Question

Let X1, ..., Xn be a sample from an exponential population with parameter λ.

(a) Find the maximum likelihood estimator for λ. (NOT PI FUNCTION)

(b) Is the estimator unbiased?

(c) Is the estimator consistent?

Answer #1

**PLEASE
- - - - PLEASE KINDLY UP-VOTE. IT HELPS ME A LOT. THANK YOU IN
ADVANCE.**

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Let X1, ..., Xn be a random sample of an
Exponential population with parameter p. That is,
f(x|p) = pe-px , x > 0
Suppose we put a Gamma (c, d) prior on p.
Find the Bayes estimator of p if we use the loss function L(p,
a) = (p - a)2.

Let X1, X2,..., Xn be a random sample from a population with
probability density function f(x) = theta(1-x)^(theta-1), where
0<x<1, where theta is a positive unknown parameter
a) Find the method of moments estimator of theta
b) Find the maximum likelihood estimator of theta
c) Show that the log likelihood function is maximized at
theta(hat)

Suppose that X is Poisson, with unknown mean λ, and let X1, ...,
Xn be a random sample from X.
a. Find the CRLB for the variance of estimators based on X1,
..., Xn. ̂̅̂
b. Verify that ? = X is an unbiased estimator for λ, and then
show that ? is an efficient estimator for λ.

Let X1, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

6. Let θ > 1 and let X1, X2, ..., Xn be a random sample from
the distribution with probability density function f(x; θ) =
1/(xlnθ) , 1 < x < θ.
a) Obtain the maximum likelihood estimator of θ, ˆθ.
b) Is ˆθ a consistent estimator of θ? Justify your answer.

Let X1,..., Xn be a random sample from a
distribution with pdf as follows:
fX(x) = e^-(x-θ) , x > θ
0 otherwise.
Find the sufficient statistic for θ.
Find the maximum likelihood estimator of θ.
Find the MVUE of θ,θˆ
Is θˆ a consistent estimator of θ?

Let X¯ be the sample mean of a random sample X1, . . . , Xn from
the exponential distribution, Exp(θ), with density function f(x) =
(1/θ) exp{−x/θ}, x > 0. Show that X¯ is an unbiased point
estimator of θ.

Suppose that X1,..., Xn form a random sample from the
uniform distribution on the interval [0,θ], where the value of the
parameter θ is unknown (θ>0).
(1)What is the maximum likelihood estimator of θ?
(2)Is this estimator unbiased? (Indeed, show that it underestimates
the parameter.)

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 10 minutes ago

asked 12 minutes ago

asked 18 minutes ago

asked 25 minutes ago

asked 29 minutes ago

asked 31 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago