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Asymptotic Results for Spacings of Largest Order Statistics

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  • Arvydas Astrauskas

    (Institute of Data Science and Digital Technologies, Vilnius University)

Abstract

We prove o- and O-type asymptotic results in probability for properly normalized spacings (differences) of the largest order statistics based on i.i.d. random sample, as the size of the sample tends to infinity. In particular, compactness properties of sequences of spacings are investigated. These results are applied to study the corresponding asymptotic properties of the ratios of upper order statistics and the number of observations near extremes. A common distribution function of the sample is assumed to satisfy mild conditions on regular variation at its right endpoint.

Suggested Citation

  • Arvydas Astrauskas, 2025. "Asymptotic Results for Spacings of Largest Order Statistics," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 87(1), pages 65-95, February.
    • Handle: RePEc:spr:sankha:v:87:y:2025:i:1:d:10.1007_s13171-024-00340-2
    DOI: 10.1007/s13171-024-00340-2
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    1. Paul Embrechts & Marius Hofert, 2013. "A note on generalized inverses," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 423-432, June.
    2. Hashorva, Enkelejd, 2004. "Bivariate maximum insurance claim and related point processes," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 117-128, August.
    3. Athreya, J. S. & Sethuraman, S., 2001. "On the asymptotics of discrete order statistics," Statistics & Probability Letters, Elsevier, vol. 54(3), pages 243-249, October.
    4. H. Nagaraja & Karthik Bharath & Fangyuan Zhang, 2015. "Spacings around an order statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 515-540, June.
    5. Baryshnikov, Yuliy & Eisenberg, Bennett & Stengle, Gilbert, 1995. "A necessary and sufficient condition for the existence of the limiting probability of a tie for first place," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 203-209, May.
    6. Brands, J. J. A. M. & Steutel, F. W. & Wilms, R. J. G., 1994. "On the number of maxima in a discrete sample," Statistics & Probability Letters, Elsevier, vol. 20(3), pages 209-217, June.
    7. Sophie A. Ladoucette & Jef L. Teugels, 2007. "Asymptotics for Ratios with Applications to Reinsurance," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 225-242, June.
    8. Smid, B. & Stam, A. J., 1975. "Convergence in distribution of quotients of order statistics," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 287-292, July.
    9. Zoltán Megyesi, 2002. "Domains of Geometric Partial Attraction of Max-Semistable Laws: Structure, Merge and Almost Sure Limit Theorems," Journal of Theoretical Probability, Springer, vol. 15(4), pages 973-1005, October.
    10. Hashorva, Enkelejd, 2003. "On the number of near-maximum insurance claim under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 37-49, February.
    11. Hu, Zhishui & Su, Chun, 2003. "Limit theorems for the number and sum of near-maxima for medium tails," Statistics & Probability Letters, Elsevier, vol. 63(3), pages 229-237, July.
    12. Li, Y. & Pakes, Anthony G., 2001. "On the number of near-maximum insurance claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 309-323, June.
    13. Pakes, Anthony G. & Li, Yun, 1998. "Limit laws for the number of near maxima via the Poisson approximation," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 395-401, November.
    14. Arvydas Astrauskas, 2023. "Some Bounds for the Expectations of Functions on Order Statistics and Their Applications," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1116-1147, June.
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