LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

M.Sc. DEGREE  EXAMINATION  – STATISTICS

FIRST SEMESTER  – APRIL 2003

ST  1802/ S  717   SAMPLING  THEORY

08.04.2003

1.00 – 4.00                                                                                             Max: 100 Marks

SECTION – A                                (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries two marks

1. Let the sampling design be

 

               If N=3 then what is the value of p₆₈?

 

2. Given a fixed size sampling design yielding sample of size 5, what is the value

of ?

3. Under what condition the mean square error of an estimator becomes its variance?
4. List all possible balanced systematic samples of size 4 when N = 12.
5. Under usual notations order V_SRS, V_SYS,V_STR assuming the presence of linear trend.
6. Name any two methods of PPS selection.
7. When N=16 and n = 4, what will be your choice for random group sizes in random group method? Give reason.
8. Define ratio estimator for the population total.
9. Name any two randomised response techniques.
10. Explain the term: Optimum allocation.

 

SECTION  – B                                              (5 ´ 8 = 40 Marks)

Answer any FIVE.  Each carries eight marks.

11. Show that under SRS,

 

where

12. Explain any one method of PPS selection in detail with a supportive example.
13. Show that under balanced systematic sampling, the expansion estimator coincides with the population total in the presence of linear trend.
14. Derive the mean square error of and obtain the condition under which is more

efficient than .

 

 

 

 

 

15. Explain the usefulness of two phase sampling in pps sampling.
16. Describe in detail any one method of Randomised Response technique.
17. Derive under Neyman allocation.
18. Verify the following relations with an example

 

(Proof should not be given)

 

Section – C                                    (2 ´ 20 =20 Marks)

Answer any TWO questions.  Each carries twenty marks.

 

19. Describe random group method. Suggest an unbiased estimator for the

population total and derive its variance.

20. Derive the first and second order inclusion probabilities in Midzeno sampling and

show that the Yates-Grundy estimator is nonnegative

21. Develop Yates-corrected estimator.
22. (a)  Describe how double sampling is employed in ratio estimation              (10)

 

(b)  Write a descriptive note on two stage sampling.                                     (10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1802/S717 – SAMPLING THEORY

08.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION-A

 

Answer ALL  the questions.                                                                             (10×2=20 marks)

 

1. Explain probability sampling design.
2. Given N = 5, n = 3, X₂ = 2, X₃ = 3, X = 25. Compute p₂₃ under Midzuno Sampling design.
3. Distinguish between inclusion probabilities and inclusion indicators.
4. List all possible Balanced Systematic Samples when N = 30 and n = 6.
5. Define Des-Raj ordered estimator.
6. Define Horvitz-Thompson estimator.
7. Write a short note on Yates corrected estimator under Linear systematic sampling.
8. Describe two phase sampling.
9. When is stratified sampling used?
10. Define proportional allocation.

 

SECTION-B

 

Answer any FIVE questions.                                                                           (5×8=40 marks)

 

11. Derive variance of Horvitz-Thompson estimator in Yates-Grundy form.
12. Explain Lahiri’s method and show that Lahiri’s method of selection is a probability proportional to size selection method.
13. Write a note on Warner’s model.
14. Explain ratio estimator, also derive the approximate bias and mean square error of the estimator.
15. Compare Linear systematic sampling and simple random sampling in the presence of a linear trend.
16. Develop Hartly-Ross unbiased ratio type estimator.
17. Derive variances and covariances in the two cases of two phase sampling, assuming simple random sampling is used in both the phases of sampling.
18. Describe Two-stage sampling. Give the unbiased estimator and also derive the variance of the unbiased estimator.

