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2025-01-12 00:52:51 +08:00
R Under development (unstable) (2019-12-19 r77606) -- "Unsuffered Consequences"
Copyright (C) 2019 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
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> pkgname <- "boot"
> source(file.path(R.home("share"), "R", "examples-header.R"))
> options(warn = 1)
> library('boot')
>
> base::assign(".oldSearch", base::search(), pos = 'CheckExEnv')
> base::assign(".old_wd", base::getwd(), pos = 'CheckExEnv')
> cleanEx()
> nameEx("Imp.Estimates")
> ### * Imp.Estimates
>
> flush(stderr()); flush(stdout())
>
> ### Name: Imp.Estimates
> ### Title: Importance Sampling Estimates
> ### Aliases: Imp.Estimates imp.moments imp.prob imp.quantile imp.reg
> ### Keywords: htest nonparametric
>
> ### ** Examples
>
> # Example 9.8 of Davison and Hinkley (1997) requires tilting the
> # resampling distribution of the studentized statistic to be centred
> # at the observed value of the test statistic, 1.84. In this example
> # we show how certain estimates can be found using resamples taken from
> # the tilted distribution.
> grav1 <- gravity[as.numeric(gravity[,2]) >= 7, ]
> grav.fun <- function(dat, w, orig) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2] - mns[1], s2hat, (mns[2] - mns[1] - orig)/sqrt(s2hat))
+ }
> grav.z0 <- grav.fun(grav1, rep(1, 26), 0)
> grav.L <- empinf(data = grav1, statistic = grav.fun, stype = "w",
+ strata = grav1[,2], index = 3, orig = grav.z0[1])
> grav.tilt <- exp.tilt(grav.L, grav.z0[3], strata = grav1[, 2])
> grav.tilt.boot <- boot(grav1, grav.fun, R = 199, stype = "w",
+ strata = grav1[, 2], weights = grav.tilt$p,
+ orig = grav.z0[1])
> # Since the weights are needed for all calculations, we shall calculate
> # them once only.
> grav.w <- imp.weights(grav.tilt.boot)
> grav.mom <- imp.moments(grav.tilt.boot, w = grav.w, index = 3)
> grav.p <- imp.prob(grav.tilt.boot, w = grav.w, index = 3, t0 = grav.z0[3])
> unlist(grav.p)
t0 raw rat reg
1.8401182 1.0000000 0.9778246 0.9767292
> grav.q <- imp.quantile(grav.tilt.boot, w = grav.w, index = 3,
+ alpha = c(0.9, 0.95, 0.975, 0.99))
> as.data.frame(grav.q)
alpha raw rat reg
1 0.900 3.048484 1.170707 1.210631
2 0.950 3.237935 1.565928 1.576607
3 0.975 3.629448 1.895876 1.923657
4 0.990 3.629448 2.258157 2.295230
>
>
>
> cleanEx()
> nameEx("abc.ci")
> ### * abc.ci
>
> flush(stderr()); flush(stdout())
>
> ### Name: abc.ci
> ### Title: Nonparametric ABC Confidence Intervals
> ### Aliases: abc.ci
> ### Keywords: nonparametric htest
>
> ### ** Examples
>
> ## Don't show:
> op <- options(digits = 5)
> ## End(Don't show)
> # 90% and 95% confidence intervals for the correlation
> # coefficient between the columns of the bigcity data
>
> abc.ci(bigcity, corr, conf=c(0.90,0.95))
conf
[1,] 0.90 0.95815 0.99173
[2,] 0.95 0.94937 0.99307
>
> # A 95% confidence interval for the difference between the means of
> # the last two samples in gravity
> mean.diff <- function(y, w)
+ { gp1 <- 1:table(as.numeric(y$series))[1]
+ sum(y[gp1, 1] * w[gp1]) - sum(y[-gp1, 1] * w[-gp1])
+ }
> grav1 <- gravity[as.numeric(gravity[, 2]) >= 7, ]
> ## IGNORE_RDIFF_BEGIN
> abc.ci(grav1, mean.diff, strata = grav1$series)
[1] 0.95000 -6.70758 -0.39394
> ## IGNORE_RDIFF_END
> ## Don't show:
> options(op)
> ## End(Don't show)
>
>
>
> cleanEx()
> nameEx("boot")
> ### * boot
>
> flush(stderr()); flush(stdout())
>
> ### Name: boot
> ### Title: Bootstrap Resampling
> ### Aliases: boot boot.return c.boot
> ### Keywords: nonparametric htest
>
> ### ** Examples
>
> ## Don't show:
> op <- options(digits = 5)
> ## End(Don't show)
> # Usual bootstrap of the ratio of means using the city data
> ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
> boot(city, ratio, R = 999, stype = "w")
ORDINARY NONPARAMETRIC BOOTSTRAP
Call:
boot(data = city, statistic = ratio, R = 999, stype = "w")
Bootstrap Statistics :
original bias std. error
t1* 1.5203 0.049024 0.21583
>
>
> # Stratified resampling for the difference of means. In this
> # example we will look at the difference of means between the final
> # two series in the gravity data.
