university of calicut · vide paper read as (2) above, the syllabus of m.sc programme in statistics...

UNIVERSITY OF CALICUT(Abstract)

MSc programme in Statistics- under Credit Semester System (PG)-Scheme and Syllabus–approved-implemented-with effect from 2010 admission onwards-Orders issued

GENERAL & ACADEMIC BRANCH-IV ‘J’ SECTION

No. GA IV/J2/4230/10 Dated, Calicut University PO, 26.07.2010Read:1. U.O.No. GAIV/J1/1373/08 dated, 23.07.2010.

2. Item no. 1 of the minutes of the meeting of the Board of Studies in Statistics PG, held on 09.06.2010 3.Item no. III a. 30 of the minutes of the meeting of the Academic Council held on 03.07.2010

O R D E R

Credit Semester System was implemented to the PG programmes in affiliated Arts and Science Colleges of the University with effect from 2010 admission onwards,as per paper read as first,above.

The Board of Studies in Statistics PG vide paper read as second above decided to recommend the implementation of Calicut University regulations for for Credit Semester System for PG curriculum 2010 in affiliated Colleges{CUCSS-PG-2010} and to apply the Credit Semester System to the MSc Statistics programme conducted by the affiliated colleges of the University with effect from 2010 –11 academic year onwards.. The Vice Chancellor,due to exigency, approved the minutes of the meeting of the Board of Studies in Statistics PG held on 09.06.2010 subject to ratification by the Academic Council and the same was ratified by the Academic Council vide paper read as 3 above.

Sanction has therefore been accorded for implementing the scheme and Syllabus of MSc programme in Statistics under CSS (PG) with effect from 2010 admission onwards.

Orders are issued accordingly. Scheme and syllabus are appended.

Sd/-DEPUTY REGISTRAR (G & A-IV)

For REGISTRARTo

The Principals of affiliated Colleges offering MSc programme in StatisticsCopy to:

P.S to V.C PAtoRegistrar/Chairman,B/S,Statistics,PG/CE/EX/DRIII/DR,PG/EGI/Enquiry/System Administrator with a request to upload in the University website/Information Centres/G&A I `F``G`Sns/GAII,III

Forwarded/By Order

Sd/-SECTION OFFICER.

UNIVERSITY OF CALICUT(Abstract)

M.Sc Programme in Statistics – under Credit Semester System PG 2010 Scheme and Syllabus of II semester – approved implemented with effect from 2010 admission onwards – Orders issued.

GENERAL & ACADEMIC BRANCH-IV ‘J’ SECTION

No. GA IV/J2/4230/2010 Dated, Calicut University PO, 04.01.2011Read: 1. U.O.No.GAIV/J1/1373/08 dated 23.07.2010.

2. U.O.No.GAIV/J2/4230/10 dated 26.07.2010. 3. Letter dated 16.12.2010 from the Chairman, Board of Studies in Statistics PG.

O R D E R

As per paper read as (1) above, Credit Semester System PG was introduced for all PG courses in affiliated Arts and Science Colleges of the University.

Vide paper read as (2) above, the syllabus of M.Sc Programme in Statistics for the 1st semester was implemented.

The Chairman, Board of Studies in Statistics vide paper read as (3) above has forwarded the syllabus for the II semester of M.Sc programme in Statistics as per the decision of Board of Studies held on 09.06.2010.

The Vice-Chancellor, due to exigency, exercising the powers of the Academic Council approved the syllabus subject to ratification by the Academic Council.

Sanction has therefore been accorded for implementing the syllabus of II semester of M.Sc Programmes in Statistics under Credit Semester System PG for the affiliated colleges with effect from 2010 admissions.

Orders are issued accordingly. Syllabus appended.

Sd/- DEPUTY REGISTRAR(G&A IV)

For REGISTRAR

ToThe Principals of all affiliated colleges offering M.Sc Programmes in Statistics.

Copy to:PS to VC/PA to Registrar/Chairman Board of Studies in Statistics PG/CE/ DR III /EX section/ DR-PG/EG-I/ Information centres/Enquiry/System Administrator (with a request to upload in the University website)GAI ‘F’ ‘G’ Sections/GAII/GAIII/SF/FC

Forwarded/By Order

Sd/- SECTION OFFICER

M. Sc. (Statistics) Degree Programme under the Credit Semester System(CSS) for the Affiliated Colleges of University of Calicut

Programme Structure & Syllabi (With effect from the academic year 2010-2011 onwards)

Duration of programme: Two years - divided into four semesters of not less than 90 working days each.

Course Code Type Course Title Credits

I SEMESTER (Total Credits: 20)

ST1C01 Core Measure Theory and Probability 4ST1C02 Core Analytical Tools for Statistics – I 4ST1C03 Core Analytical Tools for Statistics – II 4ST1C04 Core Linear Programming and Its Applications 4ST1C05 Core Distribution Theory 4

II SEMESTER (Total Credits: 18)

ST2C06 Core Estimation Theory 4ST2C07 Core Sampling Theory 4ST2C08 Core Regression Methods 4ST2C09 Core Design and Analysis of Experiments 4ST2C10 Core Statistical Computing– I 2

III SEMESTER (Total Credits: 16)

ST3C11 Core Stochastic Processes 4ST3C12 Core Testing of Statistical Hypotheses 4ST3E-- Elective Elective-I 4ST3E-- Elective Elective-II 4

IV SEMESTER (Total Credits: 18)

ST4C13 Core Multivariate Analysis 4 ST4E-- Elective Elective-III 4ST4C14 Core Project/Dissertation and General Viva-Voce 8ST4C15 Core Practical – II 2

Total Credits: 72 (Core courses-52, Elective courses-12 and Project / Dissertation -8)

The courses Elective –I, Elective –II, and Elective –III shall be chosen from the following list.

