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M.Sc. DIIGRI]E IN
sYLrrArlI oF coulrsEs 0 IFERED rN SEMESTER-II
ST2C06: ESTIMATIOI\
Unit-l: sulllcient
THEORY
(4 Credits)
statistics ancl rninirnum variar,:e unbiased estimators.
Sutlit'ient stutistics, Factorization theorem for s fliciency (proof for discrete distrihutions only).
.iorrrt sLrfficient statistics. exponential family, r,,ininral sufflcient statistics, criteria to ljrrd ihe
tttrttittlitl sutflcient statistics, Ancillary statisticf complete statistics, complete statistics. Basu's
Ihcotern (proot' lbr discrete distributions nly), Unbiasedness, Best Linear Unbiased
Estimator(BLtlE). Mininrum Variance Unbia,'ied Estimator (MVUE). Fisher Intbrmation,
Cl'amer llao inequality and its applications, Rao-Blackwell Theorem, Lehnrann- Schelfb
theotem. necessary and suftrcient condition for F,iVUE.
tiuit:2;
Cotr.stst-ent Estirnators and Consistent
Ar rmptotically Nonral Estirnators
(-orrsistent estunatol'. Invarialrce propefly of c lrsisterrt estirnators, Method of rnonrents
irncl
pclccntiies 1o cletcnninc consistent cstinrato s, Choosing between consistent estimators.
Corrsrstent r\synlptotically Nornral (CAN) Estin. rtors.
I
lnit-3:Mcthods of Estinration.
of percentiles, M :thocl of lnaxirnum liketihoocl (MLE). MLE irr
cxponc'ntial tanrily, One paratneter Crarner I urily, Cranrcr-Iluzurbazar theolem, Bayesian
Methcrd ol- llloments, Method
rr
rci i iu,.i oi-
cstintatiutr.
tinit.4: lrrlerval
Estlrtrat ion.
l)cl'irrititu. Sltortest Expected lengLh conllden, : intcrval, large samplc confidence rurcrvals,
I jnbiasctl cor-rl-idence intervals. Bayesian and Fid cial intervals.
Ilooks
til: Study:-
L Kale,ts.K.(2005). A first course in pararm, tric infbrence. Second Edition. Narosa
Publishing House, New Delhi.
2 George Casella ancl Roger L Berger (200 ). Statistical inf'erence, Second Edition.
Duxbury. Australia.
Rcl'crcrrccs:I
Lehnrann, E.L ( 1983). Theory ol point es
2. P.rhetgi, \/.K (1975). ,A,n inrrcCuction rc'
.[of n Witey and sons, Nerv York
3. Rohatgi: V.K ( 1984). Statistical lnf'ercnct
4. Rao, C.R Q(nD. Lincar Statistical Infbre
.
mation. John Wiley and sons, New Yolk.
robabi I i ty Theory. anC Itdathe rnat i cll S rati l: ri c:;.
John Wiley and sons, New York.
ce and its applications, Second Edition,.Iohn
Wiley and sons. New York.
ir
V'
\
-)
ST2C07: Sampling 1 heory (4 credits)
Unit-I:
Ccnsus and Sarnpling-Basic conc( )ts, probability sampling and non probabilitv
sarnpliltg. slmple randont sarnpling with anci without replacemcnt- estimation of pspulation
ttlcali ar]d rotal-esLimation of sample size- e tirnatiorr of proportions. Systentatic sarnplinglirrettr lntl circular systematic sarnpling-estir ration of mean and jts variance- estirnatiop ol'
rlle.ll.l ll.l p,lpulatirrns with linear and periodic rends.
Unit-II: StraLification and stratif iecl rarrdor sarnpling. Optimunt allocations - comparisgns
of varlancc under various allocations. Au iliary variable techniques. Ratio ntethod of
esttlnation-estirnation of ratio, mean ancl
squal'e
tota
Bias and relative bias of ratio estinrator. Mcan
erl'or of ratio estimator. Unbiased ratio type estimator.
Reglession methods crf
cstilnatton- Conrpariscrn oi'ratio anci regressic r estimators rvitlr strnpic meau pcr unlI metho6.
Ratio ancl regressiott ntethod of estimation
Unit-III:
tratifiecl population"
Varying probability sampling-pps ;ampling with ancl withour replacements. Des-
Ra.i order-ed estimilLors,
;rr1(l
in
Murthy's uuordered , itilnator, Horwitz-Thompson
esti
rlators. yates
Grlrndy fotnts of variance and its esliu rtor-l. Zen-Midzuno scheme of sarnpling, npS
ranrpl ing.
