Question

Let *X*_{1},...,*X*_{n} be iid
exp(θ) rvs.

(a) Compute the pdf of X_{min}.

(b) Create an unbiased estimator for θ based on X_{min}.
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) Verify that the expected value of the exponential score function is 0 and compute the Fisher Information for θ. Is either estimator a MVUE for θ?

Answer #1

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, 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?

Suppose X1,..., Xn are iid with pdf f(x;θ) = 2x / θ2,
0 < x ≤ θ. Find I(θ) and the Cramér-Rao lower bound for the
variance of an unbiased estimator for θ.

Let X1,...,Xn be iid from Poisson(theta), Set
k(theta)=exp(-theta).
(a) What is the MLE of k(theta)? Is it unbiased?
(b) Obtain the CRLB for any unbiased estimator of k(theta).

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)...

1. Let X1, . . . , Xn be i.i.d. continuous RVs with density
pθ(x) = e−(x−θ), x ≥ θ for some unknown θ > 0. Be sure to notice
that x ≥ θ. (This is an example of a shifted Exponential
distribution.) (a) Set up the integral you would solve for find the
population mean (in terms of θ); be sure to specify d[blank]. (You
should set up the integral by hand, but you can use software to
evaluate...

The random variable X is distributed with pdf fX(x,
θ) = (2/θ^2)*x*exp(-(x/θ)2), where x>0 and
θ>0. Please note the term within the exponential is
-(x/θ)^2 and the first term includes a θ^2.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?

Let X1,…, Xn be a sample of iid
Exp(?1, ?2) random variables with common pdf
f (x; ?1, ?2) =
(1/?1)e−(x−?2)/?1 for x
> ?2 and Θ = ℝ × ℝ+.
a) Show that S = (X(1), ∑ni=1
Xi ) is jointly sufficient for (?1, ?2).
b) Determine the pdf of X(1).
c) Determine E[X(1)].
d) Determine E[X2(1) ].
e ) Is X(1) an MSE-consistent estimator of
?2?
f) Given S = (X(1), ∑ni=1
Xi )is a complete sufficient statistic...

Let {X1, ..., Xn} be i.i.d. from a distribution with pdf f(x; θ)
= θ/xθ+1 for θ > 2 and x > 1.
(a) (10 points) Calculate EX1 and V ar(X1).
(b) (5 points) Find the method of moments estimator of θ.
(c) (5 points) If we denote the method of moments estimator as
ˆθ1. What does √ n( ˆθ1 − θ) converge in distribution to? (d) (5
points) Is the method of moment estimator efficient? Verify your
answer.

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)...

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