LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5500/STA 505/S 515 – ESTIMATION THEORY

03.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

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

 

1. State the problem of point estimation.
2. Define ‘bias’ of an estimator in estimating a parametric function.
3. Define a ‘Uniformly Minimum Variance Unbiased Estimator’ (UMVUE).
4. Explain Cramer-Rao lower bound.
5. Define completeness and bounded completeness.
6. Examine if is complete.
7. Let X₁, X₂ denote a random sample of size 2 from B(1, q), 0<q<1. Show that X₁+3X₂ is sufficient of q.
8. Give an example where MLE is not unique.
9. Define BLUE
10. State Gauss – Markoff theorem.

 

SECTION-B

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

 

11. Show that the sample variance is a biased estimator of the population variance. Suggest an UBE of .
12. If T_n is asymptotically unbiased with variance approaching zero as n , show that T_n is consistent.
13. Show that UMVUE is essentially unique.
14. Show that the family of Binomial distributions is complete.
15. State and establish Lehmann – Scheffe theorem.
16. State and prove Chapman – Robbin’s inequality.
17. Give an example where MLE is not consistent.
18. Describe the linear model in the Gauss – Marboff set-up.

 

SECTION-C

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

 

19. a) Let X₁, X₂,….., X_n (n > 1) be a random sample of size n from P (q), q > 0. Show that the class of unbiased estimator of q is uncountable.
20. b) Let X₁, X₂,….., X_n denote a random sample of size n from a distribution with pdf

 

 

 

f(x) ;q) =

0              ,   other wise.

Show that X₍₁₎ is a consistent estimator of q.                                                          (10+10)

 

20. a) Obtain CRLB for estimating q, in the case of

f  based on random sample of size n.

1. b) State and establish factorization theorem in the discrete case. (8+12)
2. a) Explain the method of maximum likelihood.
3. b) Let X₁, X₂, …., X_n denote a random sample of size n from N (. Obtain MLE of q = (.                                                                                           (5+15)
4. a) Let Y = Ab + e be the linear model where E (e) = 0. Show that a necessary and sufficient condition for the linear function  of the parameters to be linearly estimable is that rank (A) = rank .
5. b) Explain Bayesian estimation procedure with an example. (10+10)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

03.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

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

 

1. Define ‘bias’ of an estimator in estimating a parametric function.
2. Explain ‘consistent estimator’
3. Describe ‘efficiency’ of an estimator.
4. Define ‘Uniformly Minimum Variance Unbiased Estimator’.
5. Define Cramer – Rao lower bound (CRLB).
6. Explain bounded completeness.
7. Define complete sufficient statistic.
8. Let X₁, X₂ denote a random sample of size 2 from B(1, q), 0< q <1. Show that X₁ + 3X₂ is sufficient for q.
9. State Chapman – Robbins inequality.
10. Give an example where MLE is not unique.

 

SECTION – B

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

 

11. Show that the sample variance is a biased estimator of the population Variance. Suggest an UBE of s².
12. State and derive Cramer – Rao inequality.
13. Let T₁ and T₂ be two unbiased estimators of a parametric function with finite variances. Obtain the best unbiased linear combination of T₁ and T₂.
14. State and establish Rao – Blackwell theorem.
15. Give an example of an UMVUE which does not take values in the range of the parametric function.
16. State and prove Bhattacharya inequality.
17. State and prove invariance property of MLE.
18. Describe the method of moments and illustrate with an example.

 

SECTION – C

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

 

19. a) If T_n is consistent for y (q) and g is continuous, show that g (T_n) is consistent for

g (y (q)).

1. b) Show that UMVUE is essentially unique. (10+10)

 

20. a) Give an example to show that bounded completeness does not imply completeness.
21. State and establish factorization theorem in the discrete case.      (10+10)

 

21. a) Explain the method of maximum likelihood.
22. b) Let X₁, X₂, …, X_n denote a random sample of size n from N (m, s²). Obtain MLE of

q = (m, s²).                                                                                                              (5+15)

 

22. a) Describe the method of minimum chi-square and method of modified minimum chi-

square.

