A practical method for analysing heavy tailed data

An important practical issue of applying heavy tailed distributions is how to choose the sample fraction or threshold, since only a fraction of upper order statistics can be employed in the inference. Recently, Guillou & Hall ( 2001 ; Journal of Royal Statistical Society B, 63, 293–305) proposed a simple way to choose the threshold in estimating a tail index. In this article, the author first gives an intuitive explanation of the approach in Guillou & Hall ( 2001 ; it Journal of Royal Statistical Society B, 63, 293–305) and then proposes an alternative method, which can be extended to other settings like extreme value index estimation and tail dependence function estimation. Further the author proposes to combine this method for selecting a threshold with a bias reduction estimator to improve the performance of the tail index estimation, interval estimation of a tail index, and high quantile estimation. Simulation studies on both point estimation and interval estimation for a tail index show that both selection procedures are comparable and bias reduction estimation with the threshold selected by either method is preferred. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Does bias reduction with external estimator of second order parameter work for endpoint?

Bias reduction estimation for tail index has been studied in the literature. One method is to reduce bias with an external estimator of the second order regular variation parameter; see Gomes and Martins [2002. Asymptotically unbiased estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31]. It is known that negative extreme value index implies that the underlying distribution has a finite right endpoint. As far as we know, there exists no bias reduction estimator for the endpoint of a distribution. In this paper, we study the bias reduction method with an external estimator of the second order parameter for both the negative extreme value index and endpoint simultaneously. Surprisingly, we find that this bias reduction method for negative extreme value index requires a larger order of sample fraction than that for positive extreme value index. This finding implies that this bias reduction method for endpoint is less attractive than that for positive extreme value index. Nevertheless, our simulation study prefers the proposed bias reduction estimators to the biased estimators in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568].     

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number k _n of largest order statistics can be represented as a statistical tail function al, that is a functional T applied to the empirical tail quantile function Q _n. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F . For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T ( Q _n) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k _pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour     

Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.     

Local Estimation of the Second-Order Parameter in Extreme Value Statistics and Local Unbiased Estimation of the Tail Index

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.     

Inference for the tail index of a GARCH(1,1) model and an AR(1) model with ARCH(1) errors

For a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors, one can estimate the tail index by solving an estimating equation with unknown parameters replaced by the quasi maximum likelihood estimation, and a profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index. However, this requires that the errors of such a model have at least a finite fourth moment. In this article, we show that the finite fourth moment can be relaxed by employing a least absolute deviations estimate for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model.     

Smooth tail-index estimation

The two parametric distribution functions appearing in the extreme-value theory – the generalized extreme-value distribution and the generalized Pareto distribution – have log-concave densities if the extreme-value index γ∈[?1, 0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density ? f _n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.     

A Semi-Parametric Quantile Function Estimator for Use in Bootstrap Estimation Procedures

In this note we develop a new quantile function estimator called the tail extrapolation quantile function estimator. The estimator behaves asymptotically exactly the same as the standard linear interpolation estimator. For finite samples there is small correction towards estimating the extreme quantiles. We illustrate that by employing this new estimator we can greatly improve the coverage probabilities of the standard bootstrap percentile confidence intervals. The method does not reqiure complicated calculations and hence it should appeal to the statistical practitioner.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

Fast Censored Linear Regression

Weighted log‐rank estimating function has become a standard estimation method for the censored linear regression model, or the accelerated failure time model. Well established statistically, the estimator defined as a consistent root has, however, rather poor computational properties because the estimating function is neither continuous nor, in general, monotone. We propose a computationally efficient estimator through an asymptotics‐guided Newton algorithm, in which censored quantile regression methods are tailored to yield an initial consistent estimate and a consistent derivative estimate of the limiting estimating function. We also develop fast interval estimation with a new proposal for sandwich variance estimation. The proposed estimator is asymptotically equivalent to the consistent root estimator and barely distinguishable in samples of practical size. However, computation time is typically reduced by two to three orders of magnitude for point estimation alone. Illustrations with clinical applications are provided.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.     

