M.Sc.PROGRAMMEINSTATISTICS

TWO-YEARFULL-TIMEPROGRAMME SEMESTERSItoIV

SCHEMEOFEXAMINATION ANDCOURSECONTENTS

DepartmentofStatistics

School of Physical and Mathematical Sciences

CENTRAL UNIVERSITY OF HARYANA

MAHENDERGARH

2013

M. Sc. STATISTICS SCHEME OF EXAMINATION

First Year: Semester I / Credits
Core Courses
Course / 101 : / Analysis / 3
Course / 102 : / Probability Theory / 3
Course / 103 : / Statistical Methods / 3
Course / 104 : / SurveySampling / 3
Course / 105 : / Practical-I
comprising the following two parts:
Part A: Statistical Computing-I
Part B: Data Analysis-I (based on papers 103 and 104) / 3
Elective Course
Course / 101: / Linear Programming / 3
Course / 102: / Operations Research / 3
An elective from outside the Department
First Year: Semester II / Credits
Core Courses
Course / 201: / LinearAlgebra / 3
Course / 202: / StochasticProcesses / 3
Course / 203: / StatisticalInference-I / 3
Course / 204: / DesignofExperiments / 3
Course / 205: / Practical-II
comprising the following two parts:
Part A: Statistical Computing-II
Part B: Data Analysis-II (based on papers 203 and 204) / 3
Elective Course
Course / 201: / Bio Statistics / 3
Course / 202: / Game Theory and Non linear Programming / 3
An elective from outside the Department
Second Year: Semester III / Credits
Core Courses
Course / 301: / StatisticalInference-II / 3
Course / 302: / MultivariateAnalysis / 3
Course / 303: / GeneralizedLinearModels / 3
Course / 304 / Bayesian Inference / 3
Course / 305: / Practical-III
comprising the following two parts:
Part A: Problem Solving Using C Language-I (based on papers 301, 302 and 303) Part B: Problem Solving Using SPSS-I (based on papers 301, 302 and 303) / 3
Elective Course
Course / 301: / Non Parametric Inference / 3
Course / 302: / Actuarial Statistics / 3
An elective from outside the Department / 3
Second Year: Semester IV / Credits
Core Courses
Course / 401 : / Econometrics and Time Series Analysis / 4
Course / 402 : / Demography, Statistical Quality
Control and Reliability / 4
Elective Courses
Course / 401 : / Applied Stochastic Processes / 3
Course / 402 : / Order Statistics / 3
Course / 403 : / Information Theory / 3
Course / 404 : / Statistical Ecology / 3
Course / 405 : / Statistical Method in Epidemiology / 3
Course / 406 / Reliability Theory and Life Testing / 3
An elective from outside the Department / 3
Paper / 405: / Practical-IV
comprising the following two parts:
Part A: Problem Solving Using C Language-II (based on papers 401 and 402)
Part B: Problem Solving Using SPSS-II (based on papers 401 and 402) / 3

M. Sc. STATISTICS

SemesterI:Examination2013andonwards

Course101:Analysis

Monotonefunctionsandfunctionsof boundedvariation.Absolutecontinuityof functions, standard properties. Uniform convergence of sequence of functions and series of functions. Cauchy’scriterionandWeirstrassM-test.Conditionsfortermwisedifferentiation andtermwise integration(statementsonly),Powerseriesandradiusofconvergence.

Multipleintegralsandtheirevaluationbyrepeatedintegration.Changeof variableinmultiple integration.Beta and gammafunctions.Differentiationunderintegralsign.Leibnitzrule.Dirichletintegral,Liouville’sextension.

Maxima-minima of functions of several variables, Constrained maxima-minima of functions.

Analytic function, Cauchy-Riemann equations. Cauchy theorem and Cauchyintegralformulawithapplications,Taylor’s series.Singularities,Laurentseries. Residue andcontourintegration.

FourierandLaplacetransforms and their basic properties.