1. b) Obtain the estimate of p based on a random sample of size n from B (1, p),0 < p <1 by the method of i) Minimum chi-square and ii) Modified Minimum Chi-Square. (10+10)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

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

 

1. State the problem of point estimation.
2. Define ‘bias’ of an estimator in estimating a parametric function.
3. Define a ‘Consistent estimator’.
4. Define ‘efficiency of an estimator.
5. Explain ‘Uniformly Minimum Variance Unbiased Estimator’.
6. What is Cramer – Rao lower bound?
7. Define ‘bounded completeness’.
8. Examine if {N (0, s²), s² > 0} is complete.
9. Let X₁, X₂ denote a random sample of size 2 from B (1, q), 0 < q < 1. Show that X₁ + 3X₂ is sufficient for q.
10. Explain BLUE.

SECTION – B

 

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

 

11. Show that the sample variance is a biased estimator of the population variance s². Suggest an UBE of s².
12. If T_n is a consistent estimator of j (q), show that there exists infinitely many consistent estimators of j (q).
13. State and derive Cramer – Rao inequality.
14. Show that UMVUE is essentially unique.
15. Give an example to show that bounded completeness does not imply completeness.
16. Show that the sample mean is a complete sufficient statistic in the case of P (q), q > 0.
17. State and establish Lehmann – Scheffe theorem.
18. State and prove ‘Invariance property’ of MLE.

 

SECTION – C

 

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

 

 

19. a) Let f (x;q) =

 

• ,  otherwise

 

Based on a random sample of size n, suggest an UBE of

1. q when s is known and
2. s when q is known          (5+5)

 

1. b) Obtain CRLB for estimating q, in the case of f(x; q) = x Î R, q Î R,

based on a random sample of size n.                                                                          (10)

20. a) State and establish factorization theorem in the discrete case.

 

1. b) Obtain a sufficient statistic for q = (m, s²) based on a random sample of size n from

N (m, s²), m Î R, s² > 0.                                                                                     (12 + 8)

 

21. a) Explain the method of maximum likelihood.

 

1. b) Let X₁, X₂, …, X_n denote a random sample of size n from N (m, s²). Obtain MLE of

q  = (m, s²).                                                                                                           (5 + 15)

 

22. a) Describe the method of minimum chi-square and the method of modified minimum

chi-square.

 

1. b) Describe the linear model in the Gauss – Markov set – up.

 

1. c) Let y = Ab + e be the linear model where E (e) = 0. Show that a necessary and

sufficient condition for the linear function  b of the parameters to be linearly

estimable is that rank (A) = rank .                                                           (10 + 6 + 4)

 

 

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            LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034                    B.Sc. DEGREE EXAMINATION – STATISTICS

AB 13

FIFTH SEMESTER – NOV 2006

ST 5500 – ESTIMATION THEORY

(Also equivalent to STA 505)

 

 

Date & Time : 25-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

 

Part A

Answer all the questions.                                                                        10 X 2 = 20

 

 

1. Define bias of an estimator in estimating the parametric function.
2. Explain efficiency of an estimator.
3. Explain Uniformly Minimum Variance Unbiased Estimator (UMVUE).
4. What is Cramer-Rao lower bound?
5. Define bounded completeness.
6. State Bhattacharyya inequality.
7. Let X_1, X₂ denote a random sample of size 2 from B (1, θ), 0 < θ < 1. Show that    X₁ + 3X₂ is sufficient for θ.
8. Describe sufficient statistic.
9. Explain Bayes estimation.
10. What is BLUE?

