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].     

A robust prediction error criterion for pareto modelling of upper tails

Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings. The upper tail of the distribution, where data are sparse, is typically fitted with a model, such as the Pareto model, from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. The authors develop a robust prediction error criterion for choosing k and estimating the Pareto index. A Monte Carlo study shows the good performance of the new estimator and the analysis of real data sets illustrates that a robust procedure for selection, and not just for estimation, is needed.     

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     

Extreme value index estimator using maximum likelihood and moment estimation

ABSTRACT

When a distribution function is in the max domain of attraction of an extreme value distribution, its tail can be well approximated by a generalized Pareto distribution. Based on this fact we use a moment estimation idea to propose an adapted maximum likelihood estimator for the extreme value index, which can be understood as a combination of the maximum likelihood estimation and moment estimation. Under certain regularity conditions, we derive the asymptotic normality of the new estimator and investigate its finite sample behavior by comparing with several classical or competitive estimators. A simulation study shows that the new estimator is competitive with other estimators in view of average bias, average MSE, and coefficient of variance of the new device for the optimal selection of the threshold.     

Divergence based robust estimation of the tail index through an exponential regression model

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.     

Utilization of higher order moments of scrambling variables in randomized response sampling

In this paper, a new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced. The proposed estimator makes use of higher order moments of the scrambling variable at the estimation stage. The proposed estimator has been found to be more efficient than the estimator due to Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] and Franklin [1989. A comparison of estimators for randomized response sampling with continuous distributions from a dichotomous population. Comm. Statist. Theory Methods 18, 489–505] type estimators in randomized response sampling. Recently, Guerriero and Sandri [2007. A note on the comparison of some randomized response procedures. J. Statist. Plann. Inference 137, 2184–2190] have shown that the family of randomized response models proposed by Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] is better than the Simmons’ family in terms of efficiency and protection.     

Bias reduction for high quantiles

High quantile estimation is of importance in risk management. For a heavy-tailed distribution, estimating a high quantile is done via estimating the tail index. Reducing the bias in a tail index estimator can be achieved by using either the same order or a larger order of number of the upper order statistics in comparison with the theoretical optimal one in the classical tail index estimator. For the second approach, one can either estimate all parameters simultaneously or estimate the first and second order parameters separately. Recently, the first method and the second method via external estimators for the second order parameter have been applied to reduce the bias in high quantile estimation. Theoretically, the second method obviously gives rise to a smaller order of asymptotic mean squared error than the first one. In this paper we study the second method with simultaneous estimation of all parameters for reducing bias in high quantile estimation.     

A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment

We use bias-reduced estimators of high quantiles of heavy-tailed distributions, to introduce a new estimator for the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked in a simulation study, by four of the most popular goodness-of-fit tests. The accuracy of the resulting confidence intervals is evaluated as well. We also investigate the finite sample behavior and compare our estimator with some versions of Peng's estimator of the mean (namely those based on Hill, t-Hill and Huisman et al. extreme value index estimators). Moreover, we discuss the robustness of the tail index estimators used in this paper. Finally, our estimation procedure is applied to the well-known Danish fire insurance claims data set, to provide confidence bounds for the means of weekly and monthly maximum losses over a period of 10 years.     

Hill estimator of projections of functional data on principal components

Functional principal component scores are commonly used to reduce mathematically infinitely dimensional functional data to finite dimensional vectors. In certain applications, most notably in finance, these scores exhibit tail behaviour consistent with the assumption of regular variation. Knowledge of the index of the regular variation, α, is needed to apply methods of extreme value theory. The most commonly used method of the estimation of α is the Hill estimator. We derive conditions under which the Hill estimator computed from the sample scores is consistent for the tail index of the unobservable population scores.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

Multivariate spatial U-quantiles: A Bahadur–Kiefer representation,a Theil–Sen estimator for multiple regression,and a robust dispersion estimator

A leading multivariate extension of the univariate quantiles is the so-called “spatial” or “geometric” notion, for which sample versions are highly robust and conveniently satisfy a Bahadur–Kiefer representation. Another extension of univariate quantiles has been to univariate U-quantiles, on the basis of which, for example, the well-known Hodges–Lehmann location estimator has a natural formulation. Generalizing both extensions, we introduce multivariate spatial U-quantiles and develop a corresponding Bahadur–Kiefer representation. New statistics based on spatial U-quantiles are presented for nonparametric estimation of multiple regression coefficients, extending the classical Theil–Sen nonparametric simple linear regression slope estimator, and for robust estimation of multivariate dispersion. Some other applications are mentioned as well.     

