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Simultaneous closeness among order statistics to population quantiles. (English) Zbl 1188.62164
Summary: We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of $$p$$ in the percentile and the sample size, $$n$$. The results are finally illustrated with the estimation of percentiles for normal and exponential distributions.

##### MSC:
 62G30 Order statistics; empirical distribution functions 62G05 Nonparametric estimation 62F10 Point estimation
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##### References:
 [1] Ahmadi, J.; Balakrishnan, N., Pitman closeness of record values to population quantiles, Statistics & probability letters, 79, 2037-2044, (2009) · Zbl 1171.62033 [2] Ahmadi, J., Balakrishnan, N., 2010. On Pitman’s measure of closeness of k-records, Journal of Statistical Computation and Simulation, to appear. · Zbl 1221.62074 [3] Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 2008. A first course in order statistics. In: Classics in Applied Mathematics, vol. 54. Society for Industrial and Applied Mathematics, Philadelphia. · Zbl 1172.62017 [4] Balakrishnan, N.; Davies, K.; Keating, J.P., Pitman closeness of order statistics to population quantiles, Communications in statistics—theory & methods, 38, 802-820, (2009) · Zbl 1290.62025 [5] Balakrishnan, N., Davies, K., Keating, J.P., Mason, R.L., 2010a. Pitman closeness of best linear unbiased and invariant predictors for exponential distribution in one- and two-sample situations. Communications in Statistics—Theory & Methods, to appear. · Zbl 1244.62145 [6] Balakrishnan, N., Davies, K., Keating, J.P., Mason, R.L., 2010b. Pitman closeness, monotonicity and consistency of best linear unbiased and invariant estimators for exponential distribution under Type II censoring. Journal of Statistical Computation and Simulation, to appear. · Zbl 1219.62038 [7] Balakrishnan, N.; Iliopoulos, G.; Keating, J.P.; Mason, R.L., Pitman closeness of sample median to population Median, Statistics & probability letters, 79, 1759-1766, (2009) · Zbl 1169.62324 [8] Banks, D., 1997. Some geometry for location estimators. Classification Society of North American Newsletter, Issue 49 April; available at $$\langle$$http://www.public.iastate.edu/larsen/csna/previous/csnanews49-97apr.htm〉. [9] Blyth, C.R., Some probability paradoxes in choice from among random alternatives, Journal of the American statistical association, 67, 366-381, (1972) · Zbl 0245.62009 [10] Fountain, R.L.; Keating, J.P.; Maynard, H.B., The simultaneous comparison of estimators, Mathematical methods of statistics, 5, 187-198, (1996) · Zbl 0860.62024 [11] Karunaratne, H.S.I.; Hadjicostas, P., Comparison of estimators using banks’ criterion, Mathematical inequalities & applications, 12, 455-472, (2009) · Zbl 1186.62031 [12] Keating, J.P.; Gupta, R.C., Simultaneous comparison of scale estimators, Sankhyā, series B, 46, 275-280, (1984) · Zbl 0562.62025 [13] Keating, J.P.; Mason, R.L., Pitman’s measure of closeness, Sankhyā, series B, 47, 22-32, (1985) · Zbl 0581.62028 [14] 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 Philadelphia · Zbl 0779.62019 [15] Møller, J., Lectures on random Voronoi tessellations, (1994), Springer New York · Zbl 0812.60016
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