radon-Nikodym theorem (statement only),

product measures and iterated integrals, statement of Fubini’s theorem.

8L

5L

6L

Introduction of probability space, random variables and random vectors,

expectations, moments, convergence in probability, convergence almost

surely in context of units 1 and 2.

7L

Books Recommended:

1. Ash, Robert (1972). Real Analysis and Probability. Academic Press.

2. Basu, A.K. (1999). Measure Theory and Probability. Prentice Hall of

India.

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3. Bhat, B.R. (2000). Modern Probability Theory, New Age International

Publication.

4. Billingsley, P. (1986). Probability and Measure. Wiley.

5. Burrill, C.W. (19 ). Measure, Probability and Integration.

6. Kingman, J.F.C. and Taylor, S.J. (1966). Introduction to Measure and

Probability. Cambridge University Press.

7. Parthsarathy, K.R. (1967). Probability Measures on Metric Spaces.

Academic Press.

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S 402 MATRIX ALGEBRA

Vector Spaces, Subspaces, linear independence, spanning set and basis,

dimension, linear transformation, kernel, range, gram-schmidt

orthogonalization, Orthogonal Projection, Matrix Representation of a

linear transformation.

6L

Algebra of Matrices, canonical forms, diagonal form, Triangular form,

Jordon Form, Inner-product spaces, Trace and Rank of a Matrix and

their properties, Determinants, Inverse, Determinant and Inverse of

special matrices, orthogonal and idempotent matrices and their

properties.

8L

Kronecker Product and Hadamard Product 2L

Matrix Factorizations 8L

Eigen Value and Eigen vectors, spectral decomposition, singular value

decomposition, Quadratic forms, definiteness and related results with

proofs.

6L

Generalized inverses(g-inverses) and related results with proofs,

Methods of constructing g-inverses, general solution to a system of

linear equations,

5L

Matrix derivatives and Jocobians 2L

Matrix inequalities 2L

Applications in Statistics 3L

Books Recommended:

1. Basilevsky, A.(1983).Applied Matrix Algebra in the Statistical Sciences,

North Holland.

2. Graybill, F.A.(1969).Introduction to matrices with applications in

Statistics, John Wiley.

3. Khatri, C.G.(1971).Matrix Algebra(In Gujarathi), Gujarat Granth

Nirman Board.

4. Searl, S.R.(1982).Matrix Algebra Useful for Statictics, John Wiley.

5. A.Ramachandra Rao and P.Bhimasankaran(1992). Linear Algebra, Tata

McGraw Hill, New Delhi.

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S 403 DISTRIBUTION THEORY

Brief review of basic distribution theory. Joint, marginal and conditional

distributions. Standard discrete and continuous distributions.

Transformation of variables.

8L

Compound, truncated and mixture distributions.

Regression, multiple and partial correlation coefficients and their inter-

relationship, multinomial distribution, marginal and conditional

distributions, it’s multiple and partial correlation coefficients.

8L

Approximating distributions. Delta method and its applications.

Approximating distributions of sample moments. Transformation of

statistics.

4L

Sampling distributions of statistics from univariate normal random

samples, Non-central χ2, t and F distributions and their properties.

Bivariate and multivariate normal distributions. Sampling from bivariate

normal distributions. Distributions of linear and quadratic functions

under normality and related distribution theory. Fisher-Cochran theorem.

Distributions of correlation coefficient and regression coefficient.

13L

Order statistics: their distributions and properties. Joint and marginal

distributions of order statistics. Extreme value and their asymptotic

distributions (statement only) with applications. Probability integral

transformation, Rank orders and their exact null distributions. One and

two sample examples of rank statistics such as sign statistic, Wilcoxon

signed rank statistics, Wilcoxon two sample statistics etc.

12L

Books Recommended:

1. Hogg, R. V. and Craig, T. T. (1978) Introduction to Mathematical

Statistics (Fourth Edition) (Collier-McMillan

2. Rohatgi, V. K. (1988) Introduction to Probability Theory and

Mathematical Statistics (Wiley Eastern)

3. C. R. Rao (1995) Linear Statistical Inference and Its Applications (Wiley

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Eastern) Second Edition

4. H, Cramer (1946) Mathematical Methods of Statistics, ( Prinecton).

5. J. D. Gibbons & S. Chakraborti (1992) Nonparametric statistical

Inference (Third Edition) Marcel Dekker, New York

6. Stuart, A. and Ord, J.K. (1995) Kendall’s Advanced Theory of Statistics

(VI Ed.) Distribution Theory, Vol. I, Griffin, London.

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S 404 STATISTICAL INFERENCE I

Sufficiency; factorization criterion, minimal sufficiency; completeness;

Exponential family of distributions and its properties. Non-regular

families admitting complete sufficient statistics.

