1. Let x1, ..., xn be a random sample from the exponential distribution f(x) = (1 / theta)e^(-x / theta) for x > 0.
(a) Find the mle of theta
## can use R code
(b) Find the Fisher information I(theta)
## can use R code
1. Let x1, ..., xn be a random sample from the exponential distribution f(x) = (1...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
2. Let X1, X2, ... , Xn represent a random sample from a distribution whose probability density function (pdf) is given by f(x: 0) = 69173-210 for r >0 and 0 > 0. Using the fact that E(X;) = 40, find the Fisher information In(0) in the random sample. (Hint: There are two different ways to compute the Fisher Information.)
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known number. For i 1,2 п, let ..1 -{: : if Xic 1 Y otherwise Suppose that you get to observe Yı, Y2,... , Y,n but you do not get to observe X1, X2,... , X,n п. Find the MLE for X based on this information
4. Let X1, X2, ..., Xn be a random sample from an Exponential(1) distribution. (a) Find the pdf of the kth order statistic, Y = X(k). (b) Determine the distribution of U = e-Y.
3. Let X1, ..., Xn, ... be iid random variables from the shifted exponential distribution: Se-(2-0) f( x0) = л VI (a) Find the MLE for 0. (b) Find the MLE for ø= EX. (c) Find the MOM estimator for 0.
5. Let X1, X2, ..., Xn be a random sample from a distribution with pdf of f(x) = (@+1)xº,0<x<1. a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for @ ?