Advertisement

Fisher Information and Tukey’s Linear Sensitivity Measure Based on Ordered Ranked Set Samples

  • N. Balakrishnan
  • T. Li
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

(1995) derived the Fisher information and discussed the maximum likelihood estimation (MLE) of the parameters of a location-scale family \( F\left( {\tfrac{{x - \mu }} {\sigma }} \right) \) based on the ranked set sample (RSS). She found that a RSS provided more information about both μ and σ than a simple random sample (SRS) of the same size. We also focus here on the location-scale family. We use the idea of order statistics from independent and nonidentical random variables (INID) to propose an ordered ranked set sample (ORSS) and develop the Fisher information and the maximum likelihood estimation based on such an ORSS. We use logistic, normal, and one-parameter exponential distributions as examples and conclude that in all these three cases, the ORSS does not provide as much Fisher information as the RSS, and consequently the MLEs based on the ORSS (MLE-ORSS) are not as efficient as the MLEs based on the RSS (MLE-RSS). In addition to the MLEs, we are also interested in best linear unbiased estimators (BLUE). For this purpose, we apply another measure of information, viz., Tukey’s linear sensitivity. Tukey (1965) proposed linear sensitivity to measure information contained in an ordered sample. We use logistic, normal, one- and two-parameter exponential, two-parameter uniform, and right triangular distributions as examples and show that in all these cases except the one-parameter the RSS, and consequently the BLUEs based on the ORSS (BLUE-ORSS) are more efficient than the BLUEs based on the RSS (BLUE-RSS). In the case of one-parameter exponential, the ORSS has only slightly less information than the RSS with the relative efficiency being very close to 1.

Keywords and phrases

Ranked set samples ordered ranked set samples Fisher information linear sensitivity measure best linear unbiased estimators 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balakrishnan, N. (Ed.) (1992). Handbook of the Logistic Distribution, Marcel Dekker, New York.zbMATHGoogle Scholar
  2. 2.
    Barnett, V., and Barreto, M. C. M. (2001). Estimators for a Poisson parameter using ranked set sampling, Journal of Applied Statistics, 28, 929–941.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barreto, M. C. M., and Barnett, V. (1999). Best linear unbiased estimators for the simple linear regression model using ranked set sampling, Environmental and Ecological Statistics, 6, 119–133.CrossRefGoogle Scholar
  4. 4.
    Bhoj, D. S., and Ahsanullah, M. (1996). Estimation of parameters of the generalized geometric distribution using ranked set sampling, Biometrics, 52, 685–694.zbMATHCrossRefGoogle Scholar
  5. 5.
    Chandrasekar, B., and Balakrishnan, N. (2002). On a multiparameter version of Tukey’s linear sensitivity measure and its properties, Annals of the Institute of Statistical Mathematics, 54, 796–805.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, Z. (1999). Density estimation using ranked set sampling data, Environmental and Ecological Statistics, 6, 135–146.CrossRefGoogle Scholar
  7. 7.
    Chen, Z. (2000). The efficiency of ranked-set sampling relative to simple random sampling under multiparameter families, Statistica Sinica, 10, 247–263.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Chuiv, N. N., and Sinha, B. K. (1998). On some aspects of ranked set sampling in parametric estimation, In Handbook of Statistics (Eds., N. Balakrishnan and C. R. Rao) Vol. 17, pp. 337–377, Elsevier, Amsterdam.Google Scholar
  9. 9.
    David, H. A., and Nagaraja, H. N. (2003). Order Statistics, Third edition, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  10. 10.
    Dell, T. R., and Clutter, J. L. (1972). Ranked set sampling theory with order statistics background, Biometrics, 28, 545–555.CrossRefGoogle Scholar
  11. 11.
    Hossain, S. S., and Muttlak, H. A. (2000). MVLUE of population parameters based on ranked set sampling, Applied Mathematics and Computation, 108, 167–176.zbMATHCrossRefGoogle Scholar
  12. 12.
    Kim, Y., and Arnold, B. C. (1999). Parameter estimation under generalized ranked set sampling, Statistics_& Probability Letters, 42, 353–360.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kvam, P. H., and Tiwari, R. C. (1999). Bayes estimation of a distribution function using ranked set samples, Environmental and Ecological Statistics, 6, 11–22.CrossRefGoogle Scholar
  14. 14.
    Lloyd, E. H. (1952). Least squares estimation of location and scale parameters using order statistics, Biometrika, 39, 88–95.zbMATHMathSciNetGoogle Scholar
  15. 15.
    McIntyre, G. A. (1952). A method for unbiased selective sampling using ranked sets, Australian Journal of Agricultural Research, 3, 385–390.CrossRefGoogle Scholar
  16. 16.
    Nagaraja, H. N. (1994). Tukey’s linear sensitivity and order statistics, Annals of the Institute of Statistical Mathematics, 46, 757–768.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Perron, F., and Sinha, B. K. (2004). Estimation of variance based on a ranked set sample, Journal of Statistical Planning and Inference, 120, 21–28.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stokes, S. L. (1977). Ranked set sampling with concomitant variables, Communications in Statistics—Theory and Methods, 6, 1207–1211.CrossRefGoogle Scholar
  19. 19.
    Stokes, S. L. (1980a). Estimation of variance using judgement ordered ranked set samples, Biometrics, 36, 35–42.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Stokes, S. L. (1995). Parametric ranked set sampling, Annals of the Institute of Statistical Mathematics, 47, 465–482.zbMATHMathSciNetGoogle Scholar
  22. 22.
    Stokes, S. L., and 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.MathSciNetGoogle Scholar
  23. 23.
    Takahasi, K., and 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.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Tietjen, G. L., Kahaner, D. K., and Beckman, R. J. (1977). Variances and covariances of the normal order statistics for sample sizes 2 to 50, In Selected Tables in Mathematical Statistics Vol. 5, American Mathematical Society, Providence, RI.Google Scholar
  25. 25.
    Tukey, J. W. (1965). Which part of the sample contains the information? Proceedings of the National Academy of Sciences of the USA, 53, 127–134.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Zheng, G., and Al-Saleh, M. F. (2003). Improving the best linear unbiased estimator for the scale parameter of symmetric distribution by using the absolute value of ranked set samples, Journal of Applied Statistics, 30, 253–265.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • N. Balakrishnan
    • 1
  • T. Li
    • 1
  1. 1.McMaster UniversityHamiltonCanada

Personalised recommendations