duane: Maximum Likelihood estimation of mean value function for...
In Reliability: Functions for estimating parameters in software reliability models

Description Usage Arguments Details Value Author(s) References See Also Examples

duane computes the Maximum Likelihood estimates for the parameters rho and theta of the mean value function for the Duane model.

1	duane(t, init = c(1, 1), method = "Nelder-Mead", maxit = 10000, ...)

`t`	time between failure data
`init`	initial values for Maximum Likelihood fit of the mean value function for the Duane model.
`method`	the method to be used for optimization, see `optim` for details.
`maxit`	the maximum number of iterations, see `optim` for details.
`...`	control parameters and plot parameters optionally passed to the optimization and/or plot function. Parameters for the optimization function are passed to components of the `control` argument of `optim`.

This function estimates the parameters rho and theta of the mean value function for the Duane model. With Maximum Likelihood estimation one gets the following equations, which have to be minimized. This is

equation_1 := ρ - \frac{n}{t_n^{θ}} = 0

and

equation_2 := θ - \frac{n}{∑_{i = 1}^{n - 1}(\log(t_n / t_i))} = 0.

Where t is the time between failure data and n is the length or in other words the size of the time between failure data. So the simultaneous minimization of these equations happens by minimization of the equation

equation_1^2 + equation_2^2 = 0.

A list containing following components:

`rho`	Maximum Likelihood estimate for `rho`
`theta`	Maximum Likelihood estimate for `theta`

Andreas Wittmann andreas\_wittmann@gmx.de

J.D. Musa, A. Iannino, and K. Okumoto. Software Reliability: Measurement, Prediction, Application. McGraw-Hill, 1987.

Michael R. Lyu. Handbook of Software Realibility Engineering. IEEE Computer Society Press, 1996. http://www.cse.cuhk.edu.hk/~lyu/book/reliability/

duane.plot, mvf.duane

    
# time between-failure-data from DACS Software Reliability Dataset
# homepage, see system code 1. Number of failures is 136.
t <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24,
       108, 88, 670, 120, 26, 114, 325, 55, 242, 68, 422, 180,
       10, 1146, 600, 15, 36, 4, 0, 8, 227, 65, 176, 58, 457,
       300, 97, 263, 452, 255, 197, 193, 6, 79, 816, 1351, 148,
       21, 233, 134, 357, 193, 236, 31, 369, 748, 0, 232, 330,
       365, 1222, 543, 10, 16, 529, 379, 44, 129, 810, 290, 300,
       529, 281, 160, 828, 1011, 445, 296, 1755, 1064, 1783, 
       860, 983, 707, 33, 868, 724, 2323, 2930, 1461, 843, 12,
       261, 1800, 865, 1435, 30, 143, 108, 0, 3110, 1247, 943,
       700, 875, 245, 729, 1897, 447, 386, 446, 122, 990, 948,
       1082, 22, 75, 482, 5509, 100, 10, 1071, 371, 790, 6150,
       3321, 1045, 648, 5485, 1160, 1864, 4116)
      
duane(t)