Beta kernel quantile estimators of heavy-tailed loss distributions

In this paper we suggest several nonparametric quantile estimators based on Beta kernel. They are applied to transformed data by the generalized Champernowne distribution initially fitted to the data. A Monte Carlo based study has shown that those estimators improve the efficiency of the traditional ones, not only for light tailed distributions, but also for heavy tailed, when the probability level is close to 1. We also compare these estimators with the Extreme Value Theory Quantile applied to Danish data on large fire insurance losses.     

Pitfalls in using Weibull tailed distributions

By assuming that the underlying distribution belongs to the domain of attraction of an extreme value distribution, one can extrapolate the data to a far tail region so that a rare event can be predicted. However, when the distribution is in the domain of attraction of a Gumbel distribution, the extrapolation is quite limited generally in comparison with a heavy tailed distribution. In view of this drawback, a Weibull tailed distribution has been studied recently. Some methods for choosing the sample fraction in estimating the Weibull tail coefficient and some bias reduction estimators have been proposed in the literature. In this paper, we show that the theoretical optimal sample fraction does not exist and a bias reduction estimator does not always produce a smaller mean squared error than a biased estimator. These are different from using a heavy tailed distribution. Further we propose a refined class of Weibull tailed distributions which are more useful in estimating high quantiles and extreme tail probabilities.     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.     

The small sample performance of estimators of the standard errors of structural equation models

In this paper, we compare five asymptotically, under a correctly specified likelihood, equivalent estimators of the standard errors for parameters in structural equation models. The estimators are evaluated under different conditions regarding (i) sample size, varying between N=50 and 3200, (ii) distributional assumption of the latent variables and the disturbance terms, namely normal, and heavy tailed (t), and (iii) the complexity of the model. For the assessment of the five estimators we use overall performance, relative bias, MSE and coverage of confidence intervals. The analysis reveals substantial differences in the performance of the five asymptotically equal estimators. Most diversity was found for t distributed, i.e. heavy tailed, data.     

Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution

Inverse Gamma-Pareto composite distribution is considered as a model for heavy tailed data. The maximum likelihood (ML), smoothed empirical percentile (SM), and Bayes estimators (informative and non-informative) for the parameter θ, which is the boundary point for the supports of the two distributions are derived. A Bayesian predictive density is derived via a gamma prior for θ and the density is used to estimate risk measures. Accuracy of estimators of θ and the risk measures are assessed via simulation studies. It is shown that the informative Bayes estimator is consistently more accurate than ML, Smoothed, and the non-informative Bayes estimators.     

Generalized Kernel Estimators for the Weibull-Tail Coefficient

We introduce new families of estimators for the Weibull-tail coefficient, obtained from a weighted sum of a power transformation of excesses over a high random threshold. Asymptotic normality of the estimators is proven for an intermediate sequence of upper order statistics, and under classical regularity conditions for L-statistics and a second-order condition on the tail behavior of the underlying distribution. The small sample performance of two specific examples of kernel functions is evaluated in a simulation study.     

L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation,goodness of fit and application to earthquake data

Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum–Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962–2015, is carried out. This application identifies the extreme value Birnbaum–Saunders distribution as a better model than classic extreme value distributions for describing seismic events.     

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.     

Correction of Density Estimators that are not Densities

Abstract. Several old and new density estimators may have good theoretical performance, but are hampered by not being bona fide densities; they may be negative in certain regions or may not integrate to 1. One can therefore not simulate from them, for example. This paper develops general modification methods that turn any density estimator into one which is a bona fide density, and which is always better in performance under one set of conditions and arbitrarily close in performance under a complementary set of conditions. This improvement-for-free procedure can, in particular, be applied for higher-order kernel estimators, classes of modern h ⁴ bias kernel type estimators, superkernel estimators, the sinc kernel estimator, the k -NN estimator, orthogonal expansion estimators, and for various recently developed semi-parametric density estimators.     

Nonparametric regression estimation of conditional tails: the random covariate case

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.     

