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Sc ;ilil;il. ilffi; il
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Sc
N{'Sc' Statistics
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UNIVERSITY OF CALICUT
(Abstract)
Sy'llabus under choice Based credit semester
System programme
from 2008 admission _ ordersissued.
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No. GA
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I/12/2902/06
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Dated, Calicut University p.O,6/[email protected]
Read : l.U.O.No.G AI/ I t/ t3T/Oqdated
0t/07 t20lg.
of the meeting of the Board of studies
in staristics (pG)
:t|I,#,ir;[;i"minutes
'
t
t
';;7ril;ir(xxiii)
l!fro)"rfl
of the minures of the meering of
the Facurty of Science of
A(22) of the minutes of the meeting
of the Academic councir herd on
ORDER
As per University order read as
I " abovs, choice Based
credit semester System was
introduced fbr all regular PG
Programmes in Teaching
Departments/Schoors of this
Iniversitr.
I
The Board of Studies in statistics
(PG), vide paper (2) has
resorved to ap-prove the
Syllabus fbr the newly introduced
choice Based credit semester
System in the Deparrment
of Statistics' Unive.sity of calicut'
The Faculty of Science vide
paperread as (3) er:ciorseci
the decision ol the Board of
Studies rvhich was approved
by the Academic councir vide
paper read as (4) above.
Sanction has therefore been
accorded for implementing
the Syllabus for l\{.;.
Statistics under flhoice Based
credit semester system at the
University Departnient cf
Statistics ivith eff-ect liom 2008
admissio. onwards. T'he Syilabus
is appended herewith.
Orders are issued accordingly.
sd/_
Deputy Registrar G &A
For Registrar
To
tr
r he Iiead. Deparrment of Statistics,
University of Caricut.
Cop.t to:
Controll
Tabulati
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(Pc)/
ion Centres/
Forwarded,/Ely Order
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STAl0l - Cl: Mathematical Methods for Statistics - I
'
(4 Credits)
tJnit I: Reimann Stielt.ies Integral- Definition, existence and properties.
lntegratior-r by parts. Change of variable - Step functions as integrators.
Reduction to finite sum. Monotone increasing integrators. Riemann's conditior-r.
Integrators of bounded variations. Mean value theorems. Improper integrals.
Gamma and beta functions.
Unit II: Sequenoes and Series of Futrciiuns - Poirii wise corrvergence and
uniform convergence. Tests for uniform convergence. Properties ofuniform convergence.
\\/eirstrass theorem.
Unit III: Multivariable functions. Limit and continuity of multivariable
functions. Derir,'atives. directional derivatives and continuity'. Total deril,ative in
terms of partial derivatives, Taylor's theorem. Inverse and implicit tunctions.
Optima of multivariable functions.
L-lnit IV: Elementarv matrices. Rank of matrix, inverse. Diagonal reduction.
Deternrinants. Transformatior-rs. Idempotent matrices. Generalized inverse.
SolLrtion of liner equations. Special product of matrices. Characteristic roots and
vectors. Definition and properties. Algebraic and geometric multiplicitv of
characteristic roots. Spectral decomposition. Quadratic forms. Classification and
reduction of quadratic forms.
Text Books
I
.
Apostol (1974). Mathernatical Analysis. Second edition. Narosa, Nerv-Delhi
Chapter 7 & 9.
2. Khuri, A. T. (1993). Advar-rced Calculus with Applications in Statistics. John
Wiley, New York, Chapter 7.
-3. Rao, C.R. (2002). Linear Statistical Inference & its applications. Second Edition.
John Wile;-, Neu -York.
4. Graybill. F. A. (1983). Matrices with Applications in Statistics. John Wiley,
Nelr'-York.
Reference Book
1. Malik, S.C. & Arora,
2.
S (2006). Matl-rematical Analysis, Second
edition. New
Age International.
Len'is, D. W. (1995). Matrix Theory. ^A.llied Publishers, Bangalore.
1
STAl02 -C2: Mathcmatical Methods for Statistics
(4 Credits)
Unit
-I:
-II
of Sets - Field of
sets. sigma field, monotone class and rninimal
sigr.na field. Borel sigma field and Borel sets in R and R^p. Set functions. Additivitl'
and sigma additivity - Measures - examples and properties. Outer measure. Lebesgue
measure in R and R^p. Lebesgue-Steiltjes measure.
Classes
Unit - II : Measurable function. Properties. Sequence of nteasurable tunctions,
coll\'ergence, Egoroff's theorent. Integrals of simple. non-negative and arbitrarv
t-neasurable functions. Convergence of integrals. Monotone convergence theoren-).
donrinated convergence theorem ard Fatou's lemnta.
Unit - III : Product space and product measure. Multiple integral. Fubini's theorem
(without proof). Absolute continuity and singularity of measures. Radon-Nikodyrn
theorem (without proof) and its apltlications.
Unit IV: Vector space with real and complex scalars. Subspaces. liner dependence
and ir-rdependence, basis. dimension. Linear transformations and matrices. Jacobian
nratrix transformations, functions of matrix argument.
Rovden, H. L. (lee5). Real Analysis.
h:1:?:ffin
of
Prentice Hall of India. New-york.
Bartle, R.G. (1996). The Elements of Integration. John Wiley and Sons. New York
Lew'is, D.W. (1996), Matrix Theory. Allied Publishers, Bangalore.
Ilao,C.R. & Bhimsankar (1992). Linear Algebra. Tata-Mcgraw Hill, New-Delhi.
Rao, C.R. (2002). Linear Statistical Inference and Its Applications. Second Edr-r. Johl
Wile1,. New-York.
Mathai, A. M. (200 ). Application of Matrix and Determinants - Module
3.
Reference Books
Kingman, J.F.C and Taylor, S.J. (1973). Introduction to Measure and Probability.
Cambridge university press.
Bapat,R.B (1993). Linear Algebra and Liner Models. Hindustan Book Agency.
STAI03 - C3: Probability Theory
(4 Credits)
-I
Unit - I : Probability rneasLrre. measure. probability space, randonr variable.
Inverse function and properties. Sequence of random variables and limit.
Extension of probability rneasure - Caratheodory extension theorern (witl-rout
proof. Distribution function. decomposition of distribution function. Vector
valued random variables arrd its distribution function. Induced probability space
oi-a randour r uriable.
Unit - Il: Mathernatical expectation of simple, non-negative and arbitrary
random variables - properties of expectation. Moment generating functionsmoments. lnequalities. Cr inequality, Jenson's inequality, Basic inequality,
Markov inequality.
