Calculation of noncrossing probabilities for Poisson processes and its corollaries

Abstract

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

Detectability of nonparametric signals: higher criticism versus likelihood ratio

“…Since the noise distribution is known, standard Monte-Carlo simulations can be used to estimate the α-quantile of HC n for finite sample size. Alternatively, you can find finite recursion formulas for the exact finite distribution in the paper of Khmaladze and Shinjikashvili [27].…”

On distribution-free goodness-of-fit testing of exponentiality

“…The limit distribution of d n is given, e.g., in Shiryaev (1999, p. 251) and Borodin and Salminen (2002), and was calculated by Shinjikashvili using the numerical method of Khmaladze and Shinjikashvili (2001). For tables of the limit distribution of o 2 n see Orlov (1972) or Martynov (1977), while for A 2 n see Deheuvels and Martynov (2003).…”

Nonparametric inference in multiplicative intensity model by discrete time observation

“…Due to this lemma, we have the asymptotically distribution free test. See Khmaladze and Shinjikashvili (2001) and references therein for explicite/ numerical expression for the distribution of the limit sup t∈[0,1] |B t | which is necessary for computing p-values.…”