LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 38

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)

 

1. Define a parameter and a statistic.  Give an example for both.
2. Give an example for an estimator which is unbiased under a sampling design.
3. 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 .

1. 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.
2. Prove that E _p (  s _y ­²  )  =  S _y ²    under  SRSWOR  Design.
3. Define Midzuno Sampling Design.  Verify whether or not this design is a probability sampling design.
4. Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
5. List all possible Modified Systematic Samples of size 8 when the population size is 40.
6. Check whether _LR is more efficient than   _R .
7. 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)

     

1. Write the unit drawing mechanism for implementing SRSWOR Design and show that this mechanism implements the design.

 

1. 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.

 

1. Check whether or not _LSS is more efficient  than _SRS for population with linear trend.

 

1. Show that the usual expansion estimator is unbiased for the population total in CSS when there is a linear trend in the population.
2. Check whether the estimated variance v( _HT  ) is  non-negative under MSD for all “ s ” receiving positive probabilities.

 

1. Explain Simmon’s unrelated randomized response model and obtain the estimate of Π_A when Π_Y is unknown.

 

1. Derive the estimated variance of _DR.
2. 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 )

21. 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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