Part B

Answer any five questions.                                                                           5 X 8 = 40

 

1. Let X_1, X₂, … , X_n denote a random sample of size n from B(1, p), 0 < p < 1. Suggest an unbiased estimator of (i)  p and  (ii).  p (1- p).
2. If T_n asymptotically unbiased with variance approaching zero as n ® ¥ then show that T_n is consistent.
3. State and establish Factorization Theorem in the discrete case.
4. Show that the family of Bernoulli distributions { B(1,p), 0< p < 1} is complete.
5. State and establish Lehmann-Scheffe theorem.
6. Let X₁, X₂, … , X_n denote a random sample of size n from a distribution with p.d.f.                                  e^{– (x-q)}, x ³q, q Î Â

f(x; q) =     0      , otherwise.

 

Obtain UMVUE of q.

1. Give an example where MLE is not unique.
2. Explain Gauss-Markov model.

 

 

 

 

 

Part C

Answer any two questions.                                                                         2 X 20 = 40

 

1. a). Let X₁, X₂, … ,.X_n  denote a random sample of size n from P(q), q >0. Suggest

an unbiased estimator of i)  q        ii) 5q + 7.

b). If T_n is consistent estimator for and g is continuous then show that g(T_n)

is consistent for g().                                                                        (10 +10)

 

1. a). Show that UMVUE is essentially unique.

b). Obtain CRLB for estimating q in case of

1

f(x ; q) =                                   , – µ < x < µ and – µ < q < µ,

• [1 + (x – q)² ]

 

based on a random sample of size n.                                                            (10 +10)

1. a). State and establish Chapman – Robbins inequality.

b). Describe the method of moments with an illustration.                            (12 + 8)

1. a). Let X_1, X₂, … , X_n denote a random sample of size n from N (m, s²). Obtain

MLE of q = (m, s²).

b). Illustrate the method of moments with the help of G (a, p).                   (12 + 8)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

AC 16

DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5500 – ESTIMATION THEORY

 

 

Date & Time: 27/04/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

 

 

Part A

Answer all the questions.                                                                  10 X 2 = 20

1. State the problem of point estimation.
2. Define asymptotically unbiased estimator.
3. Define a consistent estimator and give an example.
4. Explain minimum variance bound estimator.
5. What is Fisher information?
6. Write a note on bounded completeness.
7. Examine if { N (0, σ²), σ² > 0 } is complete.
8. Let X₁, X₂ denote a random sample of size 2 from P(q),q > 0. Show that X₁ + 2X₂ is not sufficient for q.
9. State Chapman – Robbins inequality.
10. Explain linear estimation.

 

Part B

Answer any five  questions.                                                              5 X 8 = 40

 

 

1. Let X₁, X₂, … ,X_n denote a random sample of size n from B(1,q), 0<q<1. Show that  is an unbiased estimator of q², where T = .
2. If T_n  is consistent for Y(q) and g is continuous, show that g(T_n ) is consistent for g{Y(q)}.
3. State and establish Cramer – Rao inequality.
4. Show that the family of binomial distributions { B (n, p), 0 < p < 1, n fixed } is complete.
5. State and establish Rao – Blackwell theorem.
6. Let X₁, X₂, … , X_n denote a random sample of size n from U (0, q), q > 0.

Obtain the UMVUE of q.

1. Give an example for each of the following

1. MLE, which is not unbiased.
2. MLE, which is not sufficient.

1. Describe the method of minimum chi-square and the method of modified minimum

chi-square.

Part C

Answer any two questions.                                                                     2 X 20 = 40

 

1. a). Show that the sample variance is a biased estimator of the population variance.

Suggest an unbiased estimator of s².

b). If T_n is asymptotically unbiased with variance approaching zero as n

approaches infinity then show that T_n is consistent.                              (10 + 10)

 

 

1. a). Let X₁, X₂, … , X_n denote a random sample of size n from U (q-1, q +1).

Show that the mid – range U = is an unbiased estimator of q.

 

b). Obtain the estimator of p based on a random sample of size n from

B(1, p), 0 < p < 1by the method of

i). Minimum chi-square

ii). Modified minimum chi-square.                                                    (12 + 8)

 

1. a). Give an example to show that bounded completeness does not imply completeness.

