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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 謝南瑞(Narn-Rueih Shieh) | |
| dc.contributor.author | Yu-Shain Chen | en |
| dc.contributor.author | 陳鈺賢 | zh_TW |
| dc.date.accessioned | 2021-06-15T00:14:56Z | - |
| dc.date.available | 2009-07-03 | |
| dc.date.copyright | 2009-07-03 | |
| dc.date.issued | 2009 | |
| dc.date.submitted | 2009-06-23 | |
| dc.identifier.citation | [1] Billingsley, P., 1968. Convergence of probability measures. Wiley, New York.
[2] Bingham, N. H., Goldie, C. M., Teugels, J. L., Regular variation, Encyclopedia Math. Appl. 27 (Cambridge University Press, Cambridge, 1987) [3] Csörgö, M., Csörgö, S., Horváth, L., Mason, D., 1986. Weighted empirical and quantile process. Ann. Probab. 14, 31-85. [4] Csörgö, S., Mason, D., 1985. Central limit theorems for sums of extreme values. Math. Proc. Cam. Phil. Soc. 98, 547-558. [5] Davis, R., Resnick, S., 1984. Tail estimates motivated by extreme value theory. Ann. Statist. 12, 1467- 1487. [6] de Haan, L., Resnick, S., 1996. Second-order regular variation and rates of convergence in extreme value theory. Ann. Probab. 24, 97-124. [7] de Haan, L., Resnick, S.,1998. On asymptotic normality of the Hill estimator. Stochastic Models, 14 (1998), 849-867. [8] de Haan, L., Stadtmüller, U., 1996. Generalized regular varation of second order. J. Austral. Math. Soc. Ser. A 61, 381-395. [9] Durret, R., 2005. Probability: Theory and Examples. Tomson, Belmont, CA. [10] Dress, H., de Han, L., and Resnick, S., How to make a Hill plot. Ann. statist., 28-1 (2000), 254-274. [11] Einmahl, J., 1992. Limit theorems for tail processes with application to intermediate quantile estimation. J. Statist. Plann. Inference 32, 137-145. [12] Feller, W., 1971. An introduction to Probability Theory and its applications. Vol. II, 2nd edn. Wiley, New York. [13] Geluk, J., de Haan, L., Regular variation, Extensions and Tauberian Theorems, CWI Tracts, Vol. 40, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. [14] Geluk, J., de Haan, L., Resnick, S., Stâricâ, C., 1997. Second-order regular variation, convolution and the central limit theorem, Stochastic Process. Appl., 69-2, 139-159. [15] Mason, D., 1982. Laws of large numbers of extreme values. Ann. Probab., 10, 754-764. [16] Resnick, S., 1986. Point process, regular variation and weak convergence. Adv Appl. Probab. 18, 66-138. [17] Resnick, S., 1987. Extreme Values, Regular Variation, and Point processes. Springer, New York. [18] Resnick, S., Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336, (2002). ISSN 1386-1999. [19] Resnick, S., 2006. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York. [20] Resnick, S. and Stâricâ, C., 1995. Consistency of Hill’s estimatorfordependent data. J. Appl. Probab. 32 139–167. [21] Resnick, S., Stâricâ, C., 1997. Smoothing the Hill estimator. Adv Appl. Probab. 29, 271-293. [22] Resnick, S., 1997. Heavy tail modeling and teletraffic data. Ann. Statist. 25 1805–1869. [23] Seneta, E., Regularly varying functions, Lecture Notes in Math. 508 (Springer, Berlin, 1976). [24] Vervaat, W., Functional central limit theorems for process with positive drift and their inverses, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 23 (1972), 245-253. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/41261 | - |
| dc.description.abstract | 本論文首先探討一階正規變化函數的存在性與其精確的極限函數，為了統計上之應用我們更進一
