Question

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) = ∑ Xi/n

Answer #1

Let
X1 and X2 be IID exponential with parameter λ > 0. Determine the
distribution of Y = X1/(X1 + X2).

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)...

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?

Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?

Let X1, X2, . . . Xn be iid
exponential random variables with unknown mean β. Find the method
of moments estimator of β

Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
I have the pdf
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the...

Let X1,...,Xn be iid
exp(θ) rvs.
(a) Compute the pdf of Xmin.
(b) Create an unbiased estimator for θ based on Xmin.
Compute the variance of the resulting estimator.
(c) Perform a Monte Carlo simulation of N= 10,0000 samples of
your unbiased estimator from part (b) using θ = 2 and n = 100 to
validate your answer. Include a histogram of the samples.
(d) Which is more efficient: your estimator from part (b) or the
MLE for θ?
(e)...

Let X1, X2, …, Xn be iid with pdf ?(?|?) = ? −(?−?)? −? −(?−?) ,
−∞ < ? < ∞. Find a C.S.S of θ

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

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