Bias-reduced estimators for bivariate tail modelling

Ledford and Tawn (1997) introduced a flexible bivariate tail model based on the coefficient of tail dependence and on the dependence of the extreme values of the random variables. In this paper, we extend the concept by specifying the slowly varying part of the model as done by Hall (1982) with the univariate case. Based on Beirlant et al. (2009), we propose a bias-reduced estimator for the coefficient of tail dependence and for the estimation of small tail probabilities. We discuss the properties of these estimators via simulations and a real-life example. Furthermore, we discuss some theoretical asymptotic aspects of this approach.     

回归模型的同方差检验

本文利用局部经验似然和WNW方法对条件分布函数和条件分位数进行估计,并利用条件分位数的方法对回归模型中的误差方差进行了同方差假设检验,获得了零假设下检验统计量的渐近分布为X2分布．模拟计算表明同方差假设检验的条件分位数方法具有较好的功效．     

On the favorable estimation for fitting heavy tailed data

Assessment of heavy tailed data and its compound sums has many applications in insurance, auditing and operational risk capital assessment among others. In this paper, we compare the classical estimators (maximum likelihood, QQ and moment estimators) with the recently introduced robust estimators of “generalized median”, “trimmed mean” and estimators based on t-score moments. We derive the exact distribution of the likelihood ratio tests of homogeneity and simple hypothesis on the tail index of a two-parameter Pareto model. Such exact tests support the assessment of the performance of estimators. In particular, we discuss some problems that one can encounter when misemploying the log-normal assumption based methods supported by the Basel II framework. Real data and simulated examples illustrate the methods.     

Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called “maximum likelihood” estimators of the tail index γ. In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar “maximum likelihood” estimator of a positive tail index γ, based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study. Research partially supported by FCT/POCTI and POCI/FEDER.     

Weak properties and robustness of t-Hill estimators

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.     

半参数回归非线性小波误差密度估计

In this paper,a semiparametric regression model in which errors are i.i.d random variables from an unknown density f(·) is considered.Based on Hall et al.(1995),a nonlinear wavelet estimation of f(·) without restrictions of continuity everywhere on f(·) is given,and the convergence rate of the estimators in L2 is obtained.     

On an improvement of Hill and some other estimators

In the paper, we propose a new idea in the tail-index estimation. This idea allows us to improve the asymptotic performance of the classical Hill estimator and other most popular estimators over the range of the parameters present in the second-order regular-variation condition. We prove the asymptotic normality of the introduced estimators and provide a comparison (using the asymptotic mean-squared error) with other estimators of the tail index.     

Heteroscedastic Laplace mixture of experts regression models and applications

Mixture of Experts(MoE) regression models are widely studied in statistics and machine learning for modeling heterogeneity in data for regression, clustering and classification.Laplace distribution is one of the most important statistical tools to analyze thick and tail data. Laplace Mixture of Linear Experts(LMoLE) regression models are based on the Laplace distribution which is more robust. Similar to modelling variance parameter in a homogeneous population, we propose and study a new novel class of models: heteroscedastic Laplace mixture of experts regression models to analyze the heteroscedastic data coming from a heterogeneous population in this paper. The issues of maximum likelihood estimation are addressed. In particular, Minorization-Maximization(MM) algorithm for estimating the regression parameters is developed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo simulations. Results from the analysis of two real data sets are presented.     

Additive hazards regression with random effects for clustered failure times

Additive hazards model with random effects is proposed for modelling the correlated failure time data when focus is on comparing the failure times within clusters and on estimating the correlation between failure times from the same cluster, as well as the marginal regression parameters. Our model features that, when marginalized over the random effect variable, it still enjoys the structure of the additive hazards model. We develop the estimating equations for inferring the regression parameters. The proposed estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions. Furthermore, the estimator of the baseline hazards function is proposed and its asymptotic properties are also established. We propose a class of diagnostic methods to assess the overall fitting adequacy of the additive hazards model with random effects. We conduct simulation studies to evaluate the finite sample behaviors of the proposed estimators in various scenarios. Analysis of the Diabetic Retinopathy Study is provided as an illustration for the proposed method.     

Asymptotic properties of generalized DPR statistic

We investigate asymptotic properties of the so-called generalized Davydov, Paulauskas, and Račkauskas statistic that was introduced in [V. Paulauskas and M. Vaičiulis, Several modifications of DPR estimator of the tail index, Lith. Math. J., 51(1):36–50, 2011]. We obtain an asymptotics of the bias of this statistic and show that this statistic is asymptotically normal. As an application, we provide a new class of semiparametric heavy-tailed estimators and compare these estimators with other estimators, including the popular Hill and Pickands estimators.     

Robust Autocorrelation Estimation

In this article, we introduce a new class of robust autocorrelation estimators based on interpreting the sample autocorrelation function as a linear regression. We investigate the efficiency and robustness properties of the estimators that result from employing three common robust regression techniques. We discuss the construction of robust autocovariance and positive definite autocorrelation estimates, and their application to AR model fitting. We perform simulation studies with various outlier configurations to compare the different estimators.     

Likelihood Based Confidence Intervals for the Tail Index

For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.     

Mean estimation in the presence of change points

In this study we address the problem of the mean estimation of the IBEX-35 index stock quotes in the presence of change points. We rely on nonparametric regression methods for detecting and estimating changes points, and for estimating the discontinuous regression function. Model-assisted and model-based estimators and their jump-preserving counterparts are used for mean estimation and an empirical comparison between the methods is performed.     