Unit-IV: Clusler sarnpling witlr equal and unr lual clusters. Estirration of
rneau and var.iance"
relative elliciency, optimutn cluster size. va ying probability cluster santpling. Multi stasc
ancl rnultiphasc samplitrg. Norr-santpling crror
;.
Referenccs
L coclrran w.G ( 1992) Sanrpling'fcchniclur ;. wiley Easreru, New york
2' D. Singh ancl F.S. Chowdhary 'fheory a d Analysis of Sarnple Survey Designs. Wiley
Eastcrrr(New Age Intcrnatioltal), NewDcl
P-V.Sukhatme et.al. (1984) Sarnpling
-l'
i"
Tlx rry of Surveys witlr Applicatior.rs. IOWA
Universrty Press. USr\
I2
Srate
.ia
ST2C08: nBGRESSION Mi ITIIODS
(4 Credits)
Unit-l: Simple and rnultiple regression"
lntroduction to regression. Simple linear regressio t- least square estimation of parameters.
I{ypothesis testing on slope and intercept, Interval estimation. Predictron of uew observations.
Coef Tic-ienf of clelermiiiefi on-;RegiesSlonthrough 'rigin, Estimation by maxinrum likelihood.
case where x is random.
Multiple Linear Regression- Estimation of rnodel 'aralnetels, Hypothesis testing in rnultiplc
lrnear reglessiot-r, Confrdence lnterval rn multiple :gression, Predictron of new observatrons.
Unit- 2: Model Adequacy Checking, Transformat :n and rveighting to col'rect rnodel
Inadequacies.
Residual analysis. the press statistics, detection of reatrrent of outliers" lack of fit of the
regression model. Variance -stabilizing translbrn ttions. Transfornration to Iinearize the moclel,
Analytical rnethods for sclecting a transfonnation Generalized and weighted least squares.
Unit- 3: I'olynornial regression rnodel and urodel ruilcling.
Polynomial models in one variable. Nonparametri regression, Polynomial models in trvo or
more variables. orthogonal variables. Indicator va iables. Regression approach to analysis ol
variance. Model building problem, computational echniques for variable selection.
Unit-4:
Generalizerl Linear Models.
[,ogistic regression model, Poisson regression, Th gcneralized litrear nrodels- link futrctit-rtr :rnci
Iineal predictols, pararnel,er estimation and inf'erer ;c in GLM, prediction and esl-imation iri
Cil,M. r'esidual analysis in GLM over dispersion.
Books for Study:-
l.
Montgomery ,D.C., Peck, E.A.. Vining
Regression Analysis. John Wiley & Sons.
i
Geofferey (2003). Introduction
to
Linear
Refcrences:-
l.
Clhatterjec, S & B. Price (1977). Regress
2. Draper, N.R & H. Smith (19E8). Applie
York.
----J--$*errG.A..F (197 7 ). Li n ear Regtes s i o n,
4 Searle " S.R (197-1). L.inear Mode-l Wiley
t3
on analysis by exanrple, Wiley. Ncw York.
Regression Analysis. 3''r E,rlition, Wiley. Nerv
nalysis. Wiley, Ncw York.
Neu, York
t)
ST2C09: Design and Analysir of
Experiments
(4 credits)
Unit- 1: Linear Model. Esl.imable Functions r rd Best Estimate. Normal Equations, Sum ol'
Squa'es, Distribution of Sum of Squares, Estin rte and Error Sunr of Squares. Test of Linear
I{ypothesis. Basic Priucrples and Planning of iixperiments, Experiments with Single F:rctorANOVA;Analysrs of Fi.red Effects Model, Mo< :l Adequacy Checking, Choice of Sample Sizc.
ANOVA Regression Approach. Non parametric rethod in analysis of variance.
Unit- 2: Colnplete Block Desigrts. Cornpletely [ .rndomized Design, Randomized Block Dcsrgn,
Lalin Square Design, Greaco Latin Square Desig. . AnalysiS with Missing Values, ANCOVA.
Unit- 3: Incomplete Block Dcsigns-BIBD, Re overing of Intra Block Information in BIBD.
Construction of BIBD, PBIBD, Youden Squarc, ,attice Desigrr.
Unit- 4: Factorial Designs-Basic Definitions nd Principles, Two Factor Fatctorial
DesignGertcral Factorial Dcsign, 2k Factorial Design-C, ntbuncling and Partial Confbunding, Two Level
Fractional Factorial. Split Plot Design.
Books for study
l) Joshi D.D. (l9tt7) Linear Estirnation and
New Delhi
2t
Montsomery'
Sons-Ncw Yolk
l)C.
f
:sign of Expcrirnents. Wiley B,astern l-td.-
(2001) Design ancl Anal sis of Experiments.
