# A note on the estimation of extreme value distributions using maximum product of spacings

**ABSTRACT** The maximum product of spacings (MPS) is employed in the estimation of the Generalized Extreme Value Distribution (GEV) and the Generalized Pareto Distribution (GPD). Efficient estimators are obtained by the MPS for all $\gamma$. This outperforms the maximum likelihood method which is only valid for $\gamma<1$. It is then shown that the MPS gives estimators closer to the true parameters compared to the maximum likelihood estimates (MLE) in a simulation study. In cases where sample sizes are small, the MPS performs stably while the MLE does not. The performance of MPS estimators is also more stable than those of the probability-weighted moment (PWM) estimators. Finally, as a by-product of the MPS, a goodness of fit statistic, Moran's statistic, is available for the extreme value distributions. Empirical significance levels of Moran's statistic calculated are found to be satisfactory with the desired level.

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**ABSTRACT:**Extreme value methodology is being increasingly used by practitioners from a wide range of fields. The importance of accurately modeling extreme events has intensified, particularly in environmental science where such events can be seen as a barometer for climate change. These analyses require tools that must be simple to use, but must also implement complex statistical models and produce resulting inferences. This document presents a review of the software that is currently available to scientists for the statistical modeling of extreme events. We discuss all software known to the authors, both proprietary and open source, targeting different data types and application areas. It is our intention that this article will simplify the process of understanding the available software, and will help promote the methodology to an expansive set of scientific disciplines.Extremes 16(1). · 1.40 Impact Factor

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arXiv:math/0702830v1 [math.ST] 27 Feb 2007

IMS Lecture Notes–Monograph Series

Time Series and Related Topics

Vol. 52 (2006) 272–283

c ? Institute of Mathematical Statistics, 2006

DOI: 10.1214/074921706000001102

A note on the estimation of extreme value

distributions using maximum product

of spacings

T. S. T. Wong1and W. K. Li1

The University of Hong Kong

Abstract: The maximum product of spacings (MPS) is employed in the es-

timation of the Generalized Extreme Value Distribution (GEV) and the Gen-

eralized Pareto Distribution (GPD). Efficient estimators are obtained by the

MPS for all γ. This outperforms the maximum likelihood method which is only

valid for γ < 1. It is then shown that the MPS gives estimators closer to the

true parameters compared to the maximum likelihood estimates (MLE) in a

simulation study. In cases where sample sizes are small, the MPS performs sta-

bly while the MLE does not. The performance of MPS estimators is also more

stable than those of the probability-weighted moment (PWM) estimators. Fi-

nally, as a by-product of the MPS, a goodness of fit statistic, Moran’s statistic,

is available for the extreme value distributions. Empirical significance levels of

Moran’s statistic calculated are found to be satisfactory with the desired level.

1. Introduction

The GEV and the GPD (Pickands, [13]) distributions are widely-adopted in extreme

value analysis. As is well known the maximum likelihood estimates (MLE) may fail

to converge owing to the existence of an unbounded likelihood function. In some

cases, MLE can be obtained but converges at a slower rate when compared to that

of the classical MLE under regular conditions.

Recent studies (e.g. Juarez & Schucany, [9]) show that maximum likelihood esti-

mation and other common estimation techniques lack robustness. In addition, the

influence curve of the MLE is shown unstable when the sample size is small. Al-

though new methods (Juarez & Schucany, [9]; Peng and Welsh, [12]; Dupuis, [6])

were proposed, arbitrary parameters are sometimes involved, resulting in more in-

tensive computation which is in general undesirable. There have been studies in

overcoming the difficulties of the MLE in extreme value analysis but none has con-

sidered the MPS. Furthermore, a goodness-of-fit test on the fitted GEV or GPD is

rarely considered.

In this study, the MPS method will first be considered for the purpose of finding

estimators which may not be obtained by the maximum likelihood method. As a

by product, the Moran’s statistic, a function of product of spacings, can be treated

as a test statistics for model checking. This is one of the nice outcomes of MPS

which Cheng and Stephens [4] demonstrated but is overlooked by the extreme value

analysis literature.

