The Birnbaum-Saunders (BS) model is a life distribution that has recently been largely studied and applied. A random variable following the BS distribution can be defined through a simple transformation of a standard normal. The BS model can thus be generalized by switching the standard normal distribution of the basis random variable, allowing the construction of more general classes of models. Among those models, we mention the extreme value Birnbaum-Saunders (EVBS) models, recently introduced in the literature, and
based on results from extreme value theory. A real application to athletics data will be used to illustrate the methodology and to provide the way this model and related models can link with traditional extreme value analysis methods.Este trabalho é financiado por Fundos FEDER através do Programa  Operacional Factores de Competitividade - COMPETE e por Fundos Nacionais através da FCT - Fundação para a Ciência e a Tecnologia no
 âmbito do projecto PEst-C/MAT/UI0013/2011.Este trabalho é financiado por Fundos FEDER através do Programa  Operacional Factores de Competitividade - COMPETE e por Fundos  Nacionais através da FCT - Fundação para a Ciência e a Tecnologia no  âmbito do projecto PEst-C/MAT/UI0013/2011

Ferreira, Marta Susana

Gomes, M. Ivette

Leiva, Víctor

Universidade do Minho: RepositoriUM

The Extreme Value Birnbaum-Saunders Model in AthleticsM. Ivette Gomes1 Marta Ferreira2 and Vı´ctor Leiva31 CEAUL and DEIO, FCUL, Universidade de Lisboa, Portugal2 Departamento de Matema´tica, Universidade do Minho, Braga, Portugal3 Departamento de Estadı´stica, Universidad de Valparaı´so, Valparaı´so, ChileCorresponding author: M. Ivette Gomes, e-mail: ivette.gomes@fc.ul.ptAbstract: The Birnbaum-Saunders (BS) model is a life distribution that has recently beenlargely studied and applied. A random variable following the BS distribution can be definedthrough a simple transformation of a standard normal. The BS model can thus be general-ized by switching the standard normal distribution of the basis random variable, allowingthe construction of more general classes of models. Among those models, we mention theextreme value Birnbaum-Saunders (EVBS) models, recently introduced in the literature, andbased on results from extreme value theory. A real application to athletics data will be usedto illustrate the methodology and to provide the way this model and related models can linkwith traditional extreme value analysis methods.Key Words: Athletics, extreme value Birnbaum-Saunders models, parametric estimation,statistics of extremes.1 Introduction and preliminariesThe Birnbaum-Saunders (BS) model is a life distribution, introduced and studied in Birn-baum and Saunders (1969), that has been largely applied in recent decades. BS and standardnormal random variables (RVs), now denoted respectively by T and Z, are related by theformulaT = δ(αZ/2 +√{αZ/2}2 + 1)2, i.e. Z =(√T/δ −√δ/T)/α, (1)with α > 0 and δ > 0 shape and scale parameters, respectively. Consider the usual nota-tions φ and Φ for the standard normal probability density function (PDF) and cumulativedistribution function (CDF), respectively, and letat =(√t/δ −√δ/t)/α, so that a′t = d a(t)/dt =(√t/δ +√δ/t)/(2αt). (2)Then, the PDF and the CDF of T , in (1), are respectively fT (t) = a′t φ(at)and FT (t) =Φ(at), t > 0.The assumption of a normally distributed Z can be obviously relaxed, assuming that Zfollows any other distribution with PDF fZ . We then obtain a general BS type (BST) RV,denoted by T ∼ BST(δ, α; fZ ), with a PDF fT (t) = a′t fZ (at), for t > 0, and at and a′tas given in (2). Among those models, we mention the extreme value Birnbaum-Saunders(EVBS) case, recently introduced in Ferreira et al. (2012) and further considered in Gomeset al. (2012) and Ferreira (2013). These models are essentially based on results from extremevalue theory (EVT).In this article, after sketching the main limiting result on EVT, we introduce the EVBSmodels and make some comments on their importance. Finally, we provide an applicationto athletics data, showing the way some of these models can link with a traditional extremevalue analysis (EVA) method, like the block