IB.F: Bootstrap confidence bands for fatigue lifetime
In FCGR: Fatigue Crack Growth in Reliability

Description Usage Arguments Details Value Author(s) References Examples

It performs bootstrap confidence bands for fatigue lifetime. The lifetime matrix is calculated by bootstrap resampling by means the above mentioned methodologies (see craks.growth). The confidence bands are estimated by the quantile based method.

1 2	IB.F(z, nB, alpha = 0.05, method = c("SEP-lme_bkde", "SEP-lme_kde", "PB-nlme"))

`z`	cracks.growth object.
`nB`	Number of bootstrap resampling.
`alpha`	Confidence level.
`method`	Character string showing the distribution estimates method: "SEP-lme_bkde", "SEP-lme_kde" or "PB-nlme.

IB.F is performed from the output of cracks.growth function.

Return a list with the following values:

`Mat.F.B`	Matrix that contents the fatigue lifetimes corresponding to each bootstrap resampling.
`I.Bootstrap`	Data frame that contents the bootstrap confidence bands for lifetime distribution, at a confidence level of 95 percent (by default). It is composed by two columns corresponding to the bands limits: low, up.

Antonio Meneses antoniomenesesfreire@hotmail.com, Salvador Naya salva@udc.es, Javier Tarrio-Saavedra jtarrio@udc.es, Ignacio Lopez-Ullibarri ilu@udc.es

Meeker, W., Escobar, L. (1998) Statistical Methods for Reliability Data. John Wiley & Sons, Inc. New York.

Pinheiro JC., Bates DM. (2000) Mixed-effects models in S ans S-plus. Statistics and Computing. Springer-Verlang. New York.

Paris, P.C. and Erdogan, F. (1963) A critical analysis of crack propagation laws. J. Basic Eng., 85, 528.

## Not run: 
## Using the Alea.A dataset
data(Alea.A)
x <- Alea.A
## Critical crack length
aF <- 1.6    
## Censoring time         
T_c <- 0.12           
## cracks.growth function applied to Alea.A data
cg <- cracks.growth(x, aF, T_c, method = c("SEP-lme_bkde", "SEP-lme_kde",
                     "PB-nlme"), nBKDE = 5000, nKDE = 5000, nMC = 5000) 
## z is a cracks.growth object
z <- cg
## Number of bootstrap resamplings
nB <- 100 
## Application of IB.F function to cg object
ic.b <- IB.F(z, nB, alpha = 0.05, method = c("SEP-lme_bkde", "SEP-lme_kde",
           "PB-nlme")) 
## ic.b values obtainde by the "SEP-lme_bkde" model
names(ic.b)
# [1] "Mat.F.B"     "I.Bootstrap" 
## Chart with the empirical and estimated distribution functions,
## with bootstrap confidence bands at 95
# Observations from which the distribution function is estimated 
F1.F <- z$F.est[,2]  
plot( ic.b$I.Bootstrap$low,F1.F, col=2, type="l", lty=2, lwd=2, 
          xlim=c(0.05,0.18),
          main="Plot: distributions of failure times\n  confidence intervals",
          xlab="million cycles",  ylab="probability",  cex.lab=1.7,
          cex.main=2, las=1)
lines(ic.b$I.Bootstrap$up, F1.F, col=2, lty=2, lwd=2)
points(z$F.est, pch=20)
points(z$F.emp, col=4, pch=20, cex=1.5)
legend("topleft", c("Empirical", "Estimated","Bootstrap (95 percent)"),
              col=c("blue","black","red"),  lty=c(1,1,1), pch=c(20,20,20),
              cex=1.5, bty="n")
## Graph with confidence bands
matplot(ic.b$Mat.F.B, F1.F,  main="Bootstrap resampling lines",  
         type="l", lwd=2, xlim=c(0.05,0.18), xlab="million cycles", 
         ylab="probability", cex.lab=1.7,  cex.main=2, las=1)

## End(Not run)