Quantile-based inference and estimation of heavy-tailed distributions

This thesis is divided in four chapters. The two first chapters introduce a parametric quantile-based estimation method of univariate heavy-tailed distributions and elliptical distributions, respectively. If one is interested in estimating the tail index without imposing a parametric form for the entire distribution function, but only on the tail behaviour, we propose a multivariate Hill estimator for elliptical distributions in chapter three. In the first three chapters we assume an independent and identically distributed setting, and so as a first step to a dependent setting, using quantiles, we prove in the last chapter the asymptotic normality of marginal sample quantiles for stationary processes under the S-mixing condition. The first chapter introduces a quantile- and simulation-based estimation method, which we call the Method of Simulated Quantiles, or simply MSQ. Since it is based on quantiles, it is a moment-free approach. And since it is based on simulations, we do not need closed form expressions of any function that represents the probability law of the process. Thus, it is useful in case the probability density functions has no closed form or/and moments do not exist. It is based on a vector of functions of quantiles. The principle consists in matching functions of theoretical quantiles, which depend on the parameters of the assumed probability law, with those of empirical quantiles, which depend on the data. Since the theoretical functions of quantiles may not have a closed form expression, we rely on simulations. The second chapter deals with the estimation of the parameters of elliptical distributions by means of a multivariate extension of MSQ. In this chapter we propose inference for vast dimensional elliptical distributions. Estimation is based on quantiles, which always exist regardless of the thickness of the tails, and testing is based on the geometry of the elliptical family. The multivariate extension of MSQ faces the difficulty of constructing a function of quantiles that is informative about the covariation parameters. We show that the interquartile range of a projection of pairwise random variables onto the 45 degree line is very informative about the covariation. The third chapter consists in constructing a multivariate tail index estimator. In the univariate case, the most popular estimator for the tail exponent is the Hill estimator introduced by Bruce Hill in 1975. The aim of this chapter is to propose an estimator of the tail index in a multivariate context; more precisely, in the case of regularly varying elliptical distributions. Since, for univariate random variables, our estimator boils down to the Hill estimator, we name it after Bruce Hill. Our estimator is based on the distance between an elliptical probability contour and the exceedance observations. Finally, the fourth chapter investigates the asymptotic behaviour of the marginal sample quantiles for p-dimensional stationary processes and we obtain the asymptotic normality of the empirical quantile vector. We assume that the processes are S-mixing, a recently introduced and widely applicable notion of dependence. A remarkable property of S-mixing is the fact that it doesn't require any higher order moment assumptions to be verified. Since we are interested in quantiles and processes that are probably heavy-tailed, this is of particular interest.