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.     

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].     

Semi-parametric tail inference through probability-weighted moments

In this paper, 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.     

Estimating the First- and Second-Order Parameters of a Heavy-Tailed Distribution

This paper suggests censored maximum likelihood estimators for the first‐ and second‐order parameters of a heavy‐tailed distribution by incorporating the second‐order regular variation into the censored likelihood function. This approach is different from the bias‐reduced maximum likelihood method proposed by Feuerverger and Hall in 1999. The paper derives the joint asymptotic limit for the first‐ and second‐order parameters under a weaker assumption. The paper also demonstrates through a simulation study that the suggested estimator for the first‐order parameter is better than the estimator proposed by Feuerverger and Hall although these two estimators have the same asymptotic variances.     

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.     

Rank estimation for the functional linear model

This article discusses the estimation of the parameter function for a functional linear regression model under heavy-tailed errors' distributions and in the presence of outliers. Standard approaches of reducing the high dimensionality, which is inherent in functional data, are considered. After reducing the functional model to a standard multiple linear regression model, a weighted rank-based procedure is carried out to estimate the regression parameters. A Monte Carlo simulation and a real-world example are used to show the performance of the proposed estimator and a comparison made with the least-squares and least absolute deviation estimators.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Minimax Linear Smoothers

We consider the problem of estimating the mean of a multivariate distribution. As a general alternative to penalized least squares estimators, we consider minimax estimators for squared error over a restricted parameter space where the restriction is determined by the penalization term. For a quadratic penalty term, the minimax estimator among linear estimators can be found explicitly. It is shown that all symmetric linear smoothers with eigenvalues in the unit interval can be characterized as minimax linear estimators over a certain parameter space where the bias is bounded. The minimax linear estimator depends on smoothing parameters that must be estimated in practice. Using results in Kneip (1994), this can be done using Mallows' C _L -statistic and the resulting adaptive estimator is now asymptotically minimax linear. The minimax estimator is compared to the penalized least squares estimator both in finite samples and asymptotically.     

Asymptotic Behavior of Conditional Least Squares Estimators for Unstable Integer‐valued Autoregressive Models of Order 2

In this paper, the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters, of the mean of the innovations, and of the stability parameter for unstable integer‐valued autoregressive processes of order 2 is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.     

Parameter estimation for a new distribution for the strength of brittle fibers: a simulation study

Parameter estimates of a new distribution for the strength of brittle fibers and composite materials are considered. An algorithm for generating random numbers from the distribution is suggested. Two parameter estimation methods, one based on a simple least squares procedure and the other based on the maximum likelihood principle, are studied using Monte Carlo simulation. In most cases, the maximum likelihood estimators were found to have somewhat smaller root mean squared error and bias than the least squares estimators. However, the least squares estimates are generally good and provide useful initial values for the numerical iteration used to find the maximum likelihood estimates.     

Estimation of parameters of two-dimensional sinusoidal signal in heavy-tailed errors

In this paper, we consider a two-dimensional sinusoidal model observed in an additive random field. The proposed model has wide applications in statistical signal processing. The additive noise has mean zero but the variance may not be finite. We propose the least squares estimators to estimate the unknown parameters. It is observed that the least squares estimators are strongly consistent. We obtain the asymptotic distribution of the least squares estimators under the assumption that the additive errors are from a symmetric stable distribution. Some numerical experiments are performed to see how the results work for finite samples.     

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.     

A Monte Carlo study of traditional and rank-based picked-points analyses

In analysis of covariance with heteroscedastic slopes a picked-points analysis is often performed. Least-squares based picked-points analyses often lose efficiency (at times substantial) for nonnormal error distributions. Robust rank-based picked-points analyses are developed which are optimizable for heavy-tailed and/or skewed error distributions. The results of a Monte Carlo investigation of these analyses are presented. The situations include the normal model and models which violate it in one or several ways. Empirically the rank-based analyses appear to be valid over all these situations and more powerful than the least squares analysis for all the nonnormal models, while losing little efficiency at the normal model.     

