3. The Rayleigh distribution is a continuous distribution with pdf of the form Så exp(-+) $(30)...
7.2.10 Suppose that X, .., X, are iid with the Rayleigh distribution, that is the common pdf is where θ(> 0) is the unknown parameter. Find the MLE for θ, is the MLE sufficient for θ?
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
Let X1, X2, ..., Xn be iid random variables with a "Rayleigh” density having the following pdf: 22 -12 10 f(x) = e x > 0 > 0 0 пе a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = V c) (7 points) What is the MLE of 02 + 0 – 10 ? d) (7 points) For a fact, 21–1...
Let X1, X2, ..., X, be iid random variables with a "Rayleigh” density having the following pdf: f(x) = 2x2=+*10, 2 > 0 > 0 V лв a) (3 points) Find a sufficient estimator for using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = c) (7 points) What is the MLE of 02 +0 - 10 ? d) (7 points) For a fact, IX has a Gamman, o) distribution. Using...
Let X1, X2, ..., X, be iid random variables with a "Rayleigh" density having the following pdf: f(x) = 6-2°/0, a>0, 0x0 a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 6. Small help: E(X.) = V** c) (7 points) What is the MLE of 02 +0 -10? d) (7 points) For a fact, Li-1 X? has a Gammain,6) distribution. Using this information, find a consistent...
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0, x > 0, N = (0,00). (a) Find the maximum likelihood estimator of 0. (b) Find the method of moments estimator of 0. (c) Are the estimators in a) and b) unbiased? (d) What is the variance of the estimators in a) and b)? (e) Suppose the observed sample is 2.26, 0.31, 3.75, 6.92, 9.10, 7.57, 4.79, 1.41, 2.49, 0.59. Find the maximum likelihood...
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?
2 Let X1, X2, ..., X, be independent continuous random variables from the following distribution: (*) - ar-(-) where I 21 and a > 1 You may use the fact: E[X] = -1 2.1 Show that the maximum likelihood estimator of a is â MLE - Srlos Xi 2.2 Show that the method moment estimator for a is: &mom = 1 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency?
Let X1, X2, ..., Xn be a random sample from the distribution with pdf f(3;6) = V porta exp ( 0) 10.02) for some parameter 2 > 0. (a) Find the MLE for 0. (b) Find the Cramér-Rao lower bound for the variance of all unbiased estimators of 0. (c) Find the asymptotic distribution of your MLE from part (a).