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....
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 -...
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the distribution of 2 Σ x/P Follow the procedure outlined in Exercise 6.28. 6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what...
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).
Let {} be a random sample from the distribution. (a) Find a sufficient statistic for when is known (b) Find a sufficient statistic for when is known 7l beta ( α , β ) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
Q. 4 Let X1, ... , Xn be a random sample from gamma (2,0) distribution. c) Show that it is the regular case of the exponential class of distributions. d) Find a complete, sufficient statistic for 0. e) Find the unique MVUE of 0. Justify each step.
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...