## Statistics MOD Revised PHD 1 Syllabus Mumbai University by munotes

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Course Work for M.Phil / Ph. D. in STATI STICS S yllabus

To be implemented from the Academic year 201 9-2020

UNIVERSITY OF MUMBAI

Syllabus for the Course Work for

M. Phil / Ph.D.

Program: M. Phil / Ph.D.

Course : STATISTICS

(With effect from the academic year 201 9–2020)

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Program: Pre-Ph.D. course work

Course Code: PHST 001

Course Title: Advance d Probability Theory and Measure Theory

Duration of course: Classroom Teaching 60 hours +60 notional Hours = 120 hours

Credits of the course: 04 credits.

Objectives of the Course: To increase fluency of research student in Mathematical foundations,

Probability Theory and Measure Theory required for their research in Statistics.

Outcomes of the Course: Research Student will have knowledge of:

Calculus (differentiation, integration).

Modern Probability Theory.

Elementary Measure Theory.

Law of Large numbers.

Central Limit Theorem .

Pre-requisite s for the Course :

Number system, sets, bounded and unbounded set s, supremum and infimum of sets, o pen set,

closed set, limit point of a set, countable sets, uncountable sets. Sequence, convergence of a

sequence, limit poin t of a sequence, limit inferior and limit superior of a sequence, non -

convergent sequence, Cauchy principle, algebra of sequence., subsequences, Monotone

sequence. Infinite series, convergence, tests for convergence, alternate series, absolute

convergence.

Course contents:

Functions, inverse function, Limit of a function, continuous and discontinuous functions,

left and right hand limits, uniform continuity. Vector valued function. Derivative, mean

value theorems, Taylor series expa nsion and its applications, extreme values,

indeterminate forms, power series. Functions of several variables: explicit and implicit

functions, limit and continuity, differentiability, partial derivative, change of order,

higher order derivatives, total de rivative, Taylor series expansion, maxima and minima.

Riemann integration, mean value theorems, integration by parts, change of variable in an

integration, derivative under integration

Classes of sets, field, sigma field, Borel field, minimal sigma field, limit of sequences of

subsets, e e non measurable functions , introduction to counting

measure, measurable space .

probability measure, random variable and random vector, distribution function

(multivariate case ), expectation and moments, independence, characteristic function,

inversion formula, Laplace transform, conditional probability and conditional

expectations, Martingales.

Probability distributions and their relations, characterizations and generalizations.

Stable distributions, infinite divisibility, Convergence of random variables, Law of large

numbers, strong law of large numbers, central limit theorem.

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References:

Apostol, T. M. (1974): Mathem atical Analysis. 2nd edition, Narosa Publishing

house.

Ash, R. B. (2000). Probability & Measure Theory. Academic Press.2nd Edition.

Athreya, K. B. and Lahiri S. (2006). Measure Theory and Probability Theory ,

Springer.

Bartle G. and Sherbe rt, D. R. (2000): Introduction to Real Analysis. 3rd edition.

Wiley.

Bhat B.R. (1999): Modern Probability Theory: An Introductory test book. 3rd

edition. New Age International.

Billingsley, P. (1995). Probability and Measure , 3rdEdition, John Wiley, New

York

Chan dra, T. and Gangopadhyay, S. (2017): Fundamentals of Probability Theory.

Narosa Publishing House.

Chung, K. L. (2001). A Course in Probability Theory , Third Edition, Academic

Press, London

Gut, A. (2005): Probability: A Graduate Course. Springer.

Kumar, A and Kumaresan S. (2015): A Basic course in Real analysis. CRC Press.

Malik, S. C. and Arora, S. (2017): Mathematical Analysis. 5th edition. New age

International Publishers.

Program: Pre-Ph.D. course work

Course Code: PHST 002

Course Title: Advance d Statistical Inference

Duration of course: Classroom Teaching 60 hours +60 notional Hours = 120 hours

Credits of the course: 04 credits.

Objectives of the Course: To make aware research student to the recent statistical inferential

methods.

Outcomes of the Course:

Research student will be aware of basic estimation and testing of hypothesis problems.

Research student will be able to solve the research problem in inference by using,

o MCMC methods

o EM algorithm.

o Bootstrap and Jacknife methods.

Pre-requisite s for the Course:

Methods of estimation, properties of estimator, uniformly most powerful test, uniformly most

powerful unbiased test.

Course contents:

Maximum likelihood estimation (MLE) under restricted parameter space, inconsistent

MLE, MLE in discrete case, iterative procedures for MLE.

Similar tests, Neyman -Structure tests , invarian t tests. Confidence sets, U niformly Most

Accurate ( UMA), Uniformly Most Accurate Unbiased ( UMAU) confidence sets.

Bayesian inference: Point estimators, credible intervals , Bayesian Highest Posterior

Density ( HPD ) confidence intervals, testing, prediction of a future observation . Model

elec ion and h po he i e ing ba ed on objec ive probabili ie and Ba e ’ fac or large

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sample methods: Limit of posterior distribution, consistency of posterior distribution,

asymptotic normality of posterior distribution.

EM algorithm: Incomplete data problems, E and M steps, convergence of EM algorithm,

standard errors in the conte xt of EM algorithm, applications of EM algorithm, Bayesian

approach to EM algorithm.

