An extreme value theory approach to calculating minimum capital risk requirements

This paper investigates the frequency of extreme events for three LIFFE futures contracts for the calculation of minimum capital risk requirements (MCRRs). We propose a semiparametric approach where the tails are modelled by the Generalized Pareto Distribution and smaller risks are captured by the empirical distribution function. We compare the capital requirements form this approach with those calculated from the unconditional density and from a conditional density - a GARCH(1,1) model. Our primary finding is that both in-sample and for a hold-out sample, our extreme value approach yields superior results than either of the other two models which do not explicitly model the tails of the return distribution. Since the use of these internal models will be permitted under the EC-CAD II, they could be widely adopted in the near future for determining capital adequacies. Hence, close scrutiny of competing models is required to avoid a potentially costly misallocation capital resources while at the same time ensuring the safety of the financial system.

Accessing extremes of mid-latitudinal wave activity: methodology and application

A statistical methodology is proposed and tested for the analysis of extreme values of atmospheric wave activity at mid-latitudes. The adopted methods are the classical block-maximum and peak over threshold, respectively based on the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD). Time-series of the ‘Wave Activity Index’ (WAI) and the ‘Baroclinic Activity Index’ (BAI) are computed from simulations of the General Circulation Model ECHAM4.6, which is run under perpetual January conditions. Both the GEV and the GPD analyses indicate that the extremes ofWAI and BAI areWeibull distributed, this corresponds to distributions with an upper bound. However, a remarkably large variability is found in the tails of such distributions; distinct simulations carried out under the same experimental setup provide sensibly different estimates of the 200-yr WAI return level. The consequences of this phenomenon in applications of the methodology to climate change studies are discussed. The atmospheric configurations characteristic of the maxima and minima of WAI and BAI are also examined.

The cross-currency hedging performance of implied versus statistical forecasting models

This article examines the ability of several models to generate optimal hedge ratios. Statistical models employed include univariate and multivariate generalized autoregressive conditionally heteroscedastic (GARCH) models, and exponentially weighted and simple moving averages. The variances of the hedged portfolios derived using these hedge ratios are compared with those based on market expectations implied by the prices of traded options. One-month and three-month hedging horizons are considered for four currency pairs. Overall, it has been found that an exponentially weighted moving-average model leads to lower portfolio variances than any of the GARCH-based, implied or time-invariant approaches.

Non-stationary frequency analysis of extreme daily rainfall in Sao Paulo, Brazil

This work is an assessment of frequency of extreme values (EVs) of daily rainfall in the city of Sao Paulo. Brazil, over the period 1933-2005, based on the peaks-over-threshold (POT) and Generalized Pareto Distribution (GPD) approach. Usually. a GPD model is fitted to a sample of POT Values Selected With a constant threshold. However. in this work we use time-dependent thresholds, composed of relatively large p quantities (for example p of 0.97) of daily rainfall amounts computed from all available data. Samples of POT values were extracted with several Values of p. Four different GPD models (GPD-1, GPD-2, GPD-3. and GDP-4) were fitted to each one of these samples by the maximum likelihood (ML) method. The shape parameter was assumed constant for the four models, but time-varying covariates were incorporated into scale parameter of GPD-2. GPD-3, and GPD-4, describing annual cycle in GPD-2. linear trend in GPD-3, and both annual cycle and linear trend in GPD-4. The GPD-1 with constant scale and shape parameters is the simplest model. For identification of the best model among the four models WC used rescaled Akaike Information Criterion (AIC) with second-order bias correction. This criterion isolates GPD-3 as the best model, i.e. the one with positive linear trend in the scale parameter. The slope of this trend is significant compared to the null hypothesis of no trend, for about 98% confidence level. The non-parametric Mann-Kendall test also showed presence of positive trend in the annual frequency of excess over high thresholds. with p-value being virtually zero. Therefore. there is strong evidence that high quantiles of daily rainfall in the city of Sao Paulo have been increasing in magnitude and frequency over time. For example. 0.99 quantiles of daily rainfall amount have increased by about 40 mm between 1933 and 2005. Copyright (C) 2008 Royal Meteorological Society

Local Influence Analysis for Skew-Normal Linear Mixed Models

In this article, we consider local influence analysis for the skew-normal linear mixed model (SN-LMM). As the observed data log-likelihood associated with the SN-LMM is intractable, Cook`s well-known approach cannot be applied to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (2001). This approach is based on the use of an EM-type algorithm and is measurement invariant under reparametrizations. Four specific perturbation schemes are discussed. Results obtained for a simulated data set and a real data set are reported, illustrating the usefulness of the proposed methodology.