Some properties of generalized ridge estimators

Exact expressions, in the form of infinite series expansions, are given for the first and second moments of two well known generalized ridge estimators. These series expansions are then evaluated using recursive formulas and computations are verified using approximations. Results are presented for the relative mean square error and bias of these estimators as well as their relative efficiency with respect to least squares.     

Practical small sample inference for single lag subset autoregressive models

We propose a method for saddlepoint approximating the distribution of estimators in single lag subset autoregressive models of order one. By viewing the estimator as the root of an appropriate estimating equation, the approach circumvents the difficulty inherent in more standard methods that require an explicit expression for the estimator to be available. Plots of the densities reveal that the distributions of the Burg and maximum likelihood estimators are nearly identical. We show that one possible reason for this is the fact that Burg enjoys the property of estimation equation optimality among a class of estimators expressible as a ratio of quadratic forms in normal random variables, which includes Yule–Walker and least squares. By inverting a two-sided hypothesis test, we show how small sample confidence intervals for the parameters can be constructed from the saddlepoint approximations. Simulation studies reveal that the resulting intervals generally outperform traditional ones based on asymptotics and have good robustness properties with respect to heavy-tailed and skewed innovations. The applicability of the models is illustrated by analyzing a longitudinal data set in a novel manner.     

Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

Bias correction for outlier estimation in time series

The problem of outlier estimation in time series is addressed. The least squares estimators of additive and innovation outliers in the framework of linear stationary and non-stationary models are considered and their bias is evaluated. As a result, simple alternative nearly unbiased estimators are proposed both for the additive and the innovation outlier types. A simulation study confirms the theoretical results and suggests that the proposed estimators are effective in reducing the bias also for short series.     

Size distributions reconsidered

We consider tests of the hypothesis that the tail of size distributions decays faster than any power function. These are based on a single parameter that emerges from the Fisher–Tippett limit theorem, and discriminate between leading laws considered in the literature without requiring fully parametric models/specifications. We study the proposed tests taking into account the higher order regular variation of the size distribution that can lead to catastrophic distortions. The theoretical bias corrections realign successfully nominal and empirical test behavior, and inform a sensitivity analysis for practical work. The methods are used in an examination of the size distribution of cities and firms.     

Estimating parameters in a simple errors-in-variables model: a new approach base on finite sample distribution theory

This article presents a first direct application of finite sample distribution theory. The relevance of analytical finite sample research is exemplified in the framework of a simple linear errors-in-variables model (EV Model) with known or approximately known measurement error variance. Analytical results derived byRichardson/Wu (1970) are applied for constructing new approximately unbiased estimators for the slope coefficient in the EV model. The new estimators are compared with the biased least squares estimator and with asymptotic theory based corrected least squares estimators. Retaining responsibility for remaining errors the author is indebted to Prof. H. Schneewei\ and Prof. J. Gruber for helpful comments and discussions. Mrs. A. Brandtstater deserves special mention and thanks for performing the computations reported in section 4.     

Improved reduced-bias tail index and quantile estimators

In this paper, we deal with bias reduction techniques for heavy tails, trying to improve mainly upon the performance of classical high quantile estimators. High quantiles depend strongly on the tail index γ$\gamma$, for which new classes of reduced-bias estimators have recently been introduced, where the second-order parameters in the bias are estimated at a level _k1$k_1$ of a larger order than the level k at which the tail index is estimated. Doing this, it was seen that the asymptotic variance of the new estimators could be kept equal to the one of the popular Hill estimators. In a similar way, we now introduce new classes of tail index and associated high quantile estimators, with an asymptotic mean squared error smaller than that of the classical ones for all k in a large class of heavy-tailed models. We derive their asymptotic distributional properties and compare them with those of alternative estimators. Next to that, an illustration of the finite sample behavior of the estimators is also provided through a Monte Carlo simulation study and the application to a set of real data in the field of insurance.