Pitman closeness of record values to population quantiles. (English) Zbl 1171.62033

Summary: We examine the Pitman closeness of record statistics to the population quantiles of a location-scale family of distributions and study its monotonicity properties. Even though in general it depends on the parent distribution, exact expressions are derived for the required probabilities in the case of uniform (\(- 1,1\)) and exponential distributions. For the population median, it is shown that the first upper record is the Pitman-closest among all upper record values. Moreover, for the population median, in the case of symmetric distributions, the Pitman closeness probabilities of records are shown to be distribution-free and explicit expressions are also derived for these probabilities.


62G32 Statistics of extreme values; tail inference
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
Full Text: DOI


[1] Ahmadi, J.; Arghami, N. R., Nonparametric confidence and tolerance intervals from record values data, Statist. Papers, 44, 455-468 (2003) · Zbl 1050.62053
[2] Ahmadi, J.; Balakrishnan, N., Confidence intervals for quantiles in terms of record range, Statist. Probab. Lett., 68, 395-405 (2004) · Zbl 1086.62064
[3] Ahmadi, J.; Balakrishnan, N., Distribution-free confidence intervals for quantile intervals based on current records, Statist. Probab. Lett., 75, 190-202 (2005) · Zbl 1085.62050
[4] Ahmadi, J.; Razmkhah, M.; Balakrishnan, N., Current \(k\)-records and their use in distribution-free confidence intervals, Statist. Probab. Lett., 79, 29-37 (2009) · Zbl 1152.62026
[5] Arnold, B. C.; Balakrishnan, N.; Nagaraja, H. N., Records (1998), John Wiley & Sons: John Wiley & Sons New York · Zbl 0914.60007
[6] Balakrishnan, N.; Davies, K.; Keating, J. P., Pitman closeness of order statistics to population quantiles, Commun. Statist.-Simul. Comput., 38, 802-820 (2009) · Zbl 1290.62025
[8] Chandler, K. N., The distribution and frequency of record values, J. Roy. Statist. Soc. Ser. B, 14, 220-228 (1952) · Zbl 0047.38302
[9] Keating, J. P.; Mason, R. L.; Sen, P. K., Pitman’s Measure of Closeness: A Comparison of Statistical Estimators (1993), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, Pennsylvania · Zbl 0779.62019
[10] Shaked, M.; Shanthikumar, J. G., Stochastic Orders and Their Applications (1994), Academic Press: Academic Press New York · Zbl 0806.62009
[11] Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer-Verlag: Springer-Verlag New York
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.