 

b). Stat and prove invariance property of MLE.                                           (10 +10)

1. a). State and establish Bhattacharyya inequality.

b). Write short notes on Bayes estimation.                                                    (12 + 8)

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

           B.Sc. DEGREE EXAMINATION – STATISTICS

NO 24

FIFTH SEMESTER – APRIL 2008

ST 5500 – ESTIMATION THEORY

 

 

 

Date : 28-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

 

Answer ALL the questions:                                                             (10×2=20)

 

1. Define ‘bias’ of an estimator.
2. When do you say an estimator is consistent?
3. Define a sufficient statistic.
4. What do you mean by bounded completeness?
5. Describe method of moments in estimation.
6. State invariance property of maximum likelihood estimator.
7. Define Loss function and give an example.
8. Explain ‘prior distribution’ and ‘posterior distribution’.
9. Explain least square estimation.
10. Mention any two properties of least squares estimator.

 

PART-B

Answer any FIVE questions:                                                           (5×8=40)

 

1. If Tn is asymptotically unbiased with variance approaching 0 as , then show that Tn is consistent.
2. Show that is an unbiased estimate of , based on a random sample drawn from .
3. Let be a random sample of size n from  population. Examine if is complete.
4. State and prove RAo-Blackwell theorem.
5. Estimate by the method of moments in the case of Pearson’s Type III distribution with p.d.f .
6. State and establish Bhattacharya inequality.
7. Describe the method of modified minimum Chi square.
8. Write a note on Baye’s estimation.

 

PART-C

Answer any TWO questions:                                                           (10×2=20)

1. a) and is a random sample of size 3 from a population with mean value and variance .  are the estimators used to estimate mean value , where  and .

1. Are T₁ and T₂ unbiased estimators?
2. Find the value of such that T₃ a consistent estimator?

• With this value of is T₃ a consistent estimator?

1. Which is the best estimator?
2. b) If are random observations on a Bernoulli variate X taking the value 1 with probability p and the value 0 with probability (1-p), show that is a consistent estimator of p(1-p).

1. a) State and Prove cramer-Rao inequality.

1. b) Given the probability density function

Show that the Cramer-Rao-lower bound of variance of an unbiased estimator of  is 2/n, where n is the size of the random sample from this distribution. [12+8]

1. a) State and prove Lehmann – Scheffe theorem

1. b) Obtain MLE of in based on an independent sample of size n. Examine whether this estimate is sufficient for .                   [12+8]

1. a) Show that a necessary and sufficient condition for the linear parametric function to be linearly estimable is that

ank (A) = rank

where  and

1. b) Describe Gauss – Markov model [12+8]

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 10

FIFTH SEMESTER – November 2008

ST 5500 – ESTIMATION THEORY

 

 

 

Date : 03-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer ALL questions.                                                                              10 X 2 = 20

1. Define Consistent estimator. Give an example.
2. Give two examples for unbiased estimator.
3. Define UMVUE.
4. Describe the concept of Bounded completeness.
5. Describe the Method of Minimum Chi-square estimation.
6. State an MLE ofλ based on a random sample of size n form a Poisson Distribution

with parameter λ.

1. Describe the concept of Baye’s estimation.
2. Define Loss function.
3. Describe the Method of Least Squares.
4. Define BLUE.

SECTION – B

Answer Any FIVE questions.                                                                                       5 X 8 = 40

11. Derive an unbiased estimator of , based on a random sample of size n form B (1,).
12. Let { Tn = 1, 2,3, ….. } be a sequence of estimators such that

and  .Then show that T_n is

consistent for .

13. If is a random sample from P (λ),, then show that

is a sufficient statistic for .