References:

1. Apostol,T.M.(1975).MathematicalAnalysis,Addison-Wesley.

2. Bartle,R.G.(1976).ElementsofRealAnalysis,JohnWileySons.

3. Berbarian,S.K.(1998).FundamentalsofRealAnalysis,Springer-Verlag.

4. Conway,J.B.(1978).FunctionsofoneComplexVariable,Springer-Verlag.

5. Priestley,H.A.(1985).ComplexAnalysis,ClarentonPressOxford.

6. Rudin,W.(1985).PrinciplesofMathematicalAnalysis,McGrawHill.

Course102:ProbabilityTheory

Classesofsets,field,sigmafield,minimalsigmafield,Borelfield,sequenceofsets,limits ofasequenceofsets,measure,probabilitymeasure,Integrationwithrespecttomeasure.

Various definitions of Probability, Properties of probability function, Baye’s Theorem, Independence of Events.

Random Variables and Distribution Functions, Two Dimensional Random Variables- Joint, Marginal and Conditional Distributions.Moments of Random Variables – Expectation, Variance, Covariance, Conditional and Marginal Expectation.

Probability Generating Function, Moment Generating Function and their Properties.Characteristicfunction,uniquenesstheorem,continuitytheorem,inversionformula.

Markov’s,Holder’s,Minkowski’sandJensen’sinequalities.

Modes of Convergence: convergence in probability, almost surely, in the rth mean and in distribution, their relationships.

Laws oflarge numbers, Chebyshev’s and Khintchine’s WLLN, necessary andsufficient conditionfortheWLLN,Kolmogorov’sStronglawoflargenumbers andKolmogorov’stheorem.

Central limit theorem, Lindberg-Levy’s and Liapunov forms ofCLT. Statement of

Lindberg-Feller’sCLTandexamples.

References:

1.Ash,RobertB.(2000).ProbabilityandMeasureTheory,SecondEdition,AcademicPress, NewYork.

2. Bhat, B.R. (1999). Modern Probability Theory, 3rd Edition, New Age International

Publishers.

3. Billingsley,P.(1986).ProbabilityandMeasure,2ndEdition,JohnWileySons.

4. Capinski,M.andZastawniah(2001).Probabilitythroughproblems,Springer.

5. Chung,K.L.(1974).ACourseinProbabilityTheory,2nd Edition,AcademicPress,New

York.

6.Feller,W.(1968).AnIntroductiontoProbabilityTheoryanditsApplications,3rd Edition, Vol.1,JohnWileySons.

7.Goon,A.M.,Gupta,M.K.andDasgupta.B.(1985).AnOutlineofStatsticalTheory,Vol.I, WorldPress.

8. Halmos, P.R. Measure Theory

9. Loeve,M.(1978).ProbabilityTheory,4thEdition,Springer-Verlag.

10. Rohatgi, V. K. and Saleh, A.K. Md. E. (2005). An Introduction to Probability and

Statistics,SecondEdn.,JohnWiley.

Course103:StatisticalMethods

Probability distributions: Binomial, Poisson, Multinomial, Hypergeometric, Geometric. Negative Binomial, Uniform, Exponential, Laplace, Cauchy, Beta, Gamma, Weibulland Normal (Univariate and bivariate) and Lognormal distributions.

Sampling distribution of Mean and Variance, Chi-square, Student’s t, Snedecor’s F and Fisher’s-Z distribution and their applications. Chi-squaretest, Student’st- test and F test.Samplesizedeterminationfortestingandestimationprocedures.Non-central Chi-square,t andFdistributionsandtheirproperties.

Order statistics - their distributions and properties. Joint and marginal distributions of order statistics. Extreme values andtheir asymptotic distributions (statementonly) with applications. Toleranceintervals,coverageof (X(r), X(s)).

Correlation: Product moment, Spearman’s Rank and Intra-class Correlation, Correlation Ratio

Generaltheoryofregression,multipleregression, Partial and Multiple Correlations.

References:

1. Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1992). A First Course in Order

Statistics,JohnWileySons.

2. David,H.A.,andNagaraja,H.N.(2003).OrderStatistics,ThirdEdition,JohnWileyand

Sons.

3.Dudewicz, E.J. and Mishra, S.N. (1988). Modern Mathematical Statistics, Wiley, InternationalStudents’Edition.

4.Johnson, N.L., Kotz, S.andBalakrishnan, N.(2000). Discrete Univariate Distributions, JohnWiley.

5.Johnson,N.L.,Kotz,S.andBalakrishnan,N.(2000).ContinuousUnivariateDistributions, JohnWiley.