LIST OF ELECTIVES

Sl. No. Course Title Credits

E01 Advanced Operations Research 4

E02 Econometric Models 4

E03 Statistical Quality Control 4

E04 Reliability Modeling 4

E05 Advanced Probability 4

E06 Time Series Analysis 4

E07 Biostatistics 4

E08 Computer Oriented Statistical Methods 4

E09 Lifetime Data Analysis 4

E10 Statistical Decision Theory and Bayesian Analysis 4

E11 Statistical Ecology and Demography 4

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SYLLABI OF COURSES OFFERED IN SEMESTER -I

ST1C01: Measure Theory and Probability (4 Credits)

Unit- 1 . Sets, Classes of sets, Measure space, Measurable functions and Distribution functions:

Sets and sequence of sets, set operations, limit supremum, limit infimum and limit of sets, Indicater function, fields ,sigma fields, monotonic class, Borel field on the real line, set functions,Measure, measure space, probability space, examples of measures, properties of measures, measurable functions, random variables and measurable transformations, induced measure and distribution function, Jordan decomposition theorem for distribution ,multivariate distribution function, continuity theorem for additive set functions and applications, almost everywhere convergence, convergence in measure, convergence in probability, convergence almost surely, convergence in distribution.

Unit-2. Integration theory , expectation, types of convergence and limit theorems:

Definition of integrals and properties, convergence theorems for integrals and expectations-Fatou’s lemma, Lebesgue monotonic convergence theorem, Dominated convergence theorem, Slutsky’s theorem, convergence in (convergence in mean), inter relations between different types of convergence and counter examples.

Unit-3. Independence and Law of Large numbers:

Definition of independence, Borel Cantelli lemma, Borel zero one law, Kolmogrov ‘s zero one law, Weak law of large numbers(WLLN), Convergence of sums of independent random variables-Kolmogrov convergence theorem, Kolmogorov’s three-series theorem. Kolmogorov’s inequalities, Strong law of large numbers (SLLN), Kolmogorov’s Strong law of large numbers for independent random variables, Kolmogorov’s strong law large numbers for iid random variables.

Unit-4. Characteristic Function and Central limit Theorem:

Characteristic function, Moments and applications, Inversion theorem and its applications, Continuity theorem for Characteristic function (statement only),Test for characteristic functions, Polya’s theorem(statement only), Bochner’s theorem(statement only). Cenral limit theorem for i.i.d random variables, Liapounov’s Central limit theorem, Lindeberg–Feller Central limit theorem(statement only).

Book for study:-

1. A.K. Basu.(1999). Measure theory and probability. Prentice Hall of India private limited New Delhi.

References:-1. A.K.Sen.(1990), Measure and Probability .Narosa.2. Laha and Rohatgi (1979).Probability Theory. John Wiley New York.3. B.R.Bhat (1999),Modern Probability theory .Wiley Eastern ,New Delhi.4. Patrick Billingsly,(1991),Probability and Measure ,Second edition ,John Wiley .

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ST1C02: Analytical Tools for Statistics – I (4 Credits) Unit-1 .Multidimensional Calculus

Limit and continuity of a multivariable function, derivatives of a multivariable function, Taylor’s theorem for a multivariable function. Inverse and implicit function theorem, Optima of a multivariable function, Method of Lagrangian multipliers, Riemann integral of a multivariable funtion.

Unit-2. Analytical functions and complex integration

Analytical function, harmonic function, necessary condition for a function to be analytic, sufficient condition for function to be analytic, polar form of Cauchy- Riemann equation, construction of analytical function. Complex line integral, Cauchy’s theorem, Cauchy’s integral formula and its generalized form. Poisson integral formula, Morera’s theorem. Cauchy’s inequality, Lioville’s theorem, Taylor’s theorem, Laurent’s theorem.

Unit-3. Singularities and Calculus of Residues.

Zeroes of a function, singular point, different types of singularities. residue at a pole, residue at infinity, Cauchy’s residue theorem, Jordan’s lemma, integration around a unit circle, poles lie on the real axis, integration involving many valued function.

Unit- 4. Laplace transform and Fourier Transform

Laplace transform, Inverse Laplace transform. Applications to differential equations, The infinite Fourier transform, Fourier integral theorem. Different forms of Fourier integral formula, Fourier series.

Book for study

1. Andre’s I. Khuri(1993) Advanced Calculus with applications in statistics. Wiley & sons (Chapter 7)2. Pandey, H.D, Goyal, J. K & Gupta K.P(2003) Complex variables and integral transforms Pragathi

Prakashan, Meerut.3. Churchill Ruel.V.(1975), Complex variables and applications .McGraw Hill.

References.

1. Apsostol (1974) Mathematical Analysis, Second edition Norosa, New Delhi.2. Malik, S.C & Arora.S(2006) Mathematical analysis, second edition, New age international

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ST1C03: Analytical Tools for Statistics – II (4 Credits)

Unit-1.. Riemann-Stieltjes integral and uniform convergences.

Definition, existence and properties of Riemann-Stieltjes integral, integration by parts, change of variable, mean value theorems, sequence and series of functions, point wise and uniform convergences,

test of uniform convergence, consequence of uniform convergence on continuity and integrability, Weirstrass theorem.

Unit- 2. Algebra of Matrices

Linear transformations and matrices, operations on matrices, properties of matrix operations, Matrices with special structures – triangular matrix, idempotent matrix, Nilpotent matrix, symmetric Hermitian and skew Hermitian matrices unitary matrix. Row and column space of a matrix, inverse of a matrix. Rank of product of matrix, rank factorization of a matrix, Rank of a sum and projections, Inverse of a partitioned matrix, Rank of real and complex matrix, Elementary operations and reduced forms.

