LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                        ST 2808 – ESTIMATION THEORY

 

 

Date & Time : 19-04-2006/FORENOON     Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

 

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

 

1. If the class of unbiased estimators of  a parametric function is neither empty nor singleton, then show that the class is uncountable.
2. Prove or disprove the uniqueness of UMVUE.
3. If  δ  is a UMVUE and bounded, then  show that any polynomial in δ  is also a UMVUE.
4. State Chapman – Robbin’s  inequality.

5. Suppose δ  is sufficient for Р  and  Р₀  С  Р,  then show that δ  is sufficient for Р₀.

1. Let S be a sufficient statistic. If  likelihood equivalence of x and y in the support A of the random variable X implies S(x) = S(y)   x , y  Є  A , then show that S is minimal sufficient.
2. Let X ₁  , X₂   be a random sample from N ( θ ,1),   θ Є R . Verify whether or not   (X₁  , X ₂ ) is complete.
3. Give two examples of a  Location-Scale family of distributions .
4. Define Ancillary Statistic and give an example.
5. Give an example of  M- estimator T_n  of  θ  which can be thought of as a weighted

average of the sample values with the weights depending on the data.

 

PART – B

 

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

 

1. Let X be a discrete random variable with  pdf  p_θ(x)  =  θ  if  x  =  -1 and

p_θ(x)  =  (1 – θ )²θ^x   if  x  =  0,1,2,…, where 0 < θ <  1. Find the class U₀ of

unbiased estimators of ‘0’ and hence find the class U_g of unbiased estimators of

g (θ)  =  θ , 0< θ < 1.

1. Give an example where only constant estimable parametric functions have

UMVUE.

1. Give an example of a UMVUE whose variance is greater than

Chapman-Robbins’s  Lower Bound.

1. 1. Let X₁ ,…,X_n   be a random sample of size n from N(θ , 1) ,  θ Є R. Using Fisher information, show that   α _ix_i  is sufficient iff  α_i  are equal for all i .
   2. Let X₁ ,… ,X_n   be a random sample of size n from U (0,θ), θ  > 0. Then show that S() = X_(n)  is minimal sufficient.
   3. Show that a complete sufficient statistic is minimal sufficient if it exists.

 

1.  Let X₁ ,… ,X_n   be a random sample of size n from B (m,θ),  m known and

θ unknown.   Show that the joint distribution of  (X₁ ,…,X_n  ) belongs to an

exponential family. Hence find the mgf of   X_i.

18. Let X ~ N (θ , 1 ), θ Є R , and let the prior distribution of θ be N ( 0 , 1 ).

Find the Bayes estimator of θ when the loss function is

• Squared error
• Absolute error.

 

 

PART – C

 

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

 

19(a). State and prove a necessary and sufficient condition for an estimator in the class u_g

to  be a  UMVUE.  (10)

19(b). Derive Chapman – Robbin’s inequality, using covariance inequality. (10)

 

20(a). Give an example of a family which is not boundedly complete.(10)

20(b). Let X₁  ,….., X_n   be a random sample from N(μ ,σ² ), μ Є R, σ  > 0. Show that the distribution of ( X₁ ,…..,X_n  ) belongs to two-parameter exponential family. Hence by using Basu’s theorem, establish the independence of   and s². (10)

21(a). Prove that  δ^*  is  D-Optimal estimator of g(θ)  iff each component of δ^*  is a            UMVUE.  (14)

21(b). Let X₁  ,… , X_n  be a random sample from N(μ ,σ² ), μ Є R, σ  > 0. Obtain Jackknife estimator of variance   σ²  .    (6)

 

22(a). State and prove Lehmann – Scheffe Theorem for convex loss function. (8)

22(b). Let  have the p.d.f    f ( – ξ ).  If   δ  is a location equivariant estimator ,  then

show that the bias ,  risk and variance of  δ  do not depend of  ξ  .  (6)

22( c )  Let X₁  ,… ,X_n  be a random sample from N(ξ  ,1 ) ,   ξ Є R .  Find the MRE

estimator of  ξ   when the loss is squared error.  (6).

 

 

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