14. Show that the family of Binomial distributions {B (1,).0 < θ < 1} is complete.
15. Describe estimation of parameters by “Method of Maximum Likelihood”
16. Describe any two properties of MLE, with examples.
17. Explain prior and posterior distributions.
18. Derive the least square estimator of β₁ under the model Y = β₀ + β₁X+

 

SECTION – C

Answer any TWO questions.                                                                          2 X 20 = 40

19. a. State and prove Chapman-Robbin’s inequality. [12]

b Using Factorization theorem derive a sufficient statistic for μ based on a random

sample of size n from N (μ, 1), MϵR                                                                      [8]

 

20. a. State and prove a necessary and sufficient condition for an unbiased estimator to be a

UMVUE.                                                                                                                      [15]

1. If T₁ and T₂ are UMVUES of y₁(q) and  y₂(q) respectively, then show that T₁+T₂ is the UMVUE of  y₁(q) and  y₂(q).                                                                                                                [5]

 

 

21 a.  Explain the concept of estimation by the method of modified minimum chi-square.  [8]

1. Let be a random sample from a distribution with density function

f (x, θ) =                                                            [12]

Find the maximum likelihood estimator of  and examine whether it is consistent.

 

22. Explain: i) Risk function.              ii) Method of Moments

 

iii) Completeness     iv). Gauss –Markov model                           [ 4 x 5 ]

 

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           LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 24

FIFTH SEMESTER – April 2009

ST 5500 – ESTIMATION THEORY

 

 

 

Date & Time: 16/04/2009 / 9:00 – 12:00      Dept. No.                                                Max. : 100 Marks

 

 

PART A                                Answer all the questions                             [10×2=20]

 

1. Define the efficiency of an unbiased estimator. Give an example of a most

efficient estimator.

2. State the invariance property of consistent estimators..
3. State Factorization Theorem on sufficient statistic.
4. Show that exponential distribution with parameter λ is complete.
5. List out any two small sample properties of ML estimator.
6. Find ML estimator of θ in random sampling of size n from a population whose

pdf is  f(x, θ)   =          e ^{– (x – θ)},  for x > θ

=   0                      otherwise.

7. Define Loss Function. Is it a random variable? Justify.
8. When do we say that a statistic is Bayesian sufficient? Give an example.

9          Write down the normal equations of a simple linear regression model.

10. Mention the uses of Gauss-Markoff Model.

PART B                                            Answer any FIVE questions                       [5×8=40]

11. Show that the sample variance is a consistence estimator of the population

variance.

12. If X follows Binomial distribution with parameters n and p. Examine the

asymptotic unbiasedness of   T =    .

13. States and Prove Rao Blackwell Theorem.
14. Let (X₁, X₂, X₃, …X_n) is a random sample from Poisson population with parameter

λ. Use Lehman Scheffe Theorem to obtain a UMVUE of λ

15. Obtain the moment estimators of the parameters of a two-parameter gamma

distribution.

 

16. Illustrate the invariance property of ML estimator through an example
17. Explain the method of modified Chi-square estimation.
18. State and prove Gauss Markoff model on BLUE

 

PART C                                            Answer any TWO questions                       [2×20=40]

19. (a) State and prove Cramer Rao inequality in one parameter regular case. When

does the equality hold good?

(b) Establish a sufficient condition for a biased estimator to become a consistent

estimator.

20. (a) State and prove Lehman Scheffe theorem on UMVUE

(b) Obtain a joint sufficient statistic of the parameters of the bi-variate normal

population.

21. (a) Derive the moment estimators of the parameters of two parameter uniform

distribution.

(b)            Derive the ML estimators of the parameters of normal distribution by solving

simultaneous equations.

22. (a) Establish a necessary and sufficient condition for a linear parametric function to

be estimable.

(b) Let (X₁, X₂, X₃, …X_n) is a random sample of size n from Bernoulli population.

Obtain the Bayesian estimator of the parameter by taking a suitable prior

distribution..

 

 

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       LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 25

FIFTH SEMESTER – April 2009

ST 5501 – TESTING OF HYPOTHESIS

 

 

 

Date & Time: 17/04/2009 / 9:00 – 12:00       Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

 

Answer ALL questions.                                                                              10 X 2 = 20

 

1. Define Best Critical Region.
2. What is power of a test?
3. Define Monotone Likelihood Ratio property.
4. Define a UMP level α test.
5. Describe the stopping rule in SPRT.
6. State any two properties of likelihood ratio tests.
7. Write the 95 % confidence interval for population proportion based on a random sample of size n. (n > 30)
8. Describe the assumptions of t – test for testing equality of means of two independent
9. State any two assumptions of non-parametric tests.
10. Describe a test based on F- distribution.