> diff.means <- function(d, f)
+ { n <- nrow(d)
+ gp1 <- 1:table(as.numeric(d$series))[1]
+ m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
+ m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
+ ss1 <- sum(d[gp1,1]^2 * f[gp1]) - (m1 * m1 * sum(f[gp1]))
+ ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - (m2 * m2 * sum(f[-gp1]))
+ c(m1 - m2, (ss1 + ss2)/(sum(f) - 2))
+ }
> grav1 <- gravity[as.numeric(gravity[,2]) >= 7,]
> boot(grav1, diff.means, R = 999, stype = "f", strata = grav1[,2])
STRATIFIED BOOTSTRAP
Call:
boot(data = grav1, statistic = diff.means, R = 999, stype = "f",
strata = grav1[, 2])
Bootstrap Statistics :
original bias std. error
t1* -2.8462 0.007469 1.5528
t2* 16.8462 -1.475719 6.7147
>
> # In this example we show the use of boot in a prediction from
> # regression based on the nuclear data. This example is taken
> # from Example 6.8 of Davison and Hinkley (1997). Notice also
> # that two extra arguments to 'statistic' are passed through boot.
> nuke <- nuclear[, c(1, 2, 5, 7, 8, 10, 11)]
> nuke.lm <- glm(log(cost) ~ date+log(cap)+ne+ct+log(cum.n)+pt, data = nuke)
> nuke.diag <- glm.diag(nuke.lm)
> nuke.res <- nuke.diag$res * nuke.diag$sd
> nuke.res <- nuke.res - mean(nuke.res)
>
> # We set up a new data frame with the data, the standardized
> # residuals and the fitted values for use in the bootstrap.
> nuke.data <- data.frame(nuke, resid = nuke.res, fit = fitted(nuke.lm))
>
> # Now we want a prediction of plant number 32 but at date 73.00
> new.data <- data.frame(cost = 1, date = 73.00, cap = 886, ne = 0,
+ ct = 0, cum.n = 11, pt = 1)
> new.fit <- predict(nuke.lm, new.data)
>
> nuke.fun <- function(dat, inds, i.pred, fit.pred, x.pred)
+ {
+ lm.b <- glm(fit+resid[inds] ~ date+log(cap)+ne+ct+log(cum.n)+pt,
+ data = dat)
+ pred.b <- predict(lm.b, x.pred)
+ c(coef(lm.b), pred.b - (fit.pred + dat$resid[i.pred]))
+ }
>
> nuke.boot <- boot(nuke.data, nuke.fun, R = 999, m = 1,
+ fit.pred = new.fit, x.pred = new.data)
> # The bootstrap prediction squared error would then be found by
> mean(nuke.boot$t[, 8]^2)
[1] 0.084223
> # Basic bootstrap prediction limits would be
> new.fit - sort(nuke.boot$t[, 8])[c(975, 25)]
[1] 6.1488 7.2619
>
>
> # Finally a parametric bootstrap. For this example we shall look
> # at the air-conditioning data. In this example our aim is to test
> # the hypothesis that the true value of the index is 1 (i.e. that
> # the data come from an exponential distribution) against the
> # alternative that the data come from a gamma distribution with
> # index not equal to 1.
> air.fun <- function(data) {
+ ybar <- mean(data$hours)
+ para <- c(log(ybar), mean(log(data$hours)))
+ ll <- function(k) {
+ if (k <= 0) 1e200 else lgamma(k)-k*(log(k)-1-para[1]+para[2])
+ }
+ khat <- nlm(ll, ybar^2/var(data$hours))$estimate
+ c(ybar, khat)
+ }
>
> air.rg <- function(data, mle) {
+ # Function to generate random exponential variates.
