Markov Chain Monte Carlo to Study the Estimation of the Coefficient of Variation

The coefficient of variation (CV ) of a population is defined as the ratio of the population standard deviation to the population mean. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on the upper record values, we study the behavior of the CV of a random variable that follows a Lomax distribution. Specifically, we compute the maximum likelihood estimations (MLEs) and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo (MCMC) techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on a real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.

[1]  Wolfgang Reh,et al.  Significance tests and confidence intervals for coefficients of variation , 1996 .

[2]  M. Peshwani,et al.  Choice Between Weibull and Lognormal Models: A Simulation Based Bayesian Study , 2003 .

[3]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[4]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[5]  Mojtaba Ganjali,et al.  On some lifetime distributions with decreasing failure rate , 2009, Comput. Stat. Data Anal..

[6]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[7]  Marco Furini,et al.  International Journal of Computer and Applications , 2010 .

[8]  Wayne Nelson,et al.  Applied life data analysis , 1983 .

[9]  P. Hall.  Theoretical Comparison of Bootstrap Confidence Intervals , 1988 .

[10]  Ganapati P. Patil,et al.  Statistical Distributions in Scientific Work , 1981 .

[11]  K. Lomax.  Business Failures: Another Example of the Analysis of Failure Data , 1954 .

[12]  Wei Liu,et al.  On interval estimation of the coefficient of variation for the three-parameter Weibull, lognormal and gamma distribution: A simulation-based approach , 2005, Eur. J. Oper. Res..

[13]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[14]  S. Rachev,et al.  PORTFOLIO CHOICE WITH HEAVY TAILED DISTRIBUTIONS , 2004 .

[15]  M. Bryson.  Heavy-Tailed Distributions: Properties and Tests , 1974 .

[16]  W. Greene,et al.  计量经济分析 = Econometric analysis , 2009 .

[17]  B. Dhillon.  Life Distributions , 1981, IEEE Transactions on Reliability.

[18]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[19]  Kwang-Il Ahn.  On the use of coefficient of variation for uncertainty analysis in fault tree analysis , 1995 .

[20]  Stephen E. Satchell,et al.  A Re‐Examination of Sharpe's Ratio for Log‐Normal Prices , 2005 .

[21]  Marvin D. Troutt,et al.  A simulation-based approach to the study of coefficient of variation of dividend yields , 2008, Eur. J. Oper. Res..

[22]  Ahmed A. Soliman,et al.  Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches , 2006, Comput. Stat. Data Anal..

[23]  Ingram Olkin,et al.  Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families , 2007 .

[24]  Estimation of parameters of a pareto distribution by generalized order statistics , 2000 .

[25]  Peter Kuster.  The fundamentals of risk measurement , 2005 .

[26]  R. Elson,et al.  A new method of comparative bone strength measurement. , 1995, Journal of medical engineering & technology.

[27]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[28]  Bradley Efron,et al.  Nonparametric Estimates of Standard Error: The Jackknife, the Bootstrap, and Other Resampling Plans , 1983 .

[29]  Ying Li,et al.  Relationship between the Estimated Weibull Modulus and the Coefficient of Variation of the Measured Strength for Ceramics , 2004 .