Unit
- III: Independence
olevents, classes ofevents. Independence ofrandom
variables. Kohnogoror.v's ()-l law,, Borel's 0-1 criteria. Borel-Cantelli Lernma.
Unit - [V: Different modes of convergence. Convergence in probability,
convergence in distribution. rth mean convergence, almost sure convergence and
their nrr-rtuaI irnplications.
Text Book
Bhat, B.R. (1999). Modern Probability Theory. Third Editiona. Neu-age
international, New-Delhi.
Reference Books
Resnick, S.I. (1999). Probability Paths. Birkhauser. Boston.
Laha and Rohatgi (1979). Probability Theory. John Wiley and Sons, New York.
Billingsly (1995). Probability and Measure. Third Edition. John Wiley, newYork.
Basu, A.K. (1999). Measure 1-heory and Probability. Printice Hall of India, NewDelhi.
Rohatgi, V.K. (1976). An Introduction to probability Theory and Mathematical
Statistics. John-Wiley. Nen, York.
tL
STA 104-C4: Distribution Theorv
(4 Credits)
Discrete Distributions - Bernoulli, Discrete lJnilorrn. Binomia[Negative Binomial, Geometric, Hyper geometric, Poisson Logaritl'rmic Series
and rnultilomial distributions, power series distribution and their properties.
Unit
- I:
Unit
- II: Continuous Distributions -
Systems
of Distribtttions-Pearson
system
and Transformed Distributiorrs. Uniform. Exponential, Gamma, Beta. Cauchy.
Nori-nal. Piireto, S/eibull, [-aplacc. lognormal Bivnriate Normal Distributions
and their properties.
- III:
Notion of Vector of Random Variables. distribution function
n-rarginal and joint distributions in the iid case. Functions of Randorn Vectors.
Order Statistics and tlreir Distributions.
Unit
and Their Distributions- Sample Characteristics ar.rd
their distributions. Chi-Square. t and F distributions (Central and Non-Central),
Applications of Chi-sqlrare. t and F'
Unit
- IV: Sample Moments
Text Books
1
.
2.
Rohatgi, v.K. (1976). An Introductior-r to Probability Theory ar.rd
Mathematical Statistics. John - Wiley, New York. Chapter 4, Sections 2, 4
and 5. chapter 5. Sections 2,3 and 4. Chapter 7 Sections 3,4 and 5.
Krishnamoorthy, K. (2006). Hand book of Statistical Distributions with
Applications. Chaprnan & Hall. Ner.v York. Chapters 8,14,20.23 and24
Sections 1,2 and 5.
3. Johnson, N.L., Kotz. S. and Balakrishnan, N.(2004). Continuous Univariate
Distributions-Vol.l. Second Edition. John Wiley and Sons. New York.
Chapter l2 Sections 4.1.4.i.
Reference Books
1.
2.
3.
4.
,lohnson, N.L., Kotz. S. and Balakrishnan,N.(1995). Contir-ruous Unir,'ariate
Distributions-Vol. Il. Sccond Edition, John Wiley and Sons, New York..
Johnson, N.L., Kotz. S. and Kemp, A.W. (1992). Univariate Discrete
Distributions - Wiley, New York.
Kendall and Stuart, (19)
Goon Gupta and Das GuPta (19).
5
STAI05 - C5: Sampling Theory
(4 Credits)
Unit - I:
Census, Sampling, Probability sampling. and non-Probability sampling.
SRSWOR and SRSWR. Estimation of population mean. Population total and population
proportion. Variance of the estimates and standard error. Estimation of sanlple size.
Stratitled random sampling. Allocation problem. Various allocations. Construction of
strata.
- II: PPS sampling with anil .,vithout replacement.
trstimation of population ntean.
total and variance in PPS sampling with replacement. Desraj's ordered estimator.
N4urthy's unordered estimator. I{orvitz - Thomson estimator. Their variances and
standard error. Yates - Grundy estimator. Sen - Midzuno scheme of sampling. tlPS
Unit
sanrpl i ng.
- III:
Unit
Ratio estimators and Regression estimators. Comparison with simple
arithmetic mean estimator. Optimality properties of ratio and regression estimators.
Hartly
Unit
-
Ross unbiased ratio type estimator.
- IV: Circular, linear and balanced systematic sampling. Estimation of population
mean and its variance. Cluster sampling with equal and unequal clusters. Multi stage and
n'rultiphase sampling . Comparison with simple random sampling and Stratified random
sampling. Relative efficiency of cluster sampling. Two-stage sampling. Non-sampling
errors.
Text Books
Cochran (1977). Sampling Techniques. Wiley Eastern, New-Delhi.
Singh, D and Chaudhury, F.S. (1986). Theory and Analysis of Sample Survey
Desisgns. Wiley Eastern. New-Delhi.
Reference Books
Des Raj (1976). Sarnpling Theory. McGraw
Hill
Mu rthl',M.N.( I 967). Sarnpling Theory and Methods. Statistical Publishing Society.
Mukhopadhyay.P. (1999). Theory and Methods of Survey Sampling. Printice-Hall
India. New-Delhi.
t
o
STA201
- C6: ProbabilitY
TheorY
(4 Credits)
- II
Unit I: Characteristic Functions-Definition. properties, inequalities, inversion theorem,
inversion tormula for lattice distributions. Characteristic functions and moments.
Taylor's series for characteristic functions, Bochner's theorem (no proof required).
Unit II: Weak Convergence and Characteristic Functions - Helly's convergence
theorem. Helly-Bray lemma, Scheffe's theorem, convergence of distribution
functions and characteristic function, Convergence of moments.
Unit III : Laws of Large Numbers -Convergence in probability of sequence of partial
sums. Kolmogorov inequality and almoSt Sure convergence, almost sure
convergence of a series, criterion for almost sure convergence, stability of
independent randorn variables, WLLN (iid and non-iid cases), strong law of large
numbers.
Central Lirnit Thseorem (CLT) - CLT as a generalization of laws of large
numbers. Lindeberge-Levy form, Liapounov's form, Lindeberg-Feller form (with out
prool). Exanlples and relation between Liapounov's condition-
Unit
IV:
Text Book
Bhar. B. R. (1999). Modern Probability Theory. Third Edition, New Age International (P)
Limited, Bangalore. John Wiley and Sons, New York
Reference Books
l.
2.
Laha anrl Rohatgi (1979). Probability Theory, John Wiley and Sons, New
York.