6. Rao,C.R.(1973).LinearStatisticalInferenceandItsApplications(SecondEdition),John

WileyandSons.

7. Rohatgi,V.K.(1984).StatisticalInference,JohnWileyandSons.

8.Rohatgi,V.K.andSaleh,A.K.Md.E.(2005).AnIntroductiontoProbabilityandStatistics, SecondEdition,JohnWileyandSons.

Course104:Survey Sampling

Basicideasanddistinctive features ofsampling; Probability sampling designs,sampling schemes, inclusion probabilities and estimation; Reviewofimportantresultsin simpleandstratifiedrandomsampling

Samplingwithvaryingprobabilities (unequal probability sampling): PPSWR /WOR methods and related estimators of a finite population total or mean (Hansen – Hurwitz and Des Raj estimators for a general sample size and Murthy’s estimator for a sample of size 2). Horvitz – Thompson Estimator (HTE) of a finite population total /mean. Non-negativevariance estimation.

Ratio andRegression Estimators, Unbiased ratio type estimate due to Hartley and Ross, Ratio Estimate in stratified sampling.

Double (two-phase) sampling with special reference totheselection with unequal probabilities inatleastoneof thephases;Double sampling ratio and regression estimators of population mean.systematicsamplinganditsapplicationtostructured populations;Clustersampling(withvaryingsizesofclusters);Two-stagesampling(withvarying sizesoffirst-stageunits).

Non-sampling error with special reference to non-response problems.Small Area Estimation.Super–population Models.Non-Existence Theorems and Optimality of Sampling Strategies in finite population sampling.

References:

1. Chaudhuri, A. (2010). Essentials of Survey Sampling.Prentice Hall of India.

2. Chaudhari,A.andVos,J.W.E.(1988).UnifiedTheoryandStrategiesofSurvey

Sampling,North–Holland,Amsterdam.

3.Chaudhari, A.andStenger, H.(2005).SurveySampling Theoryandmethods, 2nd Edn., ChapmanandHall.

4. Cochran,W.G.(1977).SamplingTechniques,JohnWileySons,NewYork

5.Hedayat,A.S.,andSinha,B.K.(1991).DesignandInferenceinFinitePopulationSampling, Wiley,NewYork.

6.Levy,P.S.andLemeshow,S.(2008).SamplingofPopulations-Methods andApplications, Wiley.

7. Mukhopadhyay,Parimal(1997).TheoryandMethodsofSurveySampling,PrenticeHallof

India,NewDelhi.

8.Murthy, M.N. (1967). Sampling Theory and Methods, Statistical Publishing Society, Calcutta.

9. Raj,D.andChandhok,P.(1998).SampleSurveyTheory.NarosaPublishingHouse.

10. Sukhatme,P.V.,Sukhatme,B.V.,Sukhatme,S.andAsok,C.(1984).SamplingTheoryof

SurveyswithApplications,IowaStateUniversityPress,Iowa,USA.

11. Thompson,StevenK.(2002).Sampling,JohnWileyandSons,NewYork.

Paper105:Practical-I Part-A:Statistical Computing I

Programming in C; Representation of numbers, Errors. Bitwise operators,Manipulations, Operators, Fields. The C Preprocessor, Macros, Conditional Compilation, Command-lineArguments.

Stacksandtheirimplementation;Infix,Postfixand Prefixnotations.Queues,Link list, DynamicStorageManagement.Trees–Binarytrees, representations,traversal,operationsand Applications. Graphs– Introduction, representation. Sorting– Introduction, bubble sort,selection sort,insertionsort,quicksortincludinganalysis.

Randomnumbers: Pseudo-Random numbergeneration, tests.Generation ofnon-uniform random deviates– general methods, generation fromspecific distributions.

References:

  1. Gottfried,ByronS.(1998).ProgrammingwithC,TataMcGrawHillPublishingCo.Ltd.,NewDelhi.
  2. Kernighan,BrainW.andRitchie,DennisM.(1989).TheCProgrammingLanguage, PrenticeHallofIndiaPvt.Ltd.,NewDelhi.
  3. Knuth,DonaldE.(2002).TheArtofComputerProgramming,Vol.2/Seminumerical

Algorithms,PearsonEducation(Asia).

