ML estimators for Box-Cox Extreme Value distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg105180] ML estimators for Box-Cox Extreme Value distribution
• From: Petri Tötterman <petri.totterman at hanken.fi>
• Date: Mon, 23 Nov 2009 06:53:58 -0500 (EST)

Dear all,

I have a set of data, and I want to fit a distribution on this data. I
am particularily interested in the Box-Cox General Extreme Value
distribution, for which I have the CDF and PDF:
---
BoxCoxGEVDistribution /:

CDF[BoxCoxGEVDistribution[\[Mu]_, \[Sigma]_, \[Xi]_, \[Phi]_], x_] :=
1 + (((Exp[-(1 + \[Xi] ( (x - \
\[Mu])/\[Sigma]))^(-1/\[Xi])])^\[Phi]) - 1)/\[Phi];

BoxCoxGEVDistribution /:

PDF[BoxCoxGEVDistribution[\[Mu]_, \[Sigma]_, \[Xi]_, \[Phi]_], x_] =
D[CDF[BoxCoxGEVDistribution[\[Mu], \[Sigma], \[Xi], \[Phi]], x],
x];
---

I do also have a Log likelihood function from (Bali, 2007):

---
LLdBoxCoxGEV[\[Mu]_, \[Sigma]_, \[Xi]_, \[Phi]_, M_] :=
Module[{k = Length[M]}, -k Log[\[Sigma]] -
k ((1 + \[Xi])/\[Xi]) Sum[
Log[1 + \[Xi] ((M[[i]] - \[Mu])/\[Sigma])], {i, 1, k}] -
k \[Phi] Sum[(1 + \[Xi] ((M[[i]] - \[Mu])/\[Sigma]))^(-(
1/\[Xi])), {i, 1, k}] ];
---

Obviously, \[Mu]_, \[Sigma]_, \[Xi]_, \[Phi]_ are parameters which I
need to estimate, and M is a list of values from the dataset,
exceeding a predefined limit.

I would be grateful for advice, how should I continue to find the
Maximum Likelihood estimators for this distribution, using Mathematica?

Best regards,
/petri

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