Abstract
LetX 1,X 2, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ − n , μ + n be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ − n , μ + n are studied in the present paper. Some statistical applications connected with these variables are also provided.
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References
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998).Records. John Wiley & Sons, New York.
Bairamov, I.G. and Eryilmaz, S.N. (2000). Distribution properties of statistics based on minimal spacings and record exceedance statistics,Journal of Statistical Planning and Inference,90, 21–33.
Balakrishnan, N. and Stepanov, A. (2004). A note on the paper of Khmaladzeet al.Statistics & Probability Letters,68, 415–419.
Balakrishnan, N. and Stepanov, A. (2005). A note on the number of observations registered near an order statistic.Journal of Statistical Planning and Inference, to appear.
Barndorff-Nielsen, O. (1961). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables.Math. Scand.,9, 383–394.
Bingham, N.H., Goldie, C. and Teugels, J.F. (1987).Regular variation, (Encyclopaedia Math. Appl. 27). Cambridge University Press.
Feuerverger, A. and Hall, P. (1998). On statistical inference based on record values.Extremes,1 (2), 169–190.
Berred, A.M. (1992) On record values and exponent of a distribution with a regularly varying upper tail,J. Appl. Probab.,29, 575–586.
Berred, A.M. (1998). An estimator of the exponent of regular variation based on k-record values.Statistics,32, 93–109.
Hashorva, E. (2002). Remarks on domination of maxima.Statistics and Probability Letters,60, 101–109.
Khmaladze, E., Nadareishvili, M. and Nikabadze, A. (1997). Asymptotic behaviour of a number of repeated records.Statistics & Probability Letters,35, 49–58.
Li, Y. (1998). The Number of Near-Maxima of Random Variables.PhD Thesis, University of Western Australia.
Li, Y. (1999). A note on the number of records near maximum.Statist. Probab. Lett.,43, 153–158.
Li, Y. and Pakes, A. (1998). On the number of near-records after the maximum observation in a continuous sampleCommunications in Statistics—Theory and Methods,27, 673–686.
Li, Y. and Shao, Q. (2003) A chaos phenomenon on the number of near-maxima for Burr type XII distribution.Australian & New Zealand Journal of Statistics (under revision) pp25.
Nevzorov V. B. (1986). Onk-th record moments and generalizations.Zapiski Nauchn Seminar LOMI,153, 115–121 (in Russian).
Nevzorov, V. B. (2000).Records: Mathematical Theory, Translation of Mathematical Monographs. Vol.194, American Mathematical Society, Providence, Rhode Island.
Pakes, A. (2000). The number and sum of near-maxima for thin-tailed populations.Adv. Appl. Prob.,32, 1100–1116.
Pakes, A. and Steutel, F. W. (1997). On the number of records near the maximum.Austral. J. Statist.,39, 179–193.
Pakes, A. and Li, Y. (1998). Limit laws for the number of near maxima via the Poisson approximation.Statistics & Probability Letters,40, 395–401.
Rinot, Y. and Rotar, V. (2001). A remark on quadrant normal probabilities in high dimensions.Statistics and Probability Letters,51, 47–51.
Shorrock, R. W. (1972). On record values and record times.Journal of Applied Probability,9, 316–326.
Stepanov, A. (2004). Random intervals based on record values.Journal of Statistic Planning and Inference,118, 103–113.
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Stepanov, A. The number of records within a random interval of the current record value. Statistical Papers 48, 63–79 (2007). https://doi.org/10.1007/s00362-006-0316-9
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DOI: https://doi.org/10.1007/s00362-006-0316-9