Unit- 3 Eigen values, spectral representation and singular value decomposition

Characteristic roots, Cayley-Hamilton theorem, minimal polynomial, eigen values and eigen spaces, spectral representation of a semi simple matrix, algebraic and geometric multiplicities, Jordan canonical form, spectral representation of a real symmetric, Hermitian and normal matrices, singular value decomposition.

Unit -4 Linear equations generalized inverses and quadratic forms

Homogenous system, general system, Rank Nullity Theorem, generalized inverses, properties of g-inverse, Moore-Penrose inverse, properties, computation of g-inverse, definition of quadratic forms, classification of quadratic forms, rank and signature, positive definite and non negative definite matrices, extreme of quadratic forms, simultaneous diagonalisation of matrices.

Book for study:-

1. Ramachandra Rao and Bhimashankaran (1992).Linear Algebra Tata McGraw hill2. Lewis D.W (1995) Matrix theory, Allied publishers, Bangalore.3. Walter Rudin (1976).Principles of Mathematical Analysis, third edition, McGraw –hill

international book company New Delhi.

References:-

1. Suddhendu Biswas (1997) A text book of linear algebra, New age international.2. Rao C.R (2002) Linear statistical inference and its applications, Second edition, John Wiley

and Sons, New York.3. Graybill F.A (1983) Matrices with applications in statistics.

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ST1C04: Linear Programming and Its Applications (4 Credits)

Unit-1. Some basic algebraic concepts.

Definition of a vector space, subspaces, linear dependence and independence, basis and dimensions, direct sum and complement of subspaces, quotient space, inner product and orthogonality. Convex sets and hyperplanes.

Unit-2. Algebra of linear programming problems .

Introduction to linear programming problem(LPP), graphical solution, feasible, basic feasible and optimal basic feasible solution to an LPP, analytical results in general LPP, theoretical development of simplex method. Initial basic feasible solution, artificial variables, big-M method, two phase simplex method, unbounded solution, LPP with unrestricted variables, degeneracy and cycling, revised simplex method.

Unit- 3. Duality theory and its applications.

Dual of an LPP, duality theorems complementary slackness theorem, economic interpretation of duality, dual simplex method. Sensitivity analysis and parametric programming, integer programming, Gomery’s cutting plane algorithm and branch and bound techniques.

Unit- 4. Transportation problem and game theory.

Transportation problem, different methods of finding initial basic feasible solution, transportation algorithm, unbalanced transportation problem, assignment problem, travelling salesman problem. Game theory, pure and mixed strategies. Conversion of two person’s zero sum game to an Linear programming problem. Fundamental theorem of game. Solution to game through algebraic, graphical and Linear programming method.

Book for study:-

1. Ramachandra Rao and Bhimashankaran (1992).Linear Algebra Tata McGraw hill.2. Cooper and Steinberg (1975). Methods and Applications of Linear Programming, W.B.

Sounders Company, Philodelphia, London.

References:-

1. J.K.Sharma(2001).Operations Research Theory and Applications.McMillan New Delhi.2. Hadley,G.(1964).Linear Programming,Oxford &IBH Publishing Company,New Delhi. 3. Kanti Swaroop,P.K. Gupta et.al,(1985),Operation Research,Sultan Chand & Sons.4. Taha.H.A.(1982).Operation Research and Introduction ,MacMillan.

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ST1C05: Distribution Theory (4 Credits)

Unit- 1. Discrete distributions Random variables ,Moments and Moment generating functions, Probability generating functions, Discrete uniform, binomial, Poisson, geometric, negative binomial, hyper geometric and Multinomial distributions, power series distributions.

Unit- 2. Continuous distributions

Uniform , Normal, Exponential, Weibull, Pareto, Beta, Gama, Laplace, Cauchy and Log-normal distribution. Pearsonian system of distributions, location and scale families.

Unit-3. Functions of random variables.

Joint and marginal distributions, conditional distributions and independence, Bivariate transformations, covariance and correlations, bivariate normal distributions, hierarchical models and mixture distributions, multivariate distributions, inequalities and identities. Order statistics.

Unit -4 .Sampling distributions

Basic concept of random sampling, Sampling from normal distributions, properties of sample mean and variance .Chi-square distribution and its applications, t-distribution and its applications ,F-distributions- properties and applications. Noncentral Chi-square, t, and F-distributions.

Books for study:-

1. Rohatgi, V.K.(1976).Introduction to probability theory and mathematical statistics. John Wiley and sons.

2. George Casella and Roger L. Berger(2003). Statistical Inference. Wodsworth & brooks Pacefic Grove, California.

References:-

1. Johnson ,N.L.,Kotz.S. and Balakrishna.N.(1995). Continuous univariate distributions, Vol.I &Vol.II, John Wiley and Sons, New York.

2. Johnson ,N.L.,Kotz.S. and Kemp.A.W.(1992).Univarite Discrete distributions, John Wiley and Sons, New York.

3. Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics Vol I: Distribution Theory, 4th Edition

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SYLLABI OF COURSES OFFERED IN SEMESTER-II

ST2C06: Estimation Theory (4 Credits)

Unit-1: Sufficient statistics and minimum variance unbiased estimators.

Sufficient statistics, Factorization theorem for sufficiency (proof for discrete distributions only), joint sufficient statistics, exponential family, minimal sufficient statistics, criteria to find the minimal sufficient statistics, Ancillary statistics, complete statistics, complete statistics, Basu’s theorem (proof for discrete distributions only), Unbiasedness, Best Linear Unbiased Estimator(BLUE), Minimum Variance Unbiased Estimator (MVUE), Fisher Information, Cramer Rao inequality and its applications, Rao-Blackwell Theorem, Lehmann- Scheffe theorem, necessary and sufficient condition for MVUE.

Unit-2: Consistent Estimators and Consistent Asymptotically Normal Estimators.

Consistent estimator, Invariance property of consistent estimators, Method of moments and percentiles to determine consistent estimators, Choosing between consistent estimators, Consistent Asymptotically Normal (CAN) Estimators.