Robust mixture model cluster analysis using adaptive kernels

The traditional mixture model assumes that a dataset is composed of several populations of Gaussian distributions. In real life, however, data often do not fit the restrictions of normality very well. It is likely that data from a single population exhibiting either asymmetrical or heavy-tail behavior could be erroneously modeled as two populations, resulting in suboptimal decisions. To avoid these pitfalls, we generalize the mixture model using adaptive kernel density estimators. Because kernel density estimators enforce no functional form, we can adapt to non-normal asymmetric, kurtotic, and tail characteristics in each population independently. This, in effect, robustifies mixture modeling. We adapt two computational algorithms, genetic algorithm with regularized Mahalanobis distance and genetic expectation maximization algorithm, to optimize the kernel mixture model (KMM) and use results from robust estimation theory in order to data-adaptively regularize both. Finally, we likewise extend the information criterion ICOMP to score the KMM. We use these tools to simultaneously select the best mixture model and classify all observations without making any subjective decisions. The performance of the KMM is demonstrated on two medical datasets; in both cases, we recover the clinically determined group structure and substantially improve patient classification rates over the Gaussian mixture model.     

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.     

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     

Some Asymptotic Results of Kernel Density Estimators Under Random Left-Truncation and Dependent Data

Problems with truncated data arise frequently in survival analyses and reliability applications. The estimation of the density function of the lifetimes is often of interest. In this article, the estimation of density function by the kernel method is considered, when truncated data are showing some kind of dependence. We apply the strong Gaussian approximation technique to study the strong uniform consistency for kernel estimators of the density function under a truncated dependent model. We also apply the strong approximation results to study the integrated square error properties of the kernel density estimators under the truncated dependent scheme.     

Asymptotic normality of kernel estimators based upon incomplete data

In this paper, we are concerned with nonparametric estimation of the density and the failure rate functions of a random variable X which is at risk of being censored. First, we establish the asymptotic normality of a kernel density estimator in a general censoring setup. Then, we apply our result in order to derive the asymptotic normality of both the density and the failure rate estimators in the cases of right, twice and doubly censored data. Finally, the performance and the asymptotic Gaussian behaviour of the studied estimators, based on either doubly or twice censored data, are illustrated through a simulation study.     

Wavelet Threshold Estimators for Data with Correlated Noise

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable `bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an `oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short- and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.     

Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields

We investigate the asymptotic behaviour of binned kernel density estimators for dependent and locally non-stationary random fields converging to stationary random fields. We focus on the study of the bias and the asymptotic normality of the estimators. A simulation experiment conducted shows that both the kernel density estimator and the binned kernel density estimator have the same behavior and both estimate accurately the true density when the number of fields increases. We apply our results to the 2002 incidence rates of tuberculosis in the departments of France.     

Fat tail distributions and local thin tail alternatives

The behaviour of the Hill estimator for the tail index of fat tailed distributions in the presence of local alternatives which have a thin tail is investigated. The converse problem is also briefly addressed. A local thin tail alternative can severely bias the Hill statistic. The relevance of this issue for the class of stable distributions is discussed. We conduct a small simulation study to support the analysis. In the conclusion it is argued that for moderate out of sample quantile analysis the problem of local alternatives may be less pressing.     

Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using Kernel Density Method

In this article, we propose a nonparametric estimator for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method. The properties of the proposed kernel estimator are investigated and compared with well-known maximum likelihood and ordinary least squares estimators via a simulation technique. The mean squared error and the length of the bootstrap confidence interval are used as the basis criteria of the comparisons. The simulation study shows that the performance of the kernel estimator is acceptable as a general estimator. When the distribution of the data is assumed to be known, the maximum likelihood and ordinary least squares estimators perform better than the kernel estimator, while the kernel estimator is superior when the assumption of our knowledge of the data distribution is violated. A comparison among different estimators is achieved using a real data set.     

Pseudo‐likelihood estimation in ARCH models

The author presents asymptotic results for the class of pseudo‐likelihood estimators in the autoregressive conditional heteroscedastic models introduced by Engle (1982). Unlike what is required for the quasi‐likelihood estimator, some estimators in the class he considers do not require the finiteness of the fourth moment of the error density. Thus his method is applicable to heavy‐tailed error distributions for which moments higher than two may not exist.