Given a sequence of iid samples $X_1, \dots, X_n,$ where each $X_i$ comes from a Weibull distribution with shape parameter $k$ and scale parameter $\lambda$. Then it is a well-known result that the maximum $M_n:=max(X_1,\dots, X_n)$ converges weakly to a Gumbel distribution.

I am wondering: How large should $n$ be in practice such that $M_n$ can be fitted reasonably by a Gumbel distribution?

The reason why I ask the question is as follows. I performed some simulation studies in R, where I simulated Weibull distributed samples for different sample sizes, shape and scale parameters and then I fitted a Gumbel distribution to the resulting maxima. I plotted the histogram of the maxima and plotted the estimated Gumbel density to compare both. Consider the following example code:

k <- 1
lambda <- 1
n_sim <- 1e5
n <- 1e4
m <- numeric(n_sim)
for (i in 1:n_sim) {
    m[i] <- max(rweibull(n, shape = k, scale = lambda))
gumbel_fit <- fitdist(m, "gumbel", start = list("mu" = 0, "sigma" = 1))$estimate
m_hist <- hist(m, breaks = 50, freq = F)
x_seq <- seq(min(m_hist$breaks), max(m_hist$breaks), , 1000)
y <- dgumbel(x_seq, mu = gumbel_fit[1], sigma = gumbel_fit[2])
lines(x_seq, y, col = "red")

Taking $k=1, \lambda = 1, n=10^4,$ and $n_{sim}=10^5$ gives the following plot.enter image description here

In this case we see an almost perfect fit.

Now change the shape parameter to $k=0.3$ and keep all other parameters. This results in the following.enter image description here

For this scale parameter the Gumbel distribution yields no sufficient approximation. In particular the bad approximation of the right tail is troublesome, as I am mainly interested in big quantiles of the estimated distribution. Fitting a Generalized Extreme Value distribution yields much better approximations.

Now let $k=3$ and keep all other parameters. This yields the following plot. enter image description here

Again, it seems that the Gumbel distribution cannot fit the data, which can be seen nicely in the tails.

To conclude: Although the Weibull distribution belongs to the maximum domain of attraction of the Gumbel distribution, the goodness of fit of a Gumbel distribution depends heavily on the parameters and for some combinations of scale and shape parameter it seems to take a ridiculous sample size $n$ that $M_n$ can be fitted reasonably by a Gumbel distribution.

Are my conclusions correct? How can you estimate the sample size $n$ that it takes for a reasonable fit? Is it advisable in practice to use the Gumbel distribution or should one always fit a Generalized Extreme Value distribution to the data?

  • $\begingroup$ Welcome to CV. The well-known convergence result is for $[M_n - b_n]/a_n$ where the two sequences $a_n>0$ and $b_n$ matter. The determination of these sequences is quite technical, but usable sequences can be found in the classical references on Extreme-Value Theory, see my answer to this question. Yet I never heard about an analogue of the Berry-Esseen bound for the Fisher-Tippett-Gnedenko theorem... $\endgroup$
    – Yves
    Commented Aug 16, 2019 at 18:31
  • $\begingroup$ Thank you for your answer! Yeah, I looked up the results on the sequences just today in the book by Embrechts, Klüppelberg and Mikosch but I don't know how to translate them into some kind of convergence rate. If $k=1$, the sequence $a_n$ is independent of $n$, when I remenber it correctly at the moment. Hence, the convergence in this case is really good. But if $k\neq 1$, the convergence rate seems to be very slow. $\endgroup$
    – jfiedler
    Commented Aug 16, 2019 at 20:21
  • $\begingroup$ My comment (by mistake, I first I pasted it in an answer, deleted) tells that the convergence does not hold if you ignore $a_n$ and $b_n$. But as you noticed, the convergence is faster when $k \approx 1$ the tail being then nearly Gumbel. I can not answer to your question: you ask for a bound on $|F_{M_n}(x) - F_{\text{Gum}}(x)|$ $\endgroup$
    – Yves
    Commented Aug 17, 2019 at 8:25
  • $\begingroup$ Yes, such an bound would very interesting. However, until now I found some results for the normal distribution: Peter Hall showed that $\sup_{x\in\mathbb R}|F_{M_n}(x) - F_{Gum}(x)|$ converges to zero with $log(n)$ and that this is the best you can get. I fear you won't get a better result for Weibull random variables. This implies for the practice that you may need insanely large sample sizes to trust the approximation in the tails, I fear. $\endgroup$
    – jfiedler
    Commented Aug 19, 2019 at 11:42
  • $\begingroup$ Thanks for this interesting neat result. Some classical books contain parts about convergence rates: sections 3.3, 3.4 of Beirlant et al, chap. 4 of de Haan and Ferreira. In both cases, some math is still needed to find the bound:) $\endgroup$
    – Yves
    Commented Aug 19, 2019 at 12:19


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