scholarly journals Tests of a homogeneous Poisson process against clustering and other alternatives

1994 ◽  
Author(s):  
C.L. Atwood
Keyword(s):  
Poisson Process ◽  
Homogeneous Poisson Process

Applying the Anderson-Darling Test to Suicide Clusters

Crisis ◽  
2013 ◽  
Vol 34 (6) ◽  
pp. 434-437 ◽  
Author(s):  
Donald W. MacKenzie
Keyword(s):  
Poisson Process ◽  
Goodness Of Fit ◽  
Small Samples ◽  
Massachusetts Institute Of Technology ◽  
Homogeneous Poisson Process ◽  
Cornell University ◽  
The Difference ◽  
Suicide Clusters ◽  
Institute Of Technology ◽  
Anderson Darling

Background: Suicide clusters at Cornell University and the Massachusetts Institute of Technology (MIT) prompted popular and expert speculation of suicide contagion. However, some clustering is to be expected in any random process. Aim: This work tested whether suicide clusters at these two universities differed significantly from those expected under a homogeneous Poisson process, in which suicides occur randomly and independently of one another. Method: Suicide dates were collected for MIT and Cornell for 1990–2012. The Anderson-Darling statistic was used to test the goodness-of-fit of the intervals between suicides to distribution expected under the Poisson process. Results: Suicides at MIT were consistent with the homogeneous Poisson process, while those at Cornell showed clustering inconsistent with such a process (p = .05). Conclusions: The Anderson-Darling test provides a statistically powerful means to identify suicide clustering in small samples. Practitioners can use this method to test for clustering in relevant communities. The difference in clustering behavior between the two institutions suggests that more institutions should be studied to determine the prevalence of suicide clustering in universities and its causes.

scholarly journals The fighter problem: optimal allocation of a discrete commodity

2011 ◽  
Vol 43 (01) ◽  
pp. 121-130 ◽  
Author(s):  
Jay Bartroff ◽  
Ester Samuel-Cahn
Keyword(s):  
Poisson Process ◽  
Optimal Allocation ◽  
Optimal Number ◽  
Homogeneous Poisson Process ◽  
Time Period

In this paper we study the fighter problem with discrete ammunition. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter, they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g. a(j) = 1 - q j , 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: (A) the closer to the destination, the more of the n missiles one should use; (B) the more missiles one has; the more one should use; and (C) the more missiles one has, the more one should save for possible future encounters. We show that (C) holds for all 0 ≤ u ≤ 1, that (A) and (B) hold for the ‘invincible fighter’ (u = 1), and that (A) holds but (B) fails for the ‘frail fighter’ (u = 0); the latter is shown through a surprising counterexample, which is also valid for small u > 0 values.

scholarly journals The fighter problem: optimal allocation of a discrete commodity

2011 ◽  
Vol 43 (1) ◽  
pp. 121-130 ◽  
Author(s):  
Jay Bartroff ◽  
Ester Samuel-Cahn
Keyword(s):  
Poisson Process ◽  
Optimal Allocation ◽  
Optimal Number ◽  
Homogeneous Poisson Process ◽  
Time Period

In this paper we study the fighter problem with discrete ammunition. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter, they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g. a(j) = 1 - qj, 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: (A) the closer to the destination, the more of the n missiles one should use; (B) the more missiles one has; the more one should use; and (C) the more missiles one has, the more one should save for possible future encounters. We show that (C) holds for all 0 ≤ u ≤ 1, that (A) and (B) hold for the ‘invincible fighter’ (u = 1), and that (A) holds but (B) fails for the ‘frail fighter’ (u = 0); the latter is shown through a surprising counterexample, which is also valid for small u > 0 values.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (03) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

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