Tail index and second-order parameters’ semi-parametric estimation based on the log-excesses

In this paper we are interested in the derivation of the asymptotic and finite-sample distributional properties of a ‘quasi-maximum likelihood’ estimator of a ‘scale’ second-order parameter β, directly based on the log-excesses of an available sample. Such estimation is of primordial importance for the adaptive selection of the optimal sample fraction to be used in the classical semi-parametric tail index estimation as well as for the reduced-bias estimation of the tail index, high quantiles and other parameters of extreme or even rare events. An application in the area of survival analysis is provided, on the basis of a data set on males diagnosed with cancer of the tongue.     

Strong uniform consistency results of the weighted average of conditional artificial data points

In this paper, we study strong uniform consistency of a weighted average of artificial data points. This is especially useful when information is incomplete (censored data, missing data …). In this case, reconstruction of the information is often achieved nonparametrically by using a local preservation of mean criterion for which the corresponding mean is estimated by a weighted average of new data points. The present approach enlarges the possible scope for applications beyond just the incomplete data context and can also be useful to treat the estimation of the conditional mean of specific functions of complete data points. As a consequence, we establish the strong uniform consistency of the Nadaraya–Watson [Nadaraya, E.A., 1964. On estimating regression. Theory Probab. Appl. 9, 141–142; Watson, G.S., 1964. Smooth regression analysis. Sankhyā Ser. A 26, 359–372] estimator for general transformations of the data points. This result generalizes the one of Härdle et al. [Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16, 1428–1449]. In addition, the strong uniform consistency of a modulus of continuity will be obtained for this estimator. Applications of those two results are detailed for some popular estimators.     

Robust estimation in long-memory processes under additive outliers

In this paper, we introduce an alternative semiparametric estimator of the fractional differencing parameter in ARFIMA models which is robust against additive outliers. The proposed estimator is a variant of the GPH estimator [Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series model. Journal of Time Series Analysis 4, 221–238]. In particular, we use the robust sample autocorrelations of Ma, Y. and Genton, M. [2000. Highly robust estimation of the autocovariance function. Journal of Time Series Analysis 21, 663–684] to obtain an estimator for the spectral density of the process. Numerical results show that the estimator we propose for the differencing parameter is robust when the data contain additive outliers.     

Iterative regularisation in nonparametric instrumental regression

This paper considers the nonparametric regression model with an additive error that is correlated with the explanatory variables. Motivated by empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. However, the estimation of a nonparametric regression function by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function that is based on projection onto finite dimensional spaces and that includes an iterative regularisation method (the Landweber–Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both strong and weak source conditions. A Monte Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.     

On tail index estimation based on multivariate data

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, that is, of which Pareto-like marginals share the same tail index. A multivariate central limit theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. We introduce the concept of (standard) heavy-tailed random vector of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.     

Information dependency: Strong consistency of Darbellay–Vajda partition estimators

The Darbellay–Vajda partition scheme is a well known method to estimate the information dependency. This estimator belongs to a class of data-dependent partition estimators. We would like to prove that with some simple conditions, the Darbellay–Vajda partition estimator is a strong consistency for the information dependency estimation of a bivariate random vector. This result is an extension of 20 and 21 work which gives some simple conditions to confirm that the Gessaman's partition estimator and the tree-quantization partition estimator, other estimators in the class of data-dependent partition estimators, are strongly consistent.     

Calibrated linear unbiased estimators in finite population sampling

Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38] and Rao [2002. Discussion of “Exact linear unbiased estimation in survey sampling”. J. Stat. Plann. Inf. 102, 39–40] suggested some useful techniques of deriving a linear unbiased estimator of a finite population total by modifying a given linear estimator. In this paper we suggest various generalizations of their results. In particular, we search for estimators satisfying the calibration property with respect to a related auxiliary variable and obtain some new calibrated unbiased ratio-type estimators for arbitrary sampling designs. We also explore a few properties of one of the estimators suggested in Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38].     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.