Asymptotic results for \(m\)-th exponential spacings. (English) Zbl 1517.60056
Summary: In this work, we discuss \(m\)-th exponential spacings \(\triangle_{k:m:n}\) obtained from order statistics. We study limit results for such spacings when the sample size \(n\) tends to infinity and the indices \(k\) and \(m\) are either fixed or also tend to infinity. We also investigate asymptotic properties of largest exponential \(m\)-th spacing.
MSC:
| 60G70 | Extreme value theory; extremal stochastic processes |
| 62G30 | Order statistics; empirical distribution functions |
References:
| [1] | Ahsanullah, M., A characterization of the exponential distribution by higher order gap, Metrika, 31, 323-326 (1984) · Zbl 0576.62019 · doi:10.1007/BF01915219 |
| [2] | Arnold, BC; Balakrishnan, N.; Nagaraja, HN, A first course in order statistics (1992), New York: Wiley, New York · Zbl 0850.62008 |
| [3] | Balakrishnan, NK; Koutras, MV, Runs and scans with applications (2002), New York: Wiley, New York · Zbl 0991.62087 |
| [4] | Balakrishnan, N.; Stepanov, A., Generalization of Borel-Cantelli lemma, Math. Sci., 35, 61-62 (2010) · Zbl 1205.60061 |
| [5] | Barndorff-Nielsen, O., On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables, Math. Scand., 9, 383-394 (1961) · Zbl 0209.20104 · doi:10.7146/math.scand.a-10643 |
| [6] | Beirlant, J.; Zuijlen, M., The empirical distribution function and strong laws for functions of order statistics of uniform spacings, J. Multivar. Anal., 16, 300-317 (1985) · Zbl 0581.62046 · doi:10.1016/0047-259X(85)90023-5 |
| [7] | Berred, A.; Stepanov, A., Asymptotic results for lower exponential spacings, Commun. Stat. - Theory Methods, 49, 1730-1741 (2020) · Zbl 1511.62098 · doi:10.1080/03610926.2019.1565781 |
| [8] | Cressie, N., An optimal statistic based on higher order gaps, Biometrika, 66, 619-627 (1979) · Zbl 0455.62036 · doi:10.1093/biomet/66.3.619 |
| [9] | David, HA; Nagaraja, HN, Order statistics (2003), Hoboken: Wiley, Hoboken · Zbl 1053.62060 · doi:10.1002/0471722162 |
| [10] | Del Pino, GE, On the asymptotic distribution of k-spacings with applications to goodness-of-fit tests, Ann. Statist., 7, 1058-1065 (1979) · Zbl 0425.62026 · doi:10.1214/aos/1176344789 |
| [11] | Devroye, L., The largest exponential spacing, Utilitas Mathematica, 25, 303-313 (1984) · Zbl 0547.60034 |
| [12] | Ederer, F.; Meyers, MH; Mantel, N., A statistical problem in space and time: Do leukemia cases come in clusters?, Biometrics, 20, 626-636 (1964) · doi:10.2307/2528500 |
| [13] | Glaz, J and Balakrishnan, N (eds.) (1999). JScan statistics and applications, Birkhauser, Boston. · Zbl 0919.00015 |
| [14] | Hall, PG, Limit theorems for sums of general functions of m-spacings, Math. Proc. Cambridge Philos. Soc., 96, 517-532 (1984) · Zbl 0559.62013 · doi:10.1017/S0305004100062459 |
| [15] | Naus, JI, Some probabilities, expectations and variances for the side of largest clusters and smallest intervals, J. Amer. Statist. Ass., 61, 1191-1199 (1966) · Zbl 0142.16201 · doi:10.1080/01621459.1966.10482203 |
| [16] | Nevzorov, V., Records: Mathematical theory american mathematical society (2001), Rhode Island: Providence, Rhode Island |
| [17] | Pyke, R., Spacings (with discussions), J. R. Stat. Soc. Series B, 27, 395-449 (1965) · Zbl 0144.41704 |
| [18] | Riffi, MI, Distributions of gamma m-spacings, IUG J. Nat. Stud., 4, 01-06 (2017) |
| [19] | Riffi, MI, Characterizing the exponential distribution by m-spacings, J. Scientif. Eng. Res., 5, 211-214 (2018) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