maxima method.2 The main limiting result in EVT and associated parametricmodel in statistics of univariate extremesThe main limiting results in EVT date back to the papers by Fre´chet (1927), Fisher andTippett (1928), von Mises (1936) and Gnedenko (1943), who fully characterized the possi-ble non-degenerate limit laws, as n → ∞, of the sequence of maximum values, Xn:n :=max(X1, . . . , Xn), suitably normalized, with X1, . . . , Xn independent, identically (IID)RVs from an underlying CDF F . More specifically, if there are normalizing constantsan > 0, bn ∈ R and some non-degenerate CDF, G, such that, for all x ∈ C(G), the setof continuity points of G, limn→∞ P{(Xn:n − bn)/an ≤ x}= G(x), we can redefine theconstants in such a way thatG(x) ≡ Gγ(x) :={exp(−(1 + γx)−1/γ) , 1 + γx > 0, if γ 6= 0,exp(− exp(−x)), x ∈ R, if γ = 0, (3)the so-called (general) extreme value distribution (EVD), given here in the von Mises-Jenkinson form (von Mises, 1936; Jenkinson, 1955), and denoted by EVM ≡ EVM(γ).We then say that F is in the max-domain of attraction (MDA) of Gγ , in (3), and use the no-tation F ∈ DM (Gγ). The limiting CDFs, G, are then max-stable (MS), i.e., they are indeedthe unique laws S such that the functional equation Sn(αnx+ δn) = S(x), for n ≥ 1, holdsfor some αn > 0 and δn ∈ R. The EVD reduces indeed to the Fre´chet (γ > 0), Weibull(γ < 0) and Gumbel (γ = 0) CDFs, respectively. In fact, we often work with one of thethree following types:Type I (Gumbel) : Λ(x) = exp(− exp(−x)), x ∈ R (γ = 0),Type II (Fre´chet) : Φα(x) = exp(−x−α), x ≥ 0 (γ = 1/α),Type III (max-Weibull) : Ψα(x) = exp(−(−x)α), x ≤ 0 (γ = −1/α).(4)The real parameter γ in (3), the primary parameter of interest in EVA, is the so-called extremevalue index (EVI). The EVI rules the behaviour of the right-tail of F . If γ < 0, we havelight right-tails, with a finite right endpoint, all in the so-called Weibull MDA. In addition,γ = 0 corresponds to the Gumbel MDA (exponential right-tails). And if γ > 0, we havethe Fre´chet MDA corresponding to heavy right-tails (polynomial tail decay, with an infiniteright endpoint).Remark 1. All results developed for maxima can easily be reformulated for minima sinceX1:n := min{X1, . . . , Xn} = −max{−X1, . . . ,−Xn}. If we are interested in left-tails, wehave, for the linearly normalised sequence of minimum values, a limiting CDF, G∗γ∗(x) =1−Gγ∗(−x), often referred to as a EVm ≡ EVm(γ∗) models, i.e.,G∗γ∗(x) ={1− exp(−(1− γ∗x)−1/γ), 1− γ∗x > 0, if γ∗ 6= 0,1− exp(− exp(x)), x ∈ R, if γ∗ = 0. (5)The parameter γ∗, in (5), determines the left-tail behavior of F , such as the parameter γ, in(3), determines the right-tail behavior of F , being so both crucial parameters in EVT.From a statistical point of view, let us assume we have access to a sample (X1, . . . , Xn)of n IID or possibly stationary weakly dependent RVs from an underlying CDF, F . More-over, let us use the notation (X1:n ≤ X2:n ≤ · · · ≤ Xn:n) for the sample of associated as-cending order statistics. Statistics of univariate extremes (SUE) help us to learn from disas-trous or almost disatrous events, always of high relevance in society and with a high socialimpact. Its domains of application are thus quite diversified. We mention the fields of hy-drology, meteorology, insurance, finance, telecommunications, athletics and biostatistics,among others (see, e.g., Reiss and Thomas, 2001, 2007). Although it is possible to findsome historical papers with applications related to extreme events, the field dates back toGumbel, in papers from 1935 on, summarized in his book (Gumbel, 1958). He developedstatistical procedures essentially based on the aforementioned Gnedenko’s limiting resullts.Indeed, parametric inference on the right-tail of F , usually unknown, is done on the basis ofthe approximation P (Xn:n ≤ x) = Fn(x) ≈ Gγ ((x− λn)/δn) , with (λn, δn) ∈ (R,R+)a vector of unknown location and scale parameters. The limiting result given before for thenormalized sequence of maximum values validates such an approximation, and was used byGumbel, to give approximations of this type but for any of the models in (4). He suggestedthe first model in statistics of extremes, usually