Abstract
A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.
Similar content being viewed by others
References
Balakrishnan N, Stepanov A (2004) A note on the paper of Khmaladze et al. Stat Probab Lett 68:415–419
Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13:215–232
Chant D (1974) On asymptotic tests of composite hypotheses in non-standard conditions. Biometrika 61:291–298
Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events. Springer, Berlin Heidelberg New York
de Haan L (1976) Sample extremes: an elementary introduction. Stat Neerl 30:161–172
Khamaladze E, Nadareishvili M, Nikabaldze A (1997) Asymptotic behaviour of a number of repeated records. Stat Probab Lett 35:49–58
Klugman SA (1986) Loss distributions. Proc Sympos Appl Math 35:31–55
Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, Berlin Heidelberg New York
Li Y (1999) A note on the number of records near the maximum. Stat Probab Lett 43:153–158
Li Y, Pakes AG (1998) On the number of near-records after the maximum observation in a continuous sample. Commun Stat Theory Meth 27(3):673–686
Li Y, Pakes AG (2001) On the number of near-maximum insurance claims. Insur Math Econ 28:309–323
Moran AP (1971) Maximum likelihood estimation in non-standard conditions. Proc Camb Philos Soc 49:218–241
Pakes AG (2000) The number and sum of near-maxima for thin-tailed populations. Adv Appl Prob 32:1100–1116
Pakes AG (2004) Criteria for convergence of the number of near maxima for long tails. Extremes 7:123–134
Pakes AG, Li Y (1998) Limit laws for the number of near-maxima via the Poisson approximation. Stat Probab Lett 40:395–401
Pakes AG, Steutel FW (1997) On the number of records near the maximum. Aust J Stat 39(2): 179–192
Reiss RD, Thomas M (2001) Statistical analysis of extreme value, 2nd edn. Birkhäuser Verlag, Basel
Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non-standard conditions. J Am Stat Assoc 82:605–610
Shao Q (2000) Estimation for hazardous concentrations based on NOEC toxicity data: an alternative approach. Environmetrics 11:582–595
Shao Q (2002) A reparameterization method for embedded models. Commun Stat C Theory Meth 31(5):683–697
Shao Q (2004) Determination of embedded distributions. Comput Stat Data Anal 46:317–334
Shao Q, Zhou X (2004) A new parametric model for survival data with long-term survivors. Stat Med 23:3525–3543
Shao Q, Wang H, Xia J, Ip WC (2004) Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis. Hydrol Sci J 49(4):685–703
Shinjikashvili E (2001) ε-Repetitions of the maximum residuals in an AR(1) model. Aust N Z J Stat 43:359–365
Vu HTV, Zhou X (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Stat 25:897–916
Wakins AJ (1999) An algorithm for maximum likelihood estimation in the three parameter Burr XII distribution. Comput Stat Data Anal 32:19–27
Wang FK, Keats JB, Zimmer WJ (1996) Maximum likelihood estimation of the Burr XII distribution with censored and uncensored data. Microelectron Reliab 36:359–362
Wingo DR (1983) Maximum likelihood methods for fitting the Burr type XII distribution parameters to life test data. Biom J 25:77–84
Wingo DR (1993) Maximum likelihood estimation of Burr XII distribution parameters under type II censoring. Microelectron Reliab 33:1251–1257
Wood GR, Lindsay SR, Woolons RC (1996) Modelling the diameter distribution of forest stands using the Burr distribution. J Appl Stat 23:609–619
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, Y., Shao, Q. Slow convergence of the number of near-maxima for Burr XII distributions. Metrika 66, 89–104 (2007). https://doi.org/10.1007/s00184-006-0096-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-006-0096-1