Reliability (version 0.0-2)

littlewood.verall: Maximum Likelihood estimation of mean value function for Littlewood-Verall model

Description

littlewood.verall computes the Maximum Likelihood estimates for the parameters theta0, theta1 and rho of the mean value function for the Littlewood-Verall model.

Usage

littlewood.verall(t, linear = T, init = c(1, 1, 1), method = "Nelder-Mead", 
    maxit = 10000, ...)

Arguments

t

time between failure data

linear

logical. Should the linear or the quadratic form of the mean value function for the Littlewood-Verrall model be used of computation? If TRUE, which is the default, the linear form of the mean

init

initial values for Maximum Likelihood fit of the mean value function for the Littlewood-Verall 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.

Value

• A list containing following components:
• theta0
  Maximum Likelihood estimate for theta0
• theta1
  Maximum Likelihood estimate for theta1
• rho
  Maximum Likelihood estimate for rho

Details

This function estimates the parameters theta0, theta1 and rho of the mean value function in the linear or the quadratic form for the Littlewood-Verall model. First, the computation with the mean value function in the linear form is explained. With Maximum Likelihood estimation one gets the following equations, which have to be minimized. This is $$equation_1 := \frac{n}{\rho} + \sum_{i = 1}^{n} \log(\theta_0 + \theta_1 i) - \sum_{i = 1}^{n} \log(\theta_0 + \theta_1 i + t_i) = 0,$$ $$equation_2 := \rho \sum_{i = 1}^{n} \frac{1}{\theta_0 + \theta_1 i} - \rho + 1 \sum_{i = 1}^{n} \frac{1}{\theta_0 + \theta_1 i + t_i} = 0$$ and $$equation_3 := \rho \sum_{i = 1}^{n} \frac{i}{\theta_0 + \theta_1 i} - \rho + 1 \sum_{i = 1}^{n} \frac{i}{\theta_0 + \theta_1 i + t_i} = 0.$$ Second, the computation with the mean value function in the quadratic form is explained. With Maximum Likelihood estimation one gets the following equations, which have to be minimized. This is $$equation_1 := \frac{n}{\rho} + \sum_{i = 1}^{n} \log(\theta_0 + \theta_1 i^2) - \sum_{i = 1}^{n} \log(\theta_0 + \theta_1 i^2 + t_i) = 0,$$ $$equation_2 := \rho \sum_{i = 1}^{n} \frac{1}{\theta_0 + \theta_1 i^2} - \rho + 1 \sum_{i = 1}^{n} \frac{1}{\theta_0 + \theta_1 i^2 + t_i} = 0$$ and $$equation_3 := \rho \sum_{i = 1}^{n} \frac{i^2}{\theta_0 + \theta_1 i^2} - \rho + 1 \sum_{i = 1}^{n} \frac{i^2}{\theta_0 + \theta_1 i^2 + t_i} = 0.$$ 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 $$equation_1^2 + equation_2^2 + equation_3^2 = 0.$$

References

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/

See Also

littlewood.verall.plot, mvf.ver.lin, mvf.ver.quad

Examples

# 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)
      
littlewood.verall(t, linear = TRUE)
littlewood.verall(t, linear = FALSE)