Maximum likelihood estimations for the parameters of extreme value distributions are discussed in this paper using fixed point iteration. The commonly used numerical approach for addressing this problem is the Newton-Raphson approach which requires differentiation unlike the fixed point iteration which is also easier to implement. Graphical approaches are also usually proposed in the literature. We prove that these reduce in fact to the fixed point solution proposed in this paper.
Deep Dive into Fixed Point Iteration for Estimating The Parameters of Extreme Value Distributions.
Maximum likelihood estimations for the parameters of extreme value distributions are discussed in this paper using fixed point iteration. The commonly used numerical approach for addressing this problem is the Newton-Raphson approach which requires differentiation unlike the fixed point iteration which is also easier to implement. Graphical approaches are also usually proposed in the literature. We prove that these reduce in fact to the fixed point solution proposed in this paper.
Extreme value distributions are largely used in applied engineering and environmental problems (see the book by Coles [4]). Parameter estimation is the first step in the statistical analysis of parametric probability distributions. The most widely used approach is the popular maximum likelihood estimation (MLE), but usually, as for extreme value distributions, the solution is not analytic and must be then approached by numerical techniques. The commonly used one is the Newton-Raphson algorithm to determine the maximum likelihood estimates of the parameters. To employ the algorithm, the second derivatives of the log-likelihood are required. Sometimes the calculations of the derivatives based on the progressively Type II censored samples for example are complicated (see [7]). To avoid such computation, we propose to use the fixed point iteration algorithm instead. In this paper we prove that generally MLE of extreme value distributions could be expressed in a fixed point iteration form which is easier to implement and does not require differentiation. Graphical techniques were also proposed as alternatives (see Balakrishnan and Kateri [2] and Dodson [5]) which have the disadvantage of using visualization to detect the solution from graphics. We prove that these graphical solutions reduce in fact to fixed points of a suitable iteration forms.
In the following section, we propose fixed point iterations for estimating the parameters of the Gumbel and Weibull distributions from complete data. In section 3, we extend the procedure to censored samples (simple and progressive Type I and Type II) using the Type I least extreme values distribution from which estimations for Gumbel and Weibull distributions can be deduced from suitable transformations. Finally, in section 4 we illustrate the proposed approach using examples quoted from the literature.
2 Maximum Likelihood Estimations for Complete Data
The Type I extreme value distribution which is also called Gumbel distribution function is defined as:
for x ∈ R, and it has the probability density
where σ > 0 and µ ∈ R. Let x = (x 1 , …, x n ) ′ be a complete sample from the Type I extreme value distribution. The maximum likelihood estimator for σ is known to be the solution of the following equation
.
Denote the right hand side of (1) by g(σ; x). Then the equation ( 1) is in fact in a fixed point form σ = g(σ; x).
Unlike the Newton-Raphson method, which is the commonly used nonlinear numerical method for solving MLE of the extreme value distribution (see Cohen [3] p. 143), the fixed point approach does not require differentiation and then is more easier to use. Uniqueness of the solution of (1) can be proved using graphical techniques (see Balakrishnan and Kateri [2]). Indeed, the left hand side σ is monotone increasing on σ. We then have to show that g(σ; x) is monotone decreasing on σ.
It remains then to prove that
x 2 i exp -
and g * (σ; x) ≤ 0 by the Cauchy-Schwarz inequality. The last result can be deduced from the result of Balakrishnan and Kateri [2] from transformation between the Weibull and Type I extreme value distribution. It should be noted that lim σ→∞ g(σ; x) = 0 and
.
We deduce then that 0 ≤ σ ≤
and 0 ≤ g(σ; x) ≤
which also guarantees the existence of a solution (see Lemma 3.4.1 of [1]). After obtaining a solution for σ we deduce the MLE of µ from
Example 1 Consider the data about annual wind-speed maxima in km/h from 1947 to 1984 at Vancouver quoted from the software Xtremes 4.1 [10], provided with the book by Reiss and Thomas [8], stored in the file em-cwind.dat (the source is [6]). By fitting a Gumbel distribution to the data, they provide estimates for σ and µ as σ = 8.3 and µ = 60.3. The fixed point iteration (2) leads to the solution σ = 8.2891 and from (3) we obtain µ = 60.3504.
For the two parameters Weibull distribution W(θ, β) with pdf
which is the Type II extreme value distribution it can be deduced from [2] that
where g w (β; x) is given by
.
It has been proved in [2] that (g w (β; x)) -1 is a monotone increasing function then g w (β; x) is monotone decreasing which insures existence and uniqueness of the solution of the fixed point iteration (4). The MLE of the parameter θ is deduced from
3 Estimation for Censored Data
In the case of simple Type-II censored data, let r (1 < r < n) denote the number of observed lifetimes and x = (x (1) , x (2) , …, x (r) ) the ordered observed lifetimes from the Type I least extreme value distribution with probability density
If X is a random variable from a Type I greatest extreme value distribution with location parameter µ and shape parameter σ then -X follows a Type I least extreme value distribution with location parameter -µ and shape parameter σ [3]. But if we have a Type-II rigth censored data x = (x (1) , x (2) , …, x (r) ) from the Type I least extreme value distribution then y = (-x (r) , -x (r-1) , …, -x (1) ) will be a Type-II left censored data from the correspo
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