Abstract
Consider the order statistics fromNi.i.d. random variables uniformly distributed on the interval (0,1]. We present a general method for computing probabilities involving differences of the order statistics or linear combinations of the spacings between the order statistics. This method is based on repeated use of a basic recursion to break up the joint distribution of linear combinations of spacings into simpler components which are easily evaluated. LetS w denote the (continuous conditional) scan statistic with window length w. Let Cwdenote the number of m: w clumps among theNrandom points, where an m: w clump is defined as m points falling within an interval of length w. We apply our general method to compute the distribution ofS w (for smallN)and the lower-order moments of Cw. The final answers produced by our approach are piecewise polynomials (in w) whose coefficients are computed exactly. These expressions can be stored and later used to rapidly compute numerical answers which are accurate to any required degree of precision.
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References
Feller, W. (1971).An Introduction to Probability Theory and Its Applications VolumeII, New York: John Wiley&Sons.
Glaz, J. and Naus, J. (1983). Multiple clusters on the lineCommunications in Statistics—Theory and Methods 12 1961–1986.
Huffer, F. W. (1982). The moments and distributions of some quantities arising from random arcs on the circlePh.D. DissertationDepartment of Statistics, Stanford University, Stanford, CA.
Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacingsJournal of Applied Probability 25 346–354.
Huffer, F. and Lin, C. T. (1995). Approximating the distribution of the scan statistic using moments of the number of clumpsTechnical ReportDepartment of Statistics, Florida State University, Tallahassee, FL.
Huffer, F. W. and Lin, C. T. (1996). Computing the exact distribution of the extremes of linear combinations of spacingsTechnical ReportDepartment of Statistics, Florida State University, Tallahassee, FL.
Huffer, F. and Lin, C. T. (1997a). Approximating the distribution of the scan statistic using moments of the number of clumpsJournal of the American Statistical Association 921466–1475.
Huffer, F. W. and Lin, C. T. (1997b). Computing the exact distribution of the extremes of sums of consecutive spacingsComputational Statistics Data Analysis 26117–132.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1999).Continuous Multivariate Distributions-1Second edition, New York: John Wiley&Sons.
Khatri, C. G. and Mitra, S. K. (1969). Some identities and approximations concerning positive and negative multinomial distributions, InMultivariate Analysis - II(Ed., P. R. Krishnaiah), pp. 241–260, New York: Academic Press.
Lin, C. T. (1993). The computation of probabilities which involve spacings, with applications to the scan statisticPh.D. DissertationDepartment of Statistics, Florida State University, Tallahassee, FL.
Micchelli, C. A. (1980). A constructive approach to Kergin interpolation inR k :multivariate B-splines and Lagrange interpolationThe Rocky Mountain Journal of Mathematics 10485–497.
Neff, N. D. and Naus, J. I. (1980). The distribution of the size of the maximum cluster of points on a lineIMS Series of Selected Tables in Mathematical Statistics, Volume 6Providence, RI: American Mathematical Society.
Wilks, S. S. (1962).Mathematical StatisticsNew York: John Wiley&Sons.
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Huffer, F.W., Lin, CT. (1999). An Approach to Computations Involving Spacings With Applications to the Scan Statistic. In: Glaz, J., Balakrishnan, N. (eds) Scan Statistics and Applications. Statistics for Industry and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1578-3_6
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DOI: https://doi.org/10.1007/978-1-4612-1578-3_6
Publisher Name: Birkhäuser, Boston, MA
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