Unit-3:Methods of Estimation.

Method of moments, Method of percentiles, Method of maximum likelihood (MLE), MLE in exponential family, One parameter Cramer family, Cramer-Huzurbazar theorem, Bayesian method of estimation.

Unit-4: Interval Estimation.

Definition, Shortest Expected length confidence interval, large sample confidence intervals, Unbiased confidence intervals, Bayesian and Fiducial intervals.

Books for Study:-

1. Kale,B.K.(2005). A first course in parametric inference, Second Edition, Narosa Publishing House, New Delhi.

2. George Casella and Roger L Berger (2002). Statistical inference, Second Edition, Duxbury, Australia.

References:-1. Lehmann, E.L (1983). Theory of point estimation, John Wiley and sons, New York.2. Rohatgi, V.K (1976). An introduction to Probability Theory and Mathematical Statistics, John

Wiley and sons, New York.3. Rohatgi, V.K (1984). Statistical Inference, 4. Rao, C.R (2002). Linear Statistical Inference and its applications, Second Edition,

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ST2C07: Sampling Theory (4 credits)

Unit-I: Census and Sampling-Basic concepts, probability sampling and non probability sampling,

simple random sampling with and without replacement- estimation of population mean and total-

estimation of sample size- estimation of proportions. Systematic sampling-linear and circular

systematic sampling-estimation of mean and its variance- estimation of mean in populations with

linear and periodic trends.

Unit-II: Stratification and stratified random sampling. Optimum allocations , comparisons of

variance under various allocations. Auxiliary variable techniques. Ratio method of estimation-

estimation of ratio, mean and total. Bias and relative bias of ratio estimator. Mean square error of

ratio estimator. Unbiased ratio type estimator. Regression methods of estimation. Comparison of ratio

and regression estimators with simple mean per unit method. Ratio and regression method of

estimation in stratified population.

Unit-III: Varying probability sampling-pps sampling with and without replacements. Des-Raj

ordered estimators, Murthy’s unordered estimator, Horwitz-Thompson estimators, Yates and Grundy

forms of variance and its estimators, Zen-Midzuno scheme of sampling, πPS sampling.

Unit-IV: Cluster sampling with equal and unequal clusters. Estimation of mean and variance, relative

efficiency, optimum cluster size, varying probability cluster sampling. Multi stage and multiphase

sampling. Non-sampling errors.

References

1. Cochran W.G (1992) Sampling Techniques, Wiley Eastern, New York

2. D. Singh and F.S. Chowdhary Theory and Analysis of Sample Survey Designs, Wiley

Eastern(New Age International), NewDelhi.

3. P.V.Sukhatme et.al. (1984) Sampling Theory of Surveys with Applications. IOWA State

University Press, USA

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ST2C08: Regression Methods (4 Credits)

Unit-1: Simple and multiple regression.

Introduction to regression. Simple linear regression- least square estimation of parameters, Hypothesis testing on slope and intercept, Interval estimation, Prediction of new observations, Coefficient of determination, Regression through origin, Estimation by maximum likelihood, case where x is random.

Multiple Linear Regression- Estimation of model parameters, Hypothesis testing in multiple linear regression, Confidence interval in multiple regression, Prediction of new observations.

Unit- 2: Model Adequacy Checking, Transformation and weighting to correct model Inadequacies.

Residual analysis, the press statistics, detection of treatment of outliers, lack of fit of the regression model. Variance -stabilizing transformations, Transformation to linearize the model, Analytical methods for selecting a transformation, Generalized and weighted least squares.

Unit- 3: Polynomial regression model and model building.

Polynomial models in one variable, Nonparametric regression, Polynomial models in two or more variables, orthogonal variables. Indicator variables, Regression approach to analysis of variance. Model building problem, computational techniques for variable selection.

Unit-4: Generalized Linear Models.

Logistic regression model, Poisson regression, The generalized linear models- link function and linear predictors, parameter estimation and inference in GLM, prediction and estimation in GLM, residual analysis in GLM over dispersion.

Books for Study:-

1. Montgomery ,D.C., Peck, E.A., Vining G Geofferey (2003). Introduction to Linear Regression Analysis. John Wiley & Sons.

References:-

1. Chatterjee, S & B. Price (1977). Regression analysis by example, Wiley, New York.

2. Draper, N.R & H. Smith (1988). Applied Regression Analysis. 3rd Edition, Wiley, New York.

3. Seber, G.A.F (1977). Linear Regression Analysis. Wiley, New York.4. Searle , S.R (1971). Linear Model. Wiley, New York.

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ST2C09: Design and Analysis of Experiments (4 credits)

Unit- 1: Linear Model, Estimable Functions and Best Estimate, Normal Equations, Sum of Squares, Distribution of Sum of Squares, Estimate and Error Sum of Squares, Test of Linear Hypothesis, Basic Principles and Planning of Experiments, Experiments with Single Factor-ANOVA, Analysis of Fixed Effects Model, Model Adequacy Checking, Choice of Sample Size, ANOVA Regression Approach, Non parametric method in analysis of variance.

Unit- 2: Complete Block Designs, Completely Randomized Design, Randomized Block Design, Latin Square Design, Greaco Latin Square Design, Analysis with Missing Values, ANCOVA,

Unit- 3: Incomplete Block Designs-BIBD, Recovering of Intra Block Information in BIBD, Construction of BIBD, PBIBD, Youden Square, Lattice Design.

Unit- 4: Factorial Designs-Basic Definitions and Principles, Two Factor Factorial Design-General Factorial Design, 2k Factorial Design-Confounding and Partial Confounding, Two Level Fractional Factorial, Split Plot Design.

Books for study

1) Joshi D.D. (1987) Linear Estimation and Design of Experiments. Wiley Eastern Ltd., New Delhi.