+ # mle will contain the mean of the original data
+ out <- data
+ out$hours <- rexp(nrow(out), 1/mle)
+ out
+ }
>
> air.boot <- boot(aircondit, air.fun, R = 999, sim = "parametric",
+ ran.gen = air.rg, mle = mean(aircondit$hours))
>
> # The bootstrap p-value can then be approximated by
> sum(abs(air.boot$t[,2]-1) > abs(air.boot$t0[2]-1))/(1+air.boot$R)
[1] 0.446
> ## Don't show:
> options(op)
> ## End(Don't show)
>
>
>
> cleanEx()
> nameEx("boot.array")
> ### * boot.array
>
> flush(stderr()); flush(stdout())
>
> ### Name: boot.array
> ### Title: Bootstrap Resampling Arrays
> ### Aliases: boot.array
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # A frequency array for a nonparametric bootstrap
> city.boot <- boot(city, corr, R = 40, stype = "w")
> boot.array(city.boot)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1 1 0 0 0 2 1 1 3 1
[2,] 2 2 0 2 0 0 1 0 1 2
[3,] 0 1 1 1 2 1 2 1 0 1
[4,] 1 0 1 0 0 1 2 1 1 3
[5,] 0 3 0 1 0 0 2 0 2 2
[6,] 0 0 1 1 0 1 4 1 2 0
[7,] 1 3 1 2 0 0 0 1 2 0
[8,] 0 2 2 2 0 1 0 1 0 2
[9,] 3 0 1 0 2 2 1 1 0 0
[10,] 1 1 0 1 3 0 0 1 2 1
[11,] 0 1 2 1 3 1 0 0 0 2
[12,] 1 1 3 1 0 0 0 2 1 1
[13,] 0 1 0 1 1 3 1 1 1 1
[14,] 2 0 0 1 1 1 0 1 3 1
[15,] 1 0 0 0 0 0 5 0 4 0
[16,] 0 0 1 1 2 1 0 1 4 0
[17,] 1 0 1 0 5 0 0 1 0 2
[18,] 1 0 1 0 3 2 1 1 0 1
[19,] 1 2 2 2 0 1 0 1 1 0
[20,] 1 1 1 1 2 0 1 0 1 2
[21,] 2 0 2 1 1 1 0 1 0 2
[22,] 2 0 1 1 1 0 1 2 1 1
[23,] 1 2 1 1 0 1 1 1 0 2
[24,] 3 0 1 0 1 0 1 1 0 3
[25,] 1 2 1 2 1 1 0 1 1 0
[26,] 2 0 2 0 1 3 0 1 0 1
[27,] 2 1 0 2 1 1 0 1 2 0
[28,] 1 1 2 0 1 0 1 0 3 1
[29,] 1 2 1 1 1 3 1 0 0 0
[30,] 2 0 0 2 1 0 0 2 1 2
[31,] 0 1 1 1 2 0 3 1 0 1
[32,] 0 0 1 1 1 4 0 0 0 3
[33,] 1 0 3 2 0 0 0 2 1 1
[34,] 0 1 0 2 4 1 1 1 0 0
[35,] 0 1 1 0 2 1 1 0 3 1
[36,] 3 0 1 0 2 0 1 1 2 0
[37,] 2 1 1 0 1 0 2 2 1 0
[38,] 1 0 2 0 1 1 3 1 1 0
[39,] 1 1 0 2 0 1 1 1 2 1
[40,] 0 1 2 3 1 1 0 2 0 0
>
> perm.cor <- function(d,i) cor(d$x,d$u[i])
> city.perm <- boot(city, perm.cor, R = 40, sim = "permutation")
> boot.array(city.perm, indices = TRUE)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 7 8 2 6 9 5 4 3 10 1
[2,] 1 9 8 6 5 3 4 7 2 10
[3,] 10 1 6 9 4 5 2 8 3 7
[4,] 4 5 7 6 3 10 9 8 2 1
[5,] 10 3 6 7 5 2 8 9 1 4
[6,] 4 10 8 2 5 6 7 3 9 1
[7,] 5 8 10 3 2 4 7 1 6 9
[8,] 3 8 4 9 2 7 1 5 10 6
[9,] 6 5 1 3 7 2 4 10 9 8
[10,] 4 9 6 10 8 1 7 3 5 2
[11,] 5 10 1 8 6 2 3 4 9 7
[12,] 7 6 1 3 10 2 8 4 5 9
[13,] 10 8 3 6 9 7 1 5 2 4
[14,] 6 7 10 1 5 4 8 2 9 3
[15,] 9 1 5 3 2 8 6 10 7 4
[16,] 9 5 7 6 4 10 1 3 8 2
[17,] 1 8 3 10 4 5 9 7 6 2
[18,] 1 10 5 8 2 4 7 9 6 3
[19,] 8 2 7 4 9 3 10 6 5 1
[20,] 10 4 8 6 5 9 1 3 2 7
[21,] 2 1 10 4 5 7 8 3 6 9
[22,] 3 1 10 5 2 4 8 6 9 7
[23,] 8 4 1 10 9 6 2 7 5 3
[24,] 2 3 7 8 1 6 4 5 10 9
[25,] 4 8 9 5 10 6 2 7 3 1
[26,] 1 9 2 10 6 5 4 8 7 3
[27,] 9 10 7 1 3 8 6 2 4 5
[28,] 6 7 2 3 4 1 8 9 10 5
[29,] 5 2 1 9 4 8 7 6 10 3
[30,] 7 4 6 2 5 1 9 10 3 8
[31,] 8 5 1 6 7 9 4 3 2 10
[32,] 3 10 4 5 7 6 1 9 8 2
[33,] 7 4 9 10 3 5 6 1 2 8
[34,] 10 6 2 8 7 3 4 9 1 5
[35,] 7 5 6 1 10 9 2 4 3 8
[36,] 9 1 8 6 10 5 2 7 3 4
[37,] 5 6 8 2 4 10 3 9 7 1
[38,] 5 10 4 3 1 2 8 9 7 6
[39,] 6 9 7 3 5 1 10 2 8 4
[40,] 9 3 8 2 6 7 5 1 10 4
>
>
>
> cleanEx()
> nameEx("boot.ci")
> ### * boot.ci
>
> flush(stderr()); flush(stdout())
>
> ### Name: boot.ci
> ### Title: Nonparametric Bootstrap Confidence Intervals
> ### Aliases: boot.ci
> ### Keywords: nonparametric htest
>
> ### ** Examples
>
> # confidence intervals for the city data
> ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
> city.boot <- boot(city, ratio, R = 999, stype = "w", sim = "ordinary")
> boot.ci(city.boot, conf = c(0.90, 0.95),
+ type = c("norm", "basic", "perc", "bca"))
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL :
boot.ci(boot.out = city.boot, conf = c(0.9, 0.95), type = c("norm",
"basic", "perc", "bca"))
Intervals :
Level Normal Basic
90% ( 1.116, 1.826 ) ( 1.054, 1.742 )
95% ( 1.048, 1.894 ) ( 0.956, 1.775 )
Level Percentile BCa
90% ( 1.299, 1.987 ) ( 1.292, 1.957 )
95% ( 1.266, 2.085 ) ( 1.254, 2.058 )
Calculations and Intervals on Original Scale
>
> # studentized confidence interval for the two sample
> # difference of means problem using the final two series
> # of the gravity data.