-5th
edition. John Wilcy &
References
I ) Das M.N. & Giri N.S. (2002) Dcsign and
Intcrnational (P) Ltd.. New Delhi.
2) Angola Deart &
r nalysis of Experiments. 2'h cdition , New
Age
Danicl Voss (1999) Design md Analysis of Experirnents. Sprin,eer-Verlag,
Ncw York.
tt
a
S
I2C10: Statistical Conr; uting-I
(Practical )ourse,)
(2 credits)
'feaclrirrg schemc: 6 ho rs practical per week"
Statistical Contputing-I is a practical cour: :. The practical is based on the following
FIVE courses of the first and second semesters.
I. STIC0-5: Distribution Theory
J. .5'l 2CU6: Esttrnation'l'heory
3. Sl'2C07: Sampling Theory
I ST2C08: Regression Met"hods
t, -+T2€09j,De-siqn ancl-Analysis of Expenmer s
Practical is to be done using R progranrn ng
I R soltware. At least five statistical
data
oriented/supported problerns should be donel i'om cach course. Practical Record shall bc
maintained bv eaclt student and the sarne sha lL-rc submittecl lbr veritlcation at the time ol'
e,xternal examination. Students arc expected
packages
tc ercquire working knowlcdge of the statistical
- SPSS and SAS.
The Board of Examiners (BoE) shall clecidr the pattem of question paper ancl the cluration
o[ lhc external exatninalion. The external exal inal-iolt at each centre slrall be conclucted ancl
e','aluated on the sante day.iointly by two exilm rilers
-
one external and one internal, appointed
at the centre of the examination by thc Unive ;ity on the recomrnendation of the Chairrran.
BoE 'I'hc questi<ttt paper for the external exan: nirtion aL the centre will be set lry the extern:rl
cxiuttittct'in consultatiorr with the Chairman, EIE and the H/Ds of the centrc. 'fhe questions
distributed over the entirc sylla ,us. Evaluation shall be done by assessing each
candidate on thc sctentific artcl experirncnLal rr<ills, the efficiency of tlre algclrithrn/pr'ogranl
inr;tlelttented. the prescntatiorr and inter'ltretaticl
the dircct grading systent und gracle-s
will
be
fin
ts
,ol'the results. The valuation shall
lizecl ou the sarne day.
be done bv
Branch:
ST2C06: ESTIMATION THEORY
Tirnc: 3 hrs,
Max. Weightage: 36
Part A
(Answer ALL questions,
Weightage I for each question.)
I.
Dellne sutficie
2. State factorizat
3. What is meanr
exampte
unbiased for 02 ?
ator
2
of distributions
xample
r f'rom a given class of consistent estimators
estimation
N(0,0). Find rhe MLE ot' d.
) Shortest confidence interval
Part B
(Answer any EIGHT questions.
weightage 2 for each question.)
l3' Define minirnal sufficient statistic.
Based on a rand.nr sample t'rom
thc distribution
t
.l'Lr,0)=
(r(oo. Obtain aminimal sufficientstatistic
t t + 1x-61.'-m
for 0.
l4' Let x,,x r,..-,x,, be a random sampre
f --
from
e.
u(0,0),e>0.
Find acornprete statistic: fbr
15.
I-et X ,, X, be i.i.d b(1,O)random variables
and let V@) = 0(l _ e).Ohrain rhe cliiss
all unbiased estimateors of ttr(O).
l6'
Based on a random sample of size
n from Poisson distribution with nreim
MVUE of
e-'e
.f
d, ,btain
l7' State and prove invariance property of
consistent estimators.
l8' Examine the consistency oi samplJ mean
as an estimato r of o in the case ,f
Cauchy
distribution .l (x.01=
oo
+
/t a-+t+lx_a)- ,_ < ,r < oo.
l9' Let
xt'x"",xu
bea random samplefronranexponential
distribution with mea, L
0
Use method of principles to fincl a
consistent estinrator for 0.
Jr
.
q
a
20.l,et X,- X, ,...,X ,
be a random sample from gamma distribution
l'(.t,a,/i)='#o-fltra-t,0<.r<oo.EstimatetheparametersaandBbythemethodof
)a
moments.
2l . Obtain the MLE of parameter involved in one parzrmeter exponential lamily of
distrihutions.
22. Find the Bayes estimator of the parameter p in B(n, p) if the prior distribution ol'
bcta distribution B, (a,b).
p
is
23. Obtain shortest expected length confidence interval tbr the parameterd.
24. Briet'l-y explain Fiducial intervals.
Part C
(Answer any TWO questions. Weightage 4 for each question.)