8L

Chapman-Robbin’s inequality,Bhattacharya system of lower bounds in

case of single parameter; Rao-Cramer lower bound for multiparameter

case

8L

Unbiased estimators; Uniformly Minimum Variance Unbiased

Estimators(UMVUE); Rao-Black and Lehmann-Scheffee theorems.

5L

Maximum Likelihood Estimators(MLEs) ;asymptotic properties of

MLEs; Newton-Raphson Method and Method of Scoring.

8L

Equivariance; the principle of equivariance;location –scale family and

their properties; Pitman’s minimum risk equivariance estimators.

4L

Baye’s estimators: Statistical Problems viewed as problems of game

theory; loss function; risk function; prior and posterior distributions;

Bayes risk; Baye’s estimators of parameters and parametric functions for

squared error loss functions.

8L

Consistency and asymptotic normality of real and vector valued

estimators of parameters; Best asymptotic normal estimators.

5L

Books Recommended:

1. Kale, B.K. (1999) A First Course on Parametric Inference (Narosa)

2. Dudewicz, E. J. and Mishra, S.N.(1988) Modern Mathematical Statistics

(John Wiley)

3. Roussas, G. G. (1973) First Course in Mathematical Statistics (Addison

Wesley)

4. Silvey, S. D. (1975) Statistical Inference (Chapman and Hall)

5. Wilks, S. S. (1962) Mathematical Statistics (John Wiley )

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6. Lehmann, E. L. (1986) Testing of Statistical hypothesis (John Wiley)

7. Lehmann, E. L. (1988) Theory of Point Estimation (John Wiley)

8. Rohatgi, V. K. (1976) Introduction to theory of probability and

Mathematical Statistics (John Wiley & Sons)

14

S 405 STATISTICAL COMPUTING WITH C++

Introductory Concepts : Algorithm, Programming logic. 2L

Structure of C++ program, Preprocessors, Header Files. 2L

Data Types : int,char,float,bool,enumeration. 5L

Operators, I/O statements. 3L

Control Statements : if, switch, for loop, while loop, do-while loop, break

and continue statements.

8L

Arrays : one dimensional and multidimensional, array declaration, array

initialization, processing with arrays, etc.

5L

Strings as a character array, manipulation of strings. 3L

Functions : introduction, defining function, return statement, types of

functions, recursive functions, function overloading, call by value and

call by reference, using arrays as function arguments, functions having

default arguments.

8L

File Handling. 2L

Pointers : pointer declaration, pointer arithmetic, pointers and functions,

pointers and arrays.

5L

Scope and Lifetime of Variables : local, global, static, automatic,

external, register.

2L

Practicals : Writing some useful C++ programs for Statistical

Computing.

Books Recommended:

1. Robert Lafore : Object Oriented Programming With C++, Galgotia

Publishers.

2. John Hubbard (2000) : Programming with C++, 2nd

Edition, McGraw-

Hill (Schaum’s Outline Series) .

3. Cay Hortsmann (1999) : Computing Concepts with C++ Essentials, 2nd

Edition, John Wiley & Sons.

15

S 501 PROBABILITY THEORY- II

Random variables, expectations, moments.

Holder’s inequality, Minkowski’s inequality, Schwartz’s inequality,

Markov’s inequality, Jensen’s inequality.

Probability distribution of a random variable, distribution function (d.f.), the

change of variable theorem, mixtures of distributions with examples, joint

d.f.s, Jordon decomposition theorem (only statement), decomposition of

mixture d.f.s into absolutely continuous and singular parts.

Convergence of d.f., weak convergence and complete converence, weak

compactness theorem, Helly Bray theorem (only statement).

6L

6L

5L

Characteristic function (c.f.), its properties, c.f.’s and moments, inversion

theorem (only statement): Application to various distributions, continuity

theorem (only statement), composition of d.f.s, composition of c.f.’s,

compound distributions.

Independent classes and independent random variables, the multiplication

theorem, sequences of independent random variables: Borel 0-1 criterion,

Borel-Cantelli lemma, Kolmogorov’s 0-1 law.

Weak law of large numbers, Strong law of large numbers (SLLN),

Kolmogorov’s inequality, Kolmogorov’s sufficient condition for SLLN,

Kolmogorov’s SLLN (only statement).

The central limit theorem, Linberg-Levy’s theorem, Liapounov’s theorem,

statement of the Lindberg-Feller theorem.

8L

8L

6L

6L

Books Recommended:

1. Ash, Robert (1972). Real Analysis and Probability. Academic Press.

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2. Basu, A.K. (1999). Measure Theory and Probability. Prentice Hall of India.