> diff.means <- function(d, f)
+ { n <- nrow(d)
+ gp1 <- 1:table(as.numeric(d$series))[1]
+ m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
+ m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
+ ss1 <- sum(d[gp1,1]^2 * f[gp1]) - (m1 * m1 * sum(f[gp1]))
+ ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - (m2 * m2 * sum(f[-gp1]))
+ c(m1 - m2, (ss1 + ss2)/(sum(f) - 2))
+ }
> grav1 <- gravity[as.numeric(gravity[,2]) >= 7, ]
> grav1.boot <- boot(grav1, diff.means, R = 999, stype = "f",
+ strata = grav1[ ,2])
> boot.ci(grav1.boot, type = c("stud", "norm"))
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL :
boot.ci(boot.out = grav1.boot, type = c("stud", "norm"))
Intervals :
Level Normal Studentized
95% (-5.897, 0.190 ) (-7.370, -0.052 )
Calculations and Intervals on Original Scale
>
> # Nonparametric confidence intervals for mean failure time
> # of the air-conditioning data as in Example 5.4 of Davison
> # and Hinkley (1997)
> mean.fun <- function(d, i)
+ { m <- mean(d$hours[i])
+ n <- length(i)
+ v <- (n-1)*var(d$hours[i])/n^2
+ c(m, v)
+ }
> air.boot <- boot(aircondit, mean.fun, R = 999)
> boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"))
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL :
boot.ci(boot.out = air.boot, type = c("norm", "basic", "perc",
"stud"))
Intervals :
Level Normal Basic
95% ( 32.2, 184.5 ) ( 21.2, 169.4 )
Level Studentized Percentile
95% ( 45.6, 295.8 ) ( 46.8, 195.0 )
Calculations and Intervals on Original Scale
>
> # Now using the log transformation
> # There are two ways of doing this and they both give the
> # same intervals.
>
> # Method 1
> boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"),
+ h = log, hdot = function(x) 1/x)
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL :
boot.ci(boot.out = air.boot, type = c("norm", "basic", "perc",
"stud"), h = log, hdot = function(x) 1/x)
Intervals :
Level Normal Basic
95% ( 4.015, 5.491 ) ( 4.093, 5.521 )
Level Studentized Percentile
95% ( 3.922, 5.808 ) ( 3.845, 5.273 )
Calculations and Intervals on Transformed Scale
>
> # Method 2
> vt0 <- air.boot$t0[2]/air.boot$t0[1]^2
> vt <- air.boot$t[, 2]/air.boot$t[ ,1]^2
> boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"),
+ t0 = log(air.boot$t0[1]), t = log(air.boot$t[,1]),
+ var.t0 = vt0, var.t = vt)
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL :
boot.ci(boot.out = air.boot, type = c("norm", "basic", "perc",
"stud"), var.t0 = vt0, var.t = vt, t0 = log(air.boot$t0[1]),
t = log(air.boot$t[, 1]))
Intervals :
Level Normal Basic
95% ( 4.070, 5.435 ) ( 4.093, 5.521 )
Level Studentized Percentile
95% ( 3.922, 5.808 ) ( 3.845, 5.273 )
Calculations and Intervals on Original Scale
>
>
>
> cleanEx()
> nameEx("censboot")
> ### * censboot
>
> flush(stderr()); flush(stdout())
>
> ### Name: censboot
> ### Title: Bootstrap for Censored Data
> ### Aliases: censboot cens.return
> ### Keywords: survival
>
> ### ** Examples
>
> library(survival)
Attaching package: ‘survival’
The following object is masked from ‘package:boot’:
aml
> # Example 3.9 of Davison and Hinkley (1997) does a bootstrap on some
> # remission times for patients with a type of leukaemia. The patients
> # were divided into those who received maintenance chemotherapy and
> # those who did not. Here we are interested in the median remission
> # time for the two groups.
