duane {Reliability} | R Documentation |
duane
computes the Maximum Likelihood estimates for the parameters rho
and theta
of the mean value function for the Duane model.
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 := rho - frac{n}{t_n^{theta}} = 0
and
equation_2 := theta - frac{n}{sum_{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/
# 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)