Fernanda Figueiredo

Bias reduction of a tail index estimator through an external estimation of the second-order parameter

In this paper, we first consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterised in a tuning or control parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of the tail index γ with a null dominant component of asymptotic bias, and consequently with a reasonably flat mean squared error pattern, as a function of k, the number of top-order statistics considered. Such a control parameter depends on a second-order parameter ρ, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator, provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro–US

Bias reduction in risk modelling: Semi-parametric quantile estimation

exchange rates.

The EWMA median chart with estimated parameters

InStatistics of Extremes we are mainly interested in the estimation of quantities related to extreme events. In many areas of application, like for instanceInsurance Mathematics, Finance andStatistical Quality Control, a typical requirement is to find a value, high enough, so that the chance of an exceedance of that value is small. We are then interested in the estimation of ahigh quantile Xp, a value which is overpassed with a small probabilityp. In this paper we deal with the semi-parametric estimation ofXp for heavy tails. Since the classical semi-parametric estimators exhibit a reasonably high bias for low thresholds, we shall deal with bias reduction techniques, trying to improve their performance.

The median chart with estimated parameters

ABSTRACT The usual practice in control charts is to assume that the chart parameters are known or can be accurately estimated from in-control historical samples and the data are free from outliers. Both of these assumptions are not realistic in practice: a control chart may involve the estimation of process parameters from a very limited number of samples and the data may contain some outliers. In order to overcome these issues, in this article, we develop an Exponentially Weighted Moving Average (EWMA) median chart with estimated parameters to monitor the mean value of a normal process. We study the run length properties of the proposed chart using a Markov Chain approach and the performance of the proposed chart is compared to the EWMA median chart with known parameters. Several tables for the design of the proposed chart are given in order to expedite the use of the chart by practitioners. An illustrative example is also given along with some recommendations about the minimum number of initial subgroups m for different sample sizes n that must be collected for the estimation of the parameters so that the proposed chart has identical performance as the chart with known parameters. From the results we deduce that (i) there is a large difference between the known and estimated parameters cases unless the initial number of subgroups m is large; and (ii) the difference between the known and estimated parameters cases can be reduced by using dedicated chart parameters.

A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation

A modified median chart with estimated control limits is proposed for monitoring the mean value of a normal process. The estimates for the nominal process parameters used to set up the control limits in Phase I are the average of the sample medians and the average of the sample ranges of m initial subgroups of size n. The run length properties of the chart are examined, and the obtained results lead us to conclude the proposed chart present an interesting RL performance (at least for an adequate choice of the chart parameter used to determine the control limits). Control chart parameters are provided for a specific in-control average run length of 370.4 for several combinations of n and m, which facilitates the use of the chart by practitioners, and an illustrative example of implementation of the chart is presented. Some recommendations are also given about the minimum number of initial subgroups m that must be considered to implement the chart with the constant parameter used in the case of known nominal values, and in order to obtain the expected performance. [Received 24 August 2011; Revised 28 October 2011; Accepted 13 February 2012]

Modeling Extreme Events: Sample Fraction Adaptive Choice in Parameter Estimation

In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0≤q<1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.

Comparison of sampling plans by variables using the bootstrap and Monte Carlo simulations

When modeling extreme events, there are a few primordial parameters, among which we refer to the extreme value index (EVI) and the extremal index (EI). Under a framework related to large values, the EVI measures the right tail weight of the underlying distribution and the EI characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semiparametric estimators of these parameters show the same type of behavior: nice asymptotic properties but a high variance for small values of k, the number of upper order statistics used in the estimation, and a high bias for large values of k. This brings a real need for the choice of k. Choosing some well-known estimators of those two parameters, we revisit the application of a heuristic algorithm for the adaptive choice of k. A simulation study illustrates the performance of the proposed algorithm.

Acceptance-Sampling Plans for Reducing the Risk Associated with Chemical Compounds

We consider two sampling plans by variables to inspect batches of products from an industrial process under a context of unknown distribution underlying the measurements of the quality characteristic under study. Through the use of the bootstrap methodology and Monte Carlo simulations we evaluate and compare the performance of those sampling plans in terms of probability of acceptance of lots and average outgoing quality level.

The total median statistic to monitor contaminated normal data

In various manufacturing industries it is important to investigate the presence of some chemical or harmful substances in lots of raw material or final products, in order to evaluate if they are in conformity to requirements. In this work we highlight the adequacy of the inflated Pareto distribution to model measurements obtained by chromatography, and we define and evaluate acceptance-sampling plans under this distributional setup for lots of large dimension. Some technical results associated with the construction and evaluation of such sampling plans are provided as well as an algorithm for an easy implementation of the sampling plan that exhibits the best performance.

Resampling-Based Methodologies in Statistics of Extremes: Environmental and Financial Applications

Abstract Despite the advantages of the use of the normal distribution in Statistical Quality Control, the normality assumption is too restrictive for modelling real data sets, which usually exhibit asymmetry or tails heavier than the normal tails. But even in potential normal situations, there is often a small to moderate percentage of contamination in the data. In this paper, we analyze the efficiency and robustness of the total median statistic in comparison with the sample mean and the sample median to estimate the mean value of symmetric contaminated normal distributions, close to the normal, but with heavier-than-normal tails. We also compare the performance of the total median and the sample mean charts to monitor the mean value of such processes. The simulation results lead us to suggest the use of the total median statistic due to its efficiency and degree of robustness.