  1. Rubinstein,R.Y.(1981).SimulationandtheMonteCarloMethod,JohnWileySons.
  2. Tenenbaum,AaronM.,Langsam,Yedidyah,andAugenstein,MosheJ.(2004).Data

StructuresusingC,PearsonEducation,Delhi,India.

PartB:DataAnalysis-I

Computer-baseddataanalysisofproblemsfromthefollowingareas: StatisticalMethodologyandSurveySampling.

Elective Course 101: Linear Programming

Convex sets and functions with their properties. Programming problems. General linear programming problems: Formulation and their properties of solutions. Various forms of a LPP.Generation of extreme point solution.Development of minimum feasible solution.solution of LPP by graphical and simplex methods. Solution of simultaneous equations by simplex method.

Solution of LPP by artificial variable techniques: Big-M-method and Two Phase simplex method. Problem of degeneracy in LPP and its resolution.Revised simplex method and Bounded Variable Technique.

Duality in Linear Programming: Symmetric and Un-Symmetric dual Problems. Economic Interpretation of Primal and Dual Problems.Fundamental Duality Theorem.Dual simplex method.Complementary Slackness Theorem.

Sensitivity Analysis.Parametric linear programming. Integer linear programming: Gomory’s cutting plane method, Branch and Bound method. Applications of Integer Programming.

Transportation Problems: balanced and unbalanced. Initial basic feasible solution of transportation problems by North West Corner Rule, Lowest Cost Entry Method and Vogel’s Approximation Method.Optimal Solution of Transportation Problems.

Assignment problems and their solution by Hungarian assignment method.Reduction Theorem. Unbalanced assignment problem. Sensitivity in assignment problems.

Books Suggested:-

1.Gass, S.I.:Linear Programming

2.Kambo, N.S :Mathematical Programming

3.Sharma, S.D.:Operations Research

4.Hadley. G.:Linear Programming

Elective Course 102: OperationsResearch

Definition andscopeofOperationResearch, phasesinOperation Research, different typesofmodels,theirconstructionandgeneralmethodsofsolution.

Replacement problem, replacement of items that Deteriorate, replacement of items that fails completely Individual Replacement policy : Motility theorems, Groupreplacement policy, Recruitment and promotion problems.

InventoryManagement:Characteristicsofinventorysystems.Classificationof items.Deterministicinventorysystemswithand withoutlead-time.All unitsandincrementaldiscounts.Singleperiodstochasticmodels.

Job Sequencing Problems; Introduction and assumption, Processing of n jobs through twomachines(Johnson’s Algorithm) Processing of n jobs through three machines and mmachines, Processing two jobs through n machines(Graphical Method)

Simulation: Pseudorandom Number Generation, using random numbers to evaluateintegrals.Generatingdiscreterandom variables:InverseTransformMethod,Acceptance-Rejection Technique, Composition Approach. Generating continuous randomvariables:InverseTransform Algorithm,RejectionMethod.Generatinga Poissonprocess.

TheoryofNetwork–PERT/CPM: development, uses and application of PERT/CPM Techniques, Networkdiagram representation .Fulkerson 1-J rule for labeling Time estimate and determination ofcriticalPath on networkanalysis, PERT techniques, crashing.

Introduction toDecision Analysis: Pay-off tableforone-offdecisions anddiscussion of decisioncriteria,Decisiontrees.

References:

  1. Churchman Method’s of Operations Research
  2. Hadley,G.andWhitin,T.M.(1963).AnalysisofInventorySystems,PrenticeHall.
  3. Hillier, F.S. and Lieberman, G.J. (2001). Introduction to Operations Research, Seventh

Edition,Irwin.

  1. Ross,S.M.(2006).Simulation,FourthEdition,AcademicPress.
  2. Taha,H.A.(2006).OperationsResearch:AnIntroduction,EighthEdition,PrenticeHall.
  3. Wagner,B.M.(1975).PrinciplesofOR,EnglewoodCliffs,N.J.Prentice-Hall
  4. Waters,DonaldandWaters,C.D.J.(2003).InventoryControlandManagement,JohnWiley

Sons.

SemesterII:Examination2014andonwards

Course 201:LinearAlgebra

Examples ofvectorspaces, vector spaces andsubspace, independence invectorspaces, existenceofaBasis,therowandcolumnspacesofamatrix,sumandintersectionofsubspaces.