Rohatgi, v. K. (1916). An Introduction to Probability Theory and
Mathematical Statistics, John-Wiley Sons. New-York.
3.
Feller. W. (1993). An Introduction to Probability Theory and its Applications.
\[' i le1'-Eastem. New-Delhi.
4.
Ilao, C.R. (2002). Linear Statistical Inference and its Applications. Second
Edition . John Wiley and Sons. New - York.
5.
Basu, A.K. (1999). Measure Theory and Probability. Prentice Hall of India,
New Delhi.
I
a
STA202 - C7: Statistical Inference
(4 Credits)
-I
Unit - I: Fisher Information- Sufficient statistic-Minimal sufficient statisticExponential family and minimal sufficient statistic. Unbiasedness - best Linear
Ulbiased estimator - MVUE - Cramer- Rao inequality and its application - RaoBlacku,ell theorem-Completeness-Lehman-Scheffe theorem and its application.
Unit - II: Consistent estimator-examples and properties-CAN estimator-
inyariance property-asymptotic variance- Multiparameter case- choosing between
Consisfent esti nlators.
Unit - III: Method of moments-method of percentiles-method of maximum
likelihood-MLE in exponential family-solution of likelihood equations-Bayesian
method of estimation-Prior information-Loss functions (squared error absolute error
and zero-one loss functions)
functions.
-
Posterior distribution-estimators under the above loss
Unit IV:
Sl-rortest expected length confidence interval-large sample confidence
intervals-unbiased confidence intervals-examples-Bayesian and Fiducial intervals.
Text Books
Kale, B.K. (2005). A First Course on Parametric Inference. Second Edition. Narosa
Publishing. N ew-Delhi.
Casella, G. and Berger, R.L. (2002). Statistical Inferences. Second Edition. Duxbury.
Australia.
Reference Books
Rohatgi,V. K. (1976). An Introduction to Probability Theory and Mathematical
Statistics. John-Wiley and Sons, new-York
Rohatgi,V.K. (1984). Statistical Inference. John-Wiley and Sons, New-York.
Lehman, E.L. (1983). 'fheory of Point Estimation. John-Wiley and Sons. New-York
Rao, C.R. (2002). Linear Statistical Inference and lts Applications, Second Edition.
John-Wiley and Sons, New-York.
8
a
STA203
-
C8: Design and Analysis of Experiments
(4 Credits)
basic principles, guideline of design of experiments- Statistical
techniques.-Erperiments rvith single factor. ANOVA. Analysis of fixed effect models ofindividual treatment means. Random effect models. Model adequacy
"on1puiiron
checking. Choice of sample size. Regression approach ANOVA
Unit
- I: Application,
Completely Randomized Block design, randomized block design. Latin
square design. breaco-latin square design. BIBD - Recovering of intra block
ipforpatiol in BIBD - PBIBD - Youden square - Lattice design.
Unit
-II:
- III:
Factoriat designs - definition and principles. Two factor factorial design.
Random and mixed models. The general factorial designs- 2^k factorial experimentsconfounding-two Level fractional factorial design.
Unit
Unit IV: Nested or hierarchical designs
ANCOVA.
-
response surface methods and design
-
Text Book
Montgomery, D.C. (2001). Design and Analysis of Experiments.
& Sons. New-York.
5th
edition. John Wiley
Reference Book
Das, M. N. and Giri, N. S. (2002). Design and Analysis of Experimental. 2''d Edition.
New Age International (P) Ltd., New-Delhi.
,1
STA204
-
C9: Regression Methods
(4 Credits)
Unit - I: Least square estimation-properties of Ieast square estimates-unbiased
estimation - estimation
estimation ol o^2 - distribution theory - maximum likelihood
least
with linear restrictions-design mairix of less that full rank-generalized
squares.
Hypothesis testing; Likelihood ratio test-F-test - multiple correlation
estimationcoeffisierrt-Confidencc intcryals a1d regions' Siprultaneous interval
the
fbr
band
and
confidence bands for the regression surface - prediction intervals
Unit
- Il:
response.
Unit - tII: The straight line - weighted least squares for the straight lineregression
Polynontials in one vuri-ubl. - piecewise polynomial fitting - Polynomial
in several variables.
and
Unit - IV: Bias-incorrect varrance matrix-effect of outliers-Diagnosisserial
and
variance
nonconStant
remedies: residuals and hat matrix diagonals with outliersCorrelations-departures from normality - detecting and dealing
regressron'
component
diagnosing collinearitl', Ridge regression and principal
Text Books
L
2.
Seber.G. A. F. and Lee, A.J. (2003). Linear Regression Analysis, 2nd Edition.
Wilel Intersciencse- New JerseY.
Draper, N.R. and Smith, ff. (ig8gl. Applied Regression Analysis. 3'd Edition.
.lohn Wiley'& Sons Inc.. New-York'
Reference Books
l.
2.
Searle, s.R. (1997). Linear Models, wiley paperback edition.
WileY Interscience. New JerseY.
Rao.C.R.(1973). Linear Statistical Inference and Its Applications. Wiley
Eastern.
3.
4.
5.
Abraham. B. and Ledolter. J. (2005). Introduction to Regression Modeling
Duxbury Press.
Sengupia.D. and Jammalamadaka. S.R(2003). Linear Models:An Integrated
Approach, World Scientifi c.
Montgomery, D.C., Peck, F.A. and vining, G. (2001). Introduction to
Li.ear Regrlssion Analysis. 3'd Edition. John-Wiley and Sons,
Neu'-York.
lo
STA205 - C10: Practical
(2 Credits)
-I
The practicalls based on the fbllowing core papers in the second setnester:
1.
2.
3.
4.
Statistical lnference -l
Regression Methods
Design and Analysis of E'xperiments
Distribution TheorY
practicals are to be done using scientific programmable calculators or personal
computers. The question paper for the external examination will be set by the external
examiner in consultation with the chairman. The practical will be valued on the same day
the examination is held out and the marks will be finalized on the same day'
!i\
a
STA30I - C11: Statistical Inference
(4 Credits)
o
- II
Unit - I:
Tests of hypotheses - error probabilities - Most powerful tests
Pearson Lemma - Generalized Neymann - Pearson lemma
-
Neyman-
Method of Finding Tests - Likelihood ratio tests - Bayesian tests Unionintersection and intersection-union tests. Unbiased and invariant tests Similar tests and
iocaii-v nrost powerlul tesrs.