 

 

 

 

 

 

 

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

 

19. a) Derive V() for n = 2.            (15)
20. b) For any fixed size sampling design yielding samples of size n, prove that

(5)

20. a) Describe Midzuno sampling design and show that it is a sampling design. (5)
21. b) Derive the first and second order inclusion probabilities under Midzuno sampling

design.                                                                                                                          (15)

21. a) Develop Yates corrected estimator under linear systematic sampling. (10)
22. b) Suppose from a sample of n units selected with simple random sampling (SRS) a

subsample of n’ units is selected with SRS duplicated and added to the original

sample. Derive the expected value and the approximate sampling variance of , the

sample mean based on the n+n’ units.                                                                         (10)

22. a) Write a note on proportional allocation for a given cost. Also deduct V  under it

assuming SRS is used in all strata.                                                                             (10)

1. b) A sampler has two strata with relative sizes and . He believes that

S₁, S₂ can be taken as equal.  For a given cost C = C₁ n₁ + C₂ n₂,  show that (assuming

N_h is large).

(10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2004

ST 1805/1802 – SAMPLING THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

Answer ALL the questions                                                                            (10 ´ 2 = 20 marks)

 

1. Define probability Sampling Design and explain the meaning of probability Sampling.
2. Distinguish between varying size sampling design and fixed size sampling design. Give an example for each design.
3. Define the following:
4. a) Inclusion indicator
5. First and Second order inclusion probabilities.
6. Prove the following:
7. E_p [I_i (s)] = p_i ; i = 1, 2, …, N
8. E_p [I_i (s) I_j (s)] = p_ij ; i, j = 1,2, …, N ; i ¹
9. Show that an unbiased estimator for the population total can be found if an only if the first order inclusion probabilities are positive for all N units in the population.
10. Derive the formula for p_i and p_ij under Simple Random Sampling Design.
11. Describe the Linear Systematic Sampling Scheme and write its probability sampling design.
12. Derive the approximate bias of the ratio estimator for the population total Y.
13. In cluster sampling, suggest an unbiased estimator for the population total. Write the variance of the unbiased estimator.
14. Explain Multistage Sampling.

 

SECTION – B

 

Answer any FIVE questions.                                                                          (5 ´ 8 = 40 marks)

 

11. Show that an estimator can be unbiased under one design but biased under another design.

 

12. For any sampling design, prove the following:

 

13. Suggest an unit drawing mechanism for simple random sampling design and prove that the unit drawing mechanism implements the simple random sampling design.

 

14. Explain Lahiri’s method of PPS sampling. Show that Lahiri’s method of selection is a PPS selection method.

 

15. Write the reason for using Desraj ordered estimator instead of Horwitz – Thompson estimator under PPSWOR sampling scheme. Also prove that the Desraj ordered estimator is unbiased for the population total.
16. Describe the Random Group Method of Sampling. Find an unbiased estimator of population total under this method and derive its variance.
17. Compare V () and V () assuming the population values Y_i satisfy

Y_i = a + bi, i = 1, 2, …N.

[

18. Explain Warner’s randomized response technique for estimating the proportion p_A of the persons belonging to group A in a population.

 

SECTION – C

 

Answer any TWO questions                                                                          (2 ´ 20 = 40 marks)

 

19. a) Under any design P(×), derive the variance of Horwitz – Thompson estimator and find

its estimated variance.                                                                                                (16)

1. b) Define Midzuno Sampling Design and state the unit drawing mechanism for this

design.                                                                                                                          (4)

 

20. After the decision to take a simple random sample has been made, it was realized that the value of unit with level 1 would be unusually low and the value of unit with label N would be unusually high. In such cases, it is decided to use the estimator.

 

 

 

 

 

where C is a pre-determined constant.  Show that

• is unbiased for  for any C.
• Derive V ()
• Find the value of C for which is more efficient than .

