×

zbMATH — the first resource for mathematics

Confidence intervals for quantiles and tolerance intervals based on ordered ranked set samples. (English) Zbl 1106.62055
Summary: Confidence intervals for quantiles and tolerance intervals based on ordered ranked set samples (ORSS) are discussed. For this purpose, we first derive the cdf of ORSS and the joint pdf of any two ORSS. In addition, we obtain the pdf and cdf of the difference of two ORSS, viz. \(X_{s:N}^{\text{ORSS}}-X_{r:N}^{\text{ORSS}}, 1 \leq r < s \leq N\). Then, confidence intervals for quantiles based on ORSS are derived and their properties are discussed. We compare with approximate confidence intervals for quantiles given by Z. Chen [J. Stat. Plann. Inference 83, No. 1, 125–135 (2000; Zbl 0942.62009)], and show that these approximate confidence intervals are not very accurate. However, when the number of cycles in the RSS increases, these approximate confidence intervals become accurate even for small sample sizes. We also compare with intervals based on usual order statistics and find that the confidence interval based on ORSS becomes considerably narrower than the one based on usual order statistics when \(n\) becomes large. By using the cdf of \(X_{s:N}^{\text{ORSS}}-X_{r:N}^{\text{ORSS}}\), we then obtain tolerance intervals, discuss their properties, and present some tables for two-sided tolerance intervals.

MSC:
62G15 Nonparametric tolerance and confidence regions
62G30 Order statistics; empirical distribution functions
62-04 Software, source code, etc. for problems pertaining to statistics
62Q05 Statistical tables
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold B.C., Balakrishnan N., Nagaraja H.N. (1992). A First Course in Order Statistics. Wiley, New York · Zbl 0850.62008
[2] Balakrishnan N. (1988). Recurrence relations for order statistics from n independent and non-identically distributed random variables. Annals of the Institute of Statistical Mathematics 40, 273–277 · Zbl 0668.62029
[3] Balakrishnan N. (1989). Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables. Annals of the Institute of Statistical Mathematics 41, 323–329 · Zbl 0721.62049
[4] Chen Z. (1999). Density estimation using ranked set sampling data. Environmental and Ecological Statistics 6, 135–146
[5] Chen Z. (2000a). The efficiency of ranked-set sampling relative to simple random sampling under multi-parameter families. Statistica Sinica 10, 247–263 · Zbl 0970.62011
[6] Chen Z. (2000b). On ranked sample quantiles and their applications. Journal of Statistical Planning and Inference 83, 125–135 · Zbl 0942.62009
[7] Chen Z., Bai Z., Sinha B.K. (2004). Ranked Set Sampling–Theory and Application. Lecture Notes in Statistics (No. 176). Springer, Berlin Heidelberg New York · Zbl 1045.62007
[8] Chuiv N.N., Sinha B.K. (1998). On some aspects of ranked set sampling in parametric estimation. In: Balakrishnan N., Rao C.R. (eds). Handbook of Statistics. Vol 17, Elsevier, Amsterdam, pp. 337–377 · Zbl 0922.62010
[9] David H.A., Nagaraja H.N. (2003). Order Statistics (3rd ed.). Wiley, New York · Zbl 1053.62060
[10] Dell T.R., Clutter J.L. (1972). Ranked set sampling theory with order statistics background. Biometrics 28, 545–555 · Zbl 1193.62047
[11] McIntyre G.A. (1952). A method for unbiased selective sampling, using ranked sets. Australian Journal of Agricultural Research 3, 385–390
[12] Patil G.P., Sinha A.K., Taillie C. (1999). Ranked set sampling: a bibliography. Environmental and Ecological Statistics 6, 91–98
[13] Stokes S.L. (1977). Ranked set sampling with concomitant variables. Communications in Statistics–Theory and Methods 6, 1207–1211
[14] Stokes S.L. (1980a). Estimation of variance using judgement ordered ranked set samples. Biometrics 36, 35–42 · Zbl 0425.62023
[15] Stokes S.L. (1980b). Inferences on the correlation coefficient in bivariate normal populations from ranked set samples. Journal of the American Statistical Association 75, 989–995 · Zbl 0476.62011
[16] Stokes S.L. (1995). Parametric ranked set sampling. Annals of the Institute of Statistical Mathematics 47, 465–482 · Zbl 0840.62029
[17] Stokes S.L., Sager T.W. (1988). Characterization of a ranked-set sample with application to estimating distribution functions. Journal of the American Statistical Association 83, 35–42 · Zbl 0644.62050
[18] Takahasi K., Wakimoto K. (1968). On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20, 1–31 · Zbl 0157.47702
[19] Vaughan R.J., Venables W.N. (1972). Permanent expression for order statistics densities. Journal of the Royal Statistical Society, Series B 34, 308–310 · Zbl 0239.62038
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.