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

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

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

1
2
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\_wittmann@gmx.de

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

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
# 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.