2) Montgomery D.C. (2001) Design and Analysis of Experiments. 5th edition, John Wiley & Sons-New York.

References

1) Das M.N. & Giri N.S. (2002) Design and Analysis of Experiments. 2th edition , New Age International (P) Ltd., New Delhi.

2) Angola Dean & Daniel Voss (1999) Design and Analysis of Experiments. Springer-Verlag, New York.

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ST2C10: Statistical Computing-I (2 credits)(Practical Course)

Teaching scheme: 6 hours practical per week.

Statistical Computing-I is a practical course. The practical is based on the following FIVE

courses of the first and second semesters.

1. ST1C05: Distribution Theory

2. ST2C06: Estimation Theory

3. ST2C07: Sampling Theory

4. ST2C08: Regression Methods

5. ST2C09: Design and Analysis of Experiments

Practical is to be done using R programming / R software. At least five statistical data

oriented/supported problems should be done from each course. Practical Record shall be maintained by

each student and the same shall be submitted for verification at the time of external examination.

Students are expected to acquire working knowledge of the statistical packages – SPSS and SAS.

The Board of Examiners (BoE) shall decide the pattern of question paper and the duration of the

external examination. The external examination at each centre shall be conducted and evaluated on the

same day jointly by two examiners – one external and one internal, appointed at the centre of the

examination by the University on the recommendation of the Chairman, BoE. The question paper for

the external examination at the centre will be set by the external examiner in consultation with the

Chairman, BoE and the H/Ds of the centre. The questions are to be evenly distributed over the entire

syllabus. Evaluation shall be done by assessing each candidate on the scientific and experimental skills,

the efficiency of the algorithm/program implemented, the presentation and interpretation of the results.

The valuation shall be done by the direct grading system and grades will be finalized on the same day.

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MODEL QUESTION PAPER

FIRST SEMESTER M.Sc. DEGREE EXAMINATION (CSS), 2010 Branch: Statistics

ST1C01: Measure Theory and Probability

Time :3hrs MaximumWeightage:36

Part A(Answer all the questions , Weightage 1 for each question)

1. Define sigma field of subsets of a set.2. Define a measurable function .3. If P(A)=0.6,P(B)=0.4,A and B are independent then find P(AUB).4. State Dominated Convergence theorem.5. With the help of an example show that convergence in distribution need not imply convergence in

probability.6. State Fatou’s lemma.7. When do you say that two classes of events are independent.8. Verify whether

is a distribution function.

9. When do you say that a sequence { } of random variables obey the strong law of large numbers.

10. Obtain the characteristic function of an exponential random variable .11. State Bochner’s theorem for characteristic functions.12. State Lindeberg-Feller central theorem.

Part B(Answer any eight questions .Weightage 2 for each question)

13 .Prove that -field is a monotone field. Verify whether the converse is true.14 Let B be the class of subsets of N ,the set of natural numbers . Define µ: B R by

µ(A) is µ a measure on B15.Prove or disprove the statement “ almost sure convergence implies convergence in probability”. 16. Define integral of an arbitrary function starting from that of a simple function.17. State and prove monotone convergence theorem.18. Show that convergence in

19. Show that P (lim sup )=0 if < ∞.20. Establish Kolmogrov’s Strong law of large numbers for independent random variables.21. Prove that tail events of a sequence of independent random variables have their probability either zero or one.22. Show that characteristic function is uniformly continues on the real line.23.Obtain the probability density function corresponding to the characteristic function ϕ(t)= , t ϵ R.24.Establish Lindeberg -Levy central limit theorem.

Part C(Answer any two questions ,Weightage 4 for each question)

25.(a) If f and g are measurable functions then show that (i)f+g (ii)Max (f,g ) (iii) Min(f,g) are measurable.(b) If { } is sequence of measurable functions .Show that lim sup{ } is measurable.

26.(a)If X then show that If X. (b) prove or disprove the statement :convergence in probability implies convergence in mean.27.(a) Establish Borel zero one law. (b) State and prove Kolmogrov’s three series criterion28.(a)State and prove inversion theorem on characteristic functions . (b) Examine if central limit theorem holds for the sequence of independent random va riables {

}with P( = P( and P(

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ST1C02: Analytical Tools for Statistics- I

Time :3 hrs Maximum Weightage:36


1. Investigate for continuity at(1,2) of the function f(x,y)

2. Define the concept of directional derivative of a function.3. State implicit function theorem.4. When do you say that a function is analytic in a domain D.5. State a necessary condition that W=f(z) =u+iv be analytic in a domain D.6. Write down the Cauchy’s integral formula.7. When do you say that Z=a is an isolated singularity of function f(z).8. State residue theorem.9. Define pole of order m of a function f(z).10. Define Laplace transform of a function.11. If L{F(t)}=f(s),then find L{F(at)}12. Define a periodic function, Give an example.

Part B

(Answer any eight questions. Weightage 2 for each question)

13. If f(x,y)= y+ .Find and .

14. Find the value of where E is the triangle bounded by the strightlines y=x ,y=0 and y=1.15. Find maxima and minima of the function f(x,y)= -3x-12y+20.16.Show that real and imaginary part of an analytic function satisfies the Laplace equation.

17.Obtain the Taylor’s series of the function f(z)= in the region |z|<2.18.State and prove a sufficient condition for a function f(z) to be analytic19. Show that the function f(z)= has an isolated singularity at z=0.

20.Prove that if a>0 then =21. State and prove Cauchy’s residue theorem.22.Find the Laplace transform of the function cos ax.23. Find the inverse Laplace transform of .