Unit - II:
Unit - III: Non-paranetric Tests - Single sample tests the Kolrrogoro\ - Smirnor
test - the sign test - the Wilcoxon signed rank test. Two sample tests - the chi-square test
for homogeneity - the Kolmogorov - Smirnov test - the median test -the Mann Whitney'-Wilcoxon test - Test for independence
correlation coeffi ent
-
-
Kendall's tau
-
Spearman's rank
robustness.
Unit - IV:
Sequential Inference - Some lundamental ideas of'sequential sampling sequential unbiased estrmation - sequential estimation of mean ol a non-nal population
the sequential probabilitv tests (SPRT) - irnportant properties - the lundanrental identitr
of SPR'I'.
Text Books
Casella, G. and Berger, R. L. (2002). Statistical Inference Second Edition. Duxbury.
Australia.
V. K. (1976). An Introduction to Probability Theorr and Mathematical
Statistics. John - Wilev Sons. New - York.
Rohatgi,
Reference Books
Fraser, D.
A. Non -
parametric Methods in Statistics.
Lehman, E. L.(1986). T'esting
of
Statistical Hypotheses..lohn Wiley. Ner.t'- York.
Ferguson, T. S. (19671. Mathematical Statistics: A Decision - Theoretic Approach.
Academic Press. New-York.
IL
STA302
- C12: Multivariate Analysis
(4 Credits)
Multivariate Nonnal Distribution - Definition. properties. conditional
distribution, marginal distribution. Independence of a linear form and a quadratic form,
independence of two quadratic forms, distribution of quairatic fbrm of a multivariate
vector. Partial and multiple correlation coefficients. Partial regression coefficient.
LJnit
- I:
Estimation of mean vector and covariance vector - Maximum likelihood
estimation of the mean vector and dispersion matrix. The distribr,rtion of sample nrean
veclur. Inlelence colrcenriiig, ii,e i,reai, !!'ctoi *lici-, ihc dispcrsion ntatrix is ki;owir ii;r
single and two populations. Wishart distribution properties - generalized variance.
Unit - II:
Testing Problems - Mahalnobis D^2 ar-rd l{otelling's T"2 statistics.
Likelihood ratio tests - Testing the equality of mean vector. equalitv of dispersion
rnatrices. testing the independence of sub vectors. sphericitl tcst.
Unit - III:
Unit - IV:
The problem of classification - classification of one of tr,vo multivariate
normal populations when the pararneters are known and unknown. Extension of this ttr
several multivariate normal populations. Population principal componenls Summarizing sample r,'ariation by principal components - Iterative procedure to calculate
sample principal components.
Text Books
Anderson,
T.W. (1984). Multivarjate Analysis.
John
-
Wiley. Neu'York-
Rao, C.R. (2002\. Linear Statistical Int-erence and Its Applications. Second Edition..lohn
Wilel' and Sons. New - York.
Reference Books
Giri, N.C. (1996). Multivariate Statistical Analysis. Marcel Dekker. Inc.. New York.
Kshirasagar, A.M. ((1972). Multivariate Anall-sis. Marcel Dekker. Inc.. New York.
Rencher, A.C. (1998). Multivariate Statistical Analysis. John Wile1, Neu' York.
\_l
STA303 - C13: Stochastic Processes
(4 Credits)
Concept of Stochastic processes. examples, specitications. Markov chains
Chapman - Kolmogorov equations - classification of smtes - Iimiting probabilities
Gamblers ruin problem - mean time spent in transient stlrtes - branching processes
Hidden Markov chains.
Unit-I:
Unit - II:
Exponential distribution - counting process -'itlter arrival time and waiting
time distributiols. Properlies of Poissoii proccsses - Conditional distribution of arri"'al
times. Generalization of Poisson processes non-homogenous Poisson proe ess.
compound Poisson process. conditional mixed Poisson process. Continuous time
Markov Chains - Birth and death processes - transition probability function - limiting
probabilities.
- III:
Renewal processes - Iin-rit theorems and their applications. Renewal reward
process. Regenerative processes. semi-Markov process. The inspection paradox
lnsurers ruin problem.
Unit
Basic characteristics of queues - Markovian models - network of queues.
process The M/G/l system. The G/M/l model, Multi server queues. Brownian motion
hitting time - Maximum variable - variations on Brownian motion - Pricing stock
options Gaussian processes - stattonary and weakly stationan' processes'
Unit - IV:
-
Text Book
Ross, S.M. (2007). lnrroduction to Probability Models.
IXth Irdition. Academic
Press.
Reference Books
New-Dellii'
Medhi, J. (1996). Stochastic Processes. Second Edition. wiley Eastern'
Karlin and Taylor (1975). A First Course in Stochastic
Processes. Second Edition'
Academic Press. New-York.
cinlar, E. (1g75). Introduction to stochastic Processes. Prentice t'lall. New Jersey'.
New-Delhi'
Basu, A.K. (2003). Introduction to Stochastic Processes. i"larosa.
\t1
STA3O4 _
EI: TIME SERIES ANALYSIS
(4 Credits)
Unit I: Simple Descriptive
Techniques - approaches to Time series analysis.
Stationary time series, The time plot, 'fransformations, Analyzing series containing
trend, seasonal variation, autocorrelation. other tests for randomness.
Unit
-II: Probability Models for Time Series - Stationary Processes, Autocorrelation
Ftinction, N{A Processes, AR Proces;ses. |,4ixed ARN,4.^. models. lntegrated A.B-IMA
models.
Unit -III: Estin-ration in the time domain - Estimation of Autocovariance and
Autocorrelation functions, Fitting an AR process, Fitting a MA process. Estimating
the parameters of an ARMA model.
Forecasting - Univariate procedures - Extrapolation of trend curves.
Exponential smoothing. Holt-Winter's procedure. Box-Jenkins's procedr-rre. Stepwise
tlnit - IV:
i.lutoregressl
o
n
Text Book
Chatfield, C. (1989). The Analysis of Time Series - An Introduction. Fourrh Edition.
Chapman and Hall. London.
Reference Books
Box, G.E.P., Jenkins, G.M. and Reinsel, G.c. (1994). Time Series AnalysisForecasting and Control. Third Edition. Pearson Education. lnc. New - Delhi.
Brokewell, P.J. and Davis, R.A. (1987). Time Series: 'l'heory
Edition. Springer, New York.
Brokewell, P.J. and Davis, R.A. (2002). Introdr-rction
&
Methods. Second
to Time Series and
Forecasting. Second Edition. Springer, Ne'uv York.