LinearTransformations andMatrices,Kernel,Image,andIsomorphism, changeofbases, Similarity,RankandNullity.

InnerProductspaces,orthonormalsetsandtheGram-SchmidtProcess,theMethodofLeast

Squares.

Basictheoryof EigenvectorsandEigenvalues,algebraicandgeometricmultiplicityof eigenvalue,diagonalizationof matrices,application tosystemof lineardifferential equations.

Jordan canonical form, vector and matrix decomposition.

GeneralizedInversesofmatrices,Moore-Penrosegeneralizedinverse.

Realquadraticforms,reductionandclassificationof quadraticforms,indexandsignature, triangularreductionofareductionofapairofforms,singularvaluedecomposition, extremaof quadraticforms.

References:

1. Biswas,S.(1997).ATextBookofMatrixAlgebra, 2nd Edition,NewAgeInternational

Publishers.

2. Golub,G.H.andVanLoan,C.F.(1989).MatrixComputations,2nd edition,JohnHopkins

UniversityPress,Baltimore-London.

3. Nashed,M.(1976).GeneralizedInversesandApplications,AcademicPress,NewYork.

4.Rao,C.R.(1973).LinearStatisticalInferencesanditsApplications,2ndedition,JohnWiley andSons.

5.Robinson,D.J.S.(1991).ACourseinLinearAlgebrawithApplications,WorldScientific, Singapore.

6. Searle,S.R.(1982).MatrixAlgebrausefulforStatistics,JohnWileyandSons.

7.Strang,G.(1980).LinearAlgebraanditsApplication,2ndedition,AcademicPress,London- NewYork.

Course202:StochasticProcesses

Poissonprocess, Brownianmotionprocess,Two-valued processes.Modelforsystemreliability.

Meanvaluefunctionandcovariancekernelof theWienerandPoissonprocesses.Increment processofaPoissonprocess,Stationaryandevolutionaryprocesses.

Compounddistributions,Totalprogenyinbranchingprocesses.

Recurrent events, Delayed recurrent events, Renewal processes. Distribution and Asymptotic Distribution of Renewal Processes. Stopping time. Wald’s equation. Elementary Renewal Theorem. Delayed andEquilibrium Renewal Processes. Application tothetheory of successruns.Moregeneralpatternsforrecurrentevents.

One-dimensional, and two-dimensionalrandomwalks.Dualityin random walk.Gambler’sruinproblem.

Classificationof Markovchains.HighertransitionprobabilitiesinMarkovclassificationof statesandchains.Limittheorems.Irreducibleergodicchain.

Martingales, Martingale convergence theorems, Optionalstopping theorem.

References:

1. Bhat,B.R.(2000).StochasticModels-AnalysisandApplications,NewAgeInternational

Publishers.

2Feller,William(1968).AnIntroductiontoProbabilityTheoryanditsApplications,Vol.1 (ThirdEd.),JohnWiley.

3.Hoel, P.G., Port, S.C. and Stone C.J. (1972). Introduction to Stochastic Processes, HoughtonMiffinCo.

4. Karlin, S.andTaylor, H.M.(1975). AfirstcourseinStochastic Processes, Second Ed.

AcademicPress

5. Medhi,J.(1994).StochasticProcesses,2ndEdition,WileyEasternLtd.

6. Parzen,Emanuel(1962).StochasticProcesses,Holden-DayInc.

7. Prabhu, N.U. (2007). Stochastic Processes: Basic Theory and its Applications, World

Scientific.

8. Ross,SheldonM.(1983).StochasticProcesses,JohnWileyandSons,Inc.

9. Takacs,Lajos(1967).CombinatorialMethodsintheTheoryofStochasticProcesses,John

WileyandSons,Inc.

10. Williams,D.(1991).ProbabilitywithMartingales,CambridgeUniversityPress.

Course203:StatisticalInference–I

Criteria of a good estimator – unbiasedness, consistency, efficiency and sufficiency,Minimal sufficiency and ancillarity, Invariance propertyofSufficiency under one-one transformations ofsample andparameter spaces. Exponential and Pitman family of distributions.