called the block maxima model. Under thismodel, the sample of size n is divided into k sub-samples of size r (usually associated withk years, for n = r × k, and r reasonably large). Next, the maximum of the r observationsin each of the k sub-samples is considered, and one of the extremal models in (4), obviouslywith extra unknown location and scale parameters, is fitted to the sample of those k maxi-mum values. Nowadays, whenever using this approach, still quite popular in environmentalsciences, it is more common to fit to the data a univariate EVD, Gγ((x− λr)/δr), with Gγgiven in (3), (λr, δr, γr) ∈ (R,R+,R) unknown location, scale and ‘shape’ parameters. Asmentioned above, we shall use the self-explanatory notation, EVM(λ, δ, γ), for such a typeof models. We can analogously consider EVm(λ, δ, γ) models, whenever dealing with min-imum values. All statistical inference is then related to the above mentioned models. Recentsurveys on SUE can be found in Gomes et al. (2008), Beirlant et al. (2012) and McDonaldand Scarrot (2012).3 Some details on EVBS distributionsThe EVBSM (and EVBSm) distributions, based on limiting EV models for maxima, EVM,(and for minima, EVm), have been introduced in Ferreira et al. (2012). Specifically, considerthat Z follows the EV distribution for maxima, in (3), i.e., Z ∼ EVM(γ). Then, we usethe notation EVBSM(δ, α, γ) for the CDF of the RV T , in (1). Analogously, if we considerthat Z follows the EV distribution for minima, in (5), denoted by Z ∼ EVm(γ∗), and thesame expression for T , i.e., that given in (1), we use the notation T ∼ EVBSm(δ, α, γ).Obviously, and if needed, we can further introduce a location λ, and work more generallywith the RV λ + T , with λ ∈ R. We shall then use the notations EVBSM(λ, δ, α, γ) andEVBSm(λ, δ, α, γ).The EVBS models are very flexible, with extremely diversified left and right-tails. More-over, the EVBSM and EVBSm hazard rate (HR) functions present several different shapesgoing through all possible HR shape classes, and this contrarily to the EVM and EVm HRs(see Ferreira et al., 2012). These are very strong points in favour of EVBSM and EVBSmmodels, as they become quite rich and interesting for modeling purposes (see Ferreira et al.,2012, and Gomes et al., 2012).Estimation aspects and model checking for EVBS distributions have been dealt within Ferreira et al. (2012). The system of likelihood equations does not produce an explicitsolution so that a numerical procedure is necessary. To analyze data from EVBS models, anR package named evbs is being developed, and its ‘in progress’ version is already availablethrough the authors. This package contains diverse indicators, as well as methodologiesuseful for EVBS distributions, among which is the maximum likelihood (ML) estimation ofthe unknown parameters.Once the EVBS distribution parameters have been estimated, a natural question is check-ing how good is the fit of the model to the data. In order to compare the EVBS distribu-tions to other distributions, and just as in Ferreira et al. (2012) and Gomes et al. (2012),among others, we have used the following model selection criteria based on loss of infor-mation: Akaike (AIC), Schwarz’s Bayesian (BIC) and Hannan-Quinn (HQIC) informationcriteria. These criteria are given by AIC = −2`(θˆ) + 2d, BIC = −2`(θˆ) + d log (n), andHQIC = −2`(θˆ) + 2d log(log (n)), where `(θˆ) is the log-likelihood function for the pa-rameter θ associated with the model evaluated at θ = θˆ, n is the sample size, and d is thedimension of the parameter space. Generally, differences between two values of the afore-mentioned information criteria are not very noticeable. In that case, a suitable function ofthe Bayes factor (BF) B12, i.e. BF∗ := 2 log(B12), can be used to highlight such differences,if they exist. If BF∗ < 0, evidence in favor of model M1 is negative and M2 should beaccepted. If BF∗ ∈ [0, 2), there is a week (W) evidence in favor of M1. Values of BF∗ in[2, 6), [6, 10) or [10,∞) provide respectively positive (P), strong (S) or very strong (VS)evidence in favor of M1 (see Vilca et al., 2011, for details). AIC, BIC and HQIC are basedon a penalization of the likelihood