A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ

f(x;θ) = {

0 otherwise

find the method of moments estimator of θ and show that it is not bias.

Please help!

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- Nov 27th 2008, 07:45 AMJames1220Probability & Statistics
A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ

f(x;θ) = {

0 otherwise

find the method of moments estimator of θ and show that it is not bias.

Please help! - Nov 27th 2008, 10:28 PMmr fantastic
Calculate E(X) as a function of $\displaystyle \theta$ using the given pdf.

Note that the sample mean is $\displaystyle \bar{X} = \frac{x_1 + x_2}{2}$. Estimate $\displaystyle \theta$ by equating E(X) to the sample mean. I get $\displaystyle \hat{\theta} = \frac{3(x_1 + x_2)}{2} = 3 \bar{X}$.

Now show that $\displaystyle E(\hat{\theta}) = \theta$. - Nov 29th 2008, 04:47 AMJames1220
Do i get the mean E(X) by integrating? Im not managing to get it so far. I got 1 - (2x)/theta

- Nov 29th 2008, 06:28 AMselinunan
You should take this integral:

E(x)=∫xf(x)dx=∫x[(2/(Q²)(Q-x)]dx

The result is E(x)=Q/3 - Nov 29th 2008, 12:25 PMmr fantastic
- Nov 30th 2008, 01:49 AMJames1220
Thank you Mr F and selinunan. I have finally integrated and found E(X)! And yes it is θ/3. When showing that it is unbiased, theta hat = 3 xbar. How do i go about finding E(theta hat)?

- Nov 30th 2008, 01:55 AMJames1220
Ooo never mind! I've found it :) thanks!

- Dec 2nd 2008, 04:57 AMJames1220
A random sample of two functions x1, x2, is drawn from the distribution with pdf:

2/θ^2 (θ - x) 0<x<θ

f(x;θ) = {

0 otherwise

find the method of moments estimator of θ and show that it is not bias.

With help of fellow mathematicians, i found the solution to the above problem. the following question relates to this and is the second part of the question :

Show that the maximum likelihood estimator of θ is given by:

\hat{\theta} = 1/4 [3(x_1 + x_2) + Square root of (9x_1^2 - 14x_1x_2 + 9x_2^2)]

Im troubled because of the fact that there are 2 variables x_1 and x_2, so i'm having trouble finding how to fond the mle. Help if you can Pleaseee! - Dec 2nd 2008, 02:12 PMmr fantastic
Read this thread carefully: http://www.mathhelpforum.com/math-he...estimator.html

Now try the question again.

Post if you're still stuck (please be sure to say where you're stuck).