 

 

 SECTION – B

Answer Any FIVE questions.                                                                                       5 X 8 = 40

 

11. Let X have a Poisson distribution with λ {2, 4 }. To test the null

hypothesis H₀: λ = 2 against the alternative simple hypothesis H₁: λ = 4, let the

critical region be {X₁  X₁ ≤ 3}, where X₁ is the random sample of size one. Find

the power of the test.

 

12. Based on a random sample of size n (n   30), construct a 95 % confidence interval for the population mean.

 

13. If  is a random sample from B (1, ),   (,1 ) , derive the

uniformly most powerful test for testing H₀:  =  against H₁:  >

 

14. Let be a random sample from Binomial distribution with parameter. Show that the distribution has a monotone likelihood ratio in the statistic Y =

 

15. Describe the procedure of Sequential Probability Ratio Test.

 

16. Based on a random sample of size n from B(, 0 <  , derive the SPRT

for testing H₀:  against the alternative  hypothesis H₁:  at level =0.05.

 

17. Differentiate parametric and Non-Parametric testing procedures.

 

18. Explain Kolmogorov- Smirnov one sample test.

 

SECTION – C

 

Answer any TWO questions.                                                                    2 X 20 = 40

 

19. a. State and prove Neyman-Pearson theorem.       [10]

                                                                                                                             

1. Based on a random sample of size n from a distribution with pdf

f(x, ) =        0 < x < 1

• otherwise

find the best critical region for testing null hypothesis H₀: = 2 against the

alternative simple hypothesis H₁:  = 3.                                                         [10]

 

20. Based on a random sample of size n from U(0,θ), derive the likelihood ratio test

for testing  H₀:  against the alternative hypothesis H₁:  .

 

21. a. Describe the procedure of testing H_{0 :} based on a random sample of

size n, using Wilcoxon’s statistic.                                                                    [10]

1. In SPRT, under standard notations prove that and

 

22. Explain: i) Sign test for location                          ii) Level of significance

 

iii) Test of equality of two variances      iv) Randomized test.      [ 4 x 5 ]

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2010

ST 5504/ST 5500 – ESTIMATION THEORY

 

 

 

Date : 29-10-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL the questions                                                                                                              [10×2=20]

 

1. Define an unbiased estimator and give an example.
2. Define the consistency of an estimator.
3. When is an estimator called most efficient estimator?   
4. Define UMVUE.
5. State any two regularity conditions.
6. State the invariance property of ML estimator.
7. Explain ‘posterior’ distribution.
8. Define Bayes estimator.
9. State the normal equations associated with the simple regression model.
10. Define Best Linear unbiased estimator.

PART – B

 Answer any FIVE questions                                                                                                            [5×8=40]

11. If X₁,X₂, . . . X_n are random observations on a Bernoulli variate X taking the value 1 with

      probability p and the value 0 with probability (1-p), show that: is a consistent

      estimator of p(1-p).

12. If X₁,X₂, . . . X_n are random observations of Poisson population with parameter λ.

      Obtain UMVUE for λ using Lehman Scheffe Theorem.                                                                                    

13. State and prove Factorization theorem [Neymann].
14. Write short notes about the method of minimum Chi-square estimation.
15. Estimate α and p for the following distribution by the methods of moments:

        .

16. Let X₁,X₂, . . . X_n be a random sample of size n from a uniform population with p.d.f:

                         f(x; Ө) = 1; Ө-1/2 ≤ x ≤ Ө+1/2, -∞< Ө <∞.

       Obtain M.L.E for Ө.