> data(aml, package = "boot") # not the version in survival.
> aml.fun <- function(data) {
+ surv <- survfit(Surv(time, cens) ~ group, data = data)
+ out <- NULL
+ st <- 1
+ for (s in 1:length(surv$strata)) {
+ inds <- st:(st + surv$strata[s]-1)
+ md <- min(surv$time[inds[1-surv$surv[inds] >= 0.5]])
+ st <- st + surv$strata[s]
+ out <- c(out, md)
+ }
+ out
+ }
> aml.case <- censboot(aml, aml.fun, R = 499, strata = aml$group)
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
>
> # Now we will look at the same statistic using the conditional
> # bootstrap and the weird bootstrap. For the conditional bootstrap
> # the survival distribution is stratified but the censoring
> # distribution is not.
>
> aml.s1 <- survfit(Surv(time, cens) ~ group, data = aml)
> aml.s2 <- survfit(Surv(time-0.001*cens, 1-cens) ~ 1, data = aml)
> aml.cond <- censboot(aml, aml.fun, R = 499, strata = aml$group,
+ F.surv = aml.s1, G.surv = aml.s2, sim = "cond")
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
>
>
> # For the weird bootstrap we must redefine our function slightly since
> # the data will not contain the group number.
> aml.fun1 <- function(data, str) {
+ surv <- survfit(Surv(data[, 1], data[, 2]) ~ str)
+ out <- NULL
+ st <- 1
+ for (s in 1:length(surv$strata)) {
+ inds <- st:(st + surv$strata[s] - 1)
+ md <- min(surv$time[inds[1-surv$surv[inds] >= 0.5]])
+ st <- st + surv$strata[s]
+ out <- c(out, md)
+ }
+ out
+ }
> aml.wei <- censboot(cbind(aml$time, aml$cens), aml.fun1, R = 499,
+ strata = aml$group, F.surv = aml.s1, sim = "weird")
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
Warning in min(surv$time[inds[1 - surv$surv[inds] >= 0.5]]) :
no non-missing arguments to min; returning Inf
>
> # Now for an example where a cox regression model has been fitted
> # the data we will look at the melanoma data of Example 7.6 from
> # Davison and Hinkley (1997). The fitted model assumes that there
> # is a different survival distribution for the ulcerated and
> # non-ulcerated groups but that the thickness of the tumour has a
> # common effect. We will also assume that the censoring distribution
> # is different in different age groups. The statistic of interest
> # is the linear predictor. This is returned as the values at a
> # number of equally spaced points in the range of interest.
> data(melanoma, package = "boot")
> library(splines)# for ns
> mel.cox <- coxph(Surv(time, status == 1) ~ ns(thickness, df=4) + strata(ulcer),
+ data = melanoma)
> mel.surv <- survfit(mel.cox)
> agec <- cut(melanoma$age, c(0, 39, 49, 59, 69, 100))
> mel.cens <- survfit(Surv(time - 0.001*(status == 1), status != 1) ~
+ strata(agec), data = melanoma)
> mel.fun <- function(d) {
+ t1 <- ns(d$thickness, df=4)
+ cox <- coxph(Surv(d$time, d$status == 1) ~ t1+strata(d$ulcer))
+ ind <- !duplicated(d$thickness)
+ u <- d$thickness[!ind]
+ eta <- cox$linear.predictors[!ind]
+ sp <- smooth.spline(u, eta, df=20)
+ th <- seq(from = 0.25, to = 10, by = 0.25)
+ predict(sp, th)$y
+ }
> mel.str <- cbind(melanoma$ulcer, agec)
>
> # this is slow!
> mel.mod <- censboot(melanoma, mel.fun, R = 499, F.surv = mel.surv,
+ G.surv = mel.cens, cox = mel.cox, strata = mel.str, sim = "model")
> # To plot the original predictor and a 95% pointwise envelope for it
> mel.env <- envelope(mel.mod)$point
> th <- seq(0.25, 10, by = 0.25)
> plot(th, mel.env[1, ], ylim = c(-2, 2),
+ xlab = "thickness (mm)", ylab = "linear predictor", type = "n")
> lines(th, mel.mod$t0, lty = 1)
> matlines(th, t(mel.env), lty = 2)
>
>
>
> cleanEx()
detaching ‘package:splines’, ‘package:survival’
> nameEx("control")
> ### * control
>
> flush(stderr()); flush(stdout())
>
> ### Name: control
> ### Title: Control Variate Calculations
> ### Aliases: control
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # Use of control variates for the variance of the air-conditioning data
> mean.fun <- function(d, i)
+ { m <- mean(d$hours[i])
+ n <- nrow(d)
+ v <- (n-1)*var(d$hours[i])/n^2
+ c(m, v)
+ }
> air.boot <- boot(aircondit, mean.fun, R = 999)
> control(air.boot, index = 2, bias.adj = TRUE)
[1] -11.25369
> air.cont <- control(air.boot, index = 2)
> # Now let us try the variance on the log scale.
