A TEST OF FIT FOR THE GENERALIZED PARETO DISTRIBUTION BASED ON TRANSFORMS

Transcription

1 A TEST OF FIT FOR THE GENERALIZED PARETO DISTRIBUTION BASED ON TRANSFORMS Dimitrios Konstantinides, Simos G. Meintanis Department of Statistics and Acturial Science, University of the Aegean, Karlovassi, Samos, Greece and Department of Economics, National and Kapodistrian University of Athens, 8 Pesmazoglou Street, Athens, Greece Abstract. The generalized Pareto distribution is a very popular two parameter model for extreme events. In this article we develop a class of goodness of fit tests for the composite null hypothesis that the data at hand come from a Generalized Pareto distribution with unspecified parameter values. In doing so, the parameters are estimated, and the resulting estimates are employed in transforming the data, so that under the null hypothesis the transformed data approximately follow a unit exponential distribution. Then the null hypothesis of exponentiality is tested instead, by utilizing the empirical Laplace transform. The method is shown to be consistent, and the asymptotic null distribution of the test statistic is derived. The results of a Monte Carlo study are presented which include, apart from the proposed test, the standard methods based on the empirical distribution function, implemented by employing moment and maximum likelihood estimates, as well as probability weighted moments estimates. Keywords. Exterme events, Goodness of fit test, Empirical Laplace transform. AMS 2000 classification numbers: 62G10, 62G20 1

2 1 Introduction The analysis of series of observations consisting of the largest (or smallest) values was tradinionally based on the family of generalized extreme value distributions. However this approach has recently been critisized mainly because of its inefficient use of available data. One approach towards incorporating this loss of information is to consider several of the larger order statistics. Then the resulting exceedances (differences between these order statistics and a given high threshold) are typically modelled by the generalized Pareto distribution (GPD). The GPD is a two parameter model with a shape parameter α and a scale parameter c. Its distribution function is F (x; α, c) = 1 (1 (αx/c)) 1/α, α IR, c > 0, and has support x > 0 ( resp. 0 < x < c/α), if a 0 (resp. a > 0). We write GP(α, c) to denote the GPD with parameters α and c. For α < 0, the distribution is related to the P areto distribution of the second kind (see Johnson et al., 1994, p. 575), whereas for α = 0 and α = 1, respectively, the exponential with scale c and the uniform in (0,c) result. For applications of the GPD to such diverse areas of applied research as metereology, economics, ecology and reliability, the reader is referred to Choulakian and Stephens (2001), Singh and Ahmad (2004), and references therein. Recently, Choulakian and Stephens (2001) proposed classical goodness of fit tests for the null hypothesis H 0 : The law of X is GP(α, c) for some α IR and c > 0, based on independent copies {X j } n of the random variable X 0. These tests, namely the Kolmogorov Smirnov, the Crámer von Mises and the Anderson Darling test, utilize the empirical distribution function (EDF). In this article we develop a class of goodness of fit tests for the GPD based on the empirical Laplace transform. To this end consider, instead of X, the random variable Y = (1/α) log(1 (αx/c)), which under H 0 follows a unit exponential distribution. Consequently the Laplace transform (LT) L(t) = E(e ty ) of Y satisfies (1.1) (1 + t)l(t) 1 = 0, for all t > 0. 2

