Abstract
Suppose θ is the parameter of interest and λ is the nuisance parameter. When obtaining the maximum likelihood estimator (MLE) of θ in the presence of λ requires intensive computation, the pseudo MLE of θ, based on a pseudo likelihood function, can be used. Gong and Samaniego (Ann. Stat. 9:861–869, 1981) proposed a pseudo MLE (PMLE) based on simple random samples. Ranked set sampling has been applied to the bivariate variables (X,Y) where measuring one of the variables is difficult or costly. In this paper, we obtain the pseudo MLE of the correlation coefficient from a bivariate normal distribution (X,Y) based on ranked set samples, assuming that Y is difficult or more expensive to measure and that the mean and variance of Y are the nuisance parameters. The PMLE is compared with three other estimators of the correlation coefficient. Simulations show that the PMLE is more (less) efficient than other estimators, depending on value of ρ. Testing of soil contamination provides an example of the use of the methods.
Similar content being viewed by others
References
Barnett V, Green PJ, Robinson A (1976) Concomitants and correlation estimates. Biometrika 63:323–329
Chen Z, Bai Z, Sinha BK (2004) Ranked set sampling: theory and applications. Springer, New York
Dell JR, Clutter JL (1972) Ranked set sampling theory with order statistics background. Biometrics 28:545–553
Gilbert RO, Shinn JH, Essington EH, Romney EM, Moor KS, O’Farrell TP (1988) Radionuclide transport from soil to air, native vegetation, kangaroo rats and grazing cattle on the Nevada test site. Health Phys 35:869–887
Hui TP (2005) Bootstrap and likelihood based inference for ranked set samples. Unpublished PhD dissertation, Department of Statistics, The George Washington University, Washington DC
Hui TP, Modarres R, Zheng G (2005) Bootstrap confidence interval estimation of the regression mean with ranked set sampling. J Stat Comput Simul 75:543–553
Gong G, Samaniego FJ (1981) Pseudo maximum likelihood estimation: theory and application. Ann Stat 9:861–869
Johnson N, Kotz S (1972) Continuous multivariate distributions. Wiley, New York
McIntyre GA (1952) A method of unbiased selective sampling using ranked sets. Aust J Agric Res 3:385–390
Parke WR (1986) Pseudo maximum likelihood estimation: the asymptotic distribution. Ann Stat 14:355–357
Patil GP (1995) Editorial: ranked set sampling. Environ Ecol Stat 2:271–285
Sinha AK (2005) On some recent developments in ranked set sampling. Bull Inform Cybern 37:137–160
Stokes SL (1980) Inferences on the correlation coefficient in bivariate normal populations from ranked set samples. J Am Stat Assoc 75:989–995
Yu PLH, Lam K (1997) Regression estimator in ranked set sampling. Biometrics 53:1070–1080
Zheng G, Modarres R (2006) A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set sampling. J Stat Plan Inference 136:298–309
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hui, T.P., Modarres, R. & Zheng, G. Pseudo maximum likelihood estimates using ranked set sampling with applications to estimating correlation. TEST 18, 365–380 (2009). https://doi.org/10.1007/s11749-008-0096-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-008-0096-8
Keywords
- Ranked set sampling
- Concomitant variable
- Bivariate normal distribution
- Pseudo maximum likelihood estimator