Markov Chain Monte Carlo ( MCMC ) methods: Methods of generating random sample,

Metropolis -Hastings and Gibbs Sampling algorithms, convergence, applications,

Bayesian appro ach.

Bootstrap methods, estimation of sampling distribution, confidence intervals, failure of

Bootstrap, variance stabilizing transformation, Jackknife and cross -validation,

applications.

Smoothing techniques: Kernel estimators, nearest neighbor estimator s, orthogonal and

local polynomial estimators, wavelet estimators, Splines, Choice of bandwidth and other

smoothing parameters.

References:

Bolstad , W. M. (2010): Understanding computational Bayesian statistics. John Wiley.

Bolstad, W. M. (2017) : Introduction to Bayesian Statistics, 3rd Edition. John Wiley.

Congdon, P. (2006) : Bayesian Statistical Modeling, John Wiley

Davison, A.C. and Hinkley, D.V. (1997) : Bootstrap methods and their Applications.

Chapman and Hall.

Dixit, U. J. (2016): Examp les in Parametric Inference with R. Springer.

Efron, B. and Hastie, T. (2016): Computer Age Statistical Inference: Algorithms,

Evidence and Data Science. Cambridge University Press.

Gamerman , Dani (1997): Markov chain Monte Carlo: Stochastic simulation for

Bayesian inference. Chapman and Hall.

Gelman , A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2003): Bayesian Data

Analysis, second edition. Chapman and Hall.

Gilks, W. R., Richardson, S., an d Spiegelhalter, D. (eds.) (1995) : Markov Chain Monte

Carlo in Practice. Chapman and Hall.

Ghosh, J. K., Delampady M. and T. Samantha (2006) : An Introduction to Bayesian

Analysis: Theory & Methods, Springer.

Kundu, D. and Basu, A. (2009): Statistical C omputing: Existing Methods and Recent

Developments. Narosa.

Lehmann, E.L. and Casella, G. (1998) : Theory of Point Estimation . Springer. 2nd

Edition.

Lehma nn, E. L. and Romano, J. (2005): Testing Statistical Hypotheses , Springer

McLachlan, G.J. and Krishnan, T. (2008) : The EM Algorithms and Extensions. Wiley.

Rajgopalan, M. and Dhanavanthan, P. (2012) : Statistical Inference. PHI Learning

private limited.

Srivastava, M. K. and Srivastava, M. (2014) : Statistical Inference: Estimation Theory.

PHI Le arning private limited.

Srivastava, M. K. and Srivastava, M. (2014) : Statistical Inference: Testing of

Hypotheses. PHI Learning private limited.

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Examination and Evaluation pattern for the courses PHST 001 and PHST 002 :

Course will be evaluated in two par ts,

Part A] Continuous Evaluation (CE).

Part B] Course End Examination (CEE) and

CE will be of 40 marks which will in clude one mid -test of 20 marks of one hour duration and

other 20 marks are composed of any one or combinations of group discussion, viva-voce, open

book test, presentations, or assignments.

CEE will be a theory examination of 60 marks and of three hours duration based on entire

syllabus. Answer book of research students will be evaluated by the concerned teacher or

examiner appointed by Board of Studies in Statistics.

Standard of passing:

Standard of passing is as per the circular No. Exam./Thesis/Univ/VCD/947 of 2018 of Mumbai

University.

A student has to obtain at least 55% marks or equivalent grade in the UGC 7 point scale in the

CE and CEE combined.

If a research student is not able to secure minimum marks for passing then he / she has to

reappear for CEE of 100 marks.

Program: Pre-Ph.D. course work

Course Code: PHST 003

Course Title: Research Methodology

Duration of course: Classroom Teaching 60 hours +60 notional Hours = 120 hours

Credits of the course: 04 credits.

Objectives of the Course: To expose research student to the different research methodologies

for research in Statistics.

Outcomes of the Course:

Research student will understand :

Tools to solve a Research problem .

Different software .

Different types of research

Pre-requisite for the Course: -

Course contents:

Identification of research problem.

Research ethics, plagiarism, copyright.

Collection of and review the research methodologies such as qualitative, quantitative

methods in the relevant field of research in Statistics in concern with his/her research

guide.

Collection and review of published research papers and reference books in the relevant

field of research in Statistics in concern with his/her research guide.

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Statistical software such as R -Environment, SAS, SPSS, MINITAB, LATEX in the

relevant field of research in Statistics in concern with his/her research guide.

Examination and Evaluation pattern:

Course will be evaluated in two parts,

Part A] Continuous Evaluation (CE).

Part B] Course End Examination (CEE) and

CE will be of 40 marks which will include , collection of research papers , report writing based on

collected research papers , viva -voce, presentations, or assignments and will be assigned by

his/her guide .

CEE will be of 60 marks and will be a presentation based on the review of published research

articles and reference books , research student has gone through in the course.

Evaluation of CEE will be done by a following committee ,

Head of the department (Chairperson) and

Guide of research student and One / two research guides from the department.

Standard of passing:

Standard of passing is as per the circular No. Exam./Thesis/Univ/VCD/947 of 2018 of Mumbai

University.

A student has to obtain at least 55% marks or equivalent grade in the UGC 7 point scale in the

CE and CEE combined .

If a research student is not able to secure minimum marks for passing then he / she has to give

presentation again of 100 marks. Evaluation will be done by same committee before the expi ry

of registration period of that research student .

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