Minimum – variance unbiased estimators,Cramer-Raolower bound approach to MVUE.Lower bounds to variance of estimators, necessaryandsufficientconditionsforMVUE.Completestatistics, Rao Blackwell theorem. Lehman Schefe’s theorem and it’s applications in finding UMVB estimators, Cramer- Rao, Bhattacharya’s Bounds. Fisher Informationfor one andseveralparameters models.

Method of estimation- Method of Maximum Likelihood and its properties, Methods of Moments and its properties, Method of Least Square and its properties. Method of minimum chi- square and modified minimum chi- square.

Neyman-Pearson fundamental lemma and its applications, MP and UMP tests. Non-existence of UMP tests forsimple null against two-sided alternatives inone parameter exponential family. FamiliesofdistributionswithmonotonelikelihoodratioandUMPtests.

Intervalestimation, confidence level,construction ofshortestexpectedlengthconfidence interval,Uniformlymostaccurateone-sidedconfidenceIntervalanditsrelationtoUMPtestsfor one-sidednullagainstone-sidedalternativehypotheses.

References:

  1. Goon, A.M., M.K.Gupta, & B. Das Gupta(2002): Outline of Statistical Theory Vol-II World Press.
  2. Kale,B.K.(1999).AFirstCourseonParametricInference,NarosaPublishingHouse.
  3. Lehmann,E.L.(1986).TheoryofPointEstimation,JohnWileySons.
  4. Lehmann,E.L.(1986).TestingStatisticalHypotheses,JohnWileySons.
  5. Rao,C.R.(1973).LinearStatisticalInferenceandItsApplications,SecondEdition,Wiley
  6. EasternLtd.,NewDelhi.
  7. Rohatgi,V.K.andSaleh,A.K.Md.E.(2005).AnIntroductiontoProbabilityandStatistics, SecondEdition,JohnWiley.
  8. Zacks,S.(1971).TheoryofStatisticalInference,JohnWileySons.

Course204:DesignofExperiments

Reviewof linearestimationandbasicdesigns.ANOVA:Fixedeffectmodels(2-way classificationwithunequalandproportionalnumberof observationspercell),RandomandMixed effectmodels(2-wayclassificationwithm(>1)observationspercell).Tukey’s test for non- additivity.

General theory of Analysis of experimental designs; Completely randomized design, randomized block design and latin square designs, Missing plot techniques in RBD and LSD.

Symmetricalfactorialexperiments(sm,wheresisaprimeoraprimepower),Confounding insmfactorialexperiments,sk-pfractionalfactorialwheresisaprime oraprimepower.Analysis of covariance for CRD and RBD. Split plot and strip plot designs.

Incomplete Block Designs. Concepts of Connectedness, Orthogonality and Balance. IntrablockanalysisofGeneralIncompleteBlockdesign.B.I.Bdesignswithandwithoutrecovery ofinterblockinformation.PBIB Designs.

References:

1. Chakrabarti, M.C. (1962). Mathematics of Design and Analysis of Experiments, Asia

PublishingHouse,Bombay.

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

Limited.

3. Dean,A.andVoss,D.(1999).DesignandAnalysisofExperiments,Springer.FirstIndian

Reprint2006.

4. Dey,A.(1986). TheoryofBlockDesigns,JohnWileySons.

5.Hinkelmann,K.andKempthorne,O.(2005).DesignandAnalysisofExperiments,Vol.2: AdvancedExperimentalDesign,JohnWileySons.

6. John,P.W.M.(1971).StatisticalDesignandAnalysisofExperiments,MacmillanCo.,New

York.

7. Kshirsagar,A.M.(1983). ACourseinLinearModels,MarcelDekker,Inc.,N.Y.

8. Montgomery,D.C.(2005). DesignandAnalysisofExperiments,SixthEdition,JohnWiley

Sons.

9. Raghavarao, D. (1970). Construction and Combinatorial Problems in Design of

Experiments,JohnWileySons.

Paper205:Practical-II PartA:StatisticalComputing-II

MathematicalandStatisticalproblemsolvingusingsoftwarepackage:Introduction,Plotsin

2-Dand3-D.Numerical Methods: Vector andmatrixoperations, Interpolation. Numerical rootfinding,Matrixfactorization.Eigenvalueandeigenvectors,Differentiation,Integration.

