littlewood.verall: Maximum Likelihood estimation of mean value function for...

In Reliability: Functions for estimating parameters in software reliability models

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 value function is used.
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.

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}{ρ} + ∑_{i = 1}^{n} \log(θ_0 + θ_1 i) - ∑_{i = 1}^{n} \log(θ_0 + θ_1 i + t_i) = 0,

equation_2 := ρ ∑_{i = 1}^{n} \frac{1}{θ_0 + θ_1 i} - ρ + 1 ∑_{i = 1}^{n} \frac{1}{θ_0 + θ_1 i + t_i} = 0

and

equation_3 := ρ ∑_{i = 1}^{n} \frac{i}{θ_0 + θ_1 i} - ρ + 1 ∑_{i = 1}^{n} \frac{i}{θ_0 + θ_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}{ρ} + ∑_{i = 1}^{n} \log(θ_0 + θ_1 i^2) - ∑_{i = 1}^{n} \log(θ_0 + θ_1 i^2 + t_i) = 0,

equation_2 := ρ ∑_{i = 1}^{n} \frac{1}{θ_0 + θ_1 i^2} - ρ + 1 ∑_{i = 1}^{n} \frac{1}{θ_0 + θ_1 i^2 + t_i} = 0

and

equation_3 := ρ ∑_{i = 1}^{n} \frac{i^2}{θ_0 + θ_1 i^2} - ρ + 1 ∑_{i = 1}^{n} \frac{i^2}{θ_0 + θ_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.

Value

A list containing following components:

theta0	Maximum Likelihood estimate for theta0
theta1	Maximum Likelihood estimate for theta1
rho	Maximum Likelihood estimate for rho

Author(s)

Andreas Wittmann andreas\[e-mail]

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)

Example output

$theta0
[1] 44.86406

$theta1
[1] 44.54946

$rho
[1] 6.219767

$theta0
[1] 52855664

$theta1
[1] 140508630

$rho
[1] 358159111

Reliability documentation built on May 1, 2019, 9:22 p.m.