Caracterização estatística de processos sísmicos via Distribuição Generalizada de Pareto. Estudo de caso: João Câmara-RN

The work is to make a brief discussion of methods to estimate the parameters of the Generalized Pareto distribution (GPD). Being addressed the following techniques: Moments (moments), Maximum Likelihood (MLE), Biased Probability Weighted Moments (PWMB), Unbiased Probability Weighted Moments (PWMU), Mean Power Density Divergence (MDPD), Median (MED), Pickands (PICKANDS), Maximum Penalized Likelihood (MPLE), Maximum Goodness-of-fit (MGF) and the Maximum Entropy (POME) technique, the focus of this manuscript. By way of illustration adjustments were made for the Generalized Pareto distribution, for a sequence of earthquakes intraplacas which occurred in the city of João Câmara in the northeastern region of Brazil, which was monitored continuously for two years (1987 and 1988). It was found that the MLE and POME were the most efficient methods, giving them basically mean squared errors. Based on the threshold of 1.5 degrees was estimated the seismic risk for the city, and estimated the level of return to earthquakes of intensity 1.5°, 2.0°, 2.5°, 3.0° and the most intense earthquake never registered in the city, which occurred in November 1986 with magnitude of about 5.2º

Colombeau's theory and shock wave solutions for systems of PDEs

In this article we study the existence of shock wave solutions for systems of partial differential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau's theory of generalized functions. This study uses the equality in the strict sense and the association of generalized functions (that is the weak equality). The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association.

Estimativa da probabilidade do evento extremo de precipitação de janeiro de 2000 no Vale do Paraíba, baseada na distribuição generalizada de pareto

Um evento extremo de precipitação ocorreu na primeira semana do ano 2000, de 1º a 5 de janeiro, no Vale do Paraíba, parte leste do Estado de São Paulo, Brasil, causando enorme impacto socioeconômico, com mortes e destruição. Este trabalho estudou este evento em 10 estações meteorológicas selecionadas que foram consideradas como aquelas tendo dados mais homogêneos do Que outras estações na região. O modelo de distribuição generalizada de Pareto (DGP) para valores extremos de precipitação de 5 dias foi desenvolvido, individualmente para cada uma dessas estações. Na modelagem da DGP, foi adotada abordagem não-estacionaria considerando o ciclo anual e tendência de longo prazo como co-variaveis. Uma conclusão desta investigação é que as quantidades de precipitação acumulada durante os 5 dias do evento estudado podem ser classificadas como extremamente raras para a região, com probabilidade de ocorrência menor do que 1% para maioria das estações, e menor do que 0,1% em três estações.

Non-stationary frequency analysis of extreme daily rainfall in São Paulo, Brazil

This work is an assessment of frequency of extreme values (EVs) of daily rainfall in the city of São Paulo. Brazil, over the period 1933-2005, based on the peaks-over-threshold (POT) and Generalized Pareto Distribution (GPD) approach. Usually. a GPD model is fitted to a sample of POT Values Selected With a constant threshold. However. in this work we use time-dependent thresholds, composed of relatively large p quantities (for example p of 0.97) of daily rainfall amounts computed from all available data. Samples of POT values were extracted with several Values of p. Four different GPD models (GPD-1, GPD-2, GPD-3. and GDP-4) were fitted to each one of these samples by the maximum likelihood (ML) method. The shape parameter was assumed constant for the four models, but time-varying covariates were incorporated into scale parameter of GPD-2. GPD-3, and GPD-4, describing annual cycle in GPD-2. linear trend in GPD-3, and both annual cycle and linear trend in GPD-4. The GPD-1 with constant scale and shape parameters is the simplest model. For identification of the best model among the four models WC used rescaled Akaike Information Criterion (AIC) with second-order bias correction. This criterion isolates GPD-3 as the best model, i.e. the one with positive linear trend in the scale parameter. The slope of this trend is significant compared to the null hypothesis of no trend, for about 98% confidence level. The non-parametric Mann-Kendall test also showed presence of positive trend in the annual frequency of excess over high thresholds. with p-value being virtually zero. Therefore. there is strong evidence that high quantiles of daily rainfall in the city of São Paulo have been increasing in magnitude and frequency over time. For example. 0.99 quantiles of daily rainfall amount have increased by about 40 mm between 1933 and 2005. Copyright (C) 2008 Royal Meteorological Society

Disturbances, elevation, topography and spatial proximity drive vegetation patterns along an altitudinal gradient of a top biodiversity hotspot

