# Parameter estimation

• Mar 29th 2010, 06:39 PM
Anonymous1
Parameter estimation
Suppose $\displaystyle X$ has density $\displaystyle f_X (x) = \frac{ba^b}{x^{b+1}}$ if $\displaystyle x>a>0,$ otherwise $\displaystyle f_X (x)=0$

$\displaystyle (a)$Describe how to sample from this distribution using the method of inversion.
$\displaystyle (b)$Find the MLE estimates for $\displaystyle a$ and $\displaystyle b$ from $\displaystyle n$ $\displaystyle i.i.d.$ samples.
$\displaystyle (c)$Determine the asymptotic variance of the MLE estimates.
• Mar 30th 2010, 05:07 AM
CaptainBlack
Quote:

Originally Posted by Anonymous1
Suppose $\displaystyle X$ has density $\displaystyle f_X (x) = \frac{ba^b}{x^{b+1}}$ if $\displaystyle x>a>0,$ otherwise $\displaystyle f_X (x)=0$

$\displaystyle (a)$Describe how to sample from this distribution using the method of inversion.

First find the cumulative distriution function $\displaystyle F_X(x)$, then find the inverse function of this:

$\displaystyle G_X(F_X(x))=x$

where the domain of $\displaystyle G_X$ is [0,1) and range is $\displaystyle [a,\infty)$

Now generate a $\displaystyle U(0,1)$ random number $\displaystyle r$ and $\displaystyle x=G(r)$ has the required distribution, and I am absolutly certain that this is in your notes and or text book.

CB
• Mar 30th 2010, 11:21 AM
Anonymous1
My problem is mostly with $\displaystyle (b).$ I cannot simply set $\displaystyle l'(\theta)=0$ to maximize. So what do I do?
• Mar 30th 2010, 08:39 PM
matheagle
Does that mean me?
The likelihood function is...

$\displaystyle L(x_1,\cdots, x_n)=b^na^{bn}\left(\prod_{i=1}^nx_i\right)^{-(b+1)}I(x_{(1)}>a)$

so if you want to maximize this wrt a, you want a as big as possible, since bn>0.
The largest a can be is the smallest order stat, i.e., the min.
As for the max wrt b it looks like you can take the log and differentiate.
• Mar 30th 2010, 10:04 PM
Anonymous1
Quote:

Originally Posted by matheagle
Does that mean me?
The likelihood function is...

$\displaystyle L(x_1,\cdots, x_n)=b^na^{bn}\left(\prod_{i=1}^nx_i\right)^{-(b+1)}I(x_{(1)}>a)$

so if you want to maximize this wrt a, you want a as big as possible, since bn>0.
The largest a can be is the smallest order stat, i.e., the min.
As for the max wrt b it looks like you can take the log and differentiate.

Thanks!
• Mar 31st 2010, 07:35 PM
Anonymous1
Sorry if this is a stupid question, but what exactly is this about?

$\displaystyle I(x_{(1)}>a)$
• Mar 31st 2010, 08:40 PM
matheagle
Quote:

Originally Posted by Anonymous1
Sorry if this is a stupid question, but what exactly is this about?

$\displaystyle I(x_{(1)}>a)$

It's an indicator function, some call it a characteristic function.
But I use that terminology for the Fourier transform.
And I combine what some people use in a subscript with the argument.

$\displaystyle I(x_{(1)}>a)=1$ if $\displaystyle x_{(1)}>a$

while $\displaystyle I(x_{(1)}>a)=0$ if $\displaystyle x_{(1)}\le a$

Most people write $\displaystyle I(x_{(1)}>a)$ as $\displaystyle I_{(a,\infty)}(x_{(1)})$
I combine the subscript and the argument.
• Mar 31st 2010, 08:53 PM
matheagle
If $\displaystyle X\sim U(a,b)$ then instead of saying that f(x) is 0 otherwise, use...

$\displaystyle f_X(x)={1\over b-a}I(a<x<b)$

and this notation makes it easier to see how the max and min are suff stats in many problems.