5. Consider a sample of size n from Gamma(α, β). Let a be given. Find the...
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a. with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0. Show that T = (∑n i=1 Xi, ∏ n i=1 Xi) is complete and sufficient for (α, β).
6. Consider a sample of size n from Unif(a, α + β). Find the method of moment estimator of the parameter g = 2a + β
Y1, Y2, ... Yn are a random sample from the Gamma distribution with parameters α and β (a) Suppose that α-4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β = 4 is known and a is unknown. Find a complete sufficient statistic for α.
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
Let Xi,, X be a random sample from a gamma(a, B) distribution a. Identify a two-dimensional sufficient statistics for (α, β). b. Is the two-dimensional sufficient statistic in part (a) minimal sufficient?
Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho- Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho-
Let X1, ,Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β = βο. Derive Wald statistic for testing Ho using the MLE of B both in the numerator and denominator of the statistic. Let X1, ,Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β = βο. Derive Wald statistic for testing Ho using the MLE of B both in the numerator and denominator of the statistic.
(10) For a random sample of size n from a Beta(α, β) density, find a consistent estimator of β . Why is this estimator consistent? (10) For a random sample of size n from a Beta(α, β) density, find a consistent estimator of β . Why is this estimator consistent?
7.2.6. Let X1, X2....Xn be a random sample of size n from a beta d with parameters α-θ and β statistic for θ 5. Show tha the product Xi X2 . . . Xn is a sufficient oherat tious is a sufficient statistic for