It is well known that the tail behavior of a heavy-tailed distribution is controlled by a parameter called the tail index. Such a parameter is therefore of primary interest in extreme value analysis, particularly to estimate extreme quantiles. In various applications, the random variable of interest can be linked to a finite-dimensional random covariate. In such a situation, the tail index is function of the covariate and is referred to as the conditional tail index. The goal of this paper is to provide a class of estimators of this quantity. The pointwise weak consistency and asymptotic normality of these estimators are established. We illustrate the finite sample performance of our technique on a simulation study and on a real hurricane data set.