function as the model becomes more complex, i.e., withmore parameters (Sanhueza et al., 2008). Since models with more parameters should pro-vide a better fit, AIC, BIC and HQIC allow us to compare models with different numbers ofparameters due to the penalization incorporated in such criteria.4 An application to athleticsIn this section, we shall be interested in an application of EVBSM and EVM models to thebest personal marks attained at a few athletic events, in a context similar to the one usedin Henriques-Rodrigues et al. (2011), among others. We shall pay special attention to theestimation of γ, in (3). The data, already analysed in Henriques-Rodrigues et al. (2011), arerelated to two jumping events, all for men, the high jump (HJ) and the pole vault (PV). Thesources were http://www.iaaf.org/statistics/toplists/index.htmx and http://hem.bredband.net/athletics/athletics all-time best.htm. Data were collected until the end of 2007 and for anyathlete only the best mark was taken into account.Due to the fact that the observed data considered are already maxima, possibly of asmall and dependent number of marks associated with any of the n athletes, but the EVlimiting law, in (3), is “robust” to changes of the IID assumption, we have tried the fitting,through ML, of an extreme value model EVM(λ, δ, γ), comparatively with the fitting ofa EVBSM(λ, δ, α, γ). For the ML estimation of the unknown parameters (λ, δ, γ) in theEVM(λ, δ, γ) model, we have used the EVIR package in the R-software. On the basis of theaforementioned evbs R package, we have proceeded with the ML estimation of (λ, δ, α, γ),in the EVBSM(λ, δ, α, γ) model. Table 1 presents the ML estimates of (λ, δ, γ) for the EVmodel, and of (λ, δ, α, γ) for the EVBS models. In Table 2, we present the respective AIC,BIC, HQIC and BF∗ indicators.For the HJ and PV data sets, the smallest value of −` favours the EVSB model (seeTable 1). The same happens with the AIC, BIC and HQIC (see Table 2). Also, BF∗ providesa strong evidence in favor of the EVBS model for the HJ data, and a very strong evidencefor the PV data. The quality of the fitting, i.e., the good coherence between empirical andtheoretical densities and distributions, can be visualised in Figure 1.Table 1: ML estimates for the indicated models and data sets.Model λ̂ δ̂ α̂ γ̂ −`HJ EVBS 2.19271 0.04014 0.85953 -0.37179 -417.7767EV 2.24045 0.03341 – -0.09084 -414.6809PV EVBS 5.42133 0.16522 0.70185 -0.40329 -151.5884EV 5.57959 0.09477 – -0.15096 -141.2343Table 2: AIC, BIC, HQIC and BF∗ for the indicated models and data sets.Model AIC BIC HQIC BF∗HJ EVBS -827.5534 -814.2614 -822.1771 —EV -827.3618 -813.3928 -819.3296 6.1916 (S)PV EVBS -295.4686 -261.176 -289.8005 —EV -274.468 -0.64999 -269.0923 20.7082 (VS)high jumpPDF2.20 2.25 2.30 2.35 2.40024681012EVBSEV2.20 2.25 2.30 2.35 2.40 2.450.00.20.40.60.81.0xCDFEVBSEVpole vaultPDF5.4 5.6 5.8 6.0 6.201234EVBSEV5.4 5.6 5.8 6.0 6.20.00.20.40.60.81.0xCDFEVBSEVFigure 1: HJ Histogram (top, left) and CDF (top, right); PV Histogram (bottom, left) andCDF (bottom, right).Acknowledgements. Research partially supported by FCT National Funds, Centro de Mate-ma´tica da Universidade do Minho, Projects PEst-OE/MAT/UI0006/2011 (CEAUL), FCT/OE,EXTREMA, PTDC/MAT/101736/2008, Portugal, and FONDECYT, grant 1120879, Chile.References[1] Beirlant, J., Caeiro, F. and Gomes, M.I. (2012). “An overview and open research topicsin statistics of univariate extremes,” Revstat, 10, 1–3.[2] Birnbaum, Z.W. and Saunders S. (1969). “A new family of life distributions,” J. AppliedProbab., 6, 319–327.[3] Ferreira, M. (2013). “A study of exponential-type tails applied to Birnbaum-Saundersmodels,” Chilean Journal of Statistics, 4, 91–101.[4] Ferreira, M., Gomes, M.I. and Leiva, V. (2012). “On an extreme value version of theBirnbaum-Saunders distribution,” Revstat, 10, 181–210.[5] Fre´chet, M. (1927). “Sur le loi de probabilite´ de l’e´cart maximum,” Ann. Socie´te´ Polon-aise de Mathe´matique, 6, 93–116.[6] Fisher, R.A. and Tippett, L.H.C. 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The extreme value Birnbaum-Saunders model in athletics

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