3 Hence we may test H 0, by employing the data (1.2) Y j = (1/ˆα) log(1 (ˆαX j /ĉ)), j = 1, 2,..., n, where ˆα (resp. ĉ) denotes a consistent estimator of the parameter α (resp. c). Under H 0 and for large n, {Y j } n will approximately follow a unit exponential distribution. Therefore it would be natural (in view of (1.1)) to base a test for H 0 on a measure of deviation from zero of the random function D n (t) = (1 + t)l n (t) 1 on [0, ], where L n (t) = 1 n exp( ty j ), is the empirical LT of the transformed data {Y j } n. Such a test for exponentiality was developed by Henze and Meintanis (2002) and takes the form (1.3) T n,a = n 0 D 2 n(t)e at dt, a > 0. Since it was shown that the test that rejects exponentiality for large values of T n,a, is more powerful than the classical tests referred to above, it is hoped that this will be so in the present, more general, situation. It should be emphasized that the empirical LT has proved a valuable tool for statistical inference, not only in the case of testing exponentiality, but also in goodness of fit tests for the inverse Gaussian (Henze and Klar, 2002) and the gamma distribution (Henze and Meintanis, 2004), as well as in the estimation context (Csörgő and Teugels, 1990, Feuerverger, 1989), to name a few. The paper is organized as follows. In Section 2 we derive the limit null distribution of T n,a and establish the consistency of the test based on T n,a against general alternatives. The results of a Monte Carlo study are presented in Section 3, where the new tests are compared to the EDF goodness of fit tests for the GPD law. The paper concludes with real data examples given in Section 4. 2 Theoretical results We briefly review the setting for asymptotic distribution theory. For more details the reader is referred to Henze and Meintanis (2004). A convenient setting is the separable Hilbert space H = L 2 (IR +, B, e at dt) of (equivalence classes of) measurable functions 3

4 f : IR + IR + satisfying 0 f 2 (t)e at dt <. The inner product and the norm in H are denoted by < f, g > and f, respectively. Here and in what follows, the notation D P means convergence in distribution of random elements and random variables, means convergence in probability, o P (1) stands for convergence in probability to 0, O P (1) denotes boundedness in probability, and i.i.d. means independent and identically distributed. In view of the parametric bootstrap procedure, which will be carried out to obtain critical values for T n,a, we study the null asymptotic distribution of T n,a under a triangular array X n1, X n2,..., X nn, n 1, of rowwise i.i.d. random variables, where X n1 GP(α n, 1), α n IR, and lim n α n = α. For the bootstrap procedure, α n = ˆα n is chosen. We further consider only regular estimators ˆα n and ĉ n of α and c, respectively. Namely we assume that under the triangular array referred to above, n(ˆαn α n ) = 1 n n(ĉn 1) = 1 n ψ 1 (X nj ; α n ) + ɛ n,1, ψ 2 (X nj ; α n ) + ɛ n,2, where for k = 1, 2, ɛ n,k = o p (1) and ψ k has zero mean, and finite second moment which satisfies lim n Eψk 2(X n1; α n ) = Eψk 2 (X; α), with X GP(α, 1). 3 Finite sample comparisons This section presents the results of a Monte Carlo study conducted at a 10% nominal level with replications to assess the performance of the new tests. In order to avoid reliance on asymptotic critical values and since the null distribution of all test statistics considered depends on the (unknown) value of the shape parameter α we performed a parametric bootstrap to obtain the critical point p n of the test as follows: Conditionally on the observed value of ˆα n, generate 100 bootstrap samples from GP(ˆα n, 1). Calculate the value of the test statistic, say T j, (j = 1, 2,..., 100), for each bootstrap sample. Obtain p n as T(90), where T (j), j = 1, 2,..., 100 denotes the ordered values T j values. We have adapted the choice in Gürtler and Henze (2000) 4

5 and used the modified critical point p n = T(90) (T (91) T (90) ), which leads to a more accurate empirical level of the test. i) Methods of estimation: We consider the moment (MO) estimators where ˆα n = 1 2 [ ( ) 2 [ ( Xn 1], ĉ n = 1 ) 2 S n 2 X Xn n + 1], S n X n = 1 n X j, S 2 n = 1 n (X j X n ) 2. The MO estimators are regular, in the sense of (2.3), for α > 1/4 (Hosking and Wallis, 1987). Hosking and Wallis (1987) also consider a related class of estimators, which are regular for α > 1/2. These are termed probability weighted moments (PWM) estimators and may be written as ˆα n = X n X n 2w 2, ĉ n = 2w X n X n 2w, where w = 1 n (1 π j )X (j), π j = j 0.35 n, j = 1, 2,..., n, and X (1) X (2)... X (n) are the order statistics of {X j } n. The maximum likelihood (ML) estimators are regular for α < 1/2 (Smith, 1984). We have employed the routine of Grimshaw (1993) in obtaining the ML estimators, supplemented by a bisection routine to locate the maximum of the profile log likelihood, [ ] L(ϑ) = n log(1 ϑx j ) n log 1 log(1 ϑx j ), nϑ with respect to ϑ = α/c. Let ˆϑ n denote the maximizing value. Then the ML estimators of α and c, are ˆα n = 1 n log(1 ˆϑ n X j ), ĉ n = ˆα n ˆϑ n. 5

