# How to find best MLE using FindDistributionParameters

This question is an extension of this question, if we change the initial values FindDistributionParameters[data1, dist, {{a, 0.1}, {b, 0.1}, {c, 0.1}, {al, 0.1}, {be, 0.1}, {ph, 0.1}}] to FindDistributionParameters[data1, dist, {{a, 0.2}, {b, 0.1}, {c, 0.1}, {al, 0.1}, {be, 0.1}, {ph, 0.1}}] parameter estimates also change. So my question is if we have no idea about initial values then how we search best initial values or best MLE estimates which close to empirical distribution of data1. The empirical distribution with two parameters which are estimated with initial value's given above are plotted as

Not a best fit. The code of above plot is:

data1 = {275, 13, 147, 23, 181, 30, 65, 10, 300, 173, 106, 300, 300, 212, 300, 300, 300, 2, 261, 293, 88, 247, 28, 143, 300, 23, 300, 80, 245, 266}; g1 = NIntegrate[(2.51339*2.51952*3.21532)/Beta[2.87484, 0.0825351]* y^-(3.21532 + 1)*(1 + 0.0609138*2.51952*y^-3.21532)^-((2.87484*2.51339)/0.0609138 + 1)*(1 - (1 + 0.0609138*2.51952*y^-3.21532)^-(2.51339/ 0.0609138))^(0.0825351 - 1), {y, 0, x}];

g2 = NIntegrate[(1.98156*1.98026*3.2635)/Beta[2.64639, 0.090212]* y^-(3.2635 + 1)*(1 + 0.0328252*1.98026*y^-3.2635)^-((2.64639*1.98156)/0.0328252 + 1)*(1 - (1 + 0.0328252*1.98026*y^-3.2635)^-(1.98026/ 0.0328252))^(0.090212 - 1), {y, 0, x}];

Plot[{g1, g2, CDF[EmpiricalDistribution[data1], x]}, {x, 2, 400},
Exclusions -> None]

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Do you have the generating functions of your plots? –  Rod Jul 20 at 14:27
I add code in question –  Azeem Jul 20 at 14:31

Simply compare the value of loglikelihood of your estimates....

A = FindDistributionParameters[data1, dist, {{a, 0.1}, {b, 0.1}, {c, 0.1}, {al, 0.1}, {be, 0.1}, {ph,  0.1}}]

B = FindDistributionParameters[data1,  dist, {{a, 0.2}, {b, 0.1}, {c, 0.1}, {al, 0.1}, {be, 0.1}, {ph, 0.1}}]

LogLikelihood[dist /. A, data1]
-211.801

LogLikelihood[dist /. B, data1]
-212.959

So A is slightly a better fit.

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LogLikelihood tell us in two parameter estimates which one is the best, but my question is still stand which fit is best among all range of parameters `a,b,c,al,be,ph>0'. –  Azeem Jul 21 at 4:17
I would recommend you to run a table with multiple parameters as the starting values then compare the loglikelihood of all of them!! –  Morry Jul 22 at 9:44