Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Estimation of P(X > Y) when X and Y are dependent random variables using different bivariate sampling schemes

  • Samawi, Hani M. (Department of Biostatistics, Jiann-Ping Hsu College of Public Health, Georgia Southern University) ;
  • Helu, Amal (Carnegie Mellon University) ;
  • Rochani, Haresh D. (Department of Biostatistics, Jiann-Ping Hsu College of Public Health, Georgia Southern University) ;
  • Yin, Jingjing (Department of Biostatistics, Jiann-Ping Hsu College of Public Health, Georgia Southern University) ;
  • Linder, Daniel (Department of Biostatistics, Jiann-Ping Hsu College of Public Health, Georgia Southern University)
  • Received : 2016.05.14
  • Accepted : 2016.08.26
  • Published : 2016.09.30
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Abstract

The stress-strength models have been intensively investigated in the literature in regards of estimating the reliability ${\theta}$ = P(X > Y) using parametric and nonparametric approaches under different sampling schemes when X and Y are independent random variables. In this paper, we consider the problem of estimating ${\theta}$ when (X, Y) are dependent random variables with a bivariate underlying distribution. The empirical and kernel estimates of ${\theta}$ = P(X > Y), based on bivariate ranked set sampling (BVRSS) are considered, when (X, Y) are paired dependent continuous random variables. The estimators obtained are compared to their counterpart, bivariate simple random sampling (BVSRS), via the bias and mean square error (MSE). We demonstrate that the suggested estimators based on BVRSS are more efficient than those based on BVSRS. A simulation study is conducted to gain insight into the performance of the proposed estimators. A real data example is provided to illustrate the process.

