Quantiles are important performance characteristics that have been adopted in many areas for measuring the quality of service. Recently, sensitivity analysis of quantiles has attracted quite some attention. Sensitivity analysis of quantiles is particularly challenging as quantiles cannot be expressed as the expected value of some sample performance function, and it is therefore not evident how ...

We introduce an inference method based on quantiles matching, which is useful for situations where the density function does not have a closed form –but it is simple to simulate– and/or moments do not exist. Functions of theoretical quantiles, which depend on the parameters of the assumed probability law, are matched with sample quantiles, which depend on observations. Since the theoretical qua...

Engle and Manganelli (2004) propose CAViaR, a class of models suitable for estimating conditional quantiles in dynamic settings. Engle and Manganelli apply their approach to the estimation of Value at Risk, but this is only one of many possible applications. Here we extend CAViaR models to permit joint modeling of multiple quantiles, Multi-Quantile (MQ) CAViaR. We apply our new methods to estim...

A leading multivariate extension of the univariate quantiles is the so-called “spatial” or “geometric” notion, for which sample versions are highly robust and conveniently satisfy a Bahadur-Kiefer representation. Another extension of univariate quantiles has been to univariate U-quantiles, on the basis of which, for example, the well-known Hodges-Lehmann location estimator has a natural formula...

Directional quantile envelopes—essentially, depth contours—are a possible way to condense thedirectional quantile information, the information carried by the quantiles of projections. In typi-cal circumstances, they allow for relatively faithful and straightforward retrieval of the directionalquantiles, offering a straightforward probabilistic interpretation in terms of the tang...

Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically , it obtains the large sample properties of extremal (ext...

We estimate conditional and unconditional high quantiles for electricity spot prices based on a linear model with stable innovations. This approach captures the impressive peaks in such data and, as a four-parametric family captures also the assymmetry in the innovations. Moreover, it allows for explicit formulas of quantiles, which can then be calculated recursively from day to day. We also pr...

Abstract. U -quantiles are applied in robust statistics, like the Hodges-Lehmann estimator of location for example. They have been analyzed in the case of independent random variables with the help of a generalized Bahadur representation. Our main aim is to extend these results to U -quantiles of strongly mixing random variables and functionals of absolutely regular sequences. We obtain the cen...

Using quantile regressions, this paper provides evidence that the relationship between school inputs and wages varies across points in the conditional wage distribution and educational attainment levels. Although smaller classes generally have a positive return for individuals at high quantiles, they have either no impact or a negative impact at low quantiles. Similarly, while more highly paid ...