Statistics (Paper III) 2016 Question Paper

Notes 6 Pages

Indian Statistical Service

Statistics

Contributed by

Mohanlal Sant

T H .~v / i C C ’ T p T / A s t a a ■< /■
si. n o . 0005314
A-GSE-P-TUC
STATISTICS
Paper III
71/we Allowed : Three Hours Maximum Marks : 200
INSTRUCTIONS
Please read each o f the following instructions carefully
before attempting questions.
There are SIX questions divided under TWO sections.
Candidate has to attempt FIVE questions in all.
All the THREE questions in Section A are compulsoiy.
Out o f the THREE questions in Section B,
TWO questions are to be attempted.
Attempts o f questions shall be counted in sequential
order. Unless struck o ff attempt o f a question
shall be counted even i f attempted partly.
The number o f marks carried by a question /part is
indicated against it.
Unless otherwise mentioned, symbols and notations
have their usual standard meanings.
Assume suitable data, if necessary and indicate
the same clearly.
All parts and sub-parts o f a question are to be
attempted together in the answer book.
Any page or portion o f the page left blank in the
answer book must be clearly struck o ff
Answers must be written in ENGLISH only.
Page 1
Section - A
All the three questions are compulsory.
1. (a) Define Des Raj’s ordered estimator for popula
tion mean on the basis of a sample o f size 2
and show that it is unbiased. 10
(b) Let ni and Jiy ( j ^ i) be the inclusion proba
bilities o f first and second order respectively in
a simple random sample of size n, selected
from a finite population o f size N. Then show
that
N
(i) 'Zftj =n and
1=1
N
(ii) X Tty = n{n-\). 10
0*0=1
(c) Give a practical example where two-stage
sampling scheme may be adopted. For equal
size first-stage units, obtain an estimator for
population mean in two-stage sampling and its
variance. Discuss the problem o f allocation of
first and second-stage sample sizes for a fixed
cost. 20
2. (a) Describe the problem o f multicollinearity in
general linear model and explain how will you
detect it. 10
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(Contd.)
Page 2
(b) In usual notation consider the standard linear
model :Y = XP + U; U~N{0, (J2I ). Show that
the MLE P of ft have the distribution
N[p,o2{X'X yx^ assuming X X to be an
invertible matrix. 10
(c) (i) Explain the identification problem in a
system of simultaneous equations. State,
without proof, the rank and order condi
tions for identifiability o f an equation.
10
(ii) Identify the following system :
y\ ~ 3y2 ~ +x2 +U\
y2 =y3+x3+U2
; ;3 = y \ -y 2 -2 x3 + u} io
3. (a) Construct the price index number for 2010
with 2005 as base year from the following data
by using
(i) Laspeyre’s
(ii) Paasche’s and
(iii) Fisher’s method
Item Price (in Rs.)
2010 2005
Quantity
2010 2005
A
10-50 8-25 6 4
B
6-40 6-00
10 6
C
15-20 10-80 6 5
D 6-25 4-00 8 5
Verify whether the Time Reversal Test is
satisfied by the abovementioned index
numbers. 10
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(Contd.)
Page 3
(b) Define price elasticity of demand (r|/;) and
interpret when v\ is
(i) <1
(ii) >1 and
(iii) =1
If the demand function is p — 10-5 a:2, for
what value of x elasticity of demand will be
unity ? (x is the quantity demanded and p is
the price). 10
(c) Define autocorrelation of lag K of a stationary
process. Consider the time series model
defined by
X, = alX,.l + a2X,_2 + a 3Jf,_3+e,
where fs,} is white noise.
(i) Show that the autocorrelation coefficient
with lag 1 for the process is :
a , + a 7a,
Pi=T
-------
------
2‘
1

-a 2 -a la3 -a\
(ii) Consider the case where
a l=a2=cc3= 0-2 .
Comment on the stationarity o f this
model.
You may use 5 -
x - x 2 - x3 ^
(1-278-x) (3-912 + 2-278x + x2).
Calculate p, and p2. 20
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(Contd.)
Page 4
Section - B
Attempt any two questions.
4. (a) A finite population of size 100 is divided into
two strata. In the usual notations, it is given
that N ] = 60, N2 = 40, S ] = 2S2. If a sample of
size 24 is to be selected from the population,
obtain the number of units to be selected from
each of the stratum under Neymann allocation.
10
(b) For the model y = X(3+u if X and u are
correlated show that the OLS estimator for p is
not consistent. Discuss the use of instrumental
variable technique to obtain a consistent
estimator of (3. 10
(c) Consider a time series y t = Tt + C, + I t where
T, a trend, C, a cyclical component and
I, a random component. Discuss the effect
of moving averages on cyclical and random
2tt /
components assuming Q =asin
------
20
' A
5. (a) Explain Koyck’s approach to distributed
geometric lag model. 10
(b) What is a chain index number ? Discuss its
advantages and disadvantages over fixed-base
index number. 10
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(Contd.)
Page 5
(c) In what sense cluster sampling is d iffe re n t^
from simple random sampling ? Define an
unbiased estimator for population mean in
case of cluster sampling with equal cluster
size. Compare the efficiency of cluster
sampling in terms o f intra-class correlation
coefficient with respect to simple random
sampling without replacement. 20
6. (a) You believe that a set of data is the realization
of an MA(1) process X n = en+fien_\, where
the errors en are standard normal. You have
calculated the sample auto-covariance func
tion and found that 7 0= 1 and 7 ]= -0 -2 5.
Estimate the parameter p. Which value of
p do you think you should choose and why ?
10
(b) Describe the Lahiri’s method of selecting a
probability proportional to size sample from a
finite population o f size N. 10
(c) Describe lag model and distributed lag model.
What are the different lag schemes ? How
would you estimate lags by applying ordinary
least square ? 20
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