In Section 2, we discuss some problems of the MLE. In Section 3, we formulate the

MLE, the MPS and the Moran statistics. In Section 4, results of simulation studies

1Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam

Road, Hong Kong, e-mail: h0127272@hkusua.hku.hk

Keywords and phrases: generalized extreme value distribution, generalized Pareto distribution,

maximum product of spacings, maximum likelihood, Moran’s statistic.

272

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Estimation of extreme value distributions using maximum product of spacings273

are presented to evaluate the performance of the method proposed. In Section 5, we

provide some real examples in which the MPS is more convincing. A brief discussion

is presented in Section 6.

2. Problems of the MLE

The problems of the MLE in model fitting were discussed by Weiss and Wolfowitz

[15]. Related discussions in connection to the Weibull and the Gamma distributions

can be found in [2, 3, 5, 14]. Smith [14] found densities in the form

(2.1)

f(x;θ,φ) = (x − θ)α−1g(x − θ;φ),θ < x < ∞

where θ and φ are unknown parameters and g converges to a constant as x ↓ θ.

As is well-known for α > 2, the MLE is as efficient as in regular cases. For α = 2,

the estimated parameters are still asymptotically normal, but the convergence rate

is (nlogn)

MLE exists but the asymptotic efficiency problem is not solved. And the order of

convergence could be as high as O(n

GEV and the GPD encounter the above difficulties as both can be reparameterised

into the form (2.1).

As an alternative to the MLE, the MPS was established by Cheng and Amin

[2]. With the MPS, not only can problems with non-regular condition be better

solved, but models originally estimable under the MLE framework can also be

better estimated by the MPS using a much simpler algorithm. Cheng & Amin [2]

showed that the MPS estimators are asymptotically normal even for 0 < α <

1. This overcomes to a certain extent the weakness existing in the MLE. Hence,

the MPS may be one of the most robust estimation techniques and yet the least

computational expensive in extreme value analysis. The present paper employs the

MPS in the estimation of the GEV and the GPD. On the other hand, many previous

studies (Hosking, [7]; Marohn, [10]) concentrated on testing the shape parameter.

Goodness-of-fit test on the model as a whole has been very few. In this study, the

Moran’s statistic (Cheng and Stephens, [4]; Moran, [11]) arising naturally as a by

product of the MPS estimator was utilized to check the adequacy of the overall

model.

1

2 which is larger than the classical rate of n

1

2. For 1 < α < 2, the

1

α). For α < 1, MLE does not exist. Both the

3. Formulations of the MLE, the MPS and the Moran’s statistic

3.1. The MLE and the MPS

The c.d.f of the GEV and the GPD are respectively

H(x;γ,µ,σ) = exp

?

−

?

1 − γx − µ

σ

?1

γ?

,

1 − γx − µ

σ

> 0;

and

G(x;γ,σ) = 1 −

?

1 − γx

σ

?1

γ,

1 − γx

σ> 0.

where

γ ?= 0,

−∞ < µ < ∞,σ > 0

Page 3

274T. S. T. Wong and W. K. Li

Let h(x) and g(x) be the corresponding densities,

h(x) =1

σ

?

1 − γx − µ

σ

?1

γ−1

exp

?

−

?

1 − γx − µ

σ

?1

γ?

;

and

g(x) =1

σ

?

1 − γx

σ

?1

γ−1

.

The log-likelihood functions per observation are respectively

LGEV(γ,µ,σ) = −logσ +

?1

γ− 1

?

log

?

1 − γx − µ

σ

?

−

?

1 − γx − µ

σ

?1

γ

;

and

LGPD(γ,µ,σ) = −logσ +

?1

γ− 1

?

log

?

γ, the information matrix of

1 − γx

σ

?

.

Applying the same argument stated in [14], as x ↓ µ+σ

LGEV(γ,µ,σ) is infinite for γ >1

In this case, the underlying distribution is J-shaped where maximum likelihood is

bound to fail. Worse still, MLEs (Denoted byˆ ΘGEV =

?ˆ γ, ˆ σ?Trespectively for the GEV and the GPD) may not exist when γ > 1. Let

GEV : Di(θ) = H(xi,γ,µ,σ) − H(xi−1;γ,µ,σ) , (i = 1,2,...,n + 1) ;

GPD : Di(θ) = G(xi,γ,σ) − G(xi−1;γ,σ) ,

2. The same difficulty arises in the GPD as x ↓σ

γ.