A New Estimator for a Tail Index

We investigate properties of a new estimator for a tail index introduced by Davydov and co-workers. The main advantage of this estimator is the simplicity of the statistic used for the estimator. We provide results of simulation by comparing plots of our's and Hill's estimators.     

A hybrid Pareto model for asymmetric fat-tailed data: the univariate case

Density estimators that can adapt to asymmetric heavy tails are required in many applications such as finance and insurance. Extreme value theory (EVT) has developed principled methods based on asymptotic results to estimate the tails of most distributions. However, the finite sample approximation might introduce a severe bias in many cases. Moreover, the full range of the distribution is often needed, not only the tail area. On the other hand, non-parametric methods, while being powerful where data are abundant, fail to extrapolate properly in the tail area. We put forward a non-parametric density estimator that brings together the strengths of non-parametric density estimation and of EVT. A hybrid Pareto distribution that can be used in a mixture model is proposed to extend the generalized Pareto (GP) to the whole real axis. Experiments on simulated data show the following. On one hand, the mixture of hybrid Paretos converges faster in terms of log-likelihood and provides good estimates of the tail of the distributions when compared with other density estimators including the GP distribution. On the other hand, the mixture of hybrid Paretos offers an alternate way to estimate the tail index which is comparable to the one estimated with the standard GP methodology. The mixture of hybrids is also evaluated on the Danish fire insurance data set.      

重尾分布的尾部指数估计及沪深股市实证分析

重尾分布尾部指数α的估计依赖于样本中所用顺序统计量个数k的选取.本文介绍了估计α时选择k的两类不同的方法:Sum-plot方法和Bootstrap方法,并对Hall提出的Bootstrap方法作了改进,称为M-Bootstrap方法.本文利用上述方法对已知分布进行Monte-Carlo模拟,研究它们的可行性,然后对上海和深圳两市股指数据进行了实证分析.计算结果表明,上海和深圳股指收益率具有重尾性.是右偏态的,右尾厚于左尾.通过几种方法计算的结果比较发现Sum-plot方法和M-Bootstrap方法在估计重尾指数上精确性较高一些,而且不受异常值的影响.     

Alternatives to a Semi-Parametric Estimator of Parameters of Rare Events—The Jackknife Methodology*

In this paper, and in a context of regularly varying tails, we propose different alternatives to a well-known estimator of the tail index—the Hill estimator (Hill, 1975). These alternatives have essentially in mind a reduction in bias, preferably without increasing Mean Square Error, by the use of suitable Generalized Jackknife methodologies (Gray and Schucany, 1972). The first estimate obtained through this methodolgy is the one introduced by Peng (1998), under a different context. Other Generalized Jackknife estimators are linear combinations of Hill estimators at different levels. This methodology of affine combinations of Hill estimators at different levels may be easily generalized to other semi-parametric estimators of the tail index, like Pickands' estimator (Pickands, 1975) or the Moment's estimator (Dekkers et al., 1989), and consequently to a general real tail index, seeming to be a promising field of research.     

Empirical likelihood-based inferences for the area under the ROC curve with covariates

In the receiver operating characteristic (ROC) analysis,the area under the ROC curve (AUC) is a popular summary index of discriminatory accuracy of a diagnostic test.Incorporating covariates into ROC analysis can improve the diagnostic accuracy of the test.Regression model for the AUC is a tool to evaluate the effects of the covariates on the diagnostic accuracy.In this paper,empirical likelihood (EL) method is proposed for the AUC regression model.For the regression parameter vector,it can be shown that the asymptotic distribution of its EL ratio statistic is a weighted sum of independent chi-square distributions.Confidence regions are constructed for the parameter vector based on the newly developed empirical likelihood theorem,as well as for the covariate-specific AUC.Simulation studies were conducted to compare the relative performance of the proposed EL-based methods with the existing method in AUC regression.Finally,the proposed methods are illustrated with a real data set.     

Mean Integrated Squared Error of Nonlinear Wavelet-based Estimators with Long Memory Data

We consider the nonparametric regression model with long memory data that are not necessarily Gaussian and provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet-based mean regression function estimators. We show this MISE expansion, when the underlying mean regression function is only piecewise smooth, is the same as analogous expansion for the kernel estimators. However, for the kernel estimators, this MISE expansion generally fails if an additional smoothness assumption is absent. Research supported in part by the NSF grant DMS-0103939.     

The Construction and Properties of Boundary Kernels for Smoothing Sparse Multinomials

In recent years several authors have investigated the use of smoothing methods for sparse multinomial data. In particular, Hall and Titterington (1987) studied kernel smoothing in detail. It is pointed out here that the bias of kernel estimates of probabilities for cells near the boundaries of the multinomial vector can dominate the mean sum of squared error of the estimator for most true probability vectors. Fortunately, boundary kernels devised to correct boundary effects for kernel regression estimators can achieve the same result for these estimators. Properties of estimates based on boundary kernels are investigated and compared to unmodified kernel estimates and maximum penalized likelihood estimates. Monte Carlo evidence indicates that the boundary-corrected kernel estimates usually outperform uncorrected kernel estimates and are quite competitive with penalized likelihood estimates.