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Question 3. Let X1,···, Xn be a sample from the pmf Po(X = x) = α(x)0x f(0) x = 0,1,2,. " where 0 > 0, α(x) > ·0, ƒ (0) = Σa(x)0x, α(0) = 1, and let T = T(X1,···, Xn) = X. Write i=1 x = n c(t,n) = Σ П α(xi), ЄR(t) i=1 where (x1,,xn) and R(t) = {(x1,xn) EN T(x1,,xn) = t}. Show that T is a complete sufficient statistic for 0, and that the UMVUE for d(0) = 0" (r > 0 is an integer) is given by 0, if t < r, Y₁(t) = c(t,n) " c(t-r,n) if t≥ r . iid Using this result find the UMVUE of 0 when Xi ~ Poisson(0).

Question 3. Let X1,···, Xn be a sample from the pmf Po(X = x) = α(x)0x f(0) x = 0,1,2,. " where 0 > 0, α(x) > ·0, ƒ (0) = Σa(x)0x, α(0) = 1, and let T = T(X1,···, Xn) = X. Write i=1 x = n c(t,n) = Σ П α(xi), ЄR(t) i=1 where (x1,,xn) and R(t) = {(x1,xn) EN T(x1,,xn) = t}. Show that T is a complete sufficient statistic for 0, and that the UMVUE for d(0) = 0" (r > 0 is an integer) is given by 0, if t < r, Y₁(t) = c(t,n) " c(t-r,n) if t≥ r . iid Using this result find the UMVUE of 0 when Xi ~ Poisson(0).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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Question 3. Let X1,···, Xn be a sample from the pmf Po(X = x) = α(x)0x f(0) x = 0,1,2,. " where 0 > 0, α(x) > ·0, ƒ (0) = Σa(x)0x, α(0) = 1, and let T = T(X1,···, Xn) = X. Write i=1 x = n c(t,n) = Σ П α(xi), ЄR(t) i=1 where (x1,,xn) and R(t) = {(x1,xn) EN T(x1,,xn) = t}. Show that T is a complete sufficient statistic for 0, and that the UMVUE for d(0) = 0" (r > 0 is an integer) is given by 0, if t < r, Y₁(t) = c(t,n) " c(t-r,n) if t≥ r . iid Using this result find the UMVUE of 0 when Xi ~ Poisson(0).
Transcribed Image Text:Question 3. Let X1,···, Xn be a sample from the pmf Po(X = x) = α(x)0x f(0) x = 0,1,2,. " where 0 > 0, α(x) > ·0, ƒ (0) = Σa(x)0x, α(0) = 1, and let T = T(X1,···, Xn) = X. Write i=1 x = n c(t,n) = Σ П α(xi), ЄR(t) i=1 where (x1,,xn) and R(t) = {(x1,xn) EN T(x1,,xn) = t}. Show that T is a complete sufficient statistic for 0, and that the UMVUE for d(0) = 0" (r > 0 is an integer) is given by 0, if t < r, Y₁(t) = c(t,n) " c(t-r,n) if t≥ r . iid Using this result find the UMVUE of 0 when Xi ~ Poisson(0).
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