Extreme quantiles and tail index of a distribution based on kernel estimator

In this paper, we reveal the relationship between the tail exponent introduced by Parzen (1979) and tail index for a distribution function. Furthermore, we analyze the domain of attraction of the weighted sum of the distributions and its tail index. We show that the extreme quantiles can estimate directly, through knowing only the tail index of the kernel distribution function used in estimating the distribution function. Moreover, we give a smoothing parameter of extreme quantiles, which does not depend on any distribution function. The simulations and the application to reals data show that the proposed smoothed parameter gives better results for a heavy-tailed distribution, and for small sizes sample in extremes level.