Question

Suppose X_1, X_2, … X_n is a random sample from a population with density f(x) = (2/theta)*x*e^(-x^2/theta) for x greater or equal to zero.

- Find the Maximum Likelihood Estimator Theta-Hat_1 for theta.
- Find the Method of Moment Estimator Theta-Hat_2 for theta.

Answer #1

a) MLE :

b) Method of Moment:

Suppose X_1, X_2, … X_n is a random sample from a population
with density f(x) = (2/theta)*x*e^(-x^2/theta) for x greater or
equal to zero.
Determine if Theta-Hat_1 (MLE) is a minimum variance unbiased
estimator for thet
Determine if Theta-Hat_2 (MOM) is a minimum variance unbiased
estimator for theta.

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)

Let X_1, ..., X_n be a random sample from a normal
distribution, N(0, theta). Is theta_hat a UMVUE of
theta?
The above question is from chapter 9 problem 23b of Introduction
to Probability and Mathematical Statistics (for which you have a
solution posted on this website). I'm confused about the part in
the posted solution where we go from the line that says E(x^4
-2\theta * E(x^2) + E(\theta^2) to the line that says
(3\theta^2-2\theta^2+\theta^2). Could you please explain this...

Consider the random variable X with density given by f(x) = θ
2xe−θx x > 0, θ > 0 a) Derive the expression for E(X). b)
Find the method of moment estimator for θ. c) Find the maximum
likelihood estimator for θ based on a random sample of size n. Does
this estimator differ from that found in part (b)? d) Estimate θ
based on the following data: 0.1, 0.3, 0.5, 0.2, 0.3, 0.4, 0.4,
0.3, 0.3, 0.3

Let x₁ , x₂,…xₐ be a random sample from X with density f(x,?) =
α?−?−1 x>1 Find maximum likelihood
estimator of ?.

Consider the probability density function f(x) = (3θ +
1) x3θ, 0 ≤
x ≤ 1.
The random sample is 0.859, 0.008, 0.976, 0.136, 0.864, 0.449,
0.249, 0.764. The moment estimator of θ based on a random
sample of size n is .055. Please answer the following:
a) find the maximum likelihood estimator of θ based on
a random sample of size n. Then use your result to find
the maximum likelihood estimate of θ based on the given
random...

1. Remember that a Poisson Distribution has a density function
of f(x) = [e^(−k)k^x]/x! . It has a mean and variance both equal to
k.
(a) Use the method of moments to find an estimator for k.
(b) Use the maximum likelihood method to find an estimator for
k.
(c) Show that the estimator you got from the first part is an
unbiased estimator for k.
(d) (5 points) Find an expression for the variance of the
estimator you have...

Let Y1, Y2, . . ., Yn be a
random sample from a Laplace distribution with density function
f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞
where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y2) = 2θ2.
a) Find the likelihood function of the sample.
b) What is a sufficient statistic for θ?
c) Find the maximum likelihood estimator of θ.
d) Find the maximum likelihood estimator of the standard
deviation...

Let X_1,…, X_n be a random sample from the Bernoulli
distribution, say P[X=1]=θ=1-P[X=0].
and
Cramer Rao Lower Bound of θ(1-θ)
=((1-2θ)^2 θ(1-θ))/n
Find the UMVUE of θ(1-θ) if such exists.
can you proof [part (b) ] using (Leehmann Scheffe
Theorem step by step solution) to proof
[∑X1-nXbar^2 ]/(n-1) is the umvue , I have the key
solution below
x is complete and sufficient.
S^2=∑ [X1-Xbar ]^2/(n-1) is unbiased estimator of θ(1-θ) since
the sample variance is an unbiased estimator of the...

Let Y1,
Y2, …, Yndenote a random sample of size
n from a population whose density is given by
f(y) = 5y^4/theta^5
0<y<theta
0 otherwise
a) Is an unbiased estimator of
θ?
b) Find the MSE of Y bar
c) Find a function of that is an
unbiased estimator of θ.

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