# Setting upp the maximum likelihood equation

Here is what the maximum likelihood method is:

Assume we have observed $x_1,x_2,...,x_n$ from the random independent variables $X_1,X_2,...X_n$

The joint distribution for $(X_1,X_2,...X_n)$ is given by

$f_{X_1,X_2...X_n}(x_1,x_2,...x_n) = \prod\limits_{k=1}^{n}f(x_k)$

We then would like give the the parameters values of $f(\cdot)$ so that we maximize:

$L = \prod\limits_{k=1}^{n}f(x_k)$.

(all $x_k$ are given)

Okey here is my question:

consider the random variable $Z := max(X_1-u,X_2-u,...X_N -u)$ , where $u$ is some threshold, $N,X_1,X_2$ are independent the $X_i$:s are all equally distributed, N has a poisson distribution with parameter $\lambda$ ,$(X_i-u)$ are ,for all $i$, pareto distributed . Also all $X_i$ exceeds $u$.

The distribution function for $Z$ - let us call it $G(x)$, is given,after some calculations, as :

$\mathbb{P}(Z \leq x) = G(x) = e^{-\lambda(1- F(x)) }$

Where $F(x) = \mathbb{P}(X_i -u \leq x)= 1 - (\dfrac{\alpha}{\alpha + x})^{\gamma}$, (the pareto distribution with parameters $\alpha,\gamma$)

if $x_1,x_2,..x_n$ are the realizations of $X_1,X_2,..$ then the Maximum likelihood estimate of the parameters$(\alpha,\lambda,\gamma)$ can be obtained by using $x_1 - u , x_2 - u , ...x_n-u$.(all this is from the book)

I have some difficulties of setting upp the maximum likelihood equation (in terms of the definition above)

I mean i would like to think that $G'(x)$ would take the place of $f(\cdot)$ in the definition above but then we would have only one single observation from $Z$ and that is $max(x_1 - u , x_2 - u , ...x_n-u)$ and that sound strange that (the book) would propose to estimate the parameters from one observation.

What Iam missing or what is missing here?

1. If you observe all of the $X_i$ then you don't need $Z$ to estimate the parameters.
2. Are you sure $Z$ is not the sum, and that $G$ is not the probability generating function?
3. You can also denote $Y_i:=X_i-u$ and deal directly with $Y_i$ as you know their explicit distribution.
4. Now you want to write down the joint distribution of $Y_1,\ldots,Y_N,N$. Note that the last $N$ is discrete and $Y_i$ are continuous.
• thanks Liron for your answer. the answer for your first and second question is No. I think you are right about 3,4. @Liron The likelihood should look like? $\prod_i f(y_i)P(N=n)$ Where $y_i =$ is he realisation of $Y_i$ according to (3) in your comment – Danny Feb 23 '15 at 22:25