3. Bhat, B.R. (2000). Modern Probability Theory, New Age International

Publication.

4. Billingsley, P. (1986). Probability and Measure. Wiley.

5. Burrill, C.W. (19 ). Measure, Probability and Integration.

6. Kingman, J.F.C. and Taylor, S.J. (1966). Introduction to Measure and

Probability. Cambridge University Press.

7. Parthsarathy, K.R. (1967). Probability Measures on Metric Spaces.

Academic Press.

17

S 502 LINEAR MODELS AND REGRESSION ANALYSIS

Gauss-Markov set-up, Normal Equations and Least Square Estimates, Error

and estimation spaces, variances and covariances of least square estimates,

estimation of error variance, estimation with correlated observations, least

square with correlated observations, least square estimates with (a)

restriction on parameters (b) specification error, simultaneous estimates of

linear parametric functions.

12L

Tests of hypotheses for one and more than one linear parametric functions,

confidence intervals and regions, Analysis of Variance, Power of F-test,

Multiple comparison tests due to Tukey and Scheffe, simultaneous

estimates of linear parametric functions.

9L

Introduction to one-way random effects linear models and estimation

variance components.

4L

Simple linear Regression, multiple regression, fit of polynomials and use of

orthogonal polynomials.

4L

Residual and their plots as test for departure from assumptions such as

fitness of the model, normality, homogeneity of variances and detection of

outliers, Remedies.

6L

Introduction to non-linear models. 4L

Multicolinearity, Ridge regression and principle component regression,

subset selection of explanatory variables , Mallow’s Cp Statistic.

6L

Books Recommended:

1. Cook, R.D. and Weisberg, S(1982) Residual and Influence in Regression.

Chapman and Hall

2. Draper, N.R. and Smith, H.(1998).Applied Regression Analysis. Third

Edition, Wiley India Ltd.

3. Gunst,R.F. and Mason, R.L.(1980). Regression Analysis and Its

Applications –A Data Oriented Approach. Macel and Dekker.

4. Rao, C.R.(1973) Linear Statistical Inference and its Applications.Wiley

Eastern.

5. Weisberg,S.(1985). Applied Linear Regression. Wiley.

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S 503 INFERENCE II

Randomized test, randomized version of Neyman-Pearson lemma and

its generalization (proof of sufficient condition only). Uniformly most

powerful tests for one sided alternative for one parameter exponential

class of densities and extension to the distributions having monotone

likelihood ratio property.

10L

Unbiased tests, its applications to one-parameter exponential family of

distribution, Similar tests, UMP similar tests, test with Neyman

structure, UMP unbiased tests for parameters of normal distribution

10L

Confidence bounds: Neyman’s principal of confidence bounds,

uniformly most accurate and uniformly most accurate unbiased

confidence bounds.

4L

Likelihood Ratio Test (LRT), large sample properties: consistency of

tests, asymptotic distribution of LRT statistic, chi-square goodness of fit

test.

6L

Sequential Probability Ratio Test (SPRT), properties of SPRT, the

fundamental identity of SPRT and its use in derivation of OC and ASN

functions.

5L

U-statistics, properties and asymptotic distributions (in one and two

samples cases). One sample problem : Kolmogorov-Smiirnov test,

Location problem. Wilcoxon signed-rank test. Two sample problem:

Wilcoxon-Mann-Whitney test.

10L

Books Recommended:

1. Kale, B.K. (1999) A First Course on Parametric Inference (Narosa)

2. Dudewicz, E. J. and Mishra, S.N.(1988) Modern Mathematical Statistics

3. Ferguson, T. S. (1967): Mathematical statistics ( A decision theoretic

approach), Academic Press.

4. Kendall, M. G. and Stuart, A. (1979) : The Advanced Theory of

Statistics, Vol. 2, (IV edition), Griffin, London.

5. Lehmann, E. L. (1986) Testing of Statistical hypothesis (John Wiley )

6. Rohatgi, V. K. (1976) Introduction to theory of probability and

Mathematical Statistics (John Wiley & Sons)

19

S 504 THEORY OF SAMPLE SURVEYS

General principles of sample surveys. Basic ideas in estimation from

probability sampling. Review of important results in SRSWR,

SRSWOR, Stratified sampling and Systematic sampling.

8L

Varying probability sampling: PPS sampling with replacement and

without replacement, Horvitz-Thompson estimator, Random group

method, PPS systematic sampling.

8L

Use of supplementary information for estimation: Ratio and Regression

estimators with their properties and generalizations.