Keywords

References

  1. Al-Hussaini EK, Mousa MA, and Sultan KS (1997). Parametric and nonparametric simulation of P(Y < X) for finite mixtures of lognormal components, Communications in Statistics-Theory and Methods, 26, 1269-1289. https://doi.org/10.1080/03610929708831981
  2. Al-Saleh MF and Zheng G (2002). Estimation of bivariate characteristics using ranked set sampling, Australia and New Zealand Journal of Statistics, 44, 221-232. https://doi.org/10.1111/1467-842X.00224
  3. Awad AM, Azzam MM, and Hamdan MA (1981). Some inference results on Pr(Y < X) in the bivariate exponential model, Communications in Statistics-Theory and Methods, 10, 2515-2525. https://doi.org/10.1080/03610928108828206
  4. Baklizi A and Abu-Dayyeh W (2003). Shrinkage estimation of P(Y < X) in the exponential case, Communications in Statistics-Simulation and Computation, 32, 31-42. https://doi.org/10.1081/SAC-120013109
  5. Baklizi A and Eidous O (2006). Nonparametric estimation of P(X < Y) using kernel methods, Metron, 64, 47-60.
  6. Barbiero A (2012). Interval estimators for reliability: the bivariate normal case, Journal of Applied Statistics, 39, 501-512. https://doi.org/10.1080/02664763.2011.602055
  7. Domma F and Giordano S (2012). A stress-strength model with dependent variables to measure household financial fragility, Statistical Methods & Applications, 21, 375-389. https://doi.org/10.1007/s10260-012-0192-5
  8. Domma F and Giordano S (2013). A copula based approach to account for dependence in stressstrength models, Statistical Papers, 54, 807-826. https://doi.org/10.1007/s00362-012-0463-0
  9. Diaz-Frances E and Montoya JA (2013). The simplicity of likelihood based inferences for P(X < Y) and for the ratio of means in the exponential model, Statistical Papers, 54, 499-522. https://doi.org/10.1007/s00362-012-0446-1
  10. Enis P and Geisser S (1971). Estimation of the probability that Y < X, Journal of the American Statistical Association, 66, 162-168.
  11. Gupta RC, Ghitany ME, and Al-Mutairi DK (2013). Estimation of reliability from a bivariate log normal data, Journal of Statistical Computation and Simulation, 83, 1068-1081. https://doi.org/10.1080/00949655.2011.649284
  12. Johnson ME (1987). Multivariate Statistical Simulation, John Wiley & Sons, New York.
  13. Johnson NL (1975). Letter to the editor, Technometrics, 17, 393. https://doi.org/10.1080/00401706.1975.10489360
  14. Kaur A, Patil GP, Sinha AK, and Taillie C (1995). Ranked set sampling: an annotated bibliography, Environmental and Ecological Statistics, 2, 25-54. https://doi.org/10.1007/BF00452930
  15. Kotz S, Lumelskii S, and Pensky M (2003). The Stress-Strength Model and Its Generalizations: Theory and Applications, World Scientific Publishing, Singapore.
  16. Li D, Sinha BK, and Chuiv NN (1999). On estimation of P(X > c) based on a ranked set sample. In UJ Dixit and MR Satam (Eds), Statistical Inference and Design of Experiments (pp. 47-54), Alpha Science International, Oxford, UK.
  17. McIntyre GA (1952). A method for unbiased selective sampling, using ranked set, Australian Journal of Agricultural Research, 3, 385-390. https://doi.org/10.1071/AR9520385
  18. Montoya JA and Rubio FJ (2014). Nonparametric inference for P(X < Y) with paired variables, Journal of Data Science, 12, 359-375.
  19. Muttlak HA, Abu-Dayyeh WA, Saleh MF, and Al-Sawi E (2010). Estimating P(Y < X) using ranked set sampling in case of the exponential distribution, Communications in Statistics-Theory and Methods, 39, 1855-1868. https://doi.org/10.1080/03610920902912976
  20. Nadarajah S (2005). Reliability for some bivariate beta distributions, Mathematical Problems in Engineering, 2005, 101-111. https://doi.org/10.1155/MPE.2005.101
  21. Norris RC, Patil GP, and Sinha AK (1995). Estimation of multiple characteristics by ranked set sampling methods, Coenoses, 10, 95-111.
  22. Parzen E (1962). On estimation of a probability density function and mode, Annals of Mathematical Statistics, 33, 1065-1076. https://doi.org/10.1214/aoms/1177704472
  23. GP, Sinha AK, and Taillie C (1993). Relative precision of ranked set sampling: a comparison with the regression estimator, Environmetrics, 4, 399-412. https://doi.org/10.1002/env.3170040404
  24. GP, Sinha AK, and Taillie C (1994). Ranked set sampling for multiple characteristics, International Journal of Ecology and Environmental Sciences, 20, 94-109.
  25. Patil GP, Sinha AK, and Taillie C (1999). Ranked set sampling: a bibliography, Environmental and Ecological Statistics, 6, 91-98. https://doi.org/10.1023/A:1009647718555
  26. Rubio FJ and Steel MFJ (2013). Bayesian inference for P(X < Y) using asymmetric dependent distributions, Bayesian Analysis, 8, 43-62. https://doi.org/10.1214/13-BA802
  27. Samawi HM, Ahmed MS, and Abu-Dayyeh W(1996). Estimating the population mean using extreme ranked set sampling, Biometrical Journal, 38, 577-586. https://doi.org/10.1002/bimj.4710380506
  28. Samawi HM and Al-Sagheer OA (2001). On the estimation of the distribution function using extreme and median ranked set sampling, Biometrical Journal, 43, 357-373. https://doi.org/10.1002/1521-4036(200106)43:3<357::AID-BIMJ357>3.0.CO;2-Q
  29. Samawi HM and Muttlak HA (1996). Estimation of ratio using ranked set sampling, Biometrical Journal, 38, 753-764. https://doi.org/10.1002/bimj.4710380616
  30. Samawi HM and Muttlak HA (2001). On ratio estimation using median ranked set sampling, Journal of Applied Statistical Science, 10, 89-98.
  31. Sengupta S and Mukhuti S (2008). Unbiased estimation of P(X > Y) for exponential populations using order statistics with application in ranked set sampling, Communications in Statistics-Theory and Methods, 37, 898-916. https://doi.org/10.1080/03610920701693892
  32. Silverman BW (1986). Density Estimation for Statistics and Data Analysis, 26, CRC Press, New York.
  33. Tong H (1974). A note on the estimation of Pr(Y < X) in the exponential case, Technometrics, 16, 625.
  34. Ventura L and Racugno W (2011). Recent advances on Bayesian inference for P(X < Y), Bayesian Analysis, 6, 411-428.
  35. Walldius G and Jungner I (2006). The apoB/apoA-I ratio: a strong, new risk factor for cardiovascular disease and a target for lipid-lowering therapy: a review of the evidence, Journal of Internal Medicine, 259, 493-519. https://doi.org/10.1111/j.1365-2796.2006.01643.x
  36. Walldius G, Jungner I, Aastveit AH, Holme I, Furberg CD, and Sniderman AD (2004). The apoB/apoA-I ratio is better than the cholesterol ratios to estimate the balance between plasma proatherogenic and antiatherogenic lipoproteins and to predict coronary risk, Clinical Chemical Laboratory Medicine, 42, 1355-1363.
  37. Wand MP and Jones MC (1995). Kernel Smoothing, Chapman and Hall, London.
  38. Yan S, Li J, Li S, Zhang B, Du S, Gordon-Larsen P, Adair L, and Popkin B (2012). The expand- ing burden of cardiometabolic risk in China: the China Health and Nutrition Survey, Obesity Reviews, 13, 810-821. https://doi.org/10.1111/j.1467-789X.2012.01016.x
  39. Zhou W (2008). Statistical inference for P(X < Y), Statistics in Medicine, 27, 257-279. https://doi.org/10.1002/sim.2838