Cauchy functions problem for Calculus II

[Jyupiter]

So I've apparently been given an assignment on Cauchy functions (it says here on the title), but I have no idea what that means. Nevertheless, here's my attempt to solve this problem:

Given (1):
$f(x)=\frac{k}{\pi}\times\frac{1}{k^{2}+(x-\beta)^{2}}$

and (2):
$\hat{\beta}_{k}=arg\ max_{\beta} (\frac{k}{\pi})^{N}\ \prod^{N}_{i=1}\frac{1}{k^{2}+(x_{i}-\beta)^{2}}$

Describe the effect of k on (2) corresponding to (1) as shown here:http://img854.imageshack.us/img854/2724/graphjz.th.jpg . The function in (2) can be read as the values of β that would maximize the function in (1) or argument that would maximize (1).

I'm assuming that the question implies N=1 and hence k=1 for (2), and what $arg\ max$ means here is crudely what $\beta$ value would achieve the maximum point of the function. I'm convinced that $\hat{\beta}_{k}$ is somehow related to ${arg\ min}_{\beta} \sum^{N}_{i=1} |\beta-x_{i}|$ hence $x_{i}\approx\hat{\beta}_{k}$, since $|\frac{1}{a}|<|\frac{1}{b}|$ iff $|a|>|b|$ for all $a,b\in\mathbb{R}$ and $a,b\not =0$; and that $|a|<|a|+|b|$ for all $a,b\in\mathbb{R}$ and $b\not =0\ \Rightarrow\ b\rightarrow 0$.

Here's where I'm stuck, I don't see how k could affect $\hat{\beta}_{k}$, unless $\hat{\beta}_{k}$ is affected by the maximum point of the function. So how is the maximum point related to $\hat{\beta}_{k}$?

EDIT: Added the complete question and changed (2). There's a typo in the equation. Sorry for not doing it earlier.

 

[micromass]

What exactly is the question? Something like

Describe the effect of k on (2) corresponding to (1).

Doesn't make any sense to me.

Is the question to actually find $\hat{\beta_k}$?? In that case, I'd start by taking logarithms.

 

[Jyupiter]

The complete question reads like this: "Describe the effect of k on (2) corresponding to (1). The function in (2) can be read as the values of β that would maximize the function in (1) or argument that would maximize (1)."

There's an accompanying image:
http://img854.imageshack.us/img854/2724/graphjz.th.jpg

From what I gather from this, I think it wants to find how k (not K) can change $\hat{\beta}_{k}$.

 

[Jyupiter]

Major Update!

So I've spotted an error in the question, namely (2), which states k as a constant and a variable at the same time. I've reconciled this with my lecturer and it's confirmed that there's a misprint in the equation. (2) should actually look like this:
$\hat{\beta}_{k}=arg\ max_{\beta} (\frac{k}{\pi})^{N}\ \prod^{N}_{i=1}\frac{1}{k^{2}+(x_{i}-\beta)^{2}}$
The difference is in the $\prod^{N}_{i=1}$.

I've changed post #1. Problem is I'm still stuck.

 

[Jyupiter]

Correct me if I'm wrong, but the figure attached seems to hint at $K=max\ \{ (\frac{k}{\pi})^{N}\ \prod_{i=1}^{N}\ \frac{1}{k^2+(x_{i}-\beta)^2}\}$.

Then $x_{i}\rightarrow\beta\ \Rightarrow K=(\frac{1}{k\pi})^{N}\ \Rightarrow k^{N}\propto \frac{1}{K}$ and $k\not = 0$. (I'm sure this is flawed; surely not all $x_{i}\rightarrow\beta$. Is it possible to be more accurate?)

The image indicates that different $K$ do give about different $\hat{\beta}_{k}$, but how? It does seem like it has something to do with the diferent values of $\sqrt{|x_{i}-\hat{\beta}_{k}|^{2}+K^{2}}$... Perhaps something relating to $arg\ min_{\beta}\ \sum_{i=1}^{N}\ \sqrt{|x_{i}-\beta|^{2}+K^{2}}$ ?

I'm going about in circles.

EDIT: My lecturer says that one of the solutions involved the usage of Newton's Method. How I have the faintest idea.

EDIT2: The assumption is no longer k=1 in post #1, since we have changed (2). Can't edit my first post so I'll just post it here.

 

[Jyupiter]

Lecturer revealed the solution:

$\hat{\beta}_k=arg\ max_\beta\ \prod_{i=1}^{N}\ \frac{1}{k^{2}+(x_{i}-\beta)^{2}}$
$\Rightarrow \hat{\beta}_k=arg\ min_\beta\ \prod_{i=1}^{N}\ k^{2}+(x_{i}-\beta)^{2}$
$\Rightarrow \hat{\beta}_k=arg\ min_\beta\ \sum_{i=1}^{N}\ ln\ (k^{2}+(x_{i}-\beta)^{2})$
let $g(\beta)=\sum_{i=1}^{N}\ ln\ [k^{2}+(x_{i}-\beta)^{2}]$
since $f(x)=g(x)+h(x) \Rightarrow f'(x)=g'(x)+h'(x)$
then $g'(\beta)=\sum_{i=1}^{N}\ \frac{-2(x_{i}-\beta)}{k^{2}+(x_{i}-\beta)^2}$
to find the critical point, $g'(\beta)=0$
hence $\sum_{i=1}^{N}\ \frac{-2}{k^{2}+(x_{i}-\beta)^2}\cdot (x_{i}-\beta)=0$
$\Rightarrow \sum_{i=1}^{N}\ [\frac{-2}{k^{2}+(x_{i}-\beta)^2}\cdot x_{i} - \frac{-2}{k^{2}+(x_{i}-\beta)^2}\cdot \beta] = 0$
$\Rightarrow \sum_{i=1}^{N}\ [\frac{2}{k^{2}+(x_{i}-\beta)^2}\cdot \beta - \frac{2}{k^{2}+(x_{i}-\beta)^2}\cdot x_{i}] = 0$
Using Newton's Method to approximate $\hat{\beta}_{k}$,
$\hat{\beta}_{n+1}=\frac{\frac{2}{k^{2}+(x_{i}-\beta_{n})^2}\cdot x_{i}}{\frac{2}{k^{2}+(x_{i}-\beta_{n})^2}}$.

I don't follow all of the steps, especially how step 2 implies step 3, and how the Newton's method is used here, but most among these confusion is how the cheese does the solution answer how $k$ affects $\hat{\beta}_{k}$? The solution only seems to find what $\hat{\beta}_{k}$ is...