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. Co...
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.
Let X1, , xn be a random sample gamma(a, β). In parts (a)-(d) assume a is known. Consider testing Ho : β Derive Wald statistic for testing Ho using the MLE of β both in the numerator and denominator of the statistic. Let X1, , xn be a random sample gamma(a, β). In parts (a)-(d) assume a is known. Consider testing Ho : β Derive Wald statistic for testing Ho using the MLE of β both in the numerator and...
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 X1, . . . , Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0. (i) Now let’s think like a Bayesian. Consider a prior distribution of β ∼ Gamma(a, b) for some a, b > 0. Derive the posterior distribution of β given (X1, . . . , Xn) = (x1,...,xn). (j) What is the posterior Bayes estimator of β assuming squared error loss?
2. Let Xi, , Х, be a random sample gamma(a, β). In parts (a-(d) assume a is known. 30 points a. Consider testing H. : β--βο. Derive Wald statistic for testing H, using the MLE of B both in the numerator and denominator of the statistic. b. Derive a test statistic for testing H, using the asymptotic distribution of the MLE of β. What is the relation between the two statistics in parts (a) and (b)? c. Derive the Score...
7.60 Let Xi, of 1/β. , xn be iid gamma(α, β) with α known. Find the best unbiased estimator
Let Xi, known , xn be a random sample from a gamna(α, β) distribution. Find the MLE of β, assuming α s
Let X1, ..., Xn be a random sample from Gamma(1,41) distribution and Y1, ..., Ym be a random sample from Gamma(1,12) distribution. Also assume that X’s are independent of Y's. (1) Formulate the LRT for testing Ho : 11 = 12 v.s. Hy : 11 + 12; (10 points) (2) Show that the test in part (1) can be based on the following statistic (7 points) T = 21-1 Xi Dizi Xi + [2Y; = (3) Find the distribution of...
Let X1,…Xn ~ iid Gamma (α, θ) where the α is known and interested in the rate parameter θ, and we chosen a prior θ~ Gamma (3, 1). Find the posterior distribution
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 -...