17. State and prove Gauss Markoff Theorem.
18. Explain the method of Least Squares with an illustration.

 

 

PART – C

Answer any TWO questions                                                                                                     [2×20=40]

19. a) Obtain Cramer Rao Lower bound for an unbiased estimator of if f(x,θ) =

            

1. b) Establish Chapman-Robbins Inequality.
2. a) State and prove Rao Blackwell theorem and mention it’s importance.
3. b) Let X₁,X₂, . . . X_n be an iid {f _Ө, Ө Є Θ} and each Xi, i=1,2,3,4 has mean μ and

          variance σ^2. Find the minimum variance unbiased estimator among the following.

                                    T₁(x) = X₁;

T₂(x) = (X₁+X₂)/2;

T₃(x) = (X₁+2X₂+3X₃)/6;

T₄(x) =(X₁+X₂+X₃+X₄)/4

21. a) Show that UMVUE is unique.
22. b) State and prove a sufficient condition for consistency of an estimator.
23. a) Establish a necessary and sufficient condition for a linear parametric function to be

          Estimable.

1. b) In Sampling from {b(1, Ө), 0<Ө<1}., Obtain Bayesian estimator of Ө taking a suitable prior

          distribution.

 

 

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ST 5504                       / ST 5500         ESTIMATION THOERY

 

Section A

Answer all the questions                                                                            10×2=20  

 

 

1          Define an Unbiased Estimator.

2. Define consistency of an estimator. Mention any one of its properties.
3. What is the importance of factorization theorem?
4. Define a Sufficient Statistic.
5. Briefly explain the method of moment estimation.
6. Explain prior distributions and posterior distributions with reference to Bayesian Estimation.
7. List out any two properties of Maximum Likelihood estimators.
8. Define Squared Error Loss function.
9. Define Best Linear unbiased Estimation. Give an example.
10. Write down the normal equations of a simple linear regression model.

 

Section B

Answer any five questions                                                                    5×8=40 

 

11. Let X₁,X₂, ..X_n be a random sample taken from a normal population with unknown mean and unknown variance. Examine the unbiasedness and consistency of T(x) = (X_i – )²
12. State and prove a sufficient condition for an estimator to be consistent.

(P.T.O)

13 Let X₁,X₂, ..X_n be a random sample taken from a population whose probability density function is

f(x, ) =  exp{–  },  >0,  x>0

Use Factorization Theorem to obtain a sufficient statistic for .

 

14. Show that Poisson distribution is complete.

 

15. Explain the method of minimum chi-square estimation.

 

16. Obtain the Maximum Likelihood estimators of the parameters of a normal distribution.

 

17. X₁,X₂, X₃,and X₄ are four independent Poisson random variables with mean . Define T₁(x) =  {X₁ +3X₃}

                          T₂(x) ={X₁ +2X₂+3X₃}

                          T₃(x) ={X₁+X₂ +X₃+X₄}

Examine their unbiasedness and compute their variances. Which one is the best estimator among the three? Find the efficiency of T₁(x) and T₂(x) with respect to T₃(x).

 

18. Explain in detail Gauss-Markov model.

 

 

 

(P.T.O)

 

 

Section C

Answer any two questions                                                               2×20=40 

 

19        (a)       State and prove Chapman-Robbins inequality. Bring out its importance.

(b) Let X₁,X₂, ..X_n be a random sample taken from a normal population with unknown mean and unit variance. Obtain Cramer-Rao Lower bound for an unbiased estimator of the mean.                                                 (12 + 8)

 

20. (a) Show that UMVUE for a parametric function is unique.

(b) State and Prove Rao-Blackwell Theorem.                            (10 + 10)

 

21. (a) Obtain the moment estimators for Uniform distribution U(a, b).

(b) Show that Maximum Likelihood Estimator need not be unique with an example. Also, show that when MLE is unique, it is a function of the sufficient statistic.                                                                                       (10 + 10)

 

22. (a) Let X₁,X₂, ..X_n be a random sample from Bernoulli distribution b(1, q). Obtain the Bayes estimator for q by taking a suitable prior.

(b) State and prove a necessary and sufficient condition for a parametric function to be linearly estimable.                                                     (10 + 10)

 

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