(t
,A
STA305
- E2: Operations Research - I
(4 Credits)
Unit
Operations Research - defenition and scope - Linear programming problem.
simplex method, artificial basis techniques. two-phase simplex method. BigN4 method. duality concepts. duality theorems. dual simplex method"
I
Transportation and Assignment Problems - sensitirit1 analysis, parametric
programming. Sequencing and scheduling problems two machine n -iob and
three machine n.;ob problems.
Unit ll
Unit
Ciame Theory - Two person zero-sum gantcs. minittrax theorem. ganre
problem as a linear programming problem. Co-operative and competitive
lll
games.
Unit IV
Integer programming - Cutting plane method,
branch and bound technique, application of zero - olle programming.
Integer programming:
Text Book
K.V. Mital (1987). Optimization Methods in Operations Research and Systems
Analvsis. Second Edition. Wiley Eastern. New-Delhi'
Reference Books
l.
Hadley, G. (1964 ). Linear Programming, oxford
& IBH Publishing
Co., New-
Delhi.
Taha, H.A. (1982 ). Operations Research: An introduction. Macmillan'
3. Hiller F.S.. and Leiberman! G.J. (1962). Introduction to operations Research.
Holden Day.
4. Kanti Swarup, Gupta, P.K. and Singh' M.M. (1985). Operations Research,
2.
Sulthan Chand
&
Sons.
t
1
STA40l-
o
E3:
Statistical Decision Theory
(4 Credits)
Basic Concepts - Statistical decision Problem Decision rule and loss
randomized decision rule. Decision principle - sufficient statistic and convexity. Utitity
and [oss- Ioss lunctions standard loss functions - vector r,alued loss functions.
Unit - I:
Unit - Il: Prior Information and Subjective Probability - Prior information - subjective
determrnation ol prior densit.v. Non-informative priors - maxrmum entrop)' priors -the
marginai distribution to detennine the prior - the ML-ll approach to prior selectron.
Conjugate priors.
Unit - III:
-
The posterior distribution Bal,esian inference
Bayesian decision theory - empirical Bayes anall,sis - Hierarchical Bayes analysisBayesian robustness - Admissibility of Baye's rules.
Ba1'esian Analysis
- [V: Minimax Analysis - Game theory - basic concepts general techniques tbr
solving games. Ganres with finite state of nature- the supporlirrg and separating h1'per
plane theorenns. The minimax theorem. Statistical games
Unit
Text Book
Berger, O. J. (1985). Statistical Decision Theory and Bayesian Analysis. Second
Edition. Springer - Verlag.
Reference Books
Ferguson, T.S. (1967). Mathematical Statistics: A Decision - Theoretic Approach.
Academic Press, New-York.
STA402 -E.4: Lifetime Data Analysis
(4 Credits)
Basic Concepts and Models - Lil-etime distributions- continuous and discrete
models- inrportant parametric nrodels: Exponential._Weibull. l-og-nornral. Log-logistic.
Gamma, Inr,erse Gaussian distributions. I-,og location scale ntodels and mixture models.
Unit
l:
Censoring and statistical methods.
Units 2: Non-parametric Estimation of Surr,'ival Function and Quantiles - The productlintit estima.te and its properties. The Nelson-Aalen estintate. interval estinration of
tl
E
sur.;rlal probabilities. asvmptotic propcrties of estitnators. descriptire and diagrlostic
plots. estimation of hazard function. methods for truncated and interval censored data.
'Lrie tables.
- Under exponential mociei - large sample theory'. tlpe-2
of two distributions; inlerence procedures fbr Gamma
plans,
comparison
test
censored
and inverse-Gaussian distributions; models with threshold parameters, prediction
intervals. inference for log-location scale distribution: likelihood based methods; exact
methods under type-2 censoring; application to Weibull and extreme value distributions,
comparison of distributions.
Units 3: Inference procedures
Units 4: Parametric regression models - Log- location scale (Accelerated Failure time)
nrodel. Proporticnal haza.rd models N4ethods for continuous multiplicative hazard
models, time varying covariates. Semr- Parametrtc maximum likelihooci- estimattott ior
continuous observations. Incomplete data: Rank test for contparing Distributions. Logrank test, Generalized \['ilcoxon test. Multiple models of lailr.rre: Parametric and nonparametric methods. Goodness of flt test: test for exponential, Weibull extreme value
distribution and regression models. A brief discussion on multir,ariate lifetime models
and data.
fext Books
l.
Law'less, J.F. (2003). Statistical Models and Methods fbr Lif'etime
(Second Edition). John Wiley & Sons Inc", New Jersey.
2.
Kalbfiesch, J.D.and Prentice, R.L. (1980). The Statistical Analysis of Failure
Time Data, , New Jersey.
Reference Books
l. Miller, R.G. (1981). Survival Analysis, John Wile-v & Sons Inc.
2. Bain, L.G. (1978). Statistical Analysis of Reliability and Life testing Models.
3.
4.
Marcel Decker.
Nelson, W. (1982). Applied Life Data Analysis.
Cox, D.R. and Oakes, D. (1984). Analysisof Survival Data, Chapman and Hall.
Lee, Elisa, T. (1992). Statistical Methods for Survival Data Analysis. John
Wiley& Son
It
STA403
- Cl4: Practical - II
(2 credits)
The practical is based on the tbllowing elective papers in the lourth semester:
STA4OI _ E3
&
STA4O2- E4:
Practical is to be done using scientific programmable calculators or personal computer.
The question paper for the external examination lvill be set b1' the external examiner in
consultation with the chairman. The practical will be valued on the same day the
examination is carried out and the mark sheet will be given to the chairntan on the same
day.
STA 404 - Cl5: Project & Dissertation
(8 Credits)
As a part of the course work. during the fourlh semester each student has to utrdertake a
project work in a selected area ol interest under a supervisor in the departrrent. The topic
could be a theoretical work or data analysis type. At the end of the fourth semester the
student is supposed to prepare a dissertation which sumnrarizes the proiect w'ork.'fhe
project and dlssertation is of 8 credits for which the follorving er,'aluatiou will be
fbllowed:
The project report will be evaluated b1, an external examiner who is appointed
the universi$,. The following break up of total marks shall be considered.
l. Revieu of letterature, formulation
2.
3.
by'
of the problem and defining clearly the
objective: 25oh of the total marks.
Methodology and description of the techniques used: 25"h ol the total
Analysis, programming/simulation and discussion of results: 25"h of the total
amrks.
