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Problem 1: On an Application of Rao Blackwell, Lehman Schefe Theorems and Basu then- rems: Let X1,. X be IID from a Poisexon distribution with mean X, where A> 0. So the common p.m.f. is given by /(피) %3D for r=0,1,2,. (a) Is the statistic S =, X; complete and sufficient for X? Give the distribution of S. (b) What is the Uniformly Minimum Variance Unbiased Estimator (UMVUE) for 7(A) = X? Provide a reason why your estimalor is UMVUE. (c) Show that T(X1,.--- Xm) = I{X1 = 0} is an unbiasesd estimator for 7(A) =e. (d) Establish that the conditional distribution of X, given S=s is binomial with parameters s and n. TA

Problem 1: On an Application of Rao Blackwell, Lehman Schefe Theorems and Basu then- rems: Let X1,. X be IID from a Poisexon distribution with mean X, where A> 0. So the common p.m.f. is given by /(피) %3D for r=0,1,2,. (a) Is the statistic S =, X; complete and sufficient for X? Give the distribution of S. (b) What is the Uniformly Minimum Variance Unbiased Estimator (UMVUE) for 7(A) = X? Provide a reason why your estimalor is UMVUE. (c) Show that T(X1,.--- Xm) = I{X1 = 0} is an unbiasesd estimator for 7(A) =e. (d) Establish that the conditional distribution of X, given S=s is binomial with parameters s and n. TA

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Problem 1: On an Application of Rao Blackwell, Lehman Scheffe Theorems and Basu theo- rems: Let X1,. X be IID from a Poieson distribution with mesan A, where A> 0. So the common p.m.f. is given by S(r|A) = for r= 0,1,2,. (a) Is the statistic S =, X; complete and sullicient for A? Give the distribution of S. (b) What is the Uniformly Minimum Variance: Unbiasesl Estimator (UMVUE) for r(A) = X? Provide a reason why your estimator is UMVUE. (c) Show that T(X1, . Xm) = {X1 = 0} is an unbiasesd estimator for 7(A) = e. (d) Establish that the conditional distribution of X1, given S =s is binomial with paramelers s and n. (e) Us: the Rao Blackwell Theorem lw improve the estimator T in (c) by taking T(s) = E(T|S = s) and give an exact expression for T (S) (1) Is the estimator T (S) in (f) the UMVUE of 7(A) = e? Why? (g) What is the Fisher information I(A) associated with X,? (h) What is the Cramér Rao Lawer Bound (CRLB(A)) for the variance of an unbiaexd estimator of r(A) =e-? (i) Asesnuming that. the estimator T(S) in () is unbiased for e, will it achieve the CRLB(A) you obtainesd in (h)? Can you use the Lehman Scheffe theorem to conclude that T'(s) is the unique UMVUE for 7(A)?
Transcribed Image Text:Problem 1: On an Application of Rao Blackwell, Lehman Scheffe Theorems and Basu theo- rems: Let X1,. X be IID from a Poieson distribution with mesan A, where A> 0. So the common p.m.f. is given by S(r|A) = for r= 0,1,2,. (a) Is the statistic S =, X; complete and sullicient for A? Give the distribution of S. (b) What is the Uniformly Minimum Variance: Unbiasesl Estimator (UMVUE) for r(A) = X? Provide a reason why your estimator is UMVUE. (c) Show that T(X1, . Xm) = {X1 = 0} is an unbiasesd estimator for 7(A) = e. (d) Establish that the conditional distribution of X1, given S =s is binomial with paramelers s and n. (e) Us: the Rao Blackwell Theorem lw improve the estimator T in (c) by taking T(s) = E(T|S = s) and give an exact expression for T (S) (1) Is the estimator T (S) in (f) the UMVUE of 7(A) = e? Why? (g) What is the Fisher information I(A) associated with X,? (h) What is the Cramér Rao Lawer Bound (CRLB(A)) for the variance of an unbiaexd estimator of r(A) =e-? (i) Asesnuming that. the estimator T(S) in () is unbiased for e, will it achieve the CRLB(A) you obtainesd in (h)? Can you use the Lehman Scheffe theorem to conclude that T'(s) is the unique UMVUE for 7(A)?
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