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Inference on a distribution function from ranked set samples

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  • Lutz Dümbgen

    (University of Bern)

  • Ehsan Zamanzade

    (University of Isfahan)

Abstract

Consider independent observations $$(X_i,R_i)$$(Xi,Ri) with random or fixed ranks $$R_i$$Ri, while conditional on $$R_i$$Ri, the random variable $$X_i$$Xi has the same distribution as the $$R_i$$Ri-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

Suggested Citation

  • Lutz Dümbgen & Ehsan Zamanzade, 2020. "Inference on a distribution function from ranked set samples," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 157-185, February.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:1:d:10.1007_s10463-018-0680-y
    DOI: 10.1007/s10463-018-0680-y
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    References listed on IDEAS

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    1. Dinesh S. Bhoj, 2001. "Ranked Set Sampling with Unequal Samples," Biometrics, The International Biometric Society, vol. 57(3), pages 957-962, September.
    2. N. Balakrishnan & T. Li, 2006. "Confidence Intervals for Quantiles and Tolerance Intervals Based on Ordered Ranked Set Samples," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(4), pages 757-777, December.
    3. Ali Dastbaravarde & Nasser Reza Arghami & Majid Sarmad, 2016. "Some theoretical results concerning non parametric estimation by using a judgment poststratification sample," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 45(8), pages 2181-2203, April.
    4. Steven N. MacEachern & Elizabeth A. Stasny & Douglas A. Wolfe, 2004. "Judgement Post-Stratification with Imprecise Rankings," Biometrics, The International Biometric Society, vol. 60(1), pages 207-215, March.
    5. Xinlei Wang & Ke Wang & Johan Lim, 2012. "Isotonized CDF Estimation from Judgment Poststratification Data with Empty Strata," Biometrics, The International Biometric Society, vol. 68(1), pages 194-202, March.
    6. Jesse Frey & Omer Ozturk, 2011. "Constrained estimation using judgment post-stratification," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(4), pages 769-789, August.
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    Cited by:

    1. Soohyun Ahn & Xinlei Wang & Mumu Wang & Johan Lim, 2022. "On continuity correction for RSS-structured cluster randomized designs with binary outcomes," METRON, Springer;Sapienza Università di Roma, vol. 80(3), pages 383-397, December.
    2. Ehsan Zamanzade & M. Mahdizadeh & Hani M. Samawi, 2020. "Efficient estimation of cumulative distribution function using moving extreme ranked set sampling with application to reliability," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(3), pages 485-502, September.
    3. M. Mahdizadeh & Ehsan Zamanzade, 2022. "New insights on goodness-of-fit tests for ranked set samples," Statistical Papers, Springer, vol. 63(6), pages 1777-1799, December.
    4. Mozhgan Alirezaei Dizicheh & Nasrollah Iranpanah & Ehsan Zamanzade, 2021. "Bootstrap Methods for Judgment Post Stratification," Statistical Papers, Springer, vol. 62(5), pages 2453-2471, October.

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