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  • To: mathgroup at smc.vnet.net
  • Subject: [mg91163] More Inquiries
  • From: "Dewi Anggraini" <dewi_anggraini at student.rmit.edu.au>
  • Date: Fri, 8 Aug 2008 07:13:10 -0400 (EDT)

Hi All

I've tried this following formula to Gamma distribution, it estimates the unknown parameters very well

n = 69;
x1 = {3, 5, 6, 14, 11, 7, 7, 10, 11, 5, 4, 19, 9, 2, 8, 5, 6, 6, 5,
   5,
       4, 5, 5, 6, 8, 5, 8, 6, 5, 16, 5, 18, 16, 21, 6, 3, 16, 8, 3,
   8,
       11, 2, 3, 4, 8, 7, 9, 10, 8, 11, 8, 10, 9, 12, 12, 9, 6, 12,
   3, 9,
       14, 7, 4, 13, 8, 14, 5, 8, 2};
pdf = PDF[GammaDistribution[\[Lambda], \[Beta]], x1]
logl = Plus @@ Log[pdf]
maxlogl = FindMinimum[-logl, {\[Lambda], 1}, {\[Beta], 1}]
mle = maxlogl[[2]]

{190.145, {\[Lambda] -> 3.72763, \[Beta] -> 2.16947}}

{\[Lambda] -> 3.72763, \[Beta] -> 2.16947}


and also the program produces the algebraic form of Gamma distribution by the following command and it calculates the probability of non-conformance (above the specification limit). The program also can produce the pdf graph.


PDF[GammaDistribution[\[Lambda], \[Beta]], t]

(\[ExponentialE]^-t/\[Beta] t^(-1 + \[Lambda]) \
\[Beta]^-\[Lambda])/Gamma[\[Lambda]]

Integrate[
 PDF[GammaDistribution[3.7276258602234646, 2.169465718015002], t], {t,
   8, Infinity}]

0.439226

Plot[PDF[GammaDistribution[3.7276258602234646, 2.169465718015002],
  t], {t, 2, 21}]


However, when I apply the same command towards Burr distribution (the following program below), especially for the case of producing algebraic formula of the pdf, calculating the probability of non-conformance and drawing the pdf graph (the last three commands), the program does not work very well. Do I have done something wrong in the program?



n = 69;
x1 = {3, 5, 6, 14, 11, 7, 7, 10, 11, 5, 4, 19, 9, 2, 8, 5, 6, 6, 5,
   5,
       4, 5, 5, 6, 8, 5, 8, 6, 5, 16, 5, 18, 16, 21, 6, 3, 16, 8, 3,
   8,
       11, 2, 3, 4, 8, 7, 9, 10, 8, 11, 8, 10, 9, 12, 12, 9, 6, 12,
   3, 9,
       14, 7, 4, 13, 8, 14, 5, 8, 2};
BurrDistribution[x1_, c_,
  k_] := (c*k)*(x1^(c - 1)/(1 + x1^c)^(k + 1))
pdf = BurrDistribution[x1, c, k]
logl = Plus @@ Log[pdf]
maxlogl = FindMinimum[{-logl, c > 0 && k > 0}, {c, 1}, {k, 2}, 
     MaxIterations -> 1000]
mle = maxlogl[[2]]

{249.647, {c -> 37.8115, k -> 0.0135614}}

{c -> 37.8115, k -> 0.0135614}


PDF[BurrDistribution[c, k], t]

Integrate[
 PDF[BurrDistribution[37.81151579009424, 0.01356141249769735], t] {t, 
   21, Infinity}]

Plot[PDF[BurrDistribution[37.81151579009424, 0.01356141249769735], 
  t], {t, 2, 21}]


I already got assistance from some of you about gaining MLE of Burr distribution for my data and as recommended now I can run it very well.

Now, please assist me again to find where I got wrong in my second case here (to find the algebraic form of Burr, do the integration and draw the graph). Thank You.


Kindly Regards,
Dewi




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