24.Find the Fourier transform of

Part C(Answer any two questions, Weightage 4 for each question)

25. State and prove Taylor’s theorem on two variables.

26. Obtain the Taylor and Laurent’s series expansion of the function

f(z)= in the region (i)0<|z|<1(ii)1<|z|<2

27. By the method of Contour integration prove that =28.Find the Fourier series expansion of the function x in the interval

(-π, π ). Hence show that + +…

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ST1C03: Analytical Tools for Statistics- II


Part A(Answer all the questions, Weightage 1 for each question)

1. State a necessary and sufficient condition for a function to be Riemann- Stieltjes integrable.2. Give an example of a sequence of function that converges pointwise but not uniformly.3. Define normal form of an m x n matrix.4. Define Eigen values and Eigen vectors of a square matrix.5. Define minimal polynomial of a square matrix A.6. What is the reduced echelon form of a matrix with full column rank.7. Define image and kernel of a linear map.8. Give two definitions of rank of a matrix.9. Differentiate between nilpotent matrix and idempotent matrix. 10. Give an example of a Hermitian matrix11. Write down the symmetric matrix associated with the quadratic form q(x,y)=12. Define Moore Penrose g- inverse.

Part B

(Answer any eight questions. Weightage 2 for each question)

13 .If is a refinement of P ,then prove that L( , f , α ) ≥ L(P, f , α ) 14.If f is monotonic on [a,b],and if α is continuous on [a.,b],then show that f ϵ (α). 15.Prove that the following series is uniformly convergent in [-1,1]

.

16. Prove that any two characteristic vectors corresponding to two distinct characteristic roots of a hermitian matrix are orthogonal 17. Find eigen values and eigen vectors of

18.State and prove Cayley-Hamilton theorem. 19. Show that A is nilpotent if and only if all the characteristic roots of A are zero.

20. Prove or disprove : If and are n x n matrices in HCF , s in HCF 21. Explain the classification of quadratic form. 22. Let A be symmetric and B = how that and have the same signature. 23. Show that every matrix has a g inverse. If is the g- inverse of A then show that is idempotent. 24.Find the Jordan form of


25. If a sequence { } converges uniformly to f on [ a, b] and each function is

integrable, then f is integrable on [a , b] ,and the sequence { }

converges uniformly to on [ a, b].26.Define nullity of a matrix. If and are idempotent show that (i) =0 (ii) rank of an idempotent matrix A is equal to its trace.

27. State and prove a necessary and sufficient condition for a quadratic form X′A X to be positive definite. Also classify the quadratic form 9 +12

28. Compute a g-inverse

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ST1C04: Linear Programming and Its Applications



1. Define linear independence of a set of vectors.2. Verify whether S={(x,y)ϵ : x≥0}is a subspace of .3. What you mean by dimension of a vector space.4. Define a convex set.5. What are slack and surplus variables.6. What is the role of basic feasible solution in an LPP.7. Indicate a situation where artificial variable is used to solve an LPP.8. How can we handle unrestricted variable in an LPP.9. Give an example of an unbalanced transportation problem.10. State weak duality theorem.11. What you mean by a two person zero sum game.12. Define the saddle point of a rectangular game.

Part B (Answer any eight questions. Weightage 2 for each question)

13. Let V be a finite dimensional vector space show that all bases of V have same number of elements.

14. Determine whether or not the following set of vectors is linearly independent. {(1,2,6),(-1,3,4),(-1,-4,-2)} ϵ

15. Explain the algorithm for Gram- Schmidt orthogonalization process16. Show that the set of feasible solution to an LPP is a convex set.17. Distinguish between simplex method and revised simplex method.18. Write the dual of the problem. Maximize Z = + 2

Subject to :

- = 2 2 + 3 ≤ 7

19. Why sensitivity analysis is important in an LPP.20. Explain Vogel’s approximation method for finding an initial basic feasible solution of a

transportation problem.21. Solve the game graphically

2 1 0 -21 0 3 2

22. Solve the assignment problem A B C

I 19 28 31II 1

117

16

III

12 15 13

23. Discuss the dominance property in rectangular game. 24. Briefly outline the procedure of branch and bound technique for solving an integer programming

problem.


25. Let V be a vector space over R with dimension n. Show that V is isomorphic to . Player B

26. Solve the following game by LPP

27. Given the LPP. Maximize Z = + 5 Subject to + 2 18 ≤ 8 6

a) Determine an optimum solution to the LPP.b) Discuss effect on the optimum ability of the solution where the objective function is

changed to Z=

28. Discuss various steps involved in solving a transportation problem.

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FIRST SEMESTER M.Sc. DEGREE EXAMINATION (CSS), 2010Branch: Statistics

ST1C05 – Distribution Theory

Time: 3 Hours Maximum Weightage: 36

Part A(Answer all the questions. Weightage 1 for each question)

1) Define Hyper geometric distribution .

2) Write down the p.m.f of multinomial distribution and deduce that of Binomial.

3) Comment on the statement that, for a Binomial distribution mean is 2 and variance is 3.

4) Give any two real life situations where Pareto distribution is employed.

5) What is the lack of memory property? Name a continuous distribution possessing this property.

6) Describe lognormal distribution.

7) Let nXXX ,...,, 21 be i.i.d random variables from NkN

pk ,...,2,1,1 == . Find the marginal

distribution of the first order statistic )1(X .

8) If 21

)( =XE , 12

1)( =XV , find an upper limit for

>−

12

12

2

1XP using Tchebychev's

inequality.}

9) Write down Liapunov's inequality.

10) Define 2χ distribution with ' r ' degrees of freedom.

11) Give any two applications of F distribution.

12) How are 2χ and F statistics related?

Part B(Answer any eight questions. Weightage 2 for each question)

13) What is the class of Power series distribution? Mention any two major distributions belonging to this class.

14) Derive the p.g.f of Negative Binomial distribution. Hence deduce that of geometric distribution.

15) Show that for the Poisson distribution 1,11 ≥+= −+ rd

dr r

rr λµλλ µµ , where rµ is the thr

central moment.

16) State and prove the additive property of Gamma distribution.

17) If ( )2,~ σµNX obtain the moment measure of skewness and kurtosis.

