Question

- Consider a random sample
*X**1**,**X**2**, ⋯**X**n*from the pdf

*f**x;θ**=.5**1+θx**, -1≤x≤1;0,
o.w.,* where (this distribution arises in particle
physics).

- Find the method of moment estimator of θ.

- Compute the variance of your estimator. Hint: Compute the
variance of X and then apply the formula for
*X*, etc.

Answer #1

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,...,Xn be a random sample from the pdf f(x;θ) = θx^(θ−1)
, 0 ≤ x ≤ 1 , 0 < θ < ∞ Find the method of moments estimator
of θ.

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 X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

1. Let X1, X2, . . . , Xn be a random sample from a distribution
with pdf f(x, θ) = 1 3θ 4 x 3 e −x/θ , where 0 < x < ∞ and 0
< θ < ∞. Find the maximum likelihood estimator of ˆθ.

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...

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.

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, X2 · · · , Xn be a random sample from the distribution
with PDF, f(x) = (θ + 1)x^θ , 0 < x < 1, θ > −1.
Find an estimator for θ using the maximum likelihood

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.

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