LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 37

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2008

    ST 2810 – SAMPLING THEORY

 

 

 

Date : 24/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A

 

Answer  ALL  questions.  Each  carries TWO  marks.     (10 x 2 =  20 marks)

 

1. Define Population. I _i (s)   What are the assumptions made about population size?
2. Distinguish between parameter and statistic.  Give an example for both.
3. Find the following:

(i)   E _p­ [ I _i (s) ] ,  i  =  1, 2, …, N,

(ii)   E _p­ [ I _i (s) E _p­ [ I _i (s) I _j (s)] ;   i , j  =  1, 2, …, N ;   i  ≠ j .

1. Show that an unbiased estimator for the population total can be found iff the first order inclusion probabilities are positive for all units in the population.
2. In  SRSWOR ,  show that  E _p­ [ s _xy  ]  =  S _xy .
3. Define Midzuno Sampling Design and show that this method is a PPS selection method.
4. Write the estimator for population total Y under Random Group Method and show that this estimator is unbiased for Y.
5. Show  that the expansion estimator is equal to the population total under Balanced Systematic Sampling, in the presence of  linear trend.
6. Derive the approximate bias of the ratio estimator for population total.
7. Show that  _LR  is more efficient  than   _R  unless  β = R .

 

SECTION – B

 

Answer  any FIVE  questions.  Each  carries EIGHT  marks.     (5 x 8 =  40 marks)

 

1. Give an example to show that an estimator can be unbiased under one design but

biased under another design.

1. Under any design P( . ),  derive the variance of  Hurwitz – Thompson estimator for population total.
2. Describe the unit drawing mechanism for simple random sampling design and prove that the mechanism implements the design.
3. 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. Show that the estimated variance  v( _HT ) is  non-negative under Midzuno        Sampling Design for all s receiving positive probabilities.
2. Show that _LSS is more efficient  than _SRS for population with linear trend.
3. Obtain Yate’s corrected estimator under LSS in the presence of linear trend to estimate population total without error.
4. Describe Simmon’s unrelated randomized  response model and obtain the estimate of  Π_A  when Π_Y is unknown.

 

SECTION – C

 

Answer  any TWO  questions.  Each  carries TWENTY  marks.     ( 2 x 20 =  40 marks)

 

19 ( a ) After the decision to take a SRS has been made, it was realized that  Y₁ the value of unit with

label 1 would be unusually low and Y_N the value of unit with label N would be unusually high.  In

such cases it is decided to use the estimator

 

if  1    s,    N     s

*   =         if   1 s,    N  s

otherwise,

where c is a pre-determined constant.  Show that  ( i )  *   is unbiased for   for any c.

( ii ) Derive  V(*  ).  ( iii ) Find the value of c for which *  is more efficient than  .      .

( 14 )

19 ( b )  State the unit drawing mechanism for Midzuno Sampling Design and show that

the mechanism implements the design.                                                                              ( 6 )

20 ( a ) Derive the estimated variance of   _DR .                                                                     ( 10 )

20 ( b ) Show that the expansion estimator is equal to the population total under

Balanced Systematic Sampling in the presence of linear trend .                                         ( 10 )

21. Derive 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.

( 20 )

22 ( a ) Show that Hansen-Hurwitz estimator _dhh  under double sampling is unbiased

for Y and derive its variance.                                                                                               ( 12 )

22 ( b ) Explain Stratified Sampling.  Deduce the expressions for   _St ,   V (_St )   and

v (_St ) when samples are drawn independently from different strata using

( i )  SRSWOR,  and  ( ii ) PPSWR.                                                                                       ( 8 )

 

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