# Usable estimators for parameters in Gumbel distribution

The Gumbel distribution has the general form: $$F(y)=\exp\left({-\exp{\left(-\frac{y-\mu}{\sigma}\right)}}\right), \quad y\in \mathbb{R}$$ where $\mu \in \mathbb{R}$ and $\sigma >0$. Let $W_1,...,W_n$ be random variables with that distribution and iid. I want to find good estimators for $\sigma$ and $\mu$. What would be the best way to do that?

• Maximum likelihood estimation is a common method. Method of moments is another possibility. Most statistical packages implement maximum likelihood estimation which is often used to estimate the parameters of a distribution given some data. – COOLSerdash Sep 26 '13 at 22:08
• Maximum likelihood estimation would give biased estimator I think since the distribution is not symmetric so what is a good choice then? – Raxel Sep 26 '13 at 23:35
• Raxel, why is bias so important? Why would you rule out an estimator simply because it's biased, irrespective of any other consideration? Do you think all unbiased estimators - no matter what - are automatically better than any biased estimator? (Hint: When you estimate $\sigma$ for a normal distribution, what estimator do you use? Is it unbiased for $\sigma$?). In any case, if you're going to reject @COOLSerdash's answer, you should state what your criteria are so we don't waste time making suggestions that contravene your unstated criteria and so will also be rejected. – Glen_b -Reinstate Monica Sep 27 '13 at 0:02
• Further, your reasoning is unsound: consider estimating the mean parameter, $\mu$ in an exponential $f_X(X)=1/\mu\, \exp(-x/\mu)$. The distribution is skew, yet the MLE (the sample mean) is unbiased in that case. On the other hand, consider the MLE of $\theta$ for a uniform on $(-\theta,\theta)$; the r.v. is symmetric yet ML is biased. Your above reasoning says it will be biased in the first case (and, presumably, not in the second). [However, this isn't the central issue; bias or no, I worry about why you ruled out ML estimation on the basis of bias alone.] – Glen_b -Reinstate Monica Sep 27 '13 at 2:02
• Just to add to what @Glen_b said, a common way of evaluating estimators is with the mean squared error(MSE). It's a fact that the MSE of an estimator $\hat \theta$ of a parameter $\theta$ can be written as $${\rm MSE}(\hat \theta) = {\rm bias}(\hat \theta)^2 + {\rm var}(\hat \theta)$$ So it's clear that a biased estimator can perform better than an unbiased estimator if its variance is lower. This is called the "bias variance tradeoff" and is part of the logic underlying regularized estimation (e.g. penalized maximum likelihood). – Macro Sep 27 '13 at 2:32

## The Gumbel distribution

The Gumbel distribution is often used to model the distribution of extreme values. It is one of the three particular cases of the more generalized extreme value distribution (GEV), namely when the parameter of the GEV $\xi$ equals $0$ (that's why the Gumbel distribution is sometimes called "type I extreme value distribution"). The case of parameter fitting using maximum likelihood estimation (MLE) of a Gumbel distribution is discussed in Stuart Coles' book An Introduction to Statistical Modeling of Extreme Values (pages 55 ff.).

The Gumbel distribution's CDF is $$F_{X}(x)=\exp\left({-\exp{\left(-\frac{x-\mu}{\sigma}\right)}}\right), \quad x\in \mathbb{R}$$ with $\mu \in \mathbb{R}$ and $\sigma >0$. The PDF is given by the expression

$$f_{X}(x)=\frac{1}{\sigma}\exp{\left(-\frac{x-\mu}{\sigma} \right)}\cdot \exp{\left\{ -\exp{\left(-\frac{x-\mu}{\sigma}\right)}\right\}}$$

Finally, the quantile function (i.e. inverse CDF) for probability $p$ is given by $$Q(p)=\mu-\sigma\cdot\log{\left(-\log{\left(p\right)}\right)}.$$

We will need those definitions later in R.

