Hi guys,(adsbygoogle = window.adsbygoogle || []).push({});

I would like to ask you where you spot the mistake in the derivatives of the loglikelihood function of the cauchy distribution, as I am breaking my head :( I apply this to a newton optimization procedure and got correct m, but wrong scale parameter s. Thanks!

[tex]

LLF = -n\ln(pi)+n\ln(s)-\sum(\ln(s^2+(x-m)^2)),

[/tex]

First Derivatives:

[tex]

\frac {dL} {dm} = 2\sum(x-m) / \sum(s^2+(x-m)^2)

[/tex]

[tex]

\frac {dL} {ds} = n/s - 2\sum(s) / \sum(s^2+(x-m)^2)

[/tex]

Second Derivatives:

[tex]

\frac {d^2L} {dm^2} = (-2n(\sum(s^2+(x-m)^2)))+4\sum(x-m)^2)/(\sum(s^2+(x-m)^2))^2

[/tex]

[tex]

\frac {d^2L} {ds^2} =-n/s^2-2\sum(-s^2+(x-m)^2)/(\sum(s^2+(x-m)^2))^2

[/tex]

[tex]

\frac {d^2L} {dmds} =-4\sum(s(x-m))/(\sum(s^2+(x-m)^2))^2

[/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Derivatives of Cauchy Distribution

**Physics Forums | Science Articles, Homework Help, Discussion**