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...