Two-Dimensional Scan Statistics

2001 ◽  
pp. 273-300
Author(s):  
Joseph Glaz ◽  
Joseph Naus ◽  
Sylvan Wallenstein
Keyword(s):  
Scan Statistics ◽  
Two Dimensional

On the Distributions of Scan Statistics of a Two-Dimensional Poisson Process

1997 ◽  
Vol 29 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Sven Erick Alm
Keyword(s):  
Poisson Process ◽  
Higher Dimensions ◽  
Scan Statistics ◽  
Two Dimensional ◽  
Rectangular Area

Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.

scholarly journals BOUNDS FOR THE DISTRIBUTION OF TWO-DIMENSIONAL BINARY SCAN STATISTICS

2003 ◽  
Vol 17 (4) ◽  
pp. 509-525 ◽  
Author(s):  
Michael V. Boutsikas ◽  
Markos V. Koutras
Keyword(s):  
Distribution Function ◽  
Present Article ◽  
Numerical Study ◽  
Double Sequence ◽  
Reliability Theory ◽  
Scan Statistics ◽  
Asymptotic Result ◽  
Two Dimensional ◽  
Scan Statistic ◽  
Binary Trials

In the present article, we develop some efficient bounds for the distribution function of a two-dimensional scan statistic defined on a (double) sequence of independent and identically distributed (i.i.d.) binary trials. The methodology employed here takes advantage of the connection between the scan statistic problem and an equivalent reliability structure and exploits appropriate techniques of reliability theory to establish tractable bounds for the distribution of the statistic of interest. An asymptotic result is established and a numerical study is carried out to investigate the efficiency of the suggested bounds.

Large-deviation approximations to the distribution of scan statistics

1991 ◽  
Vol 23 (04) ◽  
pp. 751-771 ◽  
Author(s):  
Clive R. Loader
Keyword(s):  
Large Deviation ◽  
Two Dimensions ◽  
Unit Interval ◽  
Scan Statistics ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Likelihood Ratio Criterion ◽  
Scan Statistic ◽  
Window Width ◽  
Ratio Criterion

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.

On the Distributions of Scan Statistics of a Two-Dimensional Poisson Process

1997 ◽  
Vol 29 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Sven Erick Alm
Keyword(s):  
Poisson Process ◽  
Higher Dimensions ◽  
Scan Statistics ◽  
Two Dimensional ◽  
Rectangular Area

Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area. The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.

Large-deviation approximations to the distribution of scan statistics

1991 ◽  
Vol 23 (4) ◽  
pp. 751-771 ◽  
Author(s):  
Clive R. Loader
Keyword(s):  
Large Deviation ◽  
Two Dimensions ◽  
Unit Interval ◽  
Scan Statistics ◽  
Boundary Crossing ◽  
Two Dimensional ◽  
Likelihood Ratio Criterion ◽  
Scan Statistic ◽  
Window Width ◽  
Ratio Criterion

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.

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