Generation ofdiscrete andcontinuous random variables, Histograms andquantile-based plots.Parameterestimation–MLE,methodof moments.MonteCarlomethods–Introduction,for Statisticalinference,Bootstrapmethods.Regressionandcurvefitting.

References:

1.Gentle,J.E.,HärdleW.andMoriY.,(2004).Handbookofcomputationalstatistics— Conceptsandmethods,Springer-Verlag.

2. Knuth,DonaldE.(2002).TheArtofComputerProgramming,Vol.2/Seminumerical

Algorithms,PearsonEducation(Asia).

3. Monahan,J.M.(2001).NumericalMethodsinStatistics,Cambridge.

4. Ross,S.M.(2002).Simulation,Academicpress.

5. Rubinstein,R.Y.(1981).SimulationandtheMonteCarloMethod,JohnWileySons.

PartB:DataAnalysis-II

Computer-baseddataanalysisofproblemsfromthefollowingareas: StatisticalInference-IandDesignofExperiments.

Elective Course 201 Bio-Statistics

Functions of survival time, survival distributions and their applications viz. exponential, gamma, weibull, Rayleigh, lognormal, death density function for a distribution having bath-tub shape hazard function. Tests of goodness of fit for survival distributions (WE test for exponential distribution, W-test for lognormal distribution, Chi-square test for uncensored observations). Parametric methods for comparing two survival distributions viz. L.R test, Cox’s F-test.

Analysis of epidemiologic and clinical data: Studying association between a disease and a characteristic (a) Types of studies in epidemiology and clinical research (i) Prospective study (ii) Retrospective study (iii) cross sectional data, (b) Dichotomous response

and Dichotomous risk factor: 2X 2 Tables (c) Expressing relationship between a risk factor and a disease (d) Inference for relative risk and Odd’s Ratio for 2X2 table , Sensitivity Specificity and Predictivities, Cox Proportional Hazard Model, Type I, Type II and progressive or random censoring with biological examples, Estimation of mean survival time and variance of the estimator for type I and type II censored data with numerical examples. Competing risk theory, Indices for measurement of probability of death under competing risks and their inter-relations. Estimation of probabilities of death under competing risks by maximum likelihood and modified minimum Chi-square methods. Theory of independent and dependent risks.Bivariate normal dependent risk model. Conditional death density functions.

Stochastic epidemic models: Simple and general epidemic models (by use of random

variable technique).

Planning and design of clinical trials, Phase I, II, and III trials. Consideration in planning a clinical trial, designs for comparative trials, Sample size determination in fixed sample designs.

References:

1. Biswas, S. (1995): Applied Stochastic Processes. A Biostatistical and Population

Oriented Approach, Wiley Eastern Ltd.

2. Cox, D.R. and Oakes, D. (1984) : Analysis of Survival Data, Chapman and Hall.

3. Elandt, R.C. and Johnson (1975): Probability Models and Statistical Methods in

Genetics, John Wiley & Sons.

4. Ewens, W. J. (1979) : Mathematics of Population Genetics, Springer Verlag.

5. Ewens, W. J. and Grant, G.R. (2001): Statistical methods in Bio informatics.: An

Introduction, Springer.

6. Friedman, L.M., Furburg, C. and DeMets, D.L. (1998): Fundamentals of Clinical

Trials, Springer Verlag.

7. Gross, A. J. And Clark V.A. (1975) : Survival Distribution; Reliability

Applications in Biomedical Sciences, John Wiley & Sons.

8. Lee, Elisa, T. (1992) : Statistical Methods for Survival Data Analysis, John Wiley

& Sons.

9. Li, C.C. (1976): First Course of Population Genetics, Boxwood Press.

10. Miller, R.G. (1981): Survival Analysis, John Wiley & Sons.

Elective Course 202 Game Theory and Non-Linear Programming

Theory of Games: Characteristics of games, minimax (maximin) criterion and Optimal Strategy. Solution of games with saddle point.Equivalence of rectangular game and Liner Programming.Fundamental Theorem of Game Theory. Solution of mxn games by Linear Programming Method. Solution of 2X2 games without saddle point. Principle of dominance.Graphical solution of (2xn) and (mx2) games.

Non-Linear Programming Problems (NLPP): Kuhn-Tucker necessary and sufficient conditions of optimality, Saddle points. Formulation of NLPP and its Graphical Solution.