 

21. a) Show that Linear Regression estimator is more efficient than Ratio estimator

unless b = R.                                                                                                                (4)

 

1. b) Assuming samples are drawn using SRS in both the phases of double sampling,

suggest , and  when

• the second phase sample is a subsample of the first phase sample.
• the second phase sample is independent of the first phase sample. (16)

 

22. a) In stratified sampling, deduct , V() and () assuming

• SRS is used in all strata
• PPSWR sampling is used in all strata.            (12)

 

1. b) Obtain the variance of the following:

• Hansen – Horwitz estimator in Double Sampling.
• Estimator in Two – stage Sampling.                                            (8)

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                          ST 2810 – SAMPLING THEORY

 

 

Date & Time : 24-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section A  (10 x 2 =20)

 

Answer ALL the questions. Each carries TWO marks.

1. Define : Midzuno sampling design
2. State the identity which relates sample size of a sampling design with its first order inclusion probabilities
3. Give the formula for unbiased estimator of under Warner’s RR
4. Define balanced systematic sampling
5. Mention the situations in which product and ratio estimators can be used instead of .
6. When do you recommend “Two phase sampling”?
7. Name an estimator which uses selection probabilities.
8. Give any two limitations of “Linear Systematic Sampling “
9. Name any one sampling-estimating strategy in which no unbiased estimator for variance of estimator can be found.
10. Write the variance of Yates corrected estimator under LSS when there is a linear trend in the population

Section B (5 x 8 = 40)

 

Answer any Five. Each carries Eight  marks.

11. Prove the following identities : and verify the same in the case of following sampling design

 

 

12. From a population containing N units a sample of n units is drawn using SRS and from the drawn sample a subsample of n’ units. Suggest an unbiased estimator for the population total based on the subsample and obtain its variance
13. Describe modified systematic sampling and show that under the model
14. Describe Desraj ordered estimator and obtain an unbiased estimator of

 

15. Explain proportional allocations (1) for a given cost (2) for a given sample size. Derive the variance of under the above cases assuming simple random sampling is used in all strata.

 

16. Explain Warner’s randomized response model in detail.

 

17. Define product estimator . Obtain an expression for its bias under simple random sampling and hence develop an unbiased estimator for the population total.
18. Derive the approximate mean square error of estimators in the class also obtain the minimum mean square error in the class.

 

Section C  (2 x 20 =40)

 

Answer any TWO. Each carries TWENTY marks

 

19. Define : Horvitz-Thompson estimator. Show that it is unbiased for the population total and derive its variance in Yates-Grundy form

 

20. Derive the first and second order inclusion probabilities under Midzuno sampling scheme and show that under this design the Yates-Grundy estimator is non-negative

 

21. Develop Yates corrected estimator under Linear Systematic Samping

 

22. Develop Hartley-Ross ratio type unbiased estimator under simple random sampling.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2007

ST 2810 – SAMPLING THEORY

 

 

 

Date & Time: 21/04/2007 / 1:00 – 4:00Dept. No.                                              Max. : 100 Marks

 

 

SECTION – A

——————-

Answer ALL questions                                                     ( 10 x 2 = 20 marks)

1. Define Probability Sampling Design and mention its two types.
2. Give an example for a statistic which is unbiased under a

sampling design.

3. Define ( i )   Inclusion indicator.

( ii ) First order inclusion probability.

4. For any sampling design, find mean and variance of   I _i (s).
5. Prove that an unbiased estimator for the population total can be found iff the first order inclusion probabilities are positive for all the units in

the population.

6. Prove that E _p ( s)  =  S under Simple Random Sampling Design.
7. Define Midzuno Sampling Design. Verify whether or not this design is a probability sampling design.
8. Describe Random Group Method for selecting a sample and write the estimator for population total under this method.
9. List all possible modified systematic samples of size 8 when the population size is 40.
10. Show that _LR is more efficient than _R  unless  β = R.

 

 

 SECTION – B

          ——————-

Answer any FIVE  questions                                            ( 5 x 8 = 40 marks)

 

11. Show that the property of unbiasedness is design dependent.
12. Derive variance of Horwitz – Thompson estimator for population total under any design P .
13. Write the unit drawing mechanism for implementing Simple Random Sampling Design and show that this mechanism implements the design.
14. Show that Lahiri’s method of selection is a PPS selection method.
15. Show that v ( _HT   ) is non-negative under MSD for all “s” receiving positive probabilities.

