Let Y1,…,Yn~iid Gamma(5,β). Recall that Γ(5) = 4!
a) Find the MLE for β.
b) Is your answer to a) the MVUE? Use two methods to verify that it is unbiased.
Let Y1,…,Yn~iid Gamma(5,β). Recall that Γ(5) = 4! a) Find the MLE for β. b) Is...
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.
7.60 Let Xi, of 1/β. , xn be iid gamma(α, β) with α known. Find the best unbiased estimator
7.20 Consider Y1,...,Yn as defined in Exercise 7.19. (a) Show that Yilti is an unbiased estimator of B. (b) Calculate the exact variance of Yi/ xi and compare it to the variance of the MLE. 7.19 Suppose that the random variables Yı, ..., Yn satisfy Yi = Bli +ti, i = 1,...,n, where x1, ..., In are fixed constants, and €1,..., En are iid n(0,02), o2 unknown. (a) Find a two-dimensional sufficient statistic for (0,0%). (b) Find the MLE of...
5. Consider the gamma distribution and recall that its mean and variance are μ-αβ and σ2-032, respectively. Assume a is known. Let X1, . . , X,, ~ X where X ~ f(x; α, β). is strict. your findings to verify the additivity property in(3) = n1(3). you computed V(An). Relate V(ßn) and In(3). interval to estimate T (a) Compute the Fisher information I(8) of A (why?) and examine whether the Cramer-Rao inequality (b) Find the score of the sample...
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 -...
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of μ, σ2. Are these unbiased estimators of μ and of σ2 respectively? Aside: You can use your result in (b) to justify your answer for the bias part of the MLE estimator of σ2 (b) In this part you will show, despite that the sample variance is an unbiased estimator of σ2, that the sample standard deviation is is a biased estimator of σ....
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood estimator of β. (c) Determine whether the maximum likelihood estimator is unbiased for β. (d) Find the mean squared error of the maximum likelihood estimator of β. (e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (g) Determine the asymptotic distribution of the...
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:...
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)...