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...

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.

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.

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ).
Let Yn be the maximum of X1, X2, ..., Xn.
(a) Give the pdf of Yn.
(b) Find the mean of Yn.
(c) One estimator of θ that has been proposed is Yn. You may
note from your answer to part (b) that Yn is a biased estimator of
θ. However, cYn is unbiased for some constant c. Determine c.
(d) Find the variance of cYn,...

The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 )
Calculate the probability of X1 < X2,
i.e. P(X1 < X2, θ).

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