8L

Cluster sampling, Two-stage sampling. 8L

Double sampling for ratio and regression estimators and for

stratification

5L

Non-sampling errors, response and non-response errors and their

treatments, randomized response

8L

Books Recommended:

1. Cochran, W.G. (1984) Sampling Techniques (Wiley)

2. DesRaj and Chandhok, P. (1998) Sample Survey Theory (Narosa)

3. Mukhopadyay, p. (1998) Theory and Methods of Survey Sampling

(PHI, New- Delhi)

4. Murthy, M.N.(1977) Sampling Theory and Methods

5. Sukhatme P.V, Sukhatme, B.V., Sukhatme S. and Asok C. (1984)

Sampling Theory of Surveys with Applications (Indian Soc. for

Agricultural Statistics, New Delhi).

20

S 505 OPERATIONS RESEARCH

Linear Programming: Convex sets, supporting and separating hyperplanes,

standard linear programming problem, basic feasible solutions, simplex

algorithm and simplex method, geometry of simplex method. Duality in

linear programming, duality theorems, dual simplex method with

justification, post optimality analysis, sensitivity analysis and parametric

linear programming.

12L

Integer Linear Programming: Introduction, Gomory Cut Method, Branch

and Bound Method.

6L

Network Analysis: Definition and formulation, critical path method,

Project Evaluation and Review Technique(PERT), Optimal allocation of

resources (men-power) through time schedule.

6L

Queueing Theory: Introduction, steady state solution of M/M/c/∞/FIFO

and M/M/C/N/FIFO with associated distributions of queue length and

waiting time.(c=1 as particular case)

8L

Non-linear Programming: Quadratic Programming, Kuhn-Tucker

Conditions, Wolfe-Beal-Fletcher algorithms for solving quadratic

programming problems.

8L

Books Recommended:

1. Hadley,G. (1961).linear Programming. Addison-Wesley.

2. Hiller,F.S. and Liberman, G.J.(1995).Introduction to Operations Research.

(6th

Edition)Mcgraw-Hill Int.Ed.

3. Hiller,F.S. and Liberman, G.J.(1995).Introduction to Mathematical

Programming( 2nd

Edition)McGraw Hill Int. Ed.

4. Murty,K.G.(1983).Linear Programming (2nd

Edition)John-Wiley.

5. Taha, H.A.(1997).Operations Research(6th

Edition) Prentice-Hall India

Ltd.

6. Ravindran,A. Phillips, D.T., Solberge, J.J.(1987).Operations Research(2nd

Edition) John-Wiley.

21

S 601 DESIGN AND ANALYSIS OF EXPERIMENTS

Introduction to designed experiments; General block design and its

information matrix (C), criteria for connectedness, balance and

orthogonality; Intrablock analysis. BIBD, PBIBD(2)– recovery of

interblock information; Youden square design – intrablock analysis.

Analysis of covariance in a General –Markov model, applications to

standard designs.

17L

Missing plot technique – general theory and applications. 3L

Introduction to Galois field and finite geometries. Construction of

orthogonal Latin squares, construction of BIB designs using MOLS,

and finite geometries.

8L

General factorial experiments, factorial effects; best estimates and

testing of significance of factorial effects; study of 2n and 3

n factorial

experiments in randomized blocks; Confounding and fractional

factorial for symmetrical factorials. Split plot and split block

experiments.

12L

Application areas: Response surface experiments; clinical trials,

treatment control designs.

5L

Books Recommended:

1. Alok Dey (1986). Theory of Block Designs. Wiley Eastern.

2. Das, M.N. and Giri, N.C. (1986). Design and Analysis of

Experiments, Wiley Eastern.

3. D. C. Montgomery (1976). Design and Analysis of Experiments.

John Wiley, NY

4. Giri, N. (1986). Analysis of variance. South asian publishers.

5. Hinkelman, K. and Kempthorne, A. (1994). Design and Analysis of

Experiments. Vol. I, Introduction to experimental design.

6. Nigam, A.K., Puri, P.D. and Gupta, V.K. (1988). Characterization

and Analysis of block designs. Wiley Eastern.

7. P.W.M. John (1966). Statistical Design and Analysis of Experiments.

The Macmillan company, NY.

22

8. Raghavarao, D. (1971). Constructions and Combinatorial problems

in Design of Experiments. John Wiley, Now in paper back ed also.

9. Shah, S.M. and Kabe (1983). An Introduction to construction and

analysis of Statistical designs. Queen University Publication.

23

S-602 MULTIVARIATE ANALYSIS

Multivariate normal distribution (characterization) and its properties. Random

sampling from a multivariate normal distribution. Maximum likelihood estimation

of parameters, Distribution of the MLEs.