?ˆ γ, ˆ µ, ˆ σ?Tandˆ ΘGPD =

x1< x2< ··· < xnbe an ordered sample of size n and define spacings Di(θ) by

(i = 1,2,...,n + 1) ;

where H(x0;γ,µ,σ) ≡ G(x0;γ,σ) ≡ 0 and H(xn+1;γ,µ,σ) ≡ G(xn+1;γ,σ) ≡ 1.

MPS estimators (Denoted by˘ ΘGEV = (˘ γ, ˘ µ, ˘ σ)Tand˘ΘGPD= (˘ γ, ˘ σ)Trespec-

tively for the GEV and the GPD) are found by minimizing

M(θ) = −

n+1

?

i=1

logDi(θ).

By taking the cumulative density in the estimation, the objective function M(θ)

does not collapse for γ < 1 as x ↓ µ+σ

MLE, however, does not have such an advantage. There is in probability a solution

ˆ Θ that is asymptotically normal only for γ <1

is demonstrated by the following two theorems.

γfor the GEV or as x ↓σ

γfor the GPD. The

2. The strength of MPS over MLE

Theorem 3.1. Let Θ0GEV = (γ0,µ0,σ0)Tand Θ0GPD = (γ0,σ0)Tbe the true

parameters of the GEV and the GPD respectively. Under regularity conditions (See

for example: [14])

(i) For γ <1

2, n

1

2(ˆΘ − Θ0)

D

→ N

?

0,−E

?∂2L

D

→ Op[(nlogn)−1

∂Θ2

?−1?

;

(ii) For γ =1

2,

∂Θ2)−1), where Θ = (γ,σ)T;

?

?∂2L

?

ˆ µ +ˆ σ

ˆ γ

?

− (µ0+σ0

γ0)

2], and n

1

2(ˆΘ − Θ0)

D

→

N(0,−E(∂2L

(iii) For1

2< γ < 1,ˆ µ +ˆ σ

ˆ γ

?

−

?

µ0+σ0

γ0

?

D

→ Op(n−γ), and n

1

2(ˆΘ − Θ0)

D

→

N

?

0,−E

∂Θ2

?−1?

, where Θ is as in (ii).

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Estimation of extreme value distributions using maximum product of spacings 275

(iv) For γ ≥ 1, the MLE does not exist.

Theorem 3.2. Under the same conditions as in Theorem 3.1

→ N(0,−E(∂2L

(i) For γ <1

2, n

1

2,(˘ µ +˘ σ

1

2(˘Θ − Θ0)

D

∂Θ2)−1);

→ Op[(nlogn)−1

(ii) For γ =

˘ γ) − (µ0+σ0

γ0)

D

2], and n

1

2(˘Θ − Θ0)

D

→

N(0,−E(∂2L

∂Θ2)−1), where Θ = (γ,σ)T;

1

2, (˘ µ +˘ σ

(0,−E(∂2L

∂Θ2)−1), where Θ is as in (ii).

Proofs of Theorems 3.1 and 3.2 follow the arguments in [14] and [2] respectively

by checking the conditions therein.

It is obvious that efficient estimators can still be obtained by the MPS for γ >1

but not the MLE. From (iii) above, it is clear that the MPS still works while the

MLE fails for γ ≥ 1. It seems that it is a fact overlooked by researchers working in

the extreme value literature.

(iii) For γ >

˘ γ) − (µ0+σ0

γ0)

D

→ Op(n−γ), and n

1

2(˘Θ − Θ0)

D

→ N

2

3.2. Moran’s statistic

In the MPS estimation, M(θ) is called the Moran’s statistic which can be used as

a test for a goodness-of-fit test. Cheng and Stephens [4] showed that under the null

hypothesis, M(θ), being independent of the unknown parameters, has a normal

distribution and a chi-square approximation exists for small samples with mean

and variance approximated respectively by

µM≈ (n + 1)log(n + 1) −1

2−

1

12(n + 1),

and

σ2

M≈ (n + 1)

?π2

6

− 1

?

