Beyond tail median and conditional tail expectation: Extreme risk estimation using tail Lp-optimization

The conditional tail expectation (CTE) is an indicator of tail behavior that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A valuable alternative is the median shortfall (MS), although it only gives information about the frequency of a tail event. We construct a class of tail L^p-medians encompassing the MS and CTE. For p in (1,2), a tail L^p-median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on real fire insurance data.     

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.     

Pauvreté et convergence des consommations au Canada*

To study changes in household consumption patterns, five socio‐economic household groups were defined using a new multidimensional index (IMPR) composed of three dimensions: satisfaction of basic needs, marginalization relative to a reference population, and total disposable household income. When household incomes rose from 1969 to 1992, lower socio‐economic groups did not display the same consumption behaviours as upper‐income classes. The growth in household income during their life‐cycles did not allow them to catch up to higher‐income households. Middle‐class households experienced marked changes in consumption over their life‐cycle's, but in the 1990s, they experienced difficulties. Cinq groupes socioéconomiques de ménages sont définis à l'aide d'un nouvel indice multidimensionnel de pauvreté‐richesse (IMPR), construit à partir de trois dimensions: satisfaction des besoins de base, marginalisation par rapport à une population de référence et revenu total disponible du ménage. De 1969 à 1992, quand les revenus des ménages augmentent, les classes socioéconomiques qui occupent des positions inférieures n'ont pas les mêmes comporte‐ments de consommation que les classes supérieures. La croissance des revenus des ménages les moins riches au cours de leur cycle de vie ne leur permet pas de rattraper les positions des ménages plus riches. Dans les années 1990, la classe moyenne connaît une situation difficile.     

Estimation of the Weibull tail-coefficient with linear combination of upper order statistics

We present a new family of estimators of the Weibull tail-coefficient. The Weibull tail-coefficient is defined as the regular variation coefficient of the inverse failure rate function. Our estimators are based on a linear combination of log-spacings of the upper order statistics. Their asymptotic normality is established and illustrated for two particular cases of estimators in this family. Their finite sample performances are presented on a simulation study.     

Are time and money equally substitutable for all commodity groups in the household’s domestic production?

Review of Economics of the Household - This article uses time-use and household expenditure data to measure the substitutability between time and money within the Beckerian household production...     

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.     

Estimation of extreme quantiles from heavy and light tailed distributions

In Gardes et al. (2011), a new family of distributions is introduced, depending on two parameters τ$\tau$ and θ$\theta$, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for θ$\theta$ and extreme quantiles are also proposed, but they both depend on the unknown parameter τ$\tau$, making them useless in practical situations. In this paper, we propose an estimator of τ$\tau$ which is independent of θ$\theta$. Plugging our estimator of τ$\tau$ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.     

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.     

Bias-reduced extreme quantile estimators of Weibull tail-distributions

In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. [2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.