Loyola College M.Sc. Statistics April 2006 Estimation Theory Question Paper PDF Download

             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.
  1. 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 Tn  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  pθ(x)  =  θ  if  x  =  -1 and

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

unbiased estimators of ‘0’ and hence find the class Ug 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. Let X1 ,…,Xn   be a random sample of size n from N(θ , 1) ,  θ Є R. Using Fisher information, show that   α ixi  is sufficient iff  αare equal for all i .
    2. Let X1 ,… ,Xn   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 ,… ,Xn   be a random sample of size n from B (m,θ),  m known and

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

exponential family. Hence find the mgf of   Xi.

  1. 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 ug

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  ,….., Xn   be a random sample from N(μ ,σ2 ), μ Є R, σ  > 0. Show that the distribution of ( X1 ,…..,Xn  ) 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  ,… , Xn  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  ,… ,Xn  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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