25. ar Statc and pr(rve Basu's theorenr
b) Statc and prove a necessary and sufficient condition fbr an estimator to be MVUE,.
26. a) State and prove Lehnrann-Scheffe theorem.
b) If Xl, X,,..., X ,,'dte i.i.d random variables with p.d.f f (x,0)= s-t: i\ , x> 0.0e R.
Show that the class of linear unbiased estimators
2-t.lf
X
t,X t....,X,,is arandom sample from
;f, I,
x,- F
)'
are both consistent
of d is non-empty.
N(p,o').
show
ttrat1i.1.y, - X-)r
nA'tt
fbr o2 . which one of the two
and
rs prel'erable I
28. State the Cramer regularity conditions. Show that with probability approaching one
,? -+ 6. the likelihood equation admits a consistent solution.
as
l2
V
)
MODEL QUESTION PAPER
SECOND SEMESTER M.Sc. DEGREE EXAMINATION(CSSr. 20tt)
Branch: Statistics
ST2C07: Sampling Theory
Time:3
hours
Maximunr weightage:
Part A
Answer ALL questions. Weightage
l
.36
for each question
l.
Distinguish between sampling and census.
2. Explain non probabitity sampling.
3. [:xplain Linear systematic sample.
4. Explain plopotlional allocation in stratified sampling.
5. What do you nlean by auxiliary variablc?
6. Explain Iinear regression estimator.
7 . Explain Lahiri's method of sample selection in pps sampling.
ti. Define nPS sampling.
9. What is nreant by Zen-Midzuno scheme of sampling?
10. Distinguish between unit and clcntent of a population.
I I . Distinguish between cluster and stratunr
12. What is double sampling'? Explain.
Part B
Answcr any EIGHT questions. Weightage 2 for each question.
I3. What are the principal steps in a statistical investigation? Explain.
i4. Explain the method of estimating the proportion of units possessing arr attribute irr
population of N units, using SRSWOR.
15. lllustrate the method of selecting a circular systematic sample of size rr, using an
a f initc
exanrple.
l(r. Point out the advantages of stratified random sampling over simple random sarnpiing.
17. In ratio estimation, with usual notation, show that
irrir., ;n_i
nl
I
II
trv
nliri,
lonl
< tt, o./' i
!he ccu:-trt'ction crf Hartlel,-R-oss unbissed ratic t1,pe estirnatc:'.
19. Show that, in varying probability sampling without replacenrent, probability that
lny
specified unit is selected at the tirst draw is. in general, diltbrent frorn that ol'selecting at
any subsequent draws.
20. Dellne IJurwitz-Thompsorr estirnator tbr p<)pulation total in varying prohahility sampling.
Show thzrt this estimator is unbiased.
l3t
V
o
2l. 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 ol cluster sampling?
23. Disiinguish [,etwecn rrtuitisiage satlplirg arrd rrrultiphasc sanrpling.
24. What is non sampling error? Explain the major sources of non sampling errors.
Part C
Answer any TWO questions. Weightage 4 for each question.
25. (a) Show that. in simple random sampling without replacentent, E(s2) = 52.
(b) Discuss the any one method of estimating sampling variance of systematic sample
mcan.
26.
If terms in
V,,t
IN/,
are neglected, with usual notations show that
t )-V nnr, 2V,rurl
'
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. Expiain sub sampling wirh an example. Show, with usuai notations, that in sub salnpling
with equal firut stage units. the sample mean y2 is unbiased estimate of Y with sampling
variance. v(.vu ) =
(;-*)rr, .:(r,
+)s,",
l4l
V
q
1
MODEL QUESTION PAPER
SECoNDSEMESIBRM.SCDEGREEEXAMINATIoN(CSS)'2010
Branch: Statistics
ST2COB: REGRESSION
Max. Weightage: 36
Time: 3 hrs.
l.
2.
3.
ANALYSS
Part A
(Answer Al,l.-questions, Weightage
l for each question')
Det'ine R2 and adjusted R2
(o2 ) of the error term in the simple linear
I.-incl an unbiased'estimate of the variance
regression model.