18) If X and Y are jointly distributed with p.d.f ( ) 10,22),( <<<−−= yxyxyxf find the marginal distributions of X and Y .

19) If X is Uniformly distributed on [-1,1], find the distribution of 2XY = .

20) Obtain the distribution of thk order statistics of a random sample of size n from the p.d.f 10,1)( <<= xxf .

21) If )0,1(~ CX , derive the distribution ofX

1.

22) Define t distribution and explain its uses.

23) Find the mean of F distribution with ),( nm degrees of freedom.

24) If tX ~ distribution, Obtain the distribution of 2X .

Part C(Answer any TWO questions. Weightage 4 for each question)

25) a) Show that the Binomial distribution can be approximated to Poisson distribution.

b) If X and Y are independent random variables, verify whether ),( YXMin and YX −are independent.

26) Write an explanatory note on Pearsonian distributions.

27) State and prove Holder's inequality. Hence or otherwise deduce the Cauchy-Schwartz inequality.

28) If nXXX ,...,, 21 are i.i.d random variables from a Normal population, show that the

sample mean X and the sample variance 2S are independent.

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SECOND SEMESTER M.Sc DEGREE EXAMINATION (CSS), 2011Branch: Statistics

ST2C06: Estimation TheoryTime: 3 hrs. Max. Weightage: 36


1. Define sufficient statistic. Give an example.2. State factorization theorem for sufficiency.3. What is meant by ancillary statistic. Give an example.4. Let T be an unbiased estimator of θ . Is 2T unbiased for 2θ ?5. Define minimum variance unbiased estimator.6. Define one parameter exponential family of distributions.7. Discuss consistency of estimator with an example.8. Explain the criterion of choosing an estimator from a given class of consistent estimators.9. Describe the method of maximum likelihood estimation.10. If nXXX ,....,, 21 be a random sample from ),0( θN . Find the MLE of θ .11. Discuss Bayesian method of estimation.12. Define (i) Unbiased confidence interval (ii) Shortest confidence interval.

Part B(Answer any eight questions. Weightage 2 for each question.)

13. Define minimal sufficient statistic. Based on a random sample from the distribution

( ) .,)(1

11),(

2∞<<∞−

−+= x

xxf

θπθ

, obtain a minimal sufficient statistic for .θ

14. Let nXXX ,...,, 21 be a random sample from 0,),0( >θθU . Find a complete statistic for .θ15. Let 21 , XX be i.i.d ),1( θb random variables and let ).1()( θθθψ −= Obtain the class of all

unbiased estimators of ).(θψ16. Based on a random sample of size n from Poisson distribution with mean θ , obtain MVUE of

θ−e .17. State and prove invariance property of consistent estimators.18. Examine the consistency of sample mean as an estimator of θ in the case of Cauchy distribution

( ) .,)(1

11),(

2∞<<∞−

−+= x

xxf

θπθ

19. Let nXXX ,...,, 21 be a random sample from an exponential distribution with meanθ1

. Use

method of percentiles to find a consistent estimator for θ .

20. Let nXXX ,...,, 21 be a random sample from gamma distribution

.0,),,( 1 ∞<<= −− xxexf x αβα

αββα Estimate the parameters α and β by the method of moments.

21. Obtain the MLE of the parameter involved in one parameter exponential family of distributions.22. Find the Bayes estimator of the parameter p in ),( pnB if the prior distribution of p is beta

distribution ).,(1 baB23. Obtain shortest expected length confidence interval for the parameterθ based on a random

sample from 0,),0( >θθU .24. Briefly explain Fiducial intervals.

Part C(Answer any two questions. Weightage 4 for each question.)

1. a) State and prove Basu’s theorem.b) State and prove a necessary and sufficient condition for an estimator to be MVUE.

2. a) State and prove Lehmann-Scheffe theorem.b) If nXXX ,...,, 21 are i.i.d random variables with p.d.f Rxexf x ∈>= −− θθθ θ ,,),( )(

,

show that the class of linear unbiased estimators of θ is empty.

3. a) If nXXX ,...,, 21 is a random sample from ),( 2σµN , show that 2

1

)(1

XXn

n

ii −∑

= and

2

1

)(1

1XX

n

n

ii −

− ∑=

are both consistent for 2σ . Which one of the two is preferable?

b) Define marginal consistency and joint consistency. Establish their equivalence.4. State the Cramer regularity conditions. Show that with probability approaching one as ,∞→n the

likelihood equation admits a consistent solution.

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SECOND SEMESTER M.Sc. DEGREE EXAMINATION(CSS), 2011Branch: Statistics

ST2C07: Sampling Theory

Time: 3 hours Maximum weightage: 36


1. Distinguish between sampling and census.

2. Explain non probability sampling.

3. Explain Linear systematic sample.

4. Explain proportional allocation in stratified sampling.

5. What do you mean by auxiliary variable?

6. Explain linear regression estimator.

7. Explain Lahiri's method of sample selection in pps sampling.

8. Define πPS sampling.

9. What is meant by Zen-Midzuno scheme of sampling?

10. Distinguish between unit and element of a population.

11. Distinguish between cluster and stratum

12. What is double sampling? Explain.

Part B(Answer any eight questions. Weightage 2 for each question.)

13. What are the principal steps in a statistical investigation? Explain.

14. Explain the method of estimating the proportion of units possessing an attribute in a finite population of N units, using SRSWOR.

15. Illustrate the method of selecting a circular systematic sample of size n, using an example.

16. Point out the advantages of stratified random sampling over simple random sampling.

17. In ratio estimation, with usual notation, show that

xofcvRinbias

R

<ˆ

ˆ

σ

18. Explain the construction of Hartley-Ross unbiased ratio type estimator.

19. Show that, in varying probability sampling without replacement, probability that any specified unit is selected at the first draw is, in general, different from that of selecting at any subsequent draws.

