Debbie J. Dupuis

Exceedances over High Thresholds: A Guide to Threshold Selection

In this paper, we consider the modeling of exceedances over high thresholds.The natural distribution for such exceedances, the generalized Pareto distribution (GPD), is used and the problematic issue of threshold selection is addressed. We fit the GPD robustly to the data using techniques based on optimal bias-robust estimates. The robust procedure will assign weights between 0 and 1 to each data point. These weights are used to assess the validity of the GPD model for exceedances of the proposed threshold and thus can guide threshold selection. That is, we can initially consider a low threshold and increase it (thus reducing the number of data points) until all weights are close to one. The new approach is used to analyze two of the NERC data sets.

A hybrid estimator for generalized pareto and extreme-value distributions

The methods of moments and probability-weighted moments are the most commonly used methods for estimating the parameters of the generalized Pareto distribution and generalized extreme-value distributions. These methods, however, frequently lead to nonfeasible estimates in the sense that the supports inferred from the estimates fail to contain all observations. In this paper, we propose a hybrid estimator which is derived by incorporating a simple auxiliary constraint on feasibility into the estimates. The hybrid estimator is very easy to use, always feasible, and also has smaller bias and mean square error in many cases. Its advantages are further illustrated through the analyses of two real data sets.

Asymptotic importance sampling

Abstract An importance sampling technique is described which is based on theoretical considerations about the structure of multivariate integrands in domains having small probability content. The method is formulated in the original variable space. Sampling densities are derived for a variety of practical conditions: a single point of maximum loglikelihood; several points; points located at the intersect of several failure surfaces; and, bounded variables. Sampling in the safe domain is avoided and extensive use is made of noncartesian as well as surface coordinates. The parameters of the importance sampling densities are taylored in such a way as to yield asymptotic minimum variance unbiased estimators. The quality and the efficiency of the method improves as the failure probability decreases. Parameter sensitivies are easily computed owing to the use of local surface coordinates. Several examples are provided.

Robust weighted likelihood estimators with an application to bivariate extreme value problems

The authors achieve robust estimation of parametric models through the use of weighted maximum likelihood techniques. A new estimator is proposed and its good properties illustrated through examples. Ease of implementation is an attractive property of the new estimator. The new estimator downweights with respect to the model and can be used for complicated likelihoods such as those involved in bivariate extreme value problems. New weight functions, tailored for these problems, are constructed. The increased insight provided by our robust fits to these bivariate extreme value models is exhibited through the analysis of sea levels at two East Coast sites in the United Kingdom.

Estimating the probability of obtaining nonfeasible parameter estimates of the generalized pareto distribution

In this paper we consider the problem of estimating the parameters of the Generalized Extreme-Value distribution. The popular method of probability-weighted moments does not guarantee that estimates will be consistent with the observed data. We present a simple program to predict the probability of obtaining such nonfeasible estimates. Our estimation techniques are based on results from intensive simulations and the successful modelling of the lower tail of the distribution of the upper bound of the support. More simulations are performed to validate the new procedure.

Multivariate Extreme Value Theory And Its Usefulness In Understanding Risk

Abstract This paper gathers recent results in the analysis of multivariate extreme values and illustrates their actuarial application. We review basic and essential background on univariate extreme value theory and stochastic dependence and then provide an introduction to multivariate extreme value theory. We present important concepts for the analysis of multivariate extreme values and collect research results in this area. We draw particular attention to issues related to extremal dependence and show the importance of model selection when fitting an upper tail copula to observed joint exceedances. These ideas are illustrated on four data sets: loss amount and allocated loss adjustment expense under insurance company indemnity claims, lifetimes of pairs of joint and lastsurvivor annuitants, hurricane losses in two states, and returns on two stocks. In each case the extremal dependence structure has an important financial impact.

