Lower Confidence Bounds for System Reliability from Binary Failure Data Using Bootstrapping

Abstract

Binary failure data are collected for each of the independent components in a coherent system. Bootstrapping is used to determine a (1 −α)100 % lower confidence bound on the system reliability. When a component with perfect test results is encountered, a beta prior distribution is used to avoid an overly optimistic lower bound.

Originally published in the Journal of Quality Technology, Volume 38 in 2006, this work uses APPL as one of its technologies that enabled the research. The initial problem arose in a reliability problem posed by engineers. This is the first publication that mentions using APPL to eliminate resampling error of bootstrap studies. The code relies on the procedures Transform and Product, to in effect establish a bootstrap with B = ∞.

Appendices

Appendix 1

S-Plus code for calculating a CP (1 −α)100 % lower confidence bound for a single component for n components on test and y passes. This function was used to generate the lower confidence bounds in Example 15.1.

> confintlower <- function(n, y, alpha) {

>   if (y == 0) {

>     pl <- 0

>   }

>   if (y == n) {

>     pl <- alpha ^ (1 / n)

>   }

>   if (y > 0 && y < n) {

>     fcrit1 <- qf(alpha, 2 * y, 2 * (n - y + 1))

>     pl <- 1 / (1 + (n - y + 1) / (y * fcrit1))

>   }

>   pl

> }

Appendix 2

S-Plus code for calculating a bootstrap (1 −α)100 % lower confidence interval bound for a k-component series system of independent components using B bootstrap replications. This implements the algorithm given in Table 15.2.

> seriessystemboot <- function(n, y, alpha) {

>   k <- length(n)

>   b <- 10000

>   z <- rep(1, b)

>

>   point <- prod(y) / prod(n)

>

>   for (j in 1:b) {

>     for (i in 1:k) {

>      z[j] <- z[j] * rbinom(1, n[i], y[i] / n[i]) / n[i]

>     }

>   }

>   z <- sort(z)

>   pl <- z[floor(alpha * b)]

>   c(point, pl)

> }

Appendix 3

APPL code for calculating a bootstrap (1 −α)100 % lower confidence interval bound for a k-component series system of independent components using the equivalent of B = +∞ bootstrap replications.

> n1 := 23;

> y1 := 21;

> X1 := BinomialRV(n1, y1 / n1);

> X1 := Transform(X1, [[x -> x / n1], [-infinity, infinity]]);

> n2 := 28;

> y2 := 27;

> X2 := BinomialRV(n2, y2 / n2);

> X2 := Transform(X2, [[x -> x / n2], [-infinity, infinity]]);

> n3 := 84;

> y3 := 82;

> X3 := BinomialRV(n3, y3 / n3);

> X3 := Transform(X3, [[x -> x / n3], [-infinity, infinity]]);

> Temp := Product(X1, X2);

> T := Product(Temp, X3);

Appendix 4

S-Plus code for calculating a bootstrap (1 −α)100 % lower confidence interval bound for a k-component series system of independent component with some perfect component test results using B bootstrap replications. This implements the algorithm given in Table 15.3.

> seriessystembayesboot <- function(n, y, alpha) {

>   k <- length(n)

>   alpha1 <- 1

>   alpha2 <- 1

>   b <- 10000

>   z <- rep(1, b)

>

>   point <- prod(y) / prod(n)

>

>   for (j in 1:b) {

>     for (i in 1:k) {

>       if (y[i] == n[i]) z[j] <- z[j] * rbeta(1, alpha1 + y[i],

>           alpha2)

>       else z[j] <- z[j] * rbinom(1, n[i], y[i] / n[i]) / n[i]

>     }

>   }

>   z <- sort(z)

>   pl <- z[floor(alpha * b)]

>   c(point, pl)

> }