• To: mathgroup at smc.vnet.net
• From: Darren Glosemeyer <darreng at wolfram.com>
• Date: Wed, 6 Aug 2008 05:03:39 -0400 (EDT)
• References: <200808050803.EAA09872@smc.vnet.net>

```Dewi Anggraini wrote:
> Hi all
>
> I'm an international studentat RMIT University Melbourne. Currently, I'm
> doing my master degree in Statistics and Operations Research. This
> following 6 months will be my last semester to finish my master degree.
> Therefore, I have been doing minor thesis about "estimating unknown
> parameter of Burr distribution" by using Mathematica.
>
> I was wondering if this forum could assist me to detect where I got wrong
> with my program since it works for Burr distribution in some data but not
> with my data. Additionally, this program (I attached my program along with
> my data) also works for non-normal distribution, such as Gamma and Weibull
> since my data are non-normally distributed and they are closely to gamma
> and weibull distribution.
>
> The data are treatment time of cervical cancer patients in a hospital.
> Frequently, they come to the hospital in the late satge of cancer, thus
> the minimum time for the treatment is 2 days and the maximum treatment on
> the data is 21 days.
>
> The following is my program to run MLE of Burr distribution. However,
> it comes up with "comment" results in finding the coefficient of unknown
> parameters when I run it.
>
>
> n = 69;
> x2 = {5, 6, 5.5, 5, 8.4, 4.5, 4, 6, 7, 6, 7, 8, 5.4, 5, 4, 3.5, 4.5,
>        6.5, 5, 3.5, 5.7, 5.8, 5, 4.8, 4.5, 5.8, 3.2, 6, 4, 6, 7,
>    6.5,
>        10.7, 7, 4.7, 7, 8.3, 10, 6.5, 4.9, 3.4, 9, 6, 3.1, 5, 4.8, 4,
>    6,
>        6, 5.6, 4.2, 4.3, 4.5, 8.4, 8.6, 5.8, 6.8, 4.8, 3.4, 8, 8,
>    8.3, 8,
>        4.5, 6, 4.5, 7, 7, 8};
> BurrDistribution[x2_, c_,
>   k_] := (c*k)*(x2^(c - 1)/(1 + x2^c)^(k + 1))
> pdf = BurrDistribution[x2, c, k]
> logl = Plus @@ Log[pdf]
> maxlogl = FindMinimum[-logl, {c, 1}, {k, 2}]
> mle = maxlogl[[2]]
>
>
>
> Please assist me in finding the problem I face now. This is very important
> for my thesis.
>
>
> Regards,
> Dewi
>

The problem is that without knowledge of the constraints on the
parameter values the method steps to invalid values for the parameters
which results in a complex log likelihood. Adding the positivity
constraints on the parameters and increasing the number of iterations
gets a result.

In[1]:= n = 69;

In[2]:= x2 = {5, 6, 5.5, 5, 8.4, 4.5, 4, 6, 7, 6, 7, 8, 5.4, 5, 4, 3.5,
4.5,
6.5, 5, 3.5, 5.7, 5.8, 5, 4.8, 4.5, 5.8, 3.2, 6, 4, 6, 7, 6.5,
10.7, 7, 4.7,
7, 8.3, 10, 6.5, 4.9, 3.4, 9, 6, 3.1, 5, 4.8, 4, 6, 6, 5.6,
4.2, 4.3, 4.5, 8.4, 8.6, 5.8, 6.8, 4.8, 3.4, 8, 8, 8.3, 8,
4.5, 6,
4.5, 7, 7, 8};

In[3]:= BurrDistribution[x2_, c_,
k_] := (c*k)*(x2^(c - 1)/(1 + x2^c)^(k + 1));

In[4]:= pdf = BurrDistribution[x2, c, k];

In[5]:= logl = Plus @@ Log[pdf];

In[6]:= maxlogl =
FindMinimum[{-logl, c > 0 && k > 0}, {c, 1}, {k, 2},
MaxIterations -> 1000]

Out[6]= {226.436, {c -> 22.4199, k -> 0.0257487}}

In[7]:= mle = maxlogl[[2]]

Out[7]= {c -> 22.4199, k -> 0.0257487}

Darren Glosemeyer
Wolfram Research

```

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