LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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SECOND SEMESTER – April 2009
ST 2813 / 2810 – SAMPLING THEORY
Date & Time: 24/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION – A
Answer ALL questions. Each carries TWO marks. (10 x 2 = 20 marks)
- Define a parameter and a statistic. Give an example for both.
- Give an example for an estimator which is unbiased under a sampling design.
- Show that
(i) E p [ I i (s) ] = Π i ; i = 1, 2, …, N,
(ii) E p [ I i (s) I j (s)] = Π ij ; i , j = 1, 2, …, N ; i ≠ j .
- Prove that an unbiased estimator for the population total can be found if and only if the first order inclusion probabilities are positive for all N units in the population.
- Prove that E p ( s y 2 ) = S y 2 under SRSWOR Design.
- Define Midzuno Sampling Design. Verify whether or not this design is a probability sampling design.
- Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
- List all possible Modified Systematic Samples of size 8 when the population size is 40.
- Check whether LR is more efficient than R .
- Prove that the Desraj ordered estimator is unbiased for the population total.
SECTION – B
Answer any FIVE questions. Each carries EIGHT marks. (5 x 8 = 40 marks)
- Write the unit drawing mechanism for implementing SRSWOR Design and show that this mechanism implements the design.
- If T( s, s′ ) is a statistic based on the sets s and s′ which are samples drawn in the first phase of randomization and the second phase of randomization respectively, then prove that
V( T( s, s′ ) ) = E1 V2 ( T( s, s′ ) ) + V1 E2 ( T( s, s′ ) ) ,
where E2 is the expectation taken after fixing the subset s and E1 is the
expectation with respect to the randomization involved in the first phase.
- Check whether or not LSS is more efficient than SRS for population with linear trend.
- Show that the usual expansion estimator is unbiased for the population total in CSS when there is a linear trend in the population.
- Check whether the estimated variance v( HT ) is non-negative under MSD for all “ s ” receiving positive probabilities.
- Explain Simmon’s unrelated randomized response model and obtain the estimate of ΠA when ΠY is unknown.
- Derive the estimated variance of DR.
- Derive the formula for n h under Cost Optimum Allocation.
SECTION – C
Answer any TWO questions. Each carries TWENTY Marks (2 x 20 = 40 marks)
19 ( a ) Illustrate that an estimator can be unbiased under one design but biased under
another design. ( 10 )
( b ) Derive HT and V (HT ) using the formula for Π i and Π ij under SRSWOR
Design. ( 10 )
20 ( a ) Describe Warner’s randomized response technique and explain the procedure
For estimating the proportion Π A . ( 10 )
( b ) Deduce the expressions for St , V (St ) and v (St ) when samples are
drawn independently from different strata using ( i ) SRSWOR, and
( ii ) PPSWR Designs. ( 10 )
- Find the expressions for the approximate bias and MSE of the estimator R
and deduce their expressions under ( i ) SRSWOR, (ii) PPSWOR, and ( iii ) Midzuno Sampling Designs. ( 20 )
22 ( a ) Verify whether or not the Hansen-Hurwitz estimator dhh under double
sampling is unbiased for Y and derive its variance. ( 10 )
( b ) Find the mean and variance of TS , the estimator for population total, under
Two – Stage Sampling with SRS in both stages. ( 10 )
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