Frederico Caeiro

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{Summary: For heavy-tailed models, and working with the sample of the $k$ largest observations, we present probability weighted moments (PWM) estimators for the first order tail parameters. Under regular variation conditions on the right-tail of the underlying distribution function $F$ we prove the consistency and asymptotic normality of these estimators. Their performance, for finite sample sizes, is illustrated through a small-scale Monte Carlo simulation.}

{Summary: We consider a new class of consistent semi-parametric estimators of a negative extreme value index, based on the set of the $k$ largest observations. This class of estimators depends on a control or tuning parameter, which enables us to have access to an estimator with a null second-order component of asymptotic bias, and with a rather interesting mean squared error, as a function of $k$. We study the consistency and asymptotic normality of the proposed estimators. Their finite sample behaviour is obtained through Monte Carlo simulation.}

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{Summary: We are interested in the comparison, under a third-order framework, of classes of second-order, reduced-bias tail index estimators, giving particular emphasis to minimum-variance reduced-bias estimators of the tail index $\gamma$. The full asymptotic distributional properties of the proposed classes are derived under a third-order framework and the estimators are compared with other alternatives, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the USA Dollar is also provided.}

{Summary: In many areas of application, like, for instance, climatology, hydrology, insurance, finance, and statistical quality control, a typical requirement is to estimate a high quantile of probability $1 - p$, a value high enough so that the chance of an exceedance of that value is equal to $p$, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index $\gamma $, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for $\gamma >0$, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles' estimators, with the same asymptotic variance as the (biased) ``classical'' estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.}

{Summary: Heavy tailed-models are quite useful in many fields, like insurance, finance, telecommunications, internet traffic, among others, and it is often necessary to estimate a high quantile, i.e., a value that is exceeded with a probability $p$, small. The semiparametric estimation of this parameter relies essentially on the estimation of the tail index, the primary parameter in statistics of extremes. Classical semi-parametric estimators of extreme parameters show usually a severe bias and are known to be very sensitive to the number $k$ of top order statistics used in the estimation. For $k$ small they have a high variance, and for large $k$ a high bias. Recently, new second-order ``shape'' and ``scale'' estimators allowed the development of second-order reduced-bias estimators, which are much less sensitive to the choice of $k$. Here we study, under a third order framework, minimum-variance reduced-bias (MVRB) tail index estimators, recently introduced in the literature, and dependent on an adequate estimation of second order parameters. The improvement comes from the asymptotic variance, which is kept equal to the asymptotic variance of the classical Hill estimator [ıt B. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided that we estimate the second order parameters at a level of a larger order than the level used for the estimation of the first order parameter. The use of those MVRB tail index estimators enables us to introduce new classes of reduced-bias high quantile estimators. These new classes are compared among themselves and with previous ones through the use of a small-scale Monte Carlo simulation.}

{Let $X_i$ be i.i.d. r.v.s with heavy-tailed CDF $F(x)$ such that $$1-F(x)=(x/C)^{-1/\gamma}((1+(\beta/\rho)(x/C)^{\rho/\gamma} +\beta'(x/C)^{2\rho/\gamma}(1+o(1))),$$ where $\gamma$ is the tail index ($\gamma>0$), and $\rho<0$ and $\beta$ are the ``second order parameters''. The authors construct an estimator for $\beta$ based on the ``tail moments'' $$M_n^{(\alpha)}=(k)^{-1}\sum_{i=1}^k [łog X_{n-i+1:n}-łog X_{n-k:n}]^\alpha. $$ Consistency and asymptotic normality of the estimator are demonstrated. The small sample properties of the estimator are investigated via simulations.}

{Summary: We are interested in an adequate estimation of the dominant component of the bias of ıt B. M. Hill}\,'s estimator [Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] of a positive tail index $\gamma$, in order to remove it from the classical Hill estimator in different asymptotically equivalent ways. If the second order parameters in the bias are computed at an adequate level $k_1$ of a larger order than that of the level $k$ at which the Hill estimator is computed, there may be no change in the asymptotic variances of these reduced bias tail index estimators, which are kept equal to the asymptotic variance of the Hill estimator, i.e., equal to $\gamma^2$. The asymptotic distributional properties of the proposed estimators of $\gamma$ are derived and the estimators are compared not only asymptotically, but also for finite samples through Monte Carlo techniques.}

{Summary: We first consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parametrised in a tuning or control parameter $\alpha$. Such a control parameter enables us to have access for any available sample, to an estimator of the tail index $\gamma$ with a null dominant component of asymptotic bias and consequently with a reasonably flat mean squared error pattern, as a function of $k$, the number of top-order statistics considered.\par Such a control parameter depends on a second-order parameter $\rho$, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro-US\$ exchange rates.}

{Consider the distribution function $EV_{\gamma}(x)=\exp(-(1+\gamma x)^{- 1/\gamma}),\ \gamma>0,\ 1+\gamma x>0$, to which $\max(X_{1},łdots, X_{n})$ is attracted after suitable linear normalization. The authors consider the underlying model $F$ in the max-domain of attraction of $EV_{\gamma}$, where $ X_{i:n},\ 1łeq Iłeq n$, denotes the i-th ascending order statistic associated to the random sample $(X_{1},łdots, X_{n})$ from the unknown distribution function $F$. This article is devoted to studying semi-parametric estimators of $\gamma$ in the form $$\gamma_{n}^{(þeta,\alpha)}(k)=(\Gamma(\alpha)/M_{n}^{(\alpha- 1)}(k))łeft(M_{n}^{(þeta\alpha)}(k)/\Gamma(þeta\alpha+1)\right) ^{1/þeta},\quad \alpha\geq 1,\quad þeta>0,$$ parametrized by the parameters $\alpha$ and $þeta$, which may be controlled, where $M_{n}^{(0)}=1$ and $ M_{n}^{(\alpha)}(k)=k^{-1}\sum_{i=1}^{k}(łn X_{n-i+1:n}-łn X_{n-k:n})^{\alpha}$, $\alpha>0$, is a consistent estimator of $\Gamma(\alpha+1)\gamma^{\alpha}$, as $k\toınfty$, and $k=o(n)$, as $n\toınfty$.\par The authors derive the asymptotic distributional properties of the considered class of estimators and obtain that for $þeta>1$ it is always possible to find a control parameter $\alpha$ which makes the dominant component of the asymptotic bias of the proposed estimator null and depends on the second order parameter $\rho$. An investigation of the $\rho$-estimator is presented.}

{Summary: We consider a class of consistent semi-parametric estimators of a positive tail index $\gamma$, parameterized by a tuning or control parameter $\alpha$. Such a control parameter enables us to have access, for any available sample, to an estimator of $\gamma$ with a null dominant component of asymptotic bias, and with a reasonably flat mean squared error pattern, as a function of $k$, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relative to the classical Hill estimator [ıt B. M. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)], provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.}