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)
Let X1, …, Xn be iid Poisson(λ). Find the maximum likelihood estimator λMLE for λ, when...
Let X1, ... , Xn be iid with the Poisson(λ) distribution. What is the conditional distribution of Xi given the sample mean?
Suppose that (X1, X2,,,,Xn) are iid random variables. Find the maximum likelihood estimator of theta for the following distributions 1) Poi(theta) 2) N(Mu, theta) 3) Exp(theta)
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 from an exponential population with parameter λ. (a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased? (c) Is the estimator consistent?
2. Let X1, X2, ...,Xbe i.i.d. Poisson with parameter .. (a) Find the maximum likelihood estimator of . Is the estimator minimum variance unbi- ased? (b) Derive the asymptotic (large-sample) distribution of the maximum likelihood estimator. (c) Suppose we are interested in the probability of a zero: Q = P(Xi = 0) = exp(-). Find the maximum likelihood estimator of O and its asymptotic distribution.
Let X1, . . . , Xn ∼ iid Exp(λ) and Y1, . . . , Ym ∼ iid Exp(τ ) be independent random samples. (a) Find the restricted MLEs under the null hypothesis H0 : λ = τ . (b) Write out a formula for the LRT statistic, and describe how you could perform this test asymptotically.
Suppose X1, X2, , Xn is an iid sample from a uniform distribution over (θ, θΗθ!), where (a) Find the method of moments estimator of θ (b) Find the maximum likelihood estimator (MLE) of θ. (c) Is the MLE of θ a consistent estimator of θ? Explain.
6. Find the moment and maximum likelihood estimates of the parameter p of the negative -.. , Xn. Recall that the pmf is , and again when maximizing consider binomial distribution given an iid sample from it: X1, given by p(k) = ( )prqk-r for k = r, r + 1, the Bernoulli MLE
6. Find the moment and maximum likelihood estimates of the parameter p of the negative -.. , Xn. Recall that the pmf is , and again...
2. Asymptotic Maximum Likelihood. 25 Let X1, ..., Xn be independently Poisson distributed with parameter 1, i.e. fx, (x) = exto is X= 0, 1, 2, ... =0,1,2,... (a) Derive the maximum likelihood estimator în of 1 based on n measurements. 5 (b) Show that în is consistent. 5 (c) Is în (asymptotically) efficient? 5 (d) Derive the asymptotic distribution of vn(în – 1). 10
Suppose that X1, X2,....Xn is an iid sample of size n from a Pareto pdf of the form 0-1) otherwise, where θ > 0. (a) Find θ the method of moments (MOM) estimator for θ For what values of θ does θ exist? Why? (b) Find θ, the maximum likelihood estimator (MLE) for θ. (c) Show explicitly that the MLE depends on the sufficient statistic for this Pareto family but that the MOM estimator does not