An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage

In this article, we investigate the limitations of traditional quantile function estimators and introduce a new class of quantile function estimators, namely, the semi-parametric tail-extrapolated quantile estimators, which has excellent performance for estimating the extreme tails with finite sample sizes. The smoothed bootstrap and direct density estimation via the characteristic function methods are developed for the estimation of confidence intervals. Through a comprehensive simulation study to compare the confidence interval estimations of various quantile estimators, we discuss the preferred quantile estimator in conjunction with the confidence interval estimation method to use under different circumstances. Data examples are given to illustrate the superiority of the semi-parametric tail-extrapolated quantile estimators. The new class of quantile estimators is obtained by slight modification of traditional quantile estimators, and therefore, should be specifically appealing to researchers in estimating the extreme tails.     

Bias-calibrated estimation from sample surveys containing outliers

We discuss the problem of estimating finite population parameters on the basis of a sample containing representative outliers. We clarify the motivation for Chambers's bias-calibrated estimator of the population total and show that bias calibration is a key idea in constructing estimators of finite population parameters. We then link the problem of estimating the population total to distribution function or quantile estimation and explore a methodology based on the use of Chambers's estimator. We also propose methodology based on the use of robust estimates and a bias-calibrated form of the Chambers and Dunstan estimator of the population distribution function. This proposal leads to a bias-calibrated estimator of the population total which is an alternative to that of Chambers. We present a small simulation study to illustrate the utility of these estimators.     

Random weighting-based quantile estimation via importance resampling

Abstract

This paper presents a new method to estimate the quantiles of generic statistics by combining the concept of random weighting with importance resampling. This method converts the problem of quantile estimation to a dual problem of tail probabilities estimation. Random weighting theories are established to calculate the optimal resampling weights for estimation of tail probabilities via sequential variance minimization. Subsequently, the quantile estimation is constructed by using the obtained optimal resampling weights. Experimental results on real and simulated data sets demonstrate that the proposed random weighting method can effectively estimate the quantiles of generic statistics.     

A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

Estimating the conditional tail index by integrating a kernel conditional quantile estimator

This paper deals with the estimation of the tail index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.     

Abelian and Tauberian Theorems on the Bias of the Hill Estimator

The bias of Hill's estimator for the positive extreme value index of a distribution is investigated in relation to the convergence rate in the regular variation property of the tail function of the common distribution of the sample and the corresponding tail quantile function. Based on the theory of generalized regular variation, natural second-order conditions are proposed which both imply and are implied by convergence of the expectation of Hill's estimator to the extreme value index at certain rates. A comparison with second-order conditions encountered in the literature is made.     

Efficient Estimation of an Additive Quantile Regression Model

Abstract. In this paper, two non‐parametric estimators are proposed for estimating the components of an additive quantile regression model. The first estimator is a computationally convenient approach which can be viewed as a more viable alternative to existing kernel‐based approaches. The second estimator involves sequential fitting by univariate local polynomial quantile regressions for each additive component with the other additive components replaced by the corresponding estimates from the first estimator. The purpose of the extra local averaging is to reduce the variance of the first estimator. We show that the second estimator achieves oracle efficiency in the sense that each estimated additive component has the same variance as in the case when all other additive components were known. Asymptotic properties are derived for both estimators under dependent processes that are strictly stationary and absolutely regular. We also provide a demonstrative empirical application of additive quantile models to ambulance travel times.     

Resampling-based bias-corrected time series prediction

In this paper, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study.     

Local Smoothing Using the Bootstrap

We consider the problem of data-based choice of the bandwidth of a kernel density estimator, with an aim to estimate the density optimally at a given design point. The existing local bandwidth selectors seem to be quite sensitive to the underlying density and location of the design point. For instance, some bandwidth selectors perform poorly while estimating a density, with bounded support, at the median. Others struggle to estimate a density in the tail region or at the trough between the two modes of a multimodal density. We propose a scale invariant bandwidth selection method such that the resulting density estimator performs reliably irrespective of the density or the design point. We choose bandwidth by minimizing a bootstrap estimate of the mean squared error (MSE) of a density estimator. Our bootstrap MSE estimator is different in the sense that we estimate the variance and squared bias components separately. We provide insight into the asymptotic accuracy of the proposed density estimator.