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Maximum likelihood estimation (SMLE package)

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Dear All,

I'm trying to calculate Maximum likelihood estimators using the SMLE package but when try to do so for the weibull distribution for example (see code below) I always get the error message:
During evaluation of In[14]:= Solve::nsmet: This system cannot be solved with the methods available to Solve. >>
I do get the right result when I use Solve to find the zero for the function after differentiation with respect to beta, but why is it not working when I'm trying exactly the same thing for alpha?
 
Thank you in advance!
Tonja
 

----here the code----
 
In[8]:= SuperLog[On]
 
SuperLog is now On.
 
In[10]:= L = \!\(
\*UnderoverscriptBox[\(\[Product]\), \(i = 1\), \(n\)]
\*FractionBox[\(
\*SuperscriptBox[\(E\), \(-
\*SuperscriptBox[\((
\*FractionBox[\(x[i]\), \(\[Beta]\)])\), \(\[Alpha]\)]\)]\ \[Alpha]\
\*SuperscriptBox[\((
\*FractionBox[\(x[i]\), \(\[Beta]\)])\), \(\(-1\) + \[Alpha]\)]\), \(\[Beta]\)]\)
 
Out[10]= \!\(
\*UnderoverscriptBox[\(\[Product]\), \(i = 1\), \(n\)]
\*FractionBox[\(
\*SuperscriptBox[\(E\), \(-
\*SuperscriptBox[\((
\*FractionBox[\(x[i]\), \(\[Beta]\)])\), \(\[Alpha]\)]\)]\ \[Alpha]\
\*SuperscriptBox[\((
\*FractionBox[\(x[i]\), \(\[Beta]\)])\), \(\(-1\) + \[Alpha]\)]\), \(\[Beta]\)]\)

In[11]:= LogL = Log[L]
 
Out[11]= \[Beta]^-\[Alpha] (n \[Beta]^\[Alpha] (Log[\[Alpha]] - \
\[Alpha] Log[\[Beta]]) + (-1 + \[Alpha]) \[Beta]^\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[i]\), \(\[Alpha]\)]\))
 
In[13]:= score = {\!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Alpha]\)]LogL\), \!\(
\*SubscriptBox[\(\[PartialD]\), \(\[Beta]\)]LogL\)}

Out[13]= {-\[Beta]^-\[Alpha] Log[\[Beta]] (n \[Beta]^\[Alpha] (Log[\
\[Alpha]] - \[Alpha] Log[\[Beta]]) + (-1 + \[Alpha]) \[Beta]^\[Alpha] \
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[
i]\), \(\[Alpha]\)]\)) + \[Beta]^-\[Alpha] (n \
\[Beta]^\[Alpha] (1/\[Alpha] - Log[\[Beta]]) +
n \[Beta]^\[Alpha] Log[\[Beta]] (Log[\[Alpha]] - \[Alpha] Log[\
\[Beta]]) + \[Beta]^\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) + (-1 + \[Alpha]) \[Beta]^\[Alpha] Log[\[Beta]] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[x[i]]\
\*SuperscriptBox[\(x[
i]\), \(\[Alpha]\)]\)\)), \[Beta]^-\[Alpha] (-n \[Alpha] \
\[Beta]^(-1 + \[Alpha]) +
n \[Alpha] \[Beta]^(-1 + \[Alpha]) (Log[\[Alpha]] - \[Alpha] \
Log[\[Beta]]) + (-1 + \[Alpha]) \[Alpha] \[Beta]^(-1 + \[Alpha]) \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\)) - \[Alpha] \[Beta]^(-1 - \[Alpha]) (n \[Beta]^\
\[Alpha] (Log[\[Alpha]] - \[Alpha] Log[\[Beta]]) + (-1 + \[Alpha]) \
\[Beta]^\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[i]\), \(\[Alpha]\)]\))}
 
In[14]:= Solve[score == {0, 0}, {\[Alpha], \[Beta]}]
 
During evaluation of In[14]:= Solve::nsmet: This system cannot be solved with the methods available to Solve. >>
 
Out[14]= Solve[{-\[Beta]^-\[Alpha] Log[\[Beta]] (n \[Beta]^\[Alpha] \
(Log[\[Alpha]] - \[Alpha] Log[\[Beta]]) + (-1 + \[Alpha]) \[Beta]^\
\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[
i]\), \(\[Alpha]\)]\)) + \[Beta]^-\[Alpha] (n \[Beta]^\
\[Alpha] (1/\[Alpha] - Log[\[Beta]]) +
n \[Beta]^\[Alpha] Log[\[Beta]] (Log[\[Alpha]] - \[Alpha] Log[\
\[Beta]]) + \[Beta]^\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
x[i]]\)\) + (-1 + \[Alpha]) \[Beta]^\[Alpha] Log[\[Beta]] \
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
          x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[x[i]]\
\*SuperscriptBox[\(x[
           i]\), \(\[Alpha]\)]\)\)), \[Beta]^-\[Alpha] (-n \[Alpha] \
\[Beta]^(-1 + \[Alpha]) +
       n \[Alpha] \[Beta]^(-1 + \[Alpha]) (Log[\[Alpha]] - \[Alpha] \
Log[\[Beta]]) + (-1 + \[Alpha]) \[Alpha] \[Beta]^(-1 + \[Alpha]) \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
          x[i]]\)\)) - \[Alpha] \[Beta]^(-1 - \[Alpha]) (n \[Beta]^\
\[Alpha] (Log[\[Alpha]] - \[Alpha] Log[\[Beta]]) + (-1 + \[Alpha]) \
\[Beta]^\[Alpha] \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(Log[
          x[i]]\)\) - \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[i]\), \(\[Alpha]\)]\))} == {0,
   0}, {\[Alpha], \[Beta]}]
 
In[15]:= Solve[score[[2]] == 0, \[Beta]]
 
Out[15]= {{\[Beta] -> n^(-1/\[Alpha]) (\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\(x[i]\), \(\[Alpha]\)]\))^(1/\[Alpha])}}

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Maximum likelihood estimation (SMLE package) "Tonja Krueger" <tonja.krueger@web.de> - 2011-05-18 11:17 +0000

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