> air.cont1 <- control(air.boot, t0 = log(air.boot$t0[2]),
+ t = log(air.boot$t[, 2]))
>
>
>
> cleanEx()
> nameEx("cv.glm")
> ### * cv.glm
>
> flush(stderr()); flush(stdout())
>
> ### Name: cv.glm
> ### Title: Cross-validation for Generalized Linear Models
> ### Aliases: cv.glm
> ### Keywords: regression
>
> ### ** Examples
>
> # leave-one-out and 6-fold cross-validation prediction error for
> # the mammals data set.
> data(mammals, package="MASS")
> mammals.glm <- glm(log(brain) ~ log(body), data = mammals)
> (cv.err <- cv.glm(mammals, mammals.glm)$delta)
[1] 0.4918650 0.4916571
> (cv.err.6 <- cv.glm(mammals, mammals.glm, K = 6)$delta)
[1] 0.4911574 0.4887828
>
> # As this is a linear model we could calculate the leave-one-out
> # cross-validation estimate without any extra model-fitting.
> muhat <- fitted(mammals.glm)
> mammals.diag <- glm.diag(mammals.glm)
> (cv.err <- mean((mammals.glm$y - muhat)^2/(1 - mammals.diag$h)^2))
[1] 0.491865
>
>
> # leave-one-out and 11-fold cross-validation prediction error for
> # the nodal data set. Since the response is a binary variable an
> # appropriate cost function is
> cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
>
> nodal.glm <- glm(r ~ stage+xray+acid, binomial, data = nodal)
> (cv.err <- cv.glm(nodal, nodal.glm, cost, K = nrow(nodal))$delta)
[1] 0.1886792 0.1886792
> (cv.11.err <- cv.glm(nodal, nodal.glm, cost, K = 11)$delta)
[1] 0.2075472 0.2057672
>
>
>
> cleanEx()
> nameEx("empinf")
> ### * empinf
>
> flush(stderr()); flush(stdout())
>
> ### Name: empinf
> ### Title: Empirical Influence Values
> ### Aliases: empinf
> ### Keywords: nonparametric math
>
> ### ** Examples
>
> # The empirical influence values for the ratio of means in
> # the city data.
> ratio <- function(d, w) sum(d$x *w)/sum(d$u*w)
> empinf(data = city, statistic = ratio)
[1] -1.04367815 -0.58417763 -0.37092459 -0.18958996 0.03164142 0.10544878
[7] 0.09236345 0.20365074 1.02178280 0.73381132
> city.boot <- boot(city, ratio, 499, stype="w")
> empinf(boot.out = city.boot, type = "reg")
1 1 1 1 1 1
-1.188052737 -0.702204565 -0.447209091 -0.378938644 0.007498044 0.162261064
1 1 1 1
0.116272627 0.242543405 1.166357773 1.021472124
>
> # A statistic that may be of interest in the difference of means
> # problem is the t-statistic for testing equality of means. In
> # the bootstrap we get replicates of the difference of means and
> # the variance of that statistic and then want to use this output
> # to get the empirical influence values of the t-statistic.
> grav1 <- gravity[as.numeric(gravity[,2]) >= 7,]
> grav.fun <- function(dat, w) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2] - mns[1], s2hat)
+ }
>
> grav.boot <- boot(grav1, grav.fun, R = 499, stype = "w",
+ strata = grav1[, 2])
>
> # Since the statistic of interest is a function of the bootstrap
> # statistics, we must calculate the bootstrap replicates and pass
> # them to empinf using the t argument.
> grav.z <- (grav.boot$t[,1]-grav.boot$t0[1])/sqrt(grav.boot$t[,2])
> empinf(boot.out = grav.boot, t = grav.z)
1 1 1 1 1 1 1
-2.7187022 -0.9515832 -2.4507504 -1.3409041 0.6167772 -1.2768844 -1.3438236
1 1 1 1 1 1 2
-0.2313133 -1.0504975 -3.1949223 1.2809344 3.7306567 8.9310126 2.7195701
2 2 2 2 2 2 2
4.1111581 3.1632367 1.1507436 -2.2930450 -2.7667452 -2.3645554 -0.2002981
2 2 2 2 2
2.0526618 0.3615873 -1.9421623 -2.1031282 -1.8890234
>
>
>
> cleanEx()
> nameEx("envelope")
> ### * envelope
>
> flush(stderr()); flush(stdout())
>
> ### Name: envelope
> ### Title: Confidence Envelopes for Curves
> ### Aliases: envelope
> ### Keywords: dplot htest
>
> ### ** Examples
>
> # Testing whether the final series of measurements of the gravity data
> # may come from a normal distribution. This is done in Examples 4.7
> # and 4.8 of Davison and Hinkley (1997).
