Question

Convergeence of quantiles and distribution function

Answer 1

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The quantile process was shown by Bickel to converge in the uniform metric on intervals [a,b] with 0<a<b<1. By introducing appropriate new supremum metrics, this result is extended to all of (0, 1). Hence a natural process of ordered spacings from the uniform distribution converges in certain supremum metrics. This is used to establish the limiting normality of a large family of statistics based on ordered spacings, which can be used in testing for exponentiality. The non-null case is also considered.