Cumulative Processes: Linear Combinations of Order Statistics and Percentiles

摘要

Percentiles and linear combinations of order statistics are statistics which are sometimes preferred to averages because they can be less sensitive to the presence of a few wild observations. It is well known that for large samples, both percentiles and linear combinations of order statistics resemble averages in that the appropriately normalized statistic is approximately normally distributed, with parameters which depend on the underlying distributions. This paper shows that percentiles and linear combinations of order statistics resemble averages in a stronger sense. It is well known that if a sequence of averages is plotted against the number of observations contributing to the average, and the resulting plot is rescaled appropriately, then for long sequences the picture will act like a realization of a Brownian motion path. This paper establishes that this is still true if the averages are replaced by percentiles or by linear combinations of order statistics. (Author)