Asymptotic properties of numbers of observations near sample quantiles
G. Iliopoulos, A. Dembińska and N. Balakrishnan
In this paper, we show that the proportions of observations falling in the left and right vicinity of the $k_n$-th order statistic converge in probability to some population quantities. We then prove that suitably normalized versions of these variables are jointly asymptotically normal under some conditions. A generalization of this result to the case of proportions of observations in the vicinity of two or more central order statistics is established next. Some concluding remarks and a potential statistical application of these results are finally made.
Key words and phrases: limit theorems; order statistics; near sample quantiles observations