 

16. Derive V ( _DR   ) for n = 2.
17. Show that the usual expansion estimator is unbiased for the population total in CSS , when there is a linear trend in the population.
18. Derive the approximate Bias and Mean Square Error of the

estimator _R..

 

 SECTION – C

          ——————-

Answer any TWO questions                                            ( 2 x 20 = 40 marks)

 

19. ( a ) Derive  _HT  and  V ( _HT   ) using the formula for  Π _i

and  Π _{i j}    under SRS Design.                                 ( 10 )

         ( b ) Suppose from a sample of n units selected using SRS,  a

sub-sample of   n’   units is selected using SRS and included in

the original sample. Derive the expected value and the

approximate  sampling variance of  ‘ ,  the sample mean based

on  ( n + n’ ) units.                                                   ( 10 )

20. ( a ) Obtain Π _i  and  Π _{i j}    under MSD.                         ( 10 )

( b ) Derive estimated variance of _DR.                                        ( 10 )

21.( a ) Describe Warner’s randomized response technique and explain

the procedure for estimating the proportion Π_A .     ( 10 )

( b ) Deduct  _st  ,V ( _st   )  and  v ( _st   ) under

( i ) SRS    and

( ii )  PPSWR designs.                              ( 10 )

22 ( a ) Derive the formula for n _h  under cost optimum allocation.

( 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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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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      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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2813 – SAMPLING THEORY

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Part A

Answer  all the Questions:                                                                                                     (10 X 2 = 20)

 

1. Define mean square error of an estimator T. When does it reduce to variance?
2. Define first and second order inclusion probabilities.
3. Suggest an unbiased estimator for population proportion under SRSWOR.
4. Explain modified systematic sampling.
5. When ratio estimator is better than the expansion estimator?
6. Write the formula for n_h under Neyman allocation.
7. Explain cumulative total method.
8. What is the need for regression estimator?
9. Let V denote the distinct units drawn in SRSWR. Suggest an unbiased estimator for population mean and  write the variance based on V- distinct units.
10. Show that is unbiased for y for the populations with linear trend when k is odd.

 

 

Part B

Answer any Five questions:                                                                                                  (5 X 8 = 40)

 

11. Obtain V[I_i(s)] , Cov [I_i(s), I_j(s)]
12. Explain Midzuno’s scheme. Specify a method to draw a sample using Midzuno’s scheme and show that it actually implements the scheme.
13. Obtain the unbiased estimator and its variance for the population total when SRSWOR is used in both the stages of the two-Stage sampling method.
14. Obtain the bias and mean square error of the regression estimator.
15. Explain Warner’s method of randomized response method.
16. Suggest an unbiased estimator for the population total when PPSWR is used in all the strata. Obtain the variance of the estimator and an unbiased estimator of the variance.
17. Show that unbiasedness depends on the sampling design.
18. Explain the need for circular systematic sampling and the problems involved.

 

 

 

 

 

 

Part C

Answer any two Questions:                                                                                                 (2 X 20 = 40)

 

19. a)  Show that Horvitz – Thompson is unbiased for the population total. Obtain the variance of the estimator in the Yates- Grundy form.
20. Obtain the variance Of  .

(12 + 8)

20. a) Show that

V_ran ≥ V_prop ≥ V_opt

1. Explain Balanced systematic sampling. Show that is unbiased and write the variance of the estimator

(12 + 8)

21. a) Show that Desraj estimator is unbiased in PPSWOR and obtain its variance.
22. Derive Murthy’s estimator when n=2.

(12 + 8)

22. a) Obtain the bias of the Jackknife ratio estimator.
23. Obtain the bias and mean square error of the combined ratio estimator and separate ratio estimator in stratified random sampling. (12 + 8)

 

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