Presentation of the report, organization, linguistic style, reference etc-z 25oh
of the total.
tc(
LIST OF' ELECTIVES
1.
2^
3.
4.
5.
6.
.
8.
9.
7
Operations Research - I
Time Series Analysis
Lifetime Data Analysis
Statistical Decision Theory
Actuarial Statistics
Operations Research - [[
Reliability
Queuing Theory
Statistical Quality Assurance
Optional -l: C)perations Research
I
Operations Rescarch.-tlefinition and scope, Linear programming, sintplex
method, artificial basis techniques, tlvo phase simplex method, Big-NI
rnethod, tluality concepts, duality theorems, dual simplex methods.
Il
Transportation and assignment problents, sensitivity analysis, parametric
programming. + Sequencing and Scheduliing problems-2 machine n-.Iob
and 3- machine n-Job Prob'lems.
tlnitv
Unit
Unit
-I
III
Gaure iheory,: (rro;lersou zero surrr gaurcs, rnininritx theorcnr, garne
problem as a linear programming problem. Co-operative and competition
games.
Unit
IV lnteger
programming: Cutting plane mcthods, branch and bound
technique, application of zero
-
one programming.
Text Book
K.V.Mital and Mohan - Optimization techniques.
Reference Books
l.
2.
3.
1.
Hadley, G. (1964) - Linear Programnring, Oxford & IBH Publishing Co,
New Delhi.
Taha. H.A. (1982) : Openation Research, An Instruction, Macmillan.
Hilter FS. and Lcibcrman, GJ. (1962). lntroduction to Operations
Research, Holden Day.
Keerti Su'amp, Gupta, P.K and John, M.M.(1985): O.R., Sultan Chantl
&
Sons.
")t
Optional - 2: TIME SERIt S ANALYSIS
l.
-f
Simple Descriptive Techniques : Approaches to inre Series Anall,sis.
Stationarv time series. The time plot. 'f ransformations. Analvsing Series
containing Trend. Seasonal Variation. Autocorrelation, Other Tests tbr
Randomness.
2. Probability Models for Time
Function, MA Processes,
ARIMA models
AR
Series: Statinary Processes. Autocorrelation
Processes. Mixed ARMA models, Intregrated
in the time domain: Estimation of Autocovariance and
Autocorrelation functions. Fitting an AR process. F itting a ivlA proccss.
Estimating the parameters of an ARMA model.
3.
Estimation
4
Forecasting: Univariate procedures- Extrapolation of trend curves. Exponential
sntoolhing. Holt-Winter's procedure. Box-Jenkin's procedure,
Stepwise
autoregressron
Text Book
Chatfield, C. (1989) The Analysis of Time Series- An Introduction- Fourth Edition.
Chapman and Hall. London.
Reference Texts
Box, G.E.P., Jenkins. G.M. and Reinsel,G.C. (1994). Time Series
Forecasting and Control. Third Edition. Pearson Education. Inc. New
Brokervell, P.J. and Davis, R.A. (1987). Time Series:
New York.
-fheory
Brokewell, P.J. and Davis, R.A. (2002). Introduction
&
-
Analy'sis-
Delhi.
Ivlethods. Springer.
to Time Series and
Forecasting. Second Edition. Springer. New York.
2)
Optional - 3: Lit'etime Data
Analvsis \/
t;nit l:
[-if'etinre distributir"rns- continuous and discrctc ntodels- intportant
parametric models: Exponential._W'eibull. l-og-norntal. [-og-logistic. Gamma.
lnverse Gaussian distributions, [,og location scale models and ntixture
models. Censoring and statistical methods.
Units 2: The product- limit estimate and its properlies. -fhe Nelson-Aalen
estinrate. interval estimation ol sun ival probabilities. asvnrptotic properties of
estimators. descriptive and diagnostic plots. estimation of hazard firnctiorr.
methods fbr truncated and interval censored data. I-if-e tables.
Units 3: Undcr exponentiai muciei - iarge sarnple theory'. t-v-pe-2 censoreci tesr
plans. comparison of two distributions: inference procedures for Gamma and
inverse-Gaussian distributions: models rvith threshold pararneters. prediction
inten'als. inl'erence
fbr
log-location scale distribution: likelihood based
rnethods; exact methods under type-Z'censoring; application to Weibull and
extreme'u'alue distributions, comparison of distributions.
Units 4: Log- location scale (Accelerated Failure time) model. Proportional
hazard models. Methods for continuous multiplicati'n,e hazard models. time
varying covariates. Semi- Parametric maximum likelihood- estintation for
continuous observations. Incornplete data; Rank test fbr conrparing
Distributions. Log- rank test. Generalized Wilcoxon test. Multiple models of
f-ailure: Parametric and non- parametric methods. Goodness of fit test: test lor
exponential, Weibull extrcmc r,alue distribution and rcgression models. A
brief discussion on multivariate lifetime models and data.
Text Books
l.
Lawless, J.F. (2003). Statistical Models and Methods lor Lif-etime
(Second Edition). John Wiley & Sons Inc.. New Jersey'.
2.
Kalbfiesch, J.D. and Prentice, R.L. (1980). The Statistical Analysis
of Failure Time Data,. New Jersey.
References
l.
2.
3.
4.
5.
Miller, R.G. (1981). Survival Analysis. John Wilel'& Sons Inc.
Bain, L.G. (1978). Statistical Analysis of Reliabilitl and Lif'e testing
Models. Marcel Decker.
Nelson, W. (1982). Applied Lif'e Data Anali'sis.
Cox, D.R. and Oakes, D. (1984). Analysis o1'Survival Data.
Chapman and Hall.
Lee, Elisa, T. (1992). Statistical Methods for Survival Data Analysis,
John Wiley& Sons.
23
a
Optional - 4: Statistical Decision Theory
Upit - I:
Statistical decision Problem - Decision rule and loss randomized decisio,
rule. Decisioll principle - sufficient statistic and convexitv. t rtilit)
a11d loss- loss
.
f unctions
- slandard loss functions vector valued loss lunctions.
-
Unit -
.{
II:
Prior information - sub.iectir,'e determination of prior densitl. Non-infbnnariv,e
priors - ntaximunl entropy priors he marginal distribution to deterrnine
the prior - the
ML -ll approach to prior selection. Conjugate priors
uni! - III:
rhe posterior distribution - Ba1'esian inference - Bayesian decisio, rheorr
analysis Hierarchical Bayes analysis- Bayesian robustnessAdmissibility of Bayes rules.
empirical Bayes
Unit - IV: Game theory - basic concepts - general techniques for solving games.