8L

Wishart matrix, its distribution and properties. Distribution of generalized

variance. Null and non-null distribution of simple correlation matrix. Null

distribution of partial and multiple correlation coefficients. Application in testing

and interval estimation.

12L

Distribution of sample intra-class correlation coefficient in a random sample from

a symmetric multivariate normal distribution. Application in testing and interval

estimation.

2L

Distribution of Hotelling’s 2T statistic. Application in tests onmean vector

for one and more multivariate normal populations and also on equality of the

components of mean vector in a multivariate normal population.

6L

Wilk’s lamda distribution. Test concerning covariance matrices.

Test for identical populations.

7L

Multivariate linear regression model, estimation of parameters, tests of linear

hypotheses about regression coefficients using LRT. Multivariate analysis of

variance(MANOVA) of one and two way classified data.

10L

Books Recommended:

1. Anderson,T.W.(1958). Introduction to Multivariate Statistical Analysis, Wiley, NY

2. Giri, N C. (1977). Multivariate Statistical Inference. Academic press, NY

3. Kshirsagar, A. M. (1972). Multivariate Analysis, Marcel Dekker,NY

4. Johnson, R.A. and Wichern, D.W.(1992). Applied MultivariateStatistical Analysis

3rd

,PHI

5. Mardia, K.V,Kent, J.T, and Bibby, J.M.(1979). Multivariate Analysis, Academic

Press, NY

6. Muirhead, R. J. (1982). Aspect of Multivariate Statistical Theory,

Wiley, NY

7. Rao,C.R.(1973).Linear Statistical Inference and its Applications, 2nd ed. Wiley,NY

8. Saber, G.A.F.(1984). Multivariate Observations, Wiley, NY

9. Siotani, M., Hayakawa, T., and Fujikoshi, Y.(1985). Modern

Multivariate Statistical Analysis, American Sc. Prass, Inc.

24

S 603 -RELIABILITY AND LIFE TESTING

Reliability concepts and measures; components and systems;

coherent systems; reliability of coherent systems; cuts and paths; modular

decomposition; bounds on system reliability; structural and reliability

importance of components.

8L

Life distribution; reliability function; hazard rate; common life

distributions-Exponential, Weibull, gamma, Pareto and lognormal

distributions. Estimation of parameters and tests in these models.

10L

Notion 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.

10L

Reliability estimation based on failure times in various censored

life tests and in tests with replacement of failed items; stress-strength

reliability and its estimation

5L

Reliability Growth models; probability plotting techniques;

Hollander-Proschan and Deshpande tests for exponentiality; tests for

HPP against NHPP with repairable systems.

7L

Estimation of survival function-Actuarial Estimator, Kaplan-

Meier Estimator; Properties of K-M estimator; Estimation under the

assumption IFR/DFR.

5L

Books Recommended:

1. Cox, D.R. and Oakes, D. (1984) Analysis of Survival Data,

Chapman and Hall, New York.

2. Gross A.J. and Clark, V. A. (1975)Survival Distributions:

Reliability Applications in the Biomedical Sciences, John Wiley

and Sons.

3. Elandt - Johnson, R.E. Johnson N.L. (1980) Survival models and

Data Analysis, John Wiley and Sons

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

5. Barlow R. E. & Proschan F. (1975) Statistical Theory of

Reliability & Life testing. Holt, Rinehart & Winston Inc.

6. Zacks, S. Reliability

25

S.604 STOCHASTIC PROCESSES

Introduction to Stochastic Processes:Classification of stochastic

processes according to state space and time domain. Countable state

Markov Chains (MCs’), Chapman-Kolmogorov equations, calculation of n-

step transition probability and its limits, classification of states. Stationary

distribution, random walk and gambler’s ruin problem, Applications from

social, biological and physical sciences. Statistical inference in Markov

Chains

Discrete state space continuous time stochastic processes: Poisson

process, Generalization of Poisson process. Renewal process.

Birth and death process: Special cases of birth and death process,

Application to queues and storage problems etc.

12L

10L

10L

Branching process: Galton-Watson branching process, probability of

ultimate extinction, distribution of population size.

Stationary processes: weakly and strongly stationary processes. Moving

average and auto regressive processes.

4L

3L

Continuous time and continuous state space Markov process:

Kolmogorov-Feller differential equations, Diffusion processes with Wiener

process and Ornstein-Uhlenbeck process as particular cases. First passage

time and other problems.

6L

Books Recommended:

1. Adke, S.R. and Manjunath, S.M. (1984). An Introduction to Finite Markov

Processes. Wiley Eastern.

2. Bhat, B.R. (2000) Stochastic Models: Analysis and Applications. New Age

International Publications, New Delhi.

