Some asymptotic results for the integrated empirical process with applications to statistical tests

Abstract

The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of the integrated empirical process.

Local efficiency of integrated goodness-of-fit tests under skew alternatives

“…Let X 1 , ..., X n be a random sample from the density h(x, θ) given by (1) and depending on the known symmetric density f and symmetric distribution function G, and a real parameter θ ≥ 0. Let…”

“…These efficiencies are comparable with the efficiencies of usual distribution-free tests, but there exist also some interesting distinctions in favor of these new tests. Gradually statistical inference using integrated empirical processes becomes quite popular, see, e.g., [1], [6], [11] and [12].…”

“…It is quite interesting to calculate the efficiencies of integrated distribution-free tests mentioned above under the generalized skew alternative (1). We select the Bahadur efficiency as it is well-adapted for such calculations while other types of efficiencies such as Pitman, Chernoff or Hodges-Lehmann are not applicable or do not discriminate between two-sided tests.…”

“…These efficiencies are comparable with the efficiencies of usual distribution-free tests, but there exist also some interesting distinctions in favor of these new tests. Gradually statistical inference using integrated empirical processes becomes quite popular, see, e.g., [1], [6], [11] and [12]. However, the location alternative is a simplest alternative which is not very realistic in practice, particularly because it preserves the symmetry of the underlying distribution.…”

Strong approximations for the p-fold integrated empirical process with applications to statistical tests

“…The asymptotic behavior of weighted multivariate Cramér-von Mises-type statistics under contiguous alternatives was characterized by Bouzebda and Zari (2013). For more recent references, we refer to Durio and Nikitin (2016), Bouzebda (2016) and Alvarez-Andrade et al (2017). The main purpose of this paper is to investigate the strong approximation of the p-fold integrated empirical process.…”