Games with finite state of nature- the supporting and separating hvper plane
tietr.rr.
The minimax theorem. Statistical games
Text Book
Berger, O.J. (1985). Statistical decision Theory and Bayesian Analysis. Second
Edirion.
Springer - Verlag.
Refernce Books
Ferguson, T.S. (1967). Mathematical Statistics: A Decision - Theoretic Approach.
Academic Press. New-York.
Lehman, E.L. (1983). Theory of point Estimation. John-wiley, New-york
.
,rq
Optional - 5: Actuarial Statistics
Unit-l: [.]tilitl'theory. insurance and utilitl theorv. models lirr indivic'lual clainrs anci their
iullls. survt"'al functton. curtate future lit-etime. fi>rce oi' ntorlalitr,. [-ife tahle :rnd irs
relatton i'r'ith surt,ir,'al function. exampies. assumpticlns firr liactionai ages. some
analytical laws o1'mortality, select and ultimate tables. Multiple life functions..ioint lit'e
and last sttrvivor status. insurance and annuity benefits through rnultiple life functions
evaluation lor special morlalitl, lar.rs.
Unit-2: Multiple decrement models. deterministic and random sun,ivorship groups.
associated single decrement tables. central rates of nrultiple' decr.ement. net- single
premiums and their numerical evaluations. Distrihurion of aggregate claints. .o,rpornd
Poisson distribution and its applications.
Unit-3: Principles olcompound interest: Nominal and eff-ectir,e rates ol'interest ancl
discount, force of interest and discount, compound interest. accuntulation factor.
continuous compounding.
Life insurance: Insurance payable at the moment of death and at the end of the year
of death-level benefit insurance. endowment insurance. diferred insurance and rarying
benefit insurance. recursions. commutation lunctions.
Lif-e annLrities: Single palment. continuous lif-e annuities. discrete lit-e annuitics. lil-e
annuities with ntonthll' pal,ments. commutation functions. \'ar\ ing annuities. recursigps.
complete annuities-immediate and apportionable annuities-dr.re.
Unit-4: Nel prelniums: Continuous and discrete premiums. tnre monthlv payment
premiums. apporionable premiums, commutation functions, accumulation type benefits.
Payment premiums. apportionable premiums, commutation functions. accumulation
type benefits.
Net premium reserves: Continuous and discrete net premiutn reserve, reserves on a
semicontinuous basis. reserves based on true monthly premiums. reserves on an
apportionable or discounted continuous basis. reserves at fractional durations. allocations
of loss to policy years, recursive formulas and diflerential equations lbr reserves.
commutation functions.
Some practical considerations: Premiums that include expenses-general expenses
types o1'expenses. per policy expenses. Claim amoLlnt distributions. approximating the
individual ntodel, stop-loss insurance.
References
l. Atkinson. M.E. and Dickson. D.c.M. (2000) : An Introduction to Actuarial
Studies. Elgar Publishing.
2. Bedford, T. and Cooke, R. (2001): Probabilistic risk analysis.Cambridge.
2t
1
Bolvers.N. L..Gerher. H.ti.. Hickman.J.C.. Jones D..A.
and Nesbitt, C.J.( t986):
'Actuarial Mathcmatics,,
Society of Actuaries. lthaca. Illinois, U.S.A.. Second
Edition.
A
.+
N4edina. P. K. and Merino. S. (2003): A discrere intnrduclio,
: Marhernaticaj
ilnance and Probabilitl, Birkhauser.
). Neill. .A. (1971 ). Lif-e Clontingencies. Heineman.
(,. Philip. M. et. al (
r999): Modern Actuarial Theory and practice. chapnran
and
Hall.
7. Rolski, T., Schmidli, H., Schmidt. v.
and reugels, J. (r99g): Stochastic processes
for Insurance and Finance. Wiley.
8. Spurgeon. E.T. ( 1972): Life contingencies,
cambridge Unir,ersity press.
9. Relevant Publications of the Actuarial Education
Co.I:1. Barh Sireet, Abingdon,
Oxforclshire OX I 43FF (U.K.)
4
Optional - 6: Operations Research -
II
Unit
I
Non-linear programming, Lagrangian function, saddle point, Kuhn-Tucker
Theorem, Kuhn-Tuckerconditions, Quardratic programming, Wolfe's
algorithm for solving quadratic programming problem.
Unit
II
Dynamic and Geometric programming: A minimum path problem, single
additive constraint, additively separable returnl single multiplicative
constraint, additively separable returnl single additive constraint,
multiplicatively separable return, computational economy in DP. Concept
and examples of Geometric programming.
Unit
III Inventory management; Deterministic models, the classical economic
order
quantity, nonzero lead time, the EOQ with shortages allowed, theproduction
lot-size model. Probabilistic models. the newsboy problem., a lot size.
reorder point model.
Unit IV Replacement models; capital equipment that deteriorates with time, Items
that fail completely, mortality theorem, staffing problems, block and age
replacement policies. Simulation modeling: Monte Carlo simulation,
sampling from probability distributions. Inverse method, convolution
method, acceptance-rejection methods, generation of random numbers,
Mechanics of discrete simulation,
Text Books
I
2
3
4
Mital and Mohan. Optimization methods in operations research and systems
analysis.
M.Sasieni, A.Yaspan and L.Friendman. Operations Researchl Methods ancl
Problems.
Hamdy A. Taha. Operations Research and Introtluction.
Ravindran, Phillips and Solberg : Operations Research Principles and
Practice.
Reference Books
l:t " ixffffi,;,:H..,ff):""m1ft;1-'[
e
nfpuce'irbn''*"#':1:i
OtERhl\oNIS Reser'R.ctl : SUtJft'N Chnsrl a SoNg rNu^t,t\i
1l
Optional -7: RELIAIIt LITy
tinit-l: [tcliabilitl c()ncepls ancl nreasu.cs: c()rlp(),err.s artl s\stcnrs- c.hererl s\stcnrs:
|cliirhi lit" .l c0lrcrenl s.\ ste lns. e uts itr.tii paths: nroti ular-r.iecon.rrr.siti.n.
5,,|.,,.,(.,- ()n s\ srcllr
reliabi l rtr': srrr.rctural and reliabi
l
ity unportance
ot' conrpr)ncn1s.
