4. Find the maximuln likelihood estimator of the parameter θ of the population with the density...
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...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Bernoulli distribution with parameter θ . a) Use the method of moments to obtain an estimator of θ b) Obtain the maximum likelihood estimator (MLE) of θ.
(a) (4 points) Find the method of moments estimator for θ. (b) (4 points) Find the maximum likelihood estimator for . (c) (3 points) Show that the maximum likelihood estimator for θ is a function of a sufficient statistic. (d) (4 points) Find the Cramer-Rao lower bound for the variance of an estimator of . (e) (3 points) Identify the asymptotic distribution of the MLE. (a) (4 points) Find the method of moments estimator for θ. (b) (4 points) Find...
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?
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
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.
2. (a) Define the bias of ˆ θ as an estimator for the parameter θ. [2 marks] (b) For independent random variables X1,X2,...,Xn, assume that E(Xi) = µ and var(Xi) = σ2, i = 1,...,n. (i) Show that ˆ µ1 = {(X1+Xn)/2}is an unbiased estimator for µ and determine its variance. [3 marks] (ii) Find the relative efficiency of ˆ µ1 to the unbiased estimator ˆ µ2 = X, the sample mean. [2 marks] (iii) Is ˆ µ1 a consistent...