A new class of estimators for the shape parameter of a Pareto model

UIDB/00297/2020The Pareto model, first used in socioeconomic problems, has successfully been applied in many other areas such as astronomy, biology, bibliometrics, demography, insurance, or risk management. Although there are several variants of this distribution, the current study focuses on the classic Pareto distribution, also known as the Pareto type I distribution. We propose a new class of estimators for the Pareto shape parameter, obtained through a modification of the probability weighted moment method, called the log generalized probability weighted moments method. In addition to the asymptotic distribution, Monte Carlo simulations were performed to analyze the finite sample behavior of the proposed new estimators. A comparison with the most used estimators, such as the moment, the maximum likelihood, the least squares, and the probability weighted moments estimators was also performed. In addition, the estimators were used to construct asymptotic confidence intervals. To illustrate an application of the different estimation methods to a real data set from a clinical trial complete the article. Results indicate an overall good performance of the new proposed class.publishersversionpublishe

Reduced bias estimation of the shape parameter of the log-logistic distribution

Publisher Copyright: © 2023 The Author(s)In the literature, the log-logistic distribution is commonly presented with two parameters: one that governs the shape of the model, and the other that governs its scale. However, to make this model more suitable for data analysis, an additional location parameter can be added, resulting in the three-parameter or shifted log-logistic model. In this paper, we introduce a new estimator for the shape parameter of a three-parameter log-logistic distribution that reduces bias. We also derive various properties of the proposed estimator. Additionally, a simulation study and an application example to a real data set are conducted to examine the efficiency for finite sample sizes. The theoretical and simulated results confirm that our proposed estimation method performs significantly better than other estimation methods found in the literature.publishersversionpublishe

A class of weighted Hill estimators

Publisher Copyright: © 2021 John Wiley & Sons Ltd.In Statistics of Extremes, the estimation of the extreme value index is an essential requirement for further tail inference. In this work, we deal with the estimation of a strictly positive extreme value index from a model with a Pareto-type right tail. Under this framework, we propose a new class of weighted Hill estimators, parameterized with a tuning parameter a. We derive their non-degenerate asymptotic behavior and analyze the influence of the tuning parameter in such result. Their finite sample performance is analyzed through a Monte Carlo simulation study. A comparison with other important extreme value index estimators from the literature is also provided.authorsversionpublishe

Improvements in the estimation of the Weibull tail coefficient -- a comparative study

The Weibull tail-coefficient (WTC) plays a crucial role in extreme value statistics when dealing with Weibull-type tails. Several distributions, such as normal, Gamma, Weibull, and Logistic distributions, exhibit this type of tail behaviour. The WTC, denoted by $\theta$, is a parameter in a right-tail function of the form $\bar F(x) :=1-F(x) =: {\rm e}^{-H(x)}$, where $H(x)=-\ln(1-F(x))$ represents a regularly varying cumulative hazard function with an index of regular variation equal to 1/$\theta$, $\theta\in\mathbb{R}^{+}$. The commonly used WTC-estimators in literature are often defined as functions of the log-excesses, making them closely related to estimators of the extreme value index (EVI) for Pareto-type tails. For a positive EVI, the classical estimator is the Hill estimator. Generalized means have been successfully employed in estimating the EVI, leading to reduction of bias and of root mean square error for appropriate threshold values. In this study, we propose and investigate new classes of WTC-estimators based on power $p$ of the log-excesses within a second-order framework. The performance of these new estimators is evaluated through a large-scale Monte-Carlo simulation study, comparing them with existing WTC-estimators available in the literature

The Use of Generalized Means in the Estimation of the Weibull Tail Coefficient

Due to the specificity of the Weibull tail coefficient, most of the estimators available in the literature are based on the log excesses and are consequently quite similar to the estimators used for the estimation of a positive extreme value index. The interesting performance of estimators based on generalized means leads us to base the estimation of the Weibull tail coefficient on the power mean-of-order-. Consistency and asymptotic normality of the estimators under study are put forward. Their performance for finite samples is illustrated through a Monte Carlo simulation. It is always possible to find a negative value of (contrarily to what happens with the mean-of-order- estimator for the extreme value index), such that, for adequate values of the threshold, there is a reduction in both bias and root mean square error

Minimum-variance reduced-bias estimation of the extreme value index: A theoretical and empirical study

In extreme value (EV) analysis, the EV index (EVI), , is the primary parame- ter of extreme events. In this work, we consider positive, that is, we assume that F is heavy tailed. Classical tail parameters estimators, such as the Hill, the Moments, or the Weissman estimators, are usually asymptotically biased. Con- sequently, those estimators are quite sensitive to the number of upper order statistics used in the estimation. Minimum-variance reduced-bias (RB) estima- tors have enabled us to remove the dominant component of asymptotic bias without increasing the asymptotic variance of the new estimators. The purpose of this paper is to study a new minimum-variance RB estimator of the EVI. Under adequate conditions, we prove their nondegenerate asymptotic behavior. More- over, an asymptotic and empirical comparison with other minimum-variance RB estimators from the literature is also provided. Our results show that the proposed new estimator has the potential to be very useful in practice