Unit-2: l-ilL'disrribtttions: rcliabilit,r firnction: hazirrd ratc: c(),nnron lilL,tlisrribrrtiorsexpouetrtial' WeibLrll. Grtttrnla etc. Estinralion o1'paranrctel's ,utl
lesLs i, these nr.clels.
Notions 01'agcirtg: IIrR, IlrRA. NBLi. D\4RL. and NIltjl: ('lasscs
ancl thcir cluals.
closttrcs tlr t[tcsc classcs uttdcr tortlatiorr of cohcrcnt s\stcnrs. ctlrrrolutions
and rnirtr..cs.
Unit-3: Llrlir ariate shock rnoclels and Iil-e clistribLrrions arising out .l' tlrenr:
bir ariate
shock-rrodcls con'lnro, hir a.iatc crpt'rrcrrti.! cristr.ibuti.,q aner thcir
Itcliabilitr cstilllttlioll bascd ort tirilurc tinrcs in rariouslr,ccns()r.crl tit" tcsr. 1-rr.rpcrtics.
lrd in tcsts
rvith replacelnent ol'f'uiled items: slress-strcngtlr reliatrility and
its estinrali6n.
Unit-J: \lairltcrrancc artd rcplaccnrcnt prolicics: arailiibilitr ol'r.cpairablc s'srcrrs:
rlodcling o1'a rcpairablc svstcrl hr-1,. a non-hr)ntogcncor.ls poisson proccss.
Reliabilitr r-rowth nloclels: probabilitl plotling techniquc's: [[ollarrdcr-pr.schzrn
ancl
Deshparltlc tests lbr expotrentiirlitr': tests lirr HPP rs NHpp uith
rcpairahle srstenrs.
[]asic iclcas olaccclcr.atccl lil-c tcsting.
Ittll-'ERIN('ES
l.
2'
i.
4.
5.
6.
Barlow'll.li. and Proschan F (198-5)Statisrical lhcorr o1'llcliabilitr
arril I.itc
l-esting: I Iolt.Rinehart ant.l \\'inston.
Bain L.J. and Engclhardt (1991). Sratistical Analrsis ol lle liabilitr ancl Lile
'lcsting
lVlode ls: Marccl l)r-kkc-r.
Aven. I'. and .lensen,It. ltqgq). Stochastic Moclcls in Ilcliahilitr. Springcr_
Verlag. N-ew Yurk. Inc.
Lawless, J.F. (2003). Statistical Models and Methods for Lifetime (Second
Edition), John Wiley & Sons Inc.. New Jersey.
\elson, U'(l9til) Applicd l.ifc I)ara tural\sis:.lohn \\ ilcr.
Zacks, S. ( 1992). lntroduction to Reliability Analysis: Probability Models and
Statistics Methods. New York: Springer-Verlag,
2S
Optional - 8: QUEUEING THEORY
UNIT I: Introduction to clueueing theory. Characteristics of queueing processes.
Measures of effectiveness, Markovian queueing models, steady siate
,ol-ution of the
MlMll model- waiting time distributions. Little's formula. queues w,ith parallel channels
and truncation. Erlang's loss formula, Queues w'ith unlimited sen,ice. finite
source
queues.
UNIT II: 'lransient behaviour of M/M/l queues. transient behaviour of M/M/"o.
Busv
period analysis for M/M/l and M/M/c models. Advanced Markovian
models. grfk .ili
Mt'') I M
ll
model. Bulk service
M I MIt)
lllE,.;l
^ und E,,(lMli . A^r-..,.
brrel drscussion
ll
ot'priority
model. Erlangian
models
queues.
UNIT III: Queueing networks
- series queues. open Jackson netw,orks. closed Jackson
netu'orks- Ciyclic queues, Extettsion of Jackson networks. Non .lackscln
networks.
UNIT IV: Models with general arrival pattern, The M/G/l queueing model.
The
Pollaczek-khintchine formula. Deparlure point steady state system
size probabilities.
ergodic theory, special .ur.. M lEkll and M/D/1, waiting times, busy period
analysis.
general input and exponential service nro lels. arrival point steady
rtut. svstem size
probabilities.
References:
I
.
2'
Gross, D. and Harris, c. M. ( l9g5): Fundamentals of
eueueing Theory,
Edition, John Wiley and Sons, New york.
Kleinrock L
(
). Queueing Sysrems. Vol
York.
3.
4.
I & Vol 2. .lohn Wiley and
2,,d
Sons. New
Ross, S M.(2007). Introduction to Probability Models, 9'h Edition, Academic
Press. New York.
Bose, S.K. (2002). An Introduction to
Queueing Systems. Kluwer Academic /
Plenum Publishers. New york.
)11
Oprional
- 9: Statistical euality Assurance
[-;nit - I: <)uality anti Quality assurar]ce. Methods of Qualitr assurance. Introciucrion
to TQM. Acceptance sampling for attributes. Single sarnpling. Double sampling.
Multiple
sarnpling and Sequential sampling plans. Measuring the perfornrance ol ih.r. ,un1pling
plans.
Unit -II:
,{cceptance sampling by variables. sampling plans fbr single specification
linlit uith knou'n attd unknol,t'n variance, Sanrpling plans u,ith double specification
limits. comparison of sampling plans by variabler und-utt.ibutes. Continuous
sampling
plans I. II, IIL
Unit -III! Control charts. Basic ideas. Designing of control charts fbr the nunrber of
non-contbrmities and fraction non-conformities. Mean charts. Median charts. Ilxtrente
value charts. R-charts. and S-charts. ARL. Econornic design of control chafis.
Unit -IV:
Process capability studies. Control charts
with nremory,- CUSUM charts.
EwMA nlean charts. OC and ARL for control charts. Statistical process control,
Modeling and quality programming. orthogonal arravs and robust qualit1,.
Text Books
Montgomor)', R.C. (1985). Introduction to Statistical Qualitl Control. 4,r, edition.
Wiley, New-York.
Mittag, H.J. and Rinne, H. (1993). Statistical Methods for eualiti, Assurance. chapman
and Hall. Chapters 1.3 and 4.
Oakland, J.S. and Follorwel, R.F. (1990). Statistical Process Control. East-West press,
Chapters
li
and 14.
Schilling, E.G. (1982). Acceptance Sampling in Quality Control, Marcel Dekker.
Reference Books
Duncan, A.J. (1986). euality control and lndusrrial Statistics.
Grant, E.L. and Leaven Worth, R.S. (1980). Statistical euality Control , Mc-Graw
Hilr
Chin-Knei Chao (1987). euality
prograrnming. .tohn Wiley.
Fly UP