The correlation between vegetation patterns (species distribution and richness) and altitudinal variation has been widely reported for tropical forests, thereby providing theoretical basis for biodiversity conservation. However, this relationship may have been oversimplified, as many other factors may influence vegetation patterns, such as disturbances, topography and geographic distance. Considering these other factors, our primary question was: is there a vegetation pattern associated with substantial altitudinal variation (10-1,093 m a.s.l.) in the Atlantic Rainforest-a top hotspot for biodiversity conservation-and, if so, what are the main factors driving this pattern? We addressed this question by sampling 11 1-ha plots, applying multivariate methods, correlations and variance partitioning. The Restinga (forest on sandbanks along the coastal plains of Brazil) and a lowland area that was selectively logged 40 years ago were floristically isolated from the other plots. The maximum species richness (>200 spp. per hectare) occurred at approximately 350 m a.s.l. (submontane forest). Gaps, multiple stemmed trees, average elevation and the standard deviation of the slope significantly affected the vegetation pattern. Spatial proximity also influenced the vegetation pattern as a structuring environmental variable or via dispersal constraints. Our results clarify, for the first time, the key variables that drive species distribution and richness across a large altitudinal range within the Atlantic Rainforest. © 2013 Springer Science+Business Media Dordrecht.

Aspectos computacionais para inferência na distribuição gama generalizada

Pós-graduação em Matematica Aplicada e Computacional - FCT

A general threshold stress hybrid hazard model for lifetime data

In this paper we propose a hybrid hazard regression model with threshold stress which includes the proportional hazards and the accelerated failure time models as particular cases. To express the behavior of lifetimes the generalized-gamma distribution is assumed and an inverse power law model with a threshold stress is considered. For parameter estimation we develop a sampling-based posterior inference procedure based on Markov Chain Monte Carlo techniques. We assume proper but vague priors for the parameters of interest. A simulation study investigates the frequentist properties of the proposed estimators obtained under the assumption of vague priors. Further, some discussions on model selection criteria are given. The methodology is illustrated on simulated and real lifetime data set.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Taxonomic sufficiency of polychaete taxocenes for estuary monitoring

The polychaetes assemblage structure was used in order to investigate taxonomic sufficiency in a heavily polluted tropical bay. Species abundance was aggregated into progressively higher taxa matrices (genus, family, order) and was analyzed using univariate and multivariate techniques. Polychaetes distribution in Guanabara Bay (GB) was in accordance with a pollution gradient, probably ruled by the organic enrichment, consequent effects of hypoxia and altered redox conditions coupled with prevailing patterns of circulation. Within the sectors of GB, an increasing gradient in species richness and occurrence was observed, ranging from the azoic and impoverished stations in the inner sector to a well-structured community in terms of species composition and abundance inhabiting the outer sector. Multivariate statistical analysis showed similar results when species were aggregated into genera and families, while greater difference occurred at coarser taxonomic identification (order). The literature about taxonomic sufficiency has demonstrated that faunal patterns at different taxonomic levels tend to become similar with increased pollution. In GB, an analysis carried out solely at family level is perfectly adequate to describe the environmental gradient, considered a useful tool for a quick environmental assessment. (C) 2011 Elsevier Ltd. All rights reserved.

Characterisation of extreme winter precipitation in Mediterranean coastal sites and associated anomalous atmospheric circulation patterns

We present an analysis of daily extreme precipitation events for the extended winter season (October–March) at 20 Mediterranean coastal sites covering the period 1950–2006. The heavy tailed behaviour of precipitation extremes and estimated return levels, including associated uncertainties, are derived applying a procedure based on the Generalized Pareto Distribution, in combination with recently developed methods. Precipitation extremes have an important contribution to make seasonal totals (approximately 60% for all series). Three stations (one in the western Mediterranean and the others in the eastern basin) have a 5-year return level above 100 mm, while the lowest value (estimated for two Italian series) is equal to 58 mm. As for the 50-year return level, an Italian station (Genoa) has the highest value of 264 mm, while the other values range from 82 to 200 mm. Furthermore, six series (from stations located in France, Italy, Greece, and Cyprus) show a significant negative tendency in the probability of observing an extreme event. The relationship between extreme precipitation events and the large scale atmospheric circulation at the upper, mid and low troposphere is investigated by using NCEP/NCAR reanalysis data. A 2-step classification procedure identifies three significant anomaly patterns both for the western-central and eastern part of the Mediterranean basin. In the western Mediterranean, the anomalous southwesterly surface to mid-tropospheric flow is connected with enhanced moisture transport from the Atlantic. During ≥5-year return level events, the subtropical jet stream axis is aligned with the African coastline and interacts with the eddy-driven jet stream. This is connected with enhanced large scale ascending motions, instability and leads to the development of severe precipitation events. For the eastern Mediterranean extreme precipitation events, the identified anomaly patterns suggest warm air advection connected with anomalous ascent motions and an increase of the low- to mid-tropospheric moisture. Furthermore, the jet stream position (during ≥5-year return level events) supports the eastern basin being in a divergence area, where ascent motions are favoured. Our results contribute to an improved understanding of daily precipitation extremes in the cold season and associated large scale atmospheric features.