−1

2−

1

6(n + 1).

Define

C1= µM−

?1

2n

?1

2σM ,C2= (2n)−1

2σM .

The test statistic is

T(˘θ) =M(˘θ) +1

2k − C1

C2

which follows approximately a chi-square distribution of n degrees of freedom under

the null hypothesis. Monte Carlo simulation of the Weibull, the Gamma and the

Normal distributions in [4] showed the accuracy of the test based on T(˘θ). In the

next section, we provide further evidence supporting the use of MPS for fitting the

extreme value distributions.

4. Simulation study

A set of simulations was performed to evaluate the advantage of the MPS over the

MLE of the GEV and the GPD based on selected parameters for different sample

Page 5

276 T. S. T. Wong and W. K. Li

sizes n = (10,20,50). Empirical significance levels of Moran’s statistic were then

considered using χ2

n,αas the benchmark critical value. Finally, data were generated

from an exponential distribution and the cluster maxima of every 30 observations

were fitted to the GEV.

The subroutine DNCONF in the IMSL library was used to minimize a function.

The data analysed in the paper and the Fortran90 programs used in the computa-

tion are available upon request.

We have done extensive simulations to assess the performance of MPS estimators.

Only four simulation results in each combination of γ and n are reported. The

location and scale parameter, µ = 1 and σ = 1, were used throughout. On the

basis of the results from asymptotic normality of the MPS that were presented

in Section 3, we chose a combination of γ = (−0.2,0.2,1,1.2) to compare the

estimation performance between the maximum likelihood method and the MPS

where the last two cases should break down for the MLE. 10000 simulations of

sample sizes n = (10,20,50) were performed. Data were generated from the same

random seed and estimations were performed under the same algorithm. Define the

mean absolute error for the MLE and the MPS respectively by

1

l

??ˆ Θ

T

l− Θ01T??1

and

1

l

??˘ Θ

T

l− Θ01T??1 .

whereˆΘland˘ Θlare l × 1 vectors of the MLE and MPS estimators respectively,

|Y | means the element-wise absolute value of Y , p is the number of estimated

parameters and l = 10000 is the number of replications. The mean absolute error

measures the average deviation of estimators from the true parameters and hence

is a measure of robustness. A small mean absolute error is expected.

As suggested by a referee, the MPS was also compared to the method of

probability-weighted moment (PWM)(Hosking et al., [8]) for the GEV model. We

followed Hosking’s approach in his Table 3 and estimated the tail parameter by

Newton-Raphson’s Method. Tables 1 and 2 display the medians of the parameters

in 10000 estimations together with the mean absolute error in bracket. Both the

MPS estimates and the MLEs are in line with the true parameters but MPS tends

to give a closer result for the GEV. It can also be seen that the MPS gives much

more stable estimates than the MLE in general. For γ = −0.2 and γ = 0.2, the

PWM performed well with slightly smaller mean absolute errors than the MPS.

However, for γ = 1 and γ = 1.2, the bias of the PWM is rather severe. Note that

some of the mean absolute errors for the MLE are unacceptably large due to serious

outliers of estimated parameters. Non-regularity of the likelihood function caused

occasional non-convergence. The frequency of such problems is reported in Tables 3

and 4. Failures of convergence were detected when the magnitudes of any estimator

in an entry exceeds 100. The failure rates of MLE are relatively higher than those of

MPS. Some estimated parameters of the MLE went up to as high as 500000. This

explains the extremely large mean absolute errors of the MLE. Although there were

failures in MPS, the maximum values were less than 1000, comparably less severe

than the MLE. The PWM has zero failure rates but as mentioned above, it has a

severe bias when γ ≥ 1.It is noticed that the MLEs have smaller mean absolute

error only in cases where sample size is large. However, the MPS estimators have

virtually no fall off in its performance across sample sizes. These are in agreement

with the theoretical results in Theorems 3.1 and 3.2. Overall, the MPS seems to be

the most stable in its performance.

The Moran’s statistic, M(θ), has a chi-square distribution with n degrees of free-

dom. Monte Carlo simulations with 10000 observations per entry, each entry with