Asymptotically independent scale-free spacings with applications to discordancy testing. (English) Zbl 0612.62018

Sweeting, T. J.

Ann. Stat. 14, 1485-1496 (1986).

Let \(X_ 1,...,X_ n\) be independent observations from a distribution F with continuous f(x), positive over the range of F. For \(t\geq 1\) define the functions \(U(t)=t f(F^{-1}(1-t^{-1}))\) and \(L(t)=t f(F^{- 1}(t^{-1}))\). Let \(X_{(1)},...,X_{(n)}\) be the order statistics and define \(D_ i=c_{i,n}(X_{(i)}-X_{(i-1)})\) and \(W_ i=\sum^{i}_{j=2}D_ j\) where \(c_{i,n}=(n-i+1)U(n/(n-i+1))\). Denote the exponential distribution with mean \(\sigma\) by E(1/\(\sigma)\).
In this paper, corresponding to the results of the independent scale-free spacings for the exponential distribution E(1/\(\sigma)\), the author showed the following facts:
(i) if U is slowly varying, then there is a sequence \(\{Y_ i\}\) of independent E(1) r.v.’s such that \(D_{n-k+1}/Y_{n-k+1}\to 1\) in probability as \(n\to \infty\), where E(1) denotes the exponential distribution with mean 1.
(ii) If L and U are regularly varying with finite exponents, then \((W_{n-k+1}-W_{\ell})/(n-k-\ell +1)\to \sigma\) in probability as \(n\to \infty.\)
Further, using these results the author considered a unified approach to the problem of consecutive discordancy testing when location and scale parameters are unknown.

Reviewer: K.-I.Yoshihara

MSC:

62E20	Asymptotic distribution theory in statistics
62F05	Asymptotic properties of parametric tests

Keywords:

Monte Carlo studies; outliers; Weissman’s result; extreme value theory; regular variation; order statistics; exponential distribution; scale-free spacings; slowly varying; consecutive discordancy testing; location