Let Y1,…,Yn~iid N(μ, ?^2)
a) We showed that the maximum likelihood estimator for ?^2
is
Determine whether ? ̂??? is unbiased.
Let Y1,…,Yn~iid N(μ, ?^2) a) We showed that the maximum likelihood estimator for ?^2 is Determine...
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 the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.
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...
Let X1, ...., Xm be iid N(μ1,σ2) and Y1, ..., Yn be iid N(μ2,σ2), and X's and Y's are independent. Here -∞<μ1,μ2<∞ and 0<σ<∞ are unknown. Derive the MLE for (μ1,μ2,σ2). Is the MLE sufficient for (μ1,μ2,σ2)? Also derive the MLE for (μ1-μ2)/σ.
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Suppose Xi, X2, ,Xn is an iid N(μ, c2μ2 sample, where c2 is known. Let μ and μ denote the method of moments and maximum likelihood estimators of μ, respectively. (a) Show that ~ X and μ where ma = n-1 Σηι X? is the second sample (uncentered) moment. (b) Prove that both estimators μ and μ are consistent estimators. (c) Show that v n(μ-μ)-> N(0, σ ) and yM(^-μ)-+ N(0, σ ). Calculate σ and σ . Which estimator...
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 X1, …, Xn be iid Poisson(λ). Find the maximum likelihood estimator λMLE for λ, when it is given that λ > λ*, where λ* > 0 is a fixed constant. (Note: This is asking you to find the restricted MLE)
1. Let X b(n , 0 ), find the maximum likelihood estimate of the parameter 0 of the " corresponding binomial distribution. And prove the sample proportion is unbiased estimator of 0. 2. If are the values of a random sample from an exponential population, find the maximum likelihood estimator of its parameter 0.
1. Let X b(n , 0 ), find the maximum likelihood estimate of the parameter 0 of the " corresponding binomial distribution. And prove the sample...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...