lmomco (version 2.4.14)

stttlmomco: Scaled Total Time on Test Transform of Distributions

Description

This function computes the Scaled Total Time on Test Transform Quantile Function for a quantile function \(x(F)\) (par2qua, qlmomco). The TTT is defined by Nair et al. (2013, p. 173) as $$\phi(u) = \frac{1}{\mu}\left[(1-u)x(u) + \int_0^u x(p)\; \mathrm{d}p \right]\mbox{,}$$ where \(\phi(u)\) is the scaled total time on test for nonexceedance probability \(u\), and \(x(u)\) is a constant for \(x(F = u)\). The \(\phi(u)\) is also expressible in terms of total time on test transform quantile function (\(T(u)\), tttlmomco) as $$\phi(u) = \frac{T(u)}{\mu}\mbox{,}$$ where \(\mu\) is the conditional mean (cmlmomco) at \(u = 0\) and the later definition is the basis for implementation in lmomco. The integral in the first definition is closely related to the structure of the reversed residual mean quantile function (\(R(u)\), rrmlmomco).

Usage

stttlmomco(f, para)

Value

Scaled total time on test value for \(F\).

Arguments

f

Nonexceedance probability (\(0 \le F \le 1\)).

para

The parameters from lmom2par or vec2par.

Author

W.H. Asquith

References

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

See Also

qlmomco, tttlmomco

Examples

# It is easiest to think about residual life as starting at the origin,
# but for this example, let us set the lower limit at 100 days.
A <- vec2par(c(100, 2649, 2.11), type="gov")
f <- 0.47  # Both computations of Phi show 0.6455061
"afunc" <- function(p) { return(par2qua(p,A,paracheck=FALSE)) }
tmpa <- 1/cmlmomco(f=0, A); tmpb <- (1-f)*par2qua(f,A,paracheck=FALSE)
Phiu1 <- tmpa * ( tmpb + integrate(afunc,0,f)$value )
Phiu2 <- stttlmomco(f, A)
if (FALSE) {
# The TTT-plot (see Nair et al. (2013, p. 173))
n <- 30; X <- sort(rlmomco(n, A)); lmr <- lmoms(X)  # simulated lives and their L-moments
# recognize here that the "fit" is to the lifetime data themselves and not to special
# curves or projections of the data to other scales
"Phir" <- function(r, X, sort=TRUE) {
   n <- length(X); if(sort) X <- sort(X)
   if(r == 0) return(0) # can use 2:r as X_{0:n} is zero
   Tau.rOFn <- sapply(1:r, function(j) { Xlo <- ifelse((j-1) == 0, 0, X[(j-1)]);
                                         return((n-j+1)*(X[j] - Xlo)) })
   return(sum(Tau.rOFn))
}
Xbar <- mean(X); rOFn <- (1:n)/n # Nair et al. (2013) are clear r/n used in the Phi(u)
Phi <- sapply(1:n, function(r) { return(Phir(r,X, sort=FALSE)) }) / (n*Xbar)
layout(matrix(1:3, ncol=1))
plot(rOFn, Phi, type="b",
     xlab="NONEXCEEDANCE PROBABILITY", ylab="SCALED TOTAL TIME ON TEST")
lines(rOFn, stttlmomco(rOFn, A), lwd=2, col=8) # solid grey, the parent distribution
par1 <- pargov(lmr); par2 <- pargov(lmr, xi=min(X)) # notice attempt to "fit at minimum"
lines(pp(X), stttlmomco(rOFn, par1)) # now Weibull (i/(n+1)) being used for F via pp()
lines(pp(X), stttlmomco(rOFn, par2), lty=2) # perhaps better, but could miss short lives
F <- nonexceeds(f01=TRUE)
plot(pp(X), sort(X), xlab="NONEXCEEDANCE PROBABILITY", ylab="TOTAL TIME ON TEST (DAYS)")
lines(F, qlmomco(F, A), lwd=2, col=8) # the parent again
lines(F, qlmomco(F, par1), lty=1); lines(F, qlmomco(F, par2), lty=2) # two estimated fits
plot(F,  lrzlmomco(F, par2), col=2, type="l")  # Lorenz curve from L-moment fit (red)
lines(F, bfrlmomco(F, par2), col=3, lty=2) # Bonferroni curve from L-moment fit (green)
lines(F, lkhlmomco(F, par2), col=4, lty=4) # Leimkuhler curve from L-moment fit (blue)
lines(rOFn, Phi) # Scaled Total Time on Test
}