Clinical measurement of the static rear stability of occupied wheelchairs

OBJECTIVE To evaluate a new clinical test, platform testing of the static rear stability of wheelchairs occupied by their users (using methods adapted from the International Organization for Standardization [ISO]), from the perspective of its measurement properties, safety, and comfort. DESIGN Within-subject comparisons. SETTING Rehabilitation center. PATIENTS Ninety-seven wheelchair users. MAIN OUTCOME MEASURES Static stability (with the brakes locked and unlocked, the occupant leaning forward and back, and with antitip devices in place), dynamic stability (the criterion measure), reliability, validity, sensitivity, specificity, predictive values, and likelihood ratios. RESULTS Test-retest reliabilities (n = 18 to 24) were all >.93. The tests construct validity was demonstrated by the finding that static stability was appropriately affected by locking the brakes, body position, and antitip devices (p < .0001). Spearmans rank correlations between static and dynamic stability ranged from .29 to .65. Sensitivity ranged from 46% to 85%, specificity from 59% to 78%, positive predictive values from 76% to 86%, negative predictive values from 42% to 69%, positive likelihood ratios from 1.56 to 2.95, and negative likelihood ratios from .22 to .71. There were no adverse events, and the subjects tolerated the tests well. CONCLUSIONS In the clinical setting, the ISO platform test of static rear stability has good to excellent measurement properties, is safe, and is well tolerated. Static-stability testing in this setting should be performed in the context of a comprehensive evaluation of wheelchair safety and performance.

Large wind speeds: Modeling and outlier detection

This article addresses the problem of modeling extreme wind speeds with the aim of developing procedures that can be used to reliably identify outliers. There are several approaches to fitting extremes, including using maxima over a fixed time period or taking all observations over a threshold. Using two sets of oceanic wind data from buoys, we use robust estimation methods to estimate the parameters of the asymptotic distribution for extremes over fixed time periods and peaks over threshold. For both cases we also use a gh distribution which focuses on modeling the quantiles and propose a robust method for fitting the data to the gh distribution. Weights from the robust fits are used to identify outliers with P values being computed by resampling. We also evaluate the fits of the data to the model distributions according to several criteria concluding that the gh distribution is at least as effective in fitting the tail behavior as the more classical generalized extreme value distribution and the generalized Pareto distribution.

The influence of injector design on the decay of pre-ignition turbulence in a spherical explosion chamber

Abstract This paper reports on an experimental study to characterize the turbulent flow field inside a 20 l Siwek chamber during the pre-ignition period. An acrylonitrile–butadiene–styrene model of the chamber was constructed with optical quality windows enabling laser Doppler anemometry (LDA) to be used for turbulence measurements. Alumina (Al 2 O 3 ) particles were used as the seed dust for measuring the gas-phase turbulence. Three specific dust dispersion systems have been investigated: (1) the deflector plate (also referred to as the rebound nozzle); (2) the perforated annular nozzle; and (3) the circular “Dahoe” nozzle. It is assumed that changing the method of dust dispersion alters the turbulence characteristics. The flow field is non-stationary, i.e., the mean (or predominant fluid flow) and superimposed velocity fluctuations upon the mean decrease with time. Furthermore, there are variations from injection to injection. A procedure has been developed to analyze this non-stationary signal to extract the mean and fluctuating components of velocity, thereby paving the way for decay “laws” to be determined for a particular nozzle configuration.

Modeling Waves of Extreme Temperature: The Changing Tails of Four Cities

Heat waves are a serious threat to society, the environment, and the economy. Estimates of the recurrence probabilities of heat waves may be obtained following the successful modeling of daily maximum temperature, but working with the latter is difficult as we have to recognize, and allow for, not only a time trend but also seasonality in the mean and in the variability, as well as serial correlation. Furthermore, as the extreme values of daily maximum temperature have a different form of nonstationarity from the body, additional modeling is required to completely capture the realities. We present a time series model for the daily maximum temperature and use an exceedance over high thresholds approach to model the upper tail of the distribution of its scaled residuals. We show how a change-point analysis can be used to identify seasons of constant crossing rates and how a time-dependent shape parameter can then be introduced to capture a change in the distribution of the exceedances. Daily maximum temperature series for Des Moines, New York, Portland, and Tucson are analyzed. In-sample and out-of-sample goodness-of-fit measures show that the proposed model is an excellent fit to the data. The fitted model is then used to estimate the recurrence probabilities of runs over seasonally high temperatures, and we show that the probability of long and intense heat waves has increased considerably over 50 years. We also find that the increases vary by city and by time of year.