6 Comparative evaluation for the performance of estimators of the GPD, including the estimators considered here, may be found in Peng and Welsh (2001) and Singh and Ahmad (2004). ii) Test statistics: From (1.3) it follows by straightforward algebra that the new test statistic may conviniently be written as T n,a = 1 n j,k=1,n 1 + (Y j + Y k + a + 1) 2 (Y j + Y k + a) 3 2 with Y j, j = 1, 2,..., n defined in (1.2). 1 + Y j + a (Y j + a) 2, The new test is compared with the EDF procedures developed by Choulakian and Stephens (2001). Let ˆF (x) = F (x; ˆα n, ĉ n ). Then the Kolmogorov Smirnov (KS) statistic is where KS = max{d +, D }, D + = max { j,2,...,n n ˆF (X (j) )}, D = max { ˆF (X (j) j 1,2,...,n n )}. The Crámer von Mises (CM) statistic is CM = 1 ( 12n + ˆF (X (j) ) 2j 1 ) 2, 2n and the Anderson Darling (AD) statistic is AD = n 1 n [ (2j 1) log ˆF (X (j) ) + (2(n j) + 1) log(1 ˆF ] (X (j) )). iii) Simulation results: All calculations were done at the Department of Economics, University of Athens, using double precision arithmetic in FORTRAN and routines from the IMSL library, whenever available. Apart from the GPD, the gamma with density Γ(θ) 1 x θ 1 exp ( x), the Weibull with density θx θ 1 exp( x θ ), and the log normal distribution with density (θx) 1 (2π) 1/2 exp[ (log x) 2 /(2θ 2 )], are considered. These distributions will be denoted by Γ(θ), W (θ) and LN(θ), respectively, while the GPD with unit scale will simply be denoted by GP (α). 6

7 Table 1: Percentage of rejection for Monte Carlo samples of size n = 50 (left part) n = 75 (middle part) and n = 100 (right part). Estimation: MO (top), PWM (middle) and ML (bottom). T 0.25 T 0.5 T 0.75 T 0.25 T 0.50 T 0.75 T 0.25 T 0.50 T 0.75 GP ( 0.20) GP ( 0.10) GP (0.0) GP (0.20) GP (0.50) GP (1.0) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0) GP ( 0.40) GP ( 0.20) GP (0.0) GP (0.20) GP (0.50) GP (1.0) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0) GP ( 1.0) GP ( 0.75) GP ( 0.50) GP ( 0.25) GP (0.0) GP (0.20) GP (0.40) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0)

8 Table 2: Percentage of rejection for Monte Carlo samples of size n = 50 (left part) n = 75 (middle part) and n = 100 (right part). Estimation: MO (top), PWM (middle) and ML (bottom). CM KS AD CM KS AD CM KS AD GP ( 0.20) GP ( 0.10) GP (0.0) GP (0.20) GP (0.50) GP (1.0) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0) GP ( 0.40) GP ( 0.20) GP (0.0) GP (0.20) GP (0.50) GP (1.0) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0) GP ( 1.0) GP ( 0.75) GP ( 0.50) GP ( 0.25) GP (0.0) GP (0.20) GP (0.40) Γ(2.0) W (0.75) W (1.25) W (1.50) LN(1.0)

9 Table 1 shows results (percentage of rejection rounded to the nearest integer) for T n,a, a = 0.25, 0.50, In the table we simply write T a instead of T n,a. Corresponding results for the KS, CM, and AD tests are shown in Table 2. 9