Abstract
In applied statistics, the coefficient of variation is widely used. However, inference concerning the coefficient of variation of non-normal distributions are rarely reported. In this article, a simulation-based Bayesian approach is adopted to estimate the coefficient of variation (CV) under progressive first-failure censored data from Gompertz distribution. The sampling schemes such as, first-failure censoring, progressive type II censoring, type II censoring and complete sample can be obtained as special cases of the progressive first-failure censored scheme. The simulation-based approach will give us a point estimate as well as the empirical sampling distribution of CV. The joint prior density as a product of conditional gamma density and inverted gamma density for the unknown Gompertz parameters are considered. In addition, the results of maximum likelihood and parametric bootstrap techniques are also proposed. An analysis of a real life data set is presented for illustrative purposes. Results from simulation studies assessing the performance of our proposed method are included.
Similar content being viewed by others
References
K. Ahn, Use of coefficient of variation for uncertainty analysis in fault tree analysis, Reliability Engineering and System Safety, 47(3) (1995), 229–230.
E. K. Al-Hussaini, G. R. Al-Dayian and S. A. Adham, On finite mixture of twocomponent Gompertz lifetime model, Journal of Statistical Computation and Simulation, 67(1) (2000), 1–15.
N. Balakrishnan and R. A. Sandhu, A simple simulation algorithm for generating progressively type-II censored samples, The American Statistician, 49 (1995), 229–230.
Z. Chen, Parameter estimation of the Gompertz population, Biometrical Journal, 39 (1997), 117–124.
A. C. Davison and D.V. Hinkley, Bootstrap Methods and their Applications, 2nd, Cambridge University Press, Cambridge United Kingdom, 1997.
B. Efron and R. J. Tibshirani, An introduction to the bootstrap, New York Chapman and Hall, (1993).
R. C. Elandt-Johnson and N. L. Johnson, Survival Models and Data Analysis, John Wiley and Sons, New York, 1980.
P. H. Franses, Fitting a Gompertz curve, Journal of the Operational Research Society, 45 (1994), 109–113.
D. Gamerman, Markov chain Monte Carlo, Stochastic Simulation for Bayesian Inference, Chapman and Hall, London, 1997.
M. L. Garg, B. R. Rao and C. K. Redmond, Maximum likelihood estimation of the parameters of the Gompertz survival function, Journal of the Royal Statistical Society, 19 (1970), 152–159.
A. E. Gelfand and A. F. M. Smith, Sampling based approach to calculating marginal densities, Journal of the American Statistical Association, 85 (1990), 398–409.S.
Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Mathematical Intelligence, 6 (1984), 721–741.
W. R. Gilks, S. Richardson and D. J. Spiegelhalter, Markov chain Monte Carlo in Practices, Chapman and Hall, London, (1996).
B. Gompertz, On the nature of the function expressive of the law of human mortality and on the new mode of determining the value of life contingencies, Philosophical Transactions of the Royal Society, A, 115 (1825), 513–580.
J. Gong and Y. Li, Relationship between the estimated Weibull modulus and the coefficient of variation of the measured strength for ceramics, Journal of the American Ceramics Society, 82(2) (1999), 449–452.
A. J. Hamer, J. R. Strachan, M. M. Black, C. Ibbotson and R. A. Elson, A new method of comparative bone strength measurement, Journal of Medical Engineering and Technology, 19(1) (1995), 1–5.
W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97–109.
D. G. Hoel, A representation of mortality data by competing risks, Biometrics, 28 (1972), 475–488.
L. G. Johnson, Theory and Technique of Variation Research, Elsevier, Amsterdam, 1964.
C. H. Jun, S. Balamurali and S. H. Lee, Variables sampling plans for Weibull distributed lifetimes under sudden death testing, IEEE Transactions on Reliability, 55 (2006), 53–58.
W. C. Lee, J. W. Wu and H. Y. Yu, Statistical inference about the shape parameter of the bathtub-shaped distribution under the failure-censored sampling plan, International Journal of Information and Management Sciences, 18 (2007), 157–172.
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equations of state calculations by fast computing machines, Journal Chemical Physics, 21 (1953), 1087–1091.
E. G. Millcr and M. J. Karson, ’Testing Equality of Two Coefficients of Variation’, American statistical Association Posceedings of the Business and economics Section, Part 1 (1977), 278–283.
W. K. Pang, W. T. Y. Bosco, M. D. Troutt and H. H. Shui, A simulation-based approach to the study of coefficient of variation of dividend yields, European Journal of Operational Research, 189 (2008), 559–569.
W. K. Pang, P. K. Leung, W. K. Huang and L. Wei, On interval estimation of the coefficient of variation for the three-parameter Weibull, lognormal and gamma distribution: A simulation based approach, European Journal of Operational Research, 164 (2005), 367–377.
C. B. Read, Gompertz distribution, Encyclopedia of Statistical Sciences, Wiley, New York, 1983.
W. Reh and B. Scheffler, Significance tests and confidence intervals for coefficient of variation, Computational Statistics and Data Analysis, 22(4) (1996), 449–453.
S. J. Wu and C. Kusation based on progressive first-failure-censored sampling, Computational Statistics and Data Analysis, 53(10) (2009), 3659–3670.
J. W. Wu, W. L. Hung and C. H. Tsai, Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan, Statistics, 37 (2003), 517–525.
J. W. Wu and H. Y. Yu, Statistical inference about the shape parameter of the Burr type XII distribution under the failure-censored sampling plan, Applied Mathematics and Computation, 163 (2005), 443–482.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Soliman, A.A., Abd Ellah, A.H., Abou-Elheggag, N.A. et al. A simulation-based approach to the study of coefficient of variation of Gompertz distribution under progressive first-failure censoring. Indian J Pure Appl Math 42, 335–356 (2011). https://doi.org/10.1007/s13226-011-0022-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-011-0022-8