()f the
mean resPonse at the point x= x(|
Obtain the 100( | -u)c/o confidence interval of the
.nnede!y=Pi;+PlY.+e
4. What ii meant by prediction interval?
5. What are the maioi assumptions on the regression models?
6. Define PRESS statistic.
7. What is meant bY sPline function?
tt. Define Mallows's Cp statistic''
g. what are tne maioiconsequences of incorrect model specification
10. What is meant bY logit model?
I l. Define odds ratio.
t2. Dellne link function.
part B
(AnsweranyEIGIITquestions.Weightage2-foreachquestion.)
of a regression model?
the properties of the least square estinates
13. What are
test the signiticance of tho regression'
Explain the analysis of variance "pd;n to
the regression coefficients
l-5. Obtain the confiience interval for
16. Explain the methods fbr scaling residuals'
transformations on variubles'
lT. Discuss on" unotr,r.al procedtie for selecting
I8. Explain the Generalized least square method'
regression model'
19. Write short note on Piecewise linear
re;ression methods'
20. Briefly explain various non parametric
variable selection'
21. Discuss stepwise regression method for
22. Give a short note on GLM'
of the logistic regression model'
23. Discuss the tests on the individual coefficients
24. Give a bdef account on Poisson Regression'
Part C
14.
(AnsweranyTWOquestions.Weightage4tbreachquestion.)
puru*"t*t and show that these estimators are
29. Obtain the MLE',s of the regression
identical to the least square estimates'
of the basic
methods t'or diagnosing violations
30. Explain various residuat analysis
assurnptions'
tegression
evaluate subset regressrion models.
31. Discuss various methods used to
estimates
interpret the parameters ancl obtain the
32. Define the logistic regression model,
the Parameters.
lsl
o1
V
lo
\JA
MODEL OI.]ESTION PAPER
SEL-OND SEMESTER M.Sc. DEGREE EXAMINATION (CSSi. 20 I 0
Branch: Statistics
S1'2C(X): Design and Analysis of Experiments
Tinre:
3hrs
N,Iaximr.lm Wnioht:roe. V[
Part A
Answer ALL the questions, Weightage I for each question.
1. Define the standard Gauss-Markov
2. Mention
set up.
the guideline fbr designing expenments.
3. Define f*.tLol-Wallis
4. Distinguish
test statistic.
herween ANOVA and ANCOVA.
5. What is a Latin
Square design?
6. What dt) vou mean by missing plot technique?
7.
State the paranretric relations
8. Define Lattice
design.
9. Define a partially balanced
10. Give the advantages
I
l. Distinguish
12..
of BIBD.
incomplete block design with two associate classes.
of tactorial design.
between total and partial confounding.
Define fractional tactorial.
Part B
Answer any EIGHT questions, Weightage 2 for each question.
13. Explain Randomization, Replication and Local control.
14. L,et
ei, i=
Yt= 0r+ er,
Y: = 0z+ €.r, where 0r,02 flre unknown Pararnelers an<J
and normally distributed with 0 means anrl comnron variancc crl.
Yu = 0r+ 0:+ c3 aDd
l. 2.3 are independentlv
What is the best estimate of e I
'1.
t6t
V
TE
\-/
.t4
l-5. Explarn model adequaiy checking.
16. Derive the expression for the expected value
17.
of the mean squares in RBD'
If a single observation is missing in a GLSD, estimate the missing value'
18. Defrne efticiency
of designs. Explain how efficiency of LSD relative to RtsD
is meiusured'
19. Lay clown the procedure fbr analysis of Youden square design'
20. In a BIBD. show that b > v + r - k, Also prove Fisher's inequality.
21. Derivc' intrablock artalysis of BIBD.
ZZ.Define the terms main effects and interactions of a factorial design. Explain the analysis of
2r lactorial experiment using Yales method and prepare its ANOVA.
a
23. Obtain the confounded arrangement of 2s experiment with factors A. B, C. D, and E in whit:h
AB, BCD and ACE are confbunded. Determine inhablock subgroup. If there are r similar
replications of the experiment, outline the analysis of experiment.
t/z (27) design in blocks
24. Explain the blocking procedure for the fractional f actorial
of
2r
units.
Part C
Answer any TWO questions. Weightage 4 for each question'
25. a) Define elementary contrast of a linear model (Y, A0, o2I ). If all elementary contrast
(Y, A0, o'[; u." estimable, what is your interpretation of the mcdel.
of
b) In a linear model (Y, A0, o'I ;, A is given to be of order n x /< and of rank r <k. Then derive
(A0, ('2 i).
tl-.c disti-ibulior, i;,".hc staiisiic T= i-riin (Y-A0)' (Y-A0) uttrjer tlie assutttpiioii ti,at )'-i'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 rtf trrder k.
?7. a) Detine BIBD. Give example plans of BIBD and PBIBD.
b)
Construct a BIBD with v=16 b=20 ft={
1=-5 and
}=l
28. Detine split plot design with an example. Give the complete analysis of split plot design
t7l
tY
tL
'$.
i