3. Feller, W. (1971). An Introduction to Probability Theory and its

applications. Vol II.

4. Karlin, S. and Taylor, H.M. (1975). A first course in Stochastic Processes,

Vol.1, Academic Press.

5. Prakash Rao, B.L.S. and Bhat, B.R. (1996). Stochastic Processes and

Statistical Inference, New Age International Publications, New Delhi.

6. Ross, S.M. (1983). Stochastic Processes, Wiley.

26

S 701 COMPUTER ORIENTED STATISTICAL METHODS

Statistical Packages: Introduction of packages and their use for following Statistical

problems.

4L

Generation of random variables from: (i) Normal (ii) Gamma (iii) Weibull (iv) F, t

and χ2 (v) Binomial (vi) Poisson (vii) Multivariate Normal Distributions , and (viii)

from different Markov chains

6L

Simulation: Simulation of Probability distributions of different statistics whose

probability distributions may not be obtained theoretically.

5L

Logistic Regression Analysis: Introduction; The multiple logistic regression model;

Fitting the logistic regression model; testing for the significance of the model.

Application of logistic regression in study of Matched case control data.

5L

Cox’s regression model: Proportional Hazard Model. Estimation and tests of

parameters of the proportional hazard model. Use of this in comparison of two more

life distributions.

5L

Bootstrap techniques: Introduction; The Bootstrap estimate

of standard error of the correlation coefficient. Application of

Bootstrap in regression Models. The Bootstrap estimate of bias.

7L

Multivariate techniques: (i) Canonical Correlation (ii)

Discriminant Analysis (iii) Principal component analysis (iv)

Factor Analysis (v) Cluster Analysis.

8L

Books Recommended:

1. Fishman, G.S. (1996) Monte Carlo: Concepts, Algorithms, and

Applications.(Springer).

2. Rubinstein, R.Y. (1981); Simulation and the Monte Carlo Method. (Wiley).

3. Tanner, M.A. (1996); Tools for Statistical Inference, Third edition. (Springer.)

4. Efron, B. and Tibshirani. R.J. (1993); An Introduction to the Bootstrap. (Chapman

and Hall).

5 . Shao J. and Tu, D. (1995); The Jackknife and the Bootstrap. Springer Verlag.

6. McLachlan, G.J. and Krishnan, T. (1997) The EM Algorithms and

Extensions.(Wiley.)

7. Simonoff J.S. (1996) Smoothing Methods in Statistics. (Springer).

8. Kshirsagar, A. M. (1972). Multivariate Analysis, Marcel Dekker, NY

27

S 702 STATISTICAL QUALITY CONTROL TECHNIQUES

Quality Improvement in the Modern Business Environment: Meaning of quality

and quality improvement. Usefulness of statistical methods in quality improvement.

Total quality management. Link between quality and productivity; quality and cost.

ISO-9000 and QS-9000 quality system standards. Recent advancement in quality

management.

4L

Basic Concepts of quality control: Process control and process capability. Relation

between theory of testing hypotheses and charts. choice of control limits, rational

subgroup principle, allocating sampling effort, average run length.

4L

Statistical Process Control:Control charts for measurements and attributes. X ,R,

s, p and np charts. (revision) CUSUM charts, EWMA chart –Use of these charts for

prediction. CUSUM, EWMA for controlling process variability. Comparison of

these charts with Shewart charts. Acceptance control charts.

18L

Process Capability Indices:Purpose of capability Indices. Determining the process

capability using X , R charts. The role of normality in determining defective parts

per million. One sided specification, non-normal distributions.

7L

Some special plans: Chain sampling plans , continuous sampling plans, Dorrian-

Shainin’s lot plot method, Skip –lot sampling plans.

Process Design and Iimprovement with designed experiments: Use of Design of

Experiments in SPC: 2k- factorial design with k≥1. 2

k-p fractional factorial design.

7L

Taguchi’s contribution to Quality Engineering: Elements and principle of quality

engineering. Steps in robust design; signal to noise ratio.

5L

Books Recommended:

1 Montgomery, D. C. (1985) Introduction to Statistical Quality Control.(Wiley)

2 Montgomery, D.C. (1985) Design and Analysis of Experiments; Wiley

Rayon, T.P(1989) Statistical Methods for quality improvement. John Wiley and ons.

