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Background:

For my research I've been trying to model brittle fracture mathematically. The basic concept is that a brittle materials fail because already existing small defects such as small cracks within the material start to grow in an unstoppable fashion.

Because the cracks are too small to detect them individually, a distribution of their length $ F(a) $ is assumened. The length of the crack $ a $ determines the strength of the crack. A longer crack leads to a lower strength.

Since as soon as a crack fails under a given mechanical load the whole component containing the cracks fails, the largest crack is dominating the strength of the component.

Problem:

Let's assume there are $ n $ cracks and all their lengths $ a_1 , a_2,..., a_n $ are independent and identically distributed by the distribution $ F(a) $. The distribution of the maximum lengths $a$ can then be discribed by: $$ \Phi (a) = F(a)^n. $$ Extreme value theory now says that for large $ n $ this distribution $ \Phi (a) $ becomes one of the three extreme value distributions.

Numerically I can show that if i assume a log-normal distribution for the distribution of crack length $$ F(a) = \frac{1}{\sqrt{2 \pi} \sigma } \int_0^x \frac{1}{t} \exp \left[ - \frac{\left( \ln t- \mu \right)^2 }{2 \sigma^2} \right] d t $$ the distribution $ \Phi (a) $ convergences to the Fréchet distribution $$ Fr (a) = \exp \left[ -\left( \frac{a}{\beta} \right)^{-\alpha} \right] $$ for large, but still finite, $n$. So far it has shown that the parameters $\alpha$ and $\beta$ of $ Fr(a) $ seem to be dependent on the initial distribution of cracks $ F(a)$ as well as on the number of cracks $ n $.

Question:

Is there a possibility to derive the parameters of the Fréchet Distribution $ Fr(a) $ for given parameters of the initial log-normal distribution $F(a)$ for finite numbers of $ n $? If so, how can they be derived?

Related questions:

In this question it is asked how to derive the parameters of the Gumbel extreme value distribution from a given normal distribution which is marked as a duplicate of this question. However, if I understand correctly, both of these questions assume $ n $ to be infinite.

EDIT

To maybe make some things a bit clearer, here is a short example of my numerical analysis done so far.

For the arbitrarily chosen parameters $ \mu = 3.2 $ and $ \sigma = 0.4 $ I use Mathematica's RandomVariate[] function to create a set of $100000$ pseudo-random reals which are distributed by a log-normal distribution with said parameters.

example distribution

From those I select the largest value. This process is repeated $1000$ times to get a distribution of the largest values.

In order to test which of the three extreme value distributions fits best, I use the ProbabilityScalePlot[] function from Mathematica, which plots the cumulative distribution of these largest values scaled for the Gumbel, the Fréchet and the Weibull distribution.

extreme value output

All numerical runs so far show that the Fréchet distribution seems to fit best.

I also tried to approach this problem analytically by reading for example this: Rate of convergence towards a Fréchet type limit distribution. But since my background is in mechanical engineering I'm lacking the skills to even know what is happening beyond page 2.

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  • $\begingroup$ The log-normal distribution is in the Gumbel Domain of Attraction, not in the Fréchet DA. $\endgroup$ – Yves Aug 7 '18 at 7:36
  • $\begingroup$ @Yves , thank you for your reply. I've added an edit where I show what my numerical analysis show so far and why I think the distribution of the largest values converges towards the Fréchet distribution. $\endgroup$ – Brasol Aug 7 '18 at 12:39
  • $\begingroup$ Hum... The plot with title "Gumbel scaled" (left) seems to be the same as the one with "weibull scaled" (right). $\endgroup$ – Yves Aug 7 '18 at 14:05
  • $\begingroup$ They look quite similar so I've double checked it. There is in fact a small difference between the Gumbel and the Weibull scaled plot. If it helps I could add the Mathematica code to my post but I think it's quite long as it is right now. $\endgroup$ – Brasol Aug 7 '18 at 14:46
  • $\begingroup$ Why do you need an approximation when you have the exact distribution for any finite sample size? $\Phi(a)=F(a)^n$ is the cumulative distribution function for the maximum of a sample of size $n$.. $\endgroup$ – JimB Jan 19 '19 at 5:47
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You don't need an approximation because you can obtain the exact distribution for any desired sample size. Because you are using Mathematica, you can get this directly:

PDF[OrderDistribution[{LogNormalDistribution[\[Mu], \[Sigma]], n}, n], z]

$$\frac{2^{\frac{1}{2}-n} n e^{-\frac{(\log (z)-\mu )^2}{2 \sigma ^2}} \text{erfc}\left(\frac{\mu -\log (z)}{\sqrt{2} \sigma }\right)^{n-1}}{\sqrt{\pi } \sigma z}$$

For your particular example, here are the Mathematica commands:

x = Max[#] & /@ RandomVariate[LogNormalDistribution[3.2, 0.4], {1000, 100000}];
Show[Histogram[x, Automatic, "PDF"],
 Plot[PDF[OrderDistribution[{LogNormalDistribution[3.2, 0.4], 100000}, 100000], z], {z, 110, 200}]]

Histogram and pdf

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