## Parameter estimation

Given that $X_1,...,X_n$ are iid variables following a Gumbel distribution, the log-likelihood is $$\ell(\mu,\sigma)=-n\log{(\sigma)}-\sum_{i=1}^{n}\left(\frac{x_{i}-\mu}{\sigma}\right)-\sum_{i=1}^{n}\exp{\left\{-\left(\frac{x_{i}-\mu}{\sigma}\right) \right\}}.$$ The log-likelihood can be maximized using standard numerical optimization algorithms.

In their book Statistical Distributions, Forbes et al. (2010) provide the MLE estimates for $\mu$ and $\sigma$. Namely, the estimators $\hat{\mu},\hat{\sigma}$ are the solutions of the simultaneous equations \begin{align} \hat{\mu} &=-\hat{\sigma}\log{\left[\frac{1}{n}\sum_{i=1}^{n}\exp{\left(-\frac{x_{i}}{\hat{\sigma}}\right)} \right]}\\ \hat{\sigma} &= \bar{x}-\frac{\sum_{i=1}^{n} x_{i}\exp{\left(-\frac{x_{i}}{\hat{\sigma}}\right)}}{ \sum_{i=1}^{n} \exp{\left(-\frac{x_{i}}{\hat{\sigma}}\right)}}\\ \end{align} where $\bar{x}$ denotes the sample mean.

The maximum likelihood estimation can be done in R (other statistical packages such as Stata and SAS provide similar capacities). Because the Gumbel distribution is not available by default in R, we have to define the CDF, PDF and quantile function, which is straightforward. Here is an example R-script that does the trick:

#=================================================================================
#=================================================================================

library(fitdistrplus)

#=================================================================================
# Define the PDF, CDF and quantile function for the Gumbel distribution
#=================================================================================

dgumbel <- function(x,mu,s){ # PDF
exp((mu - x)/s - exp((mu - x)/s))/s
}

pgumbel <- function(q,mu,s){ # CDF
exp(-exp(-((q - mu)/s)))
}

qgumbel <- function(p, mu, s){ # quantile function
mu-s*log(-log(p))
}

#=================================================================================
# Some data (annual maximum mean daily flows ("annual floods")
#=================================================================================

flood.data <- c(312,590,248,670,365,770,465,545,315,115,232,260,655,675,
455,1020,700,570,853,395,926,99,680,121,976,916,921,191,
187,377,128,582,744,710,520,672,645,655,918,512,255,1126,
1386,1394,600,950,731,700,1407,1284,165,1496,809)

#=================================================================================
# Fit the Gumbel distribution using maximum likelihood estimation (MLE)
# Make some diagnostic plots
#=================================================================================

gumbel.fit <- fitdist(max.data, "gumbel", start=list(mu=5, s=5), method="mle")

summary(gumbel.fit)

Fitting of the distribution ' gumbel ' by maximum likelihood
Parameters :
estimate Std. Error
mu 471.6864   43.33664
s  298.8155   32.11813
Loglikelihood:  -385.1877   AIC:  774.3754   BIC:  778.316
Correlation matrix:
mu         s
mu 1.0000000 0.3208292
s  0.3208292 1.0000000

gofstat(gumbel.fit, discrete=FALSE) # goodness-of-fit statistics

Goodness-of-fit statistics
1-mle-gumbel
Kolmogorov-Smirnov statistic   0.09956968
Cramer-von Mises statistic     0.08826106
Anderson-Darling statistic     0.53360850

Goodness-of-fit criteria
1-mle-gumbel
Aikake's Information Criterion     774.3754
Bayesian Information Criterion     778.3160

# Plot the fit

par(cex=1.2, bg="white")
plot(gumbel.fit, lwd=2, col="steelblue")


The maximum likelihood estimates are $\hat{\mu} = 471.69, \hat{\sigma}=298.82$ with respective standard errors $\widehat{\mathrm{SE}}_{\hat{\mu}}=43.34, \widehat{\mathrm{SE}}_{\hat{\sigma}}=32.12$. The fit looks reasonable (there are some hints of systematic deviations) and the package fitdistrplus provides the estimated standard errors of the parameters and goodness-of-fit statistics.