Extended exponential model


(Marshall-Olkin)


   
   [ext_exponential_ex1]
      
   model
   {
      for( i in 1 : N )
      {
      x[i] ~ dext.exp(alpha, lambda)
      }
      
   # Prior distributions of the model parameters   
      
         alpha ~ dunif(1, 200)   
         lambda ~ dunif(0.001, 10)               
   }

The data set gives 100 observations on breaking stress of carbon fibres, Nichols and Padgett (2006).

Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), 641-652.

Nichols, M.D. and W.J. Padgett, W.J. (2006). A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International, 22, 141-151.

The MLE’s are alpha =75.67982, ; lambda = 1.67576,

Data
list( N=100, x=c(3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11,
4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90,
3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22,
3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56,
3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92,
1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59,
3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12, 1.71,
2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38,
1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80,
1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65))
Inits for chain 1
list(alpha=10.0, lambda=2.0)
   
Inits for chain 2
list(alpha=5.0, lambda=1.0)



Results
      

      mean   median   mode   sd   MC_error   val2.5pc   val97.5pc   start   sample   ESS
   alpha   97.66   91.82   81.29   37.47   0.4723   40.26   182.5   1001   50000   6294
   lambda   1.735   1.738   1.768   0.14   0.001818   1.457   1.995   1001   50000   5934

Note that the distribution of alpha is very skewed and that the mode is much closer to the MLE / MAPthan the mean or median.

MAP estimates are

[ext_exponential_ex2]

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