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 系統識別號 U0002-1807201420340700 中文論文名稱 圓周上連續間隔總和的極值之分配 英文論文名稱 The Distribution of the Extremes of Sums of Consecutive Spacings on a Circle 校院名稱 淡江大學 系所名稱(中) 數學學系碩士班 系所名稱(英) Department of Mathematics 學年度 102 學期 2 出版年 103 研究生中文姓名 呂崇道 研究生英文姓名 Chung-Tao Lu 學號 601190266 學位類別 碩士 語文別 英文 口試日期 2014-06-24 論文頁數 22頁 口試委員 指導教授-林千代委員-蔡志群委員-彭健育 中文關鍵字 圓形的掃描統計  間距符號  計算 英文關鍵字 Circular Scan Statistic  Spacings  Symbolic Computation 學科別分類 學科別＞自然科學＞數學 中文摘要 根據Huffer (1988)和Huffer and Lin (1997, 2001)所提出的遞迴關係式，我們發展出幾個特殊的演算法來計算圓周上重疊連續間隔的線性組合之聯合機率分布。我們所提出的演算法可以應用來推算多重覆蓋與掃描統計量的機率。 英文摘要 Based on the recursions in Huffer (1988) and Huffer and Lin (2001), and the marking algorithm in Huffer and Lin (1997), some specialized algorithms are developed.Our approach can evaluate the probabilities involving the joint distribution of overlapping sums of consecutive spacings on a circle. Application to the distribution of circular scan statistic and multiple coverage probabilities are discussed. 論文目次 1 Introduction 1 2 Recursions and Basic Properties 2 3 The Algorithm 5 4 Illustrative Example 8 5 Special Results 13 5.1 Explicit Expression for {A}1 ..............14 5.2 Explicit Expression for {A}2 ..............16 6 Concluding Remarks 17 References 18 Appendix 21 參考文獻 Ajne, B. (1968). A simple test for uniformity of a circular distribution. Biometrika 55, 343–354. Cressie, N. (1977). On some properties of the scan statistic on the circle and the line. Journal of Applied Probability 14, 272–283. Glaz, J. and Balakrishnan, N. (1999). Scan Statistics and Applications. Birkh‥auser, Boston. Glaz, J., Naus, J., Roos, M. and Wallenstein S. (1994). Poisson approximations for the distribution and moments of ordered m-spacings. Journal of Applied Probability 31 A, 271–281. Holst, L. (1980). On multiple covering of a circle with random arcs. Journal of Applied Probability 17, 284–290. Holst, L. (1981). On convergence of the coverage by random arcs on a circle and the largest spacing. The Annals of Probability 9, 648–655. Holst, L. (1990). A circle covering problem and DNA breakage. Statistics and Probability Letters 9, 295–298. Holst, L. and H‥usler, J. (1984). On the random coverage of the circle. Journal of Applied Probability 21, 558–566. Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability 25, 346–354. Huffer, F. W., and Lin, C. T. (1997). Computing the exact distribution of the extremes of sums of consecutive spacings. Computational Statistics and Data Analysis 26, 117–132. Huffer, F. W., and Lin, C. T. (1999). Using moments to approximate the distribution of the scan statistic. In Scan Statistics and Applications (Eds Glaz and Balakrishnan), Birkh‥auser,Boston, pp. 165–201. Huffer, F.W. and Lin, C.T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11, 1141–1157. Huffer, F. W. and Lin, C. T. (2006). Linear combinations of spacings. In Encyclopedia of Statistical Sciences, Volume 12, Second Edition (Eds., S. Kotz, N. Balakrishnan, C.B. Read and B. Vidakovic), pp. 7866–7875. John Wiley & Sons, Hoboken, New Jersey. Khatri, C. G., and Mitra, S. K. (1969). Some Identities and Approximations Concerning Positive and Negative Multinomial Distributions, in: P. R. Krishnaiah (Ed.). Multivariate Analysis – II , Academic Press, New York, 241–260. Kokic, P. N. (1987). On tests of uniformity for randomly distributed arcs on a circle. Australian Journal of Statistics 29, 179–187. Lin, C. T. (1993). The computation of probabilities which involve spacings, with applications to the scan statistic. Ph. D. Dissertation, Dept. of Statistics, Florida State University, Tallahassee, FL. Pyke, R. (1965). Spacings (with discussion). Journal of the Royal Statistical Society. Series B 7, 395–449. Pyke, R. (1972). Spacings revisited. Proceedings of the Sixth Berkeley Symposium on Mathe-matical Statistics and Probability I, 417–427. Roos, M. (1993). Compound Poisson approximations for the number of extreme spacings.Journal of Applied Probability 25, 847–874. Rothman, E. D. (1969). Properties and applications of test statistics invariant under rotation of a circle. Ph.D. Dissertation, Johns Hopkins University. Siegel, A. F. (1978a) Random space filling and moments of coverage in geometrical probability.Journal of Applied Probability 15, 340–355. Siegel, A. F. (1978b) Random arcs on the circle. Journal of Applied Probability 15, 774–789. Siegel, A. F. and Holst, L. (1982). Covering the circle with random arcs of random sizes.Journal of Applied Probability 19, 373–381. Solomon, H. (1978). Geometric Probability. SIAM, Philadelphia, PA. Wallenstein, S. R. (1971). Coincidence probabilities used in nearest neighbor problems on the line and circle. Ph. D. Thesis, Rutgers University. 論文使用權限 同意紙本無償授權給館內讀者為學術之目的重製使用，於2014-07-22公開。同意授權瀏覽/列印電子全文服務，於2014-07-22起公開。 若您有任何疑問，請與我們聯絡！圖書館： 請來電 (02)2621-5656 轉 2487 或 來信 dss@mail.tku.edu.tw 