Semi-parametric probability-weighted moments estimation revisited

PEst-OE/MAT/UI0006/2011 PEst-OE/MAT/UI0297/2011In this paper, for heavy-tailed models and through the use of probability weighted moments based on the largest observations, we deal essentially with the semi-parametric estimation of the Value-at-Risk at a level p, the size of the loss occurred with a small probability p, as well as the dual problem of estimation of the probability of exceedance of a high level x. These estimation procedures depend crucially on the estimation of the extreme value index, the primary parameter in Statistics of Extremes, also done on the basis of the same weighted moments. Under regular variation conditions on the right-tail of the underlying distribution function F, we prove the consistency and asymptotic normality of the estimators under consideration in this paper, through the usual link of their asymptotic behaviour to the one of the extreme value index estimator they are based on. The performance of these estimators, for finite samples, is illustrated through Monte-Carlo simulations. An adaptive choice of thresholds is put forward. Applications to a real data set in the field of insurance as well as to simulated data are also provided.authorsversionpublishe

Treatment of intracranial solitary fibrous tumor with stereotactic radiosurgery : a case report and review of literature

Trabalho Final do Curso de Mestrado Integrado em Medicina, Faculdade de Medicina, Universidade de Lisboa, 2016Background: Intracranial solitary fibrous tumors (SFT) are rare neoplasms of the brain with a typical benign and slow-growing behavior. The gold-standard of treatment is gross total resection (GTR). However, sometimes this approach is dangerous or not feasible because of anatomical considerations. Therefore, approaches like stereotactic radiosurgery (SRS) are currently being evaluated. Clinical Presentation: The authors present a 59-year-old male patient with a month history of language and humor disorders with headaches and right central facial paresis. Imaging studies revealed an anterior left temporal mass with edema and mass effect. GTR was performed, with histology revealing a SFT. During follow-up, regrowth with invasion of the left cavernous sinus and optical nerve compression was reported. Subtotal resection (STR) was performed leaving only an intracavernous sinus residue. Pathology once again was consistent with SFT. The tumor residue was treated with linear accelerator-based SRS. During follow-up a slow tumor regrowth was observed in the first 12 months after SRS, with posterior stabilization and shrinkage. The shrinkage was only observed 24 months post-SRS. Conclusion: our case represents the successful treatment of an SFT using SRS. It strengthens the role of SRS in managing these tumors when surgery is not an option.Introdução: os tumores fibrosos solitários intracranianos (TFS) são neoplasias raras do encéfalo com características benignas e crescimento lento. O tratamento principal passa pela ressecção total do tumor (RTT). No entanto, esta abordagem nem sempre é possível por condicionantes anatómicas. Assim, a radiocirurgia estereotácica (RCE) tem sido estudada como terapêutica adjuvante. Caso clínico: Os autores apresentam um doente de 59 anos, do sexo masculino, com história de alterações da linguagem e do humor, associadas a cefaleias e parésia facial central direita com a duração de um mês. Os estudos imagiológicos revelaram presença de um tumor na região temporal esquerda, com edema e efeito de massa. Foi efetuada RTT, com histologia compatível com TFS. Durante o período de seguimento, foi detetada nova lesão, com invasão do seio cavernoso esquerdo e compressão do nervo óptico. Foi realizada ressecção subtotal, deixando apenas um resíduo intracavernoso. O exame histológico foi novamente compatível com TFS. O resíduo tumoral foi tratado com RCE (acelerador linear de partículas). No seguimento, observou-se crescimento lento da lesão durante os primeiros 12 meses, com estabilização e posterior involução da lesão, apenas 24 meses pós-RCE. Conclusão: o caso apresentado representa o tratamento bem-sucedido de um TFS recorrendo a RCE, reforçando a sua potencial utilidade na terapêutica adjuvante de tumores deste tipo

Reduced-bias and partially reduced-bias mean-of-order-p value-at-risk estimation: a Monte-Carlo comparison and an application

On the basis of a sample of either independent, identically distributed or possibly weakly dependent and stationary random variables from an unknown model F with a heavy right-tail function, and for any small level q, the value-at-risk (VaR) at the level q, i.e. the size of the loss that occurs with a probability q, is estimated by new semi-parametric reduced-bias procedures based on the mean-of-order-p of a set of k quotients of upper order statistics, with p an adequate real number. After a brief reference to the asymptotic properties of these new VaR-estimators, we proceed to an overall comparison of alternative VaR-estimators, for finite samples, through large-scale Monte-Carlo simulation techniques. Possible algorithms for an adaptive VaR-estimation, an application to financial data and concluding remarks are also provided