步討論廣義型的一階正規變化，在是當的假設下，其極限仍有精確的型態。當處理尾估計量之極 值理論和漸進常態之行為時，我們則考慮一個較正規變化更為精細的情形-二階正規變化。對於我 們於機率上之主要應用，我們考慮另一特殊二階正規變化的情形。並且令此函數f為某分配之尾 分布函數，即f = 1F，其中F為某分布函數。由於此種f為單調函數，當其滿足二階正規變化 時，其反函數仍有相對應等價不同指數的二階正規變化結果。 接著我們給幾個特殊二階正規變化的例子，如Log Gamma分配、Hall/Wesis類型的分 配、Stable密度函數、柯西分布等等。我們也給了一個不符合二階正規變化性質的例子， 即Pareto分配。 對於分別滿足一階及二階正規變化之尾分布，我們討論對於具有一階正規變化之尾分布其和 仍然保有一階正規變化的特性；當尾分布函數滿足特殊二階正規變化時，而兩個獨立同分配非負 隨機變數之最大值仍然保有特殊二階正規變化的性質，但這兩種情形下其指數並不一定維持不 變。 我們也討論尾經驗過程的中央極限行為與特殊二階正規變化的關係，並且利用此結果去得 到Hill過程的漸進行為。藉此我們可以得到Hill估計量漸進常態的結果，此結果可於統計上建立正 規變化尾分布之指數的信賴區間。 | zh_TW |
| dc.description.provenance | Made available in DSpace on 2021-06-15T00:14:56Z (GMT). No. of bitstreams: 1 ntu-98-R96221017-1.pdf: 552113 bytes, checksum: 6299561f4b63f85adf6dfedc880c6ae3 (MD5) Previous issue date: 2009 | en |
| dc.description.tableofcontents | 1 INTRODUCTION 1
2 REGULAR VARIATION OF FIRST-ORDER AND SECOND-ORDER 5 2.1 First-order Regular Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Specialized Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Generalized Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Second-order Regular Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Generalized Second-order Regular Variation . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Specialalized Second-order Regular Variation . . . . . . . . . . . . . . . . . . . 10 2.3 Second-order Regular Variation of Tail Distributions . . . . . . . . . . . . . . . . . . . . 11 3 EXAMPLES 13 4 CONVOLUTION AND MAXIMA 17 4.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 RELATED FUNCTION SPACES AND PROBABILITY THEORY 21 5.1 The Space D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.1 The Skorohod Topology on D[0,1] . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.2 Simple Consequences of Skorohod Metric on D[0,1] . . . . . . . . . . . . . . . . 24 5.1.3 The Space D[0,¥) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.1.4 The Space D(0,¥] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2.2 Methods of Showing Weak Convergence in D[0,1] . . . . . . . . . . . . . . . . . 27 5.2.3 Methods of Showing New Weak Convergence from Old . . . . . . . . . . . . . . 28 6 SECOND-ORDER REGULAR VARIATION AND WEAK CONVERGENCE OF TAIL EMPIRICAL MEASURES IN Space D 30 6.1 Tail Empirical Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 Weak Convergence of Tail Empirical Measures with First-order Regularly Varying Tail . . 32 6.4 Weak Convergence of Tail Empirical Measure with Second-order Regularly Varying Tail . 36 7 ASYMPTOTIC BEHAVIOR OF HILL PROCESS 38 7.1 Hill Estimator and Hill Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 Basic Convergence Based on Distributions of Second-order Regularly Varying Tail . . . . 39 7.3 Asymptotic Behavior of Hill Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8 CONCLUSION ..........49 Reference 50 | |
| dc.language.iso | en | |
| dc.subject | 二階正規變化 | zh_TW |
| dc.subject | 弱收斂 | zh_TW |
| dc.subject | weak convergence | en |
| dc.subject | second-order regular variation | en |
| dc.title | 二階正規變化函數及其在機率上之應用 | zh_TW |
| dc.title | Second-order regular variation and its application in probability | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 97-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 王振男(Jenn-Nan Wang),彭柏堅(Ken Palmer) | |
| dc.subject.keyword | 二階正規變化,弱收斂, | zh_TW |
| dc.subject.keyword | second-order regular variation,weak convergence, | en |
| dc.relation.page | 51 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2009-06-24 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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