20. Define Hurwitz-Thompson estimator for population total in varying probability sampling. Show that this estimator is unbiased.

21. What do you mean by ordered estimators in varying probability sampling? Show that Des Raj ordered estimator is unbiased.

22. Define cluster sampling. What are the advantages and disadvantages of cluster sampling?

23. Distinguish between multistage sampling and multiphase sampling.

24. What is non sampling error? Explain the major sources of non sampling errors.


25. (a) Show that, in simple random sampling without replacement, 22 )( SsE = .

1. Discuss the any one method of estimating sampling variance of systematic sample mean.

26. If terms in hN

1are neglected, with usual notations show that

randpropopt VVV ≥≥ .

27. Obtain the expression of the gain in precision of the unbiased estimator of population mean in PPS sampling with replacement over simple random sampling.

28. Explain sub sampling with an example. Show, with usual notations, that in sub sampling with

equal first stage units, the sample mean 2y is unbiased estimate of Y with sampling variance,

222

11111)( wb S

MmnS

NnyV

−+

−= .

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SECOND SEMESTER M.Sc DEGREE EXAMINATION (CSS), 2011Branch: Statistics

ST2C08: Regression Methods

Time: 3 hrs. Max. Weightage: 36

Part A(Answer all the questions, Weightage 1 for each question.)

1. Define R2 and adjusted R2.

2. Find an unbiased estimate of the variance (σ2 ) of the error term in the simple linear regression model.

3. Obtain the 100(1-α)% confidence interval of the mean response at the point x= x0 of the model y = β0+ β1x+ε.

4. What is meant by prediction interval?5. What are the major assumptions on the regression models?6. Define PRESS statistic.7. What is meant by spline function?8. Define Mallows’s Cp statistic.9. What are the major consequences of incorrect model specification?10. What is meant by logit model?11. Define odds ratio.12. Define link function.

Part B

(Answer any eight questions. Weightage 2 for each question.)

13. What are the properties of the least square estimates of a regression model?14. Explain the analysis of variance approach to test the significance of the regression.15. Obtain the confidence interval for the regression coefficients.16. Explain the methods for scaling residuals.17. Discuss one analytical procedure for selecting transformations on variables.18. Explain the Generalized least square method.19. Write short note on Piecewise linear regression model.20. Briefly explain various non parametric regression methods.21. Discuss stepwise regression method for variable selection.22. Give a short note on GLM.23. Discuss the tests on the individual coefficients of the logistic regression model.24. Give a brief account on Poisson Regression.


25. Obtain the MLE’s of the regression parameters and show that these estimators are identical to the least square estimates.

26. Explain various residual analysis methods for diagnosing violations of the basic regression assumptions.

27. Discuss various methods used to evaluate subset regression models.28. Define the logistic regression model, interpret the parameters and obtain the estimates of the

parameters.……………


SECOND SEMESTER M.Sc. DEGREE EXAMINATION (CSS), 2011

Branch: Statistics

ST2C09: Design and Analysis of Experiments

Time: 3hrs Maximum Weightage: 36

Part A(Answer all the questions, Weightage 1 for each question.)

1.Define the standard Gauss-Markov set up.

2. Mention the guideline for designing experiments.

3. Define Krushkal-Wallis test statistic.

4. Distinguish between ANOVA and ANCOVA.

5.What is a Latin Square design?

6. What do you mean by missing plot technique?

7. State the parametric relations of BIBD.

8. Define Lattice design.

9. Define a partially balanced incomplete block design with two associate classes.

10. Give the advantages of factorial design.

11. Distinguish between total and partial confounding.

12. Define fractional factorial.

Part B(Answer any eight questions, Weightage 2 for each question.)

13. Explain Randomization, Replication and Local control.

14. Let Y1= θ1+ e1, Y2 = θ1+ θ2+ e2 and Y3 = θ2+ e3, where θ1, θ2 are unknown parameters and ei , i=1. 2.3 are independently and normally distributed with 0 means and common variance σ2. What is the best estimate of θ1 ?.

15. Explain model adequacy checking.

16. Derive the expression for the expected value of the mean squares in RBD.

17. If a single observation is missing in a GLSD, estimate the missing value.

18. Define efficiency of designs. Explain how efficiency of LSD relative to RBD is measured.

19. Lay down the procedure for analysis of Youden square design.

20. In a BIBD, show that b ≥ v + r - k, Also prove Fisher’s inequality.

21. Derive intrablock analysis of BIBD.

22. Define the terms main effects and interactions of a factorial design. Explain the analysis of a 23 factorial experiment using Yates method and prepare its ANOVA.

23. Obtain the confounded arrangement of 25 experiment with factors A, B, C, D, and E in which AB, BCD and ACE are confounded. Determine intrablock subgroup. If there are r similar replications of the experiment, outline the analysis of experiment.

24. Explain the blocking procedure for the fractional factorial ½ (27) design in blocks of 23 units.


25. a) Define elementary contrast of a linear model (Y, Aθ, σ2 I ). If all elementary contrast of (Y, Aθ, σ2 I) are estimable, what is your interpretation of the model.

b) In a linear model (Y, Aθ, σ2 I ), A is given to be of order n x k and of rank r <k. Then derive the distribution of the statistic T= min (Y-Aθ)’ (Y-Aθ) under the assumption that Y ~ N (Aθ, σ2 I).

26. a) Give the layout and analysis of RBD with two missing values.

b) Explain randomization in LSD. Develop the procedure for analysis of LSD of order k.

27. a) Define BIBD. Give example plans of BIBD and PBIBD.

b) Construct a BIBD with v=16 b=20 k=4 r=5 and λ=1.

28. Define split plot design with an example. Give the complete analysis of split plot design.

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university of calicut · vide paper read as (2) above, the syllabus of m.sc programme in statistics...

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