3 Ott, E.R. (1975) Process Quality Control; McGraw Hill

4 Wetherill, G.B. (1977) Sampling Inspection and Quality Control; Halsted Press

5 Phadke, M.S. (1989) Quality Engineering through Robust Design; Prentice Hall

6 Wetherill, G.B. and Brown, D.W. (1991) Statistical Process Control, Theory and

Practice;Chapman and Hall

28

SB 703 : BIOASSAYS

Types of biological assays: Direct assays; Ratio estimators, asymptotic

distributions;.Fieller’s theorem

5L

Regression approaches to estimating dose-response relationships:

Logit and probit approaches when dose-response curve for standard

preparation is unknown; Quantal responses; Methods of estimation of

parameters; Estimation of extreme quantiles; Doseallocation schemes;

Polychotomous quantal response

11L

Estimation of points on the quantal response function 5L

Sequential procedures 7L

Estimation of safe doses 6L

Bayesian approach to bioassay

5L

Books Recommended:

1 Z. Govindarajulu (2000). Statistical Techniques in Bioassay, S. Kargar

2 D. J. Finney (1971). Statistical Method in Bioassay, Griffin.

3 D. J. Finney (1971). Probit Analysis (3rd Ed.), Griffin.

4 G. B. Weatherile (1966). Sequential Methodsin Statistics, Methuen.

.

29

SB 704 CLINICAL TRIALS

Introduction to clinical trials : the need and ethics of clinical trials, bias and

random error in clinical studies, conduct of clinical trials, overview of Phase I-IV

trials, multi-center trials.

3L

Data management : data definitions, case report forms, database design,

datacollection systems for good clinical practice, protocol definition.

2L

Design of clinical trials : parallel vs. cross-over designs, cross-sectional vs.

longitudinal designs, review of factorial designs, objectives and endpoints of

clinical trials, design of Phase I trials, design of single-stage and multi-stage Phase

II trials, design and monitoring of Phase III trials with sequential stopping, design

of bioequivalence trials.

20L

Reporting and analysis : analysis of categorical outcomes from Phase I - III trials,

analysis of survival data from clinical trials. Interim analysis method, motivating

intent-to-treat analyses, Determining sample size.

15L

Surrogate endpoints : selection and design of trials with surrogate endpoints,

analysis of surrogate endpoint data

2L

Meta-analysis of clinical trials. 3L

Books Recommended:

1 S. Piantadosi (1997). Clinical Trials : A Methodologic Perspective. Wiley and Sons

2 C. Jennison and B. W. Turnbull (1999). Group Sequential Methods with Applications

to Clinical Trials, CRC Press.

3 L. M. Friedman, C. Furburg, D. L. Demets (1998). Fundamentals of Clinical Trials,

Springer Verlag.

4 J. L. Fleiss (1989). The Design and Analysis of Clinical Experiments. Wiley and

Sons.

5 E. Marubeni and M. G. Valsecchi (1994). Analyzing Survival Data from Clinical

Trials and Observational Studies, Wiley and Sons.

30

SF 703(A) ECONOMETRICS

Nature of Econometrics, Review of general linear model, use of dummy variables and

seasonal adjustments, generalized least square estimation and prediction, Hetroscedastic

disturbances.

12L

Autocorrelation and its consequences and tests, Multicolinearity Problem, its

implications and tools for handling the problem, ridge regression, use of principle

component analysis.

12L

Linear Regression with stochastic regressors, Instrumental variables, Errors in variables,

Autoregressive linear regression.

8L

Simultaneous Linear Equations Model:

(a) Examples, Identification problem, Restrictions on structural parameters, rank and

order conditions, restriction on variances and covariances

6L

(b) Estimations in simultaneous equations models, recursive system, 2SLS estimators,

limited information estimators, Full information maximum likelihood method.

7L

Books Recommended:

1. Apte PG (1990); Text book of Econometrics. Tata McGraw Hill.

2. Cramer, J.S. (1971) : Empirical Econometrics, North Holland.

3. Gujarathi, D (1979) : Basic Econometrics, McGraw Hill.

4. Intrulligator, MD (1980) : Econometric models - Techniques and applications, Prentice

Hall of India.

5. Johnston, J. (1984) : Econometric methods, Third edition, McGraw Hill.

6. Klein, L.R. (1962) : An introduction to Econometrics, Prentice Hall of India.

7. Koutsoyiannis, A (1979) : Theory of Econometrics, Macmillan Press.

8. Malinvaud, E (1966) : Statistical methods of Econometrics, North Holland.

9. Srivastava, V.K. and Giles D.A.E (1987) : Seemingly unrelated regression equations

models, Maicel Dekker.

10. Theil, H. (1982) : Introduction to the theory and practice of Econometrics, John Wiley.

11. Walters, A (1970) : An introduction to Econometrics, McMillan & Co.

12. Wetherill, G.B. (1986) : Regression analysis with applications, Chapman Hall.

31

SF 703(B): TIME SERIES ANALYSIS

Time-series as discrete parameter stochastic process. Auto covariance and

autocorrelation functions and their properties.

