A constrained optimization problem for determining the smallest Pareto confidence region under progressive Type-II censoring

Document Type : Original Paper


Department of Statistics, University of Mazandaran, Babolsar, IRAN


In this paper, a constrained optimization problem is formulated and solved to determine the smallest joint confidence region for Pareto parameters based on the progressively Type-II censored samples. The objective function is the area of the confidence region and the problem constraint is the specified confidence level. The proposed ‎ joint confidence ‎region is also valid for the complete samples and right censored samples. The area of the smallest proposed confidence region and the area of the balanced confidence region are compared. Finally, two numerical examples are presented to describe the proposed optimization method.


Main Subjects

[1] Johnson‎, ‎N.L.‎ and ‎Kotz‎, ‎Balakrishnan‎, ‎N‎. ‎(1994)‎. Continuous Uinvariare Distributions‎, ‎Vol‎. ‎1, 2nd ed. John Wiley & Sons‎, ‎New York‎.
‎[2] Fernandez‎, ‎A‎.‎J‎. ‎(2012)‎. ‎Minimizing the area of a Pareto confidence region, European Journal of Operational research, 221‎, ‎205–212‎.
‎[3] Fernandez‎, ‎A‎.J‎. ‎(2013)‎. ‎Smallest Pareto confidence regions and applications‎,‎ Computational Statistics and Data Analysis, 62‎, ‎11–25‎.
‎[4]  Fernandez‎, ‎A‎.‎J‎. ‎(2014)‎. Computing optimal confidence sets for Pareto models under progressive censoring, Journal of Computational and Applied Mathematics, 258, 168–180
‎[5] Asgharzadeh‎, ‎A.‎, ‎Abdi‎, ‎M. and ‎Kus‎, ‎C‎. ‎(2011)‎. ‎Interval estimation for the two-parameter Pareto distribution based on record values, Selçuk Journal of Applied Mathematics, 149–161‎. ‎Special Issue‎. ‎‎
‎[‎6‎]‎ Asgharzadeh, A., Fernandez, A.J. and Abdi, M. (2017). Confidence sets for two-parameter Rayleigh distribution under progressive censoring, Applied Mathematical modelling, 47, 656–667.
‎[7] Balakrishnan‎, ‎N.‎, ‎and Aggarwala‎, ‎R. ‎(2000)‎. ‎Progressive censoring‎: ‎Theory‎, ‎Methods and Applications‎, ‎Birkhauser Publishers‎, ‎Boston‎. ‎‎
‎‎‎[‎8‎‎]‎ Balakrishnan‎, ‎N. and ‎Cramer‎, ‎E‎. ‎(2014). The Art of Progressive Censoring‎, ‎Springer‎, ‎New York‎.‎‎
‎[‎9‎] Kus‎, ‎C‎. ‎and Kaya‎, ‎M.F‎. ‎(2007)‎. ‎Estimation for the parameters of the Pareto distribution under progressive censoring‎, Communications in Statistics - Theory and ‎Methods, ‎‎‎36‎, ‎1359–1365‎.
‎[‎10‎] Nelson‎, ‎W‎. ‎B‎. ‎(1970)‎. ‎Statistical methods for accelerated life test datathe inverse power law model‎, ‎General Electric Co‎. ‎Tech‎. ‎Rep‎.71-C011‎, ‎New York‎: ‎Schenectady.
‎[‎11‎] Asgharzadeh‎, ‎A.‎, ‎Mohammadpour‎, ‎M. and ‎Ganji‎, ‎Z.M‎. ‎(2014)‎. ‎Estimation and reconstruction Based on Left Censored Data from Pareto Model‎, Journal of Iranian Statistical Society‎,‎ 13, 151–175‎.
‎‎[‎12‎] ‎Parsi, ‎S., ‎Ganjal‎i, M. and Sanjari‎ Farsipour, ‎N‎‎. (2010). Simultaneous confidence intervals for the parameters of Pareto distribution under progressive censoring, Communications in Statistics-Theory and ‎Methods‎, 39, 94–106.‎
Volume 9, Issue 1
March 2019
Pages 44-57
  • Receive Date: 26 September 2017
  • Revise Date: 20 October 2018
  • Accept Date: 30 October 2018
  • First Publish Date: 21 March 2019