Estimation of the parameter of L´evy distribution using ranked set sampling

Document Type : Original Article

Authors

1 Department of Statistics, Roudehen Branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

3 Department of Statistics, Allameh Tabataba'i University, Tehran, Iran.

10.22060/ajmc.2020.18499.1037

Abstract

Ranked set sampling is a statistical technique for data collection that generally leads to more efficient estimators than competitors based on simple random sampling. In this paper, we consider estimation of scale parameter of L´evy distribution using a ranked set sample. We derive the best linear unbiased estimator and its variance, based on a ranked set sample. Also we compare numerically, variance of this estimator with mean square error of the maximum likelihood, a median based estimator and an estimator based on Laplace transform. It turns out that the best linear unbiased estimator based on ranked set sampling is more efficient than other mentioned estimators.

Keywords

Main Subjects

References

[1] P. Besbeas, B. J. T. Morgan, Efficient and robust estimation for the one-sided stable distribution of index 1 2 . Statist. Probab. Lett., 66 (2004) 251-257.
[2] Z. Chen, Z. Bai, B. K. Sinha, Ranked set sampling: Theory and applications, Springer, New York, 2004.
[3] T. R. Dell, J. L. Clutter, Ranked set sampling theory with order statistics background, Biometrics 28 (1972) 545-555.
[4] H. Fei, B. K. Sinha, Z. Wu, Estimation of parameters in two-parameter Weibull and extreme-value distributions using ranked set sample, J. Statist. Res. 28 (1994), 149-161.
[5] A. Hatefi, M. Jafari Jozani, D. Ziou, Estimation and classification for finite Mixture models under ranked set sampling, Statistica Sinica 24 (2014) 675-698.
[6] K. Lam, B. K. Sinha, Z. Wu, Estimation of parameters in a two-parameter exponential distribution using ranked set sample, Ann. Inst. Statist. Math., 46 (1994) 723-736.
[7] K. Lam, B. K. Sinha, Z. Wu, Estimation of location and scale parameters of a Logistic distribution using ranked set sample, In: Nagaraja, Sen, Morrison, (Eds), Papers in Honor of Herbert A. David. (1995) 187-197.
[8] G. A. McIntyre, A method for unbiased selective sampling using ranked-set sampling, Austral. J. Agric. Res. 3 (1952) 385-390. [9] M. Mohammadi, M. Mohammadpour, Existence of order statistics moments of α-stable distributions, Preprint, 2010.
[10] N. Ni Chuiv,B. K. Sinha, Z. Wu, Estimation of the location parameter of a Cauchy distribution using a ranked set sample, Technical Report, University of Maryland Baltimore County, 1994.
[11] N. Ni Chuiv, B. K. Sinha, On some aspects of ranked set sampling in parametric estimation, In: Balakrishnan, Rao (Eds), Handbook of Statistics, 17 (1998) 337-377.
[12] A. Piryatinska, Inference for L´evy models and their application in medicine and statistical physics, Ph.D. Thesis, Department of statistics, Case Western Reserve University, Cleveland, OH, 2005.
[13] G. Samorodnitsky, Extrema of skewed stable process, Stochastic Process. Appl. 30 (1988) 17-39.
[14] E. Scalas, K. Kim, The art of fitting financial time series with L´evy stable distributions, 2006, http://mpra.ub.uni-muenchen.de/336/.
[15] B. K. Sinha, S. Purkayastha, On some aspects of ranked set sampling for estimation of Normal and Exponential parameters, Statist. Decisions 14 (1996) 223-240.
[16] K. Takahasi, K. Wakimato, On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968) 1-31.
[17] V. V. Uchaikin, V. M. Zolotarev, Chance and Stability: stable distributions and their applications, Utrecht: VSP Press, 1999. [18] G. Zheng, R. Modarres, A robust estimate of the correlation coefficient for bivariate normal distribution using ranked set, J. Statist. Plann. Inference 136 (2006) 298-309.
AUT Journal of Mathematics and Computing
Volume 2, Issue 1
Winter and Spring 2021
Pages 53-60

Files

History

  • Receive Date: 27 May 2020
  • Revise Date: 02 December 2020
  • Accept Date: 20 December 2020

Share

How to cite

Statistics