Exploratory time Series Analysis, Tests for trend and seasonality. Exponential

and Moving average smoothing. Hot and winters smoothing. Forecasting based

on smoothing, adaptive smoothing.

9L

Detailed study of the stationary processes: (1) moving average (MA), (2) Auto

regressive (AR), (3) ARMA and (4) AR integrated MA (ARIMA) models. Box-

Jenkins models. Discussion (without proof) of estimation of mean, auto

covariance and autocorrelation functions under large sample theory. Choice of

AR and MA periods. Estimation of ARIMA models parameters. Forecasting.

Residual analysis and diagnostic checking. Use of computer packages like SPSS.

24L

Spectral analysis of weakly stationary process. Periodogram and correlogram

analyses. Computations based on Fourier transform. Spectral decomposition of

weakly AR process and representation as a one-sided MA process-necessary and

sufficient conditions.

12L

Books Recommended:

1. Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis –

Forecasting and Control, Holden-day, San Franciscor.

2. Anderson, T. W. (1971). The Statistical Analysis of Time Series, Wiley,

N.Y.

3. Montgomery, D. C. and Johnson, L. A. (1977). Forecasting and Time

Series Analysis, McGraw Hill.

4. Kendall, Sir Maurice and Ord, J. K. (1990). Time Series (Third Edition),

Edward Arnold.

5. Brockwell, P.J. and Davis, R. A. Time Series: Theory and Methods

(second Edition). Springer – Verlag.

32

Additional Books for Reference:

6. Fuller, W. A. (1976). Introduction to Statistical Time Series, John Wiley,

N. Y.

7. Granger, C. W. J. and Newbold (1984). Forecasting Econometric Time

Series, Third Edition, Academic Press.

8. Priestley, M. B. (1981). Spectral Analysis & Time Series, Griffin,

London.

9. 4. Bloomfield, P. (1976). Fourier Analysis of Time Series – An

Introduction, Wiley.

10. Granger, C. W. J. and Hatanka, M. (1964). Spectral Analysis of

Economic Time Series, Princeton University Press, N. J.

11. Koopmans, L. H. (1974). The Spectral Analysis of Time Series,

Academic Press.

12 Nelson, C. R. (1973). Applied Time Series for Managerial Forecasting,

Holden-Day.

13. Findley, D. F. (Ed.) (1981). Applied Time Series Analysis II, Academic

Press.

33

SF-704:ACTUARIAL STATISTICS

Section I – Probability Models and Life Tables.

Utility theory, insurance and utility theory, models for individual claims and their

sums, survival function, curtate future lifetime, force of mortality

3L

Life table and its relation with survival function, examples, assumptions for

fractional ages, some analytical laws of mortality, select and ultimate tables.

Multiple life functions, joint life and last survivor status, insurance and annuity

benefits through multiple life functions evaluation for special mortality laws

6L

Multiple decrement models, deterministic and random survivorship groups,

associated single decrement tables, central rates of multiple decrement, net single

premiums and their numerical evaluations.

6L

Distribution of aggregate claims, compound Poisson distribution and its

applications. Distribution of aggregate claims, compound Poisson distribution

and its applications.

3L

Section II – Insurance and Annuities

3L

Principles of compound interest: Nominal and effective rates of interest and

discount, force of interest and discount, compound interest, accumulation factor,

continuous compounding.

6L

Life insurance: Insurance payable at the moment’s of death and at the end of the

year of death-level benefit insurance, endowment insurance, differed insurance

and varying benefit insurance, recursions, commutation functions.

6L

Life annuities: Single payment, continuous life annuities, discrete life annuities,

life annuities with monthly payments, commutation functions, varying annuities,

recursions, complete annuities-immediate and apportion able annuities-due.

34

Net Premiums: Continuous and discrete premiums, true monthly payment

premiums, apportionable premiums, commutation functions, accumulation type

benefits.

Payment premiums, apportionable premiums, commutation functions

accumulation type benefits.

6L

Net premium reserves: Continuous and discrete net premium reserve, reserves

on a semi continuous basis, reserves based on true monthly premiums, reserves

on an apportion able or discounted continuous basis, reserves at fractional

durations, 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.

6L

Books Recommeded:

1. N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbitt,

(1986), Actuarial Mathematics’, Society of Actuaries, Itasca, IIIinois, U.

S. A. Second Edition (1997)

Section I – Chapters: 1, 2, 3, 8, 9, and 11

Section II – Chapters: 4, 5, 6, 7, 13, and 14

Books for Additional References:

2. Spurgeon E. T. (1972), Life Contingencies, Cambridge University Press

3. Neill, A. (1977). Life Contingencies, Heinemann