> grav1 <- gravity$g[gravity$series == 8]
> grav.z <- (grav1 - mean(grav1))/sqrt(var(grav1))
> grav.gen <- function(dat, mle) rnorm(length(dat))
> grav.qqboot <- boot(grav.z, sort, R = 999, sim = "parametric",
+ ran.gen = grav.gen)
> grav.qq <- qqnorm(grav.z, plot.it = FALSE)
> grav.qq <- lapply(grav.qq, sort)
> plot(grav.qq, ylim = c(-3.5, 3.5), ylab = "Studentized Order Statistics",
+ xlab = "Normal Quantiles")
> grav.env <- envelope(grav.qqboot, level = 0.9)
> lines(grav.qq$x, grav.env$point[1, ], lty = 4)
> lines(grav.qq$x, grav.env$point[2, ], lty = 4)
> lines(grav.qq$x, grav.env$overall[1, ], lty = 1)
> lines(grav.qq$x, grav.env$overall[2, ], lty = 1)
>
>
>
> cleanEx()
> nameEx("exp.tilt")
> ### * exp.tilt
>
> flush(stderr()); flush(stdout())
>
> ### Name: exp.tilt
> ### Title: Exponential Tilting
> ### Aliases: exp.tilt
> ### Keywords: nonparametric smooth
>
> ### ** Examples
>
> # Example 9.8 of Davison and Hinkley (1997) requires tilting the resampling
> # distribution of the studentized statistic to be centred at the observed
> # value of the test statistic 1.84. This can be achieved as follows.
> grav1 <- gravity[as.numeric(gravity[,2]) >=7 , ]
> grav.fun <- function(dat, w, orig) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2]-mns[1], s2hat, (mns[2]-mns[1]-orig)/sqrt(s2hat))
+ }
> grav.z0 <- grav.fun(grav1, rep(1, 26), 0)
> grav.L <- empinf(data = grav1, statistic = grav.fun, stype = "w",
+ strata = grav1[,2], index = 3, orig = grav.z0[1])
> grav.tilt <- exp.tilt(grav.L, grav.z0[3], strata = grav1[,2])
> boot(grav1, grav.fun, R = 499, stype = "w", weights = grav.tilt$p,
+ strata = grav1[,2], orig = grav.z0[1])
STRATIFIED WEIGHTED BOOTSTRAP
Call:
boot(data = grav1, statistic = grav.fun, R = 499, stype = "w",
strata = grav1[, 2], weights = grav.tilt$p, orig = grav.z0[1])
Bootstrap Statistics :
original bias std. error mean(t*)
t1* 2.846154 -0.3661063 1.705171 5.702944
t2* 2.392353 -0.3538294 1.002889 3.444050
t3* 0.000000 -0.5160619 1.314298 1.473456
>
>
>
> cleanEx()
> nameEx("glm.diag.plots")
> ### * glm.diag.plots
>
> flush(stderr()); flush(stdout())
>
> ### Name: glm.diag.plots
> ### Title: Diagnostics plots for generalized linear models
> ### Aliases: glm.diag.plots
> ### Keywords: regression dplot hplot
>
> ### ** Examples
>
> # In this example we look at the leukaemia data which was looked at in
> # Example 7.1 of Davison and Hinkley (1997)
> data(leuk, package = "MASS")
> leuk.mod <- glm(time ~ ag-1+log10(wbc), family = Gamma(log), data = leuk)
> leuk.diag <- glm.diag(leuk.mod)
> glm.diag.plots(leuk.mod, leuk.diag)
>
>
>
> cleanEx()
> nameEx("jack.after.boot")
> ### * jack.after.boot
>
> flush(stderr()); flush(stdout())
>
> ### Name: jack.after.boot
> ### Title: Jackknife-after-Bootstrap Plots
> ### Aliases: jack.after.boot
> ### Keywords: hplot nonparametric
>
> ### ** Examples
>
> # To draw the jackknife-after-bootstrap plot for the head size data as in
> # Example 3.24 of Davison and Hinkley (1997)
> frets.fun <- function(data, i) {
+ pcorr <- function(x) {
+ # Function to find the correlations and partial correlations between
+ # the four measurements.
+ v <- cor(x)
+ v.d <- diag(var(x))
+ iv <- solve(v)
+ iv.d <- sqrt(diag(iv))
+ iv <- - diag(1/iv.d) %*% iv %*% diag(1/iv.d)
+ q <- NULL
+ n <- nrow(v)
+ for (i in 1:(n-1))
+ q <- rbind( q, c(v[i, 1:i], iv[i,(i+1):n]) )
+ q <- rbind( q, v[n, ] )
+ diag(q) <- round(diag(q))
+ q
+ }
+ d <- data[i, ]
+ v <- pcorr(d)
+ c(v[1,], v[2,], v[3,], v[4,])
+ }
> frets.boot <- boot(log(as.matrix(frets)), frets.fun, R = 999)
> # we will concentrate on the partial correlation between head breadth
> # for the first son and head length for the second. This is the 7th
> # element in the output of frets.fun so we set index = 7
> jack.after.boot(frets.boot, useJ = FALSE, stinf = FALSE, index = 7)
>
>
>
> cleanEx()
> nameEx("k3.linear")
> ### * k3.linear
>
> flush(stderr()); flush(stdout())
>
> ### Name: k3.linear
> ### Title: Linear Skewness Estimate
> ### Aliases: k3.linear
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # To estimate the skewness of the ratio of means for the city data.
> ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
> k3.linear(empinf(data = city, statistic = ratio))
[1] 7.831452e-05
>
>
>
> cleanEx()
> nameEx("linear.approx")
> ### * linear.approx
>
> flush(stderr()); flush(stdout())
>
> ### Name: linear.approx
> ### Title: Linear Approximation of Bootstrap Replicates
> ### Aliases: linear.approx
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # Using the city data let us look at the linear approximation to the
> # ratio statistic and its logarithm. We compare these with the
> # corresponding plots for the bigcity data
>
> ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
> city.boot <- boot(city, ratio, R = 499, stype = "w")
> bigcity.boot <- boot(bigcity, ratio, R = 499, stype = "w")
> op <- par(pty = "s", mfrow = c(2, 2))
>
> # The first plot is for the city data ratio statistic.
> city.lin1 <- linear.approx(city.boot)
> lim <- range(c(city.boot$t,city.lin1))
> plot(city.boot$t, city.lin1, xlim = lim, ylim = lim,
+ main = "Ratio; n=10", xlab = "t*", ylab = "tL*")
> abline(0, 1)
>
> # Now for the log of the ratio statistic for the city data.
> city.lin2 <- linear.approx(city.boot,t0 = log(city.boot$t0),
+ t = log(city.boot$t))
> lim <- range(c(log(city.boot$t),city.lin2))
> plot(log(city.boot$t), city.lin2, xlim = lim, ylim = lim,
+ main = "Log(Ratio); n=10", xlab = "t*", ylab = "tL*")
> abline(0, 1)
>
> # The ratio statistic for the bigcity data.
> bigcity.lin1 <- linear.approx(bigcity.boot)
> lim <- range(c(bigcity.boot$t,bigcity.lin1))
> plot(bigcity.lin1, bigcity.boot$t, xlim = lim, ylim = lim,
+ main = "Ratio; n=49", xlab = "t*", ylab = "tL*")
> abline(0, 1)
>
> # Finally the log of the ratio statistic for the bigcity data.
> bigcity.lin2 <- linear.approx(bigcity.boot,t0 = log(bigcity.boot$t0),
+ t = log(bigcity.boot$t))
> lim <- range(c(log(bigcity.boot$t),bigcity.lin2))
> plot(bigcity.lin2, log(bigcity.boot$t), xlim = lim, ylim = lim,
+ main = "Log(Ratio); n=49", xlab = "t*", ylab = "tL*")
> abline(0, 1)
>
> par(op)
>
>
>
> graphics::par(get("par.postscript", pos = 'CheckExEnv'))
> cleanEx()
> nameEx("lines.saddle.distn")
> ### * lines.saddle.distn
>
> flush(stderr()); flush(stdout())
>
> ### Name: lines.saddle.distn
> ### Title: Add a Saddlepoint Approximation to a Plot
> ### Aliases: lines.saddle.distn
> ### Keywords: aplot smooth nonparametric
>
> ### ** Examples
>
> # In this example we show how a plot such as that in Figure 9.9 of
> # Davison and Hinkley (1997) may be produced. Note the large number of
> # bootstrap replicates required in this example.
> expdata <- rexp(12)
> vfun <- function(d, i) {
+ n <- length(d)
+ (n-1)/n*var(d[i])
+ }
> exp.boot <- boot(expdata,vfun, R = 9999)
> exp.L <- (expdata - mean(expdata))^2 - exp.boot$t0
> exp.tL <- linear.approx(exp.boot, L = exp.L)
> hist(exp.tL, nclass = 50, probability = TRUE)
> exp.t0 <- c(0, sqrt(var(exp.boot$t)))
> exp.sp <- saddle.distn(A = exp.L/12,wdist = "m", t0 = exp.t0)
>
> # The saddlepoint approximation in this case is to the density of
> # t-t0 and so t0 must be added for the plot.
> lines(exp.sp, h = function(u, t0) u+t0, J = function(u, t0) 1,
+ t0 = exp.boot$t0)
>
>
>
> cleanEx()
> nameEx("norm.ci")
> ### * norm.ci
>
> flush(stderr()); flush(stdout())
>
> ### Name: norm.ci
> ### Title: Normal Approximation Confidence Intervals
> ### Aliases: norm.ci
> ### Keywords: htest
>
> ### ** Examples
>
> # In Example 5.1 of Davison and Hinkley (1997), normal approximation
> # confidence intervals are found for the air-conditioning data.
> air.mean <- mean(aircondit$hours)
> air.n <- nrow(aircondit)
> air.v <- air.mean^2/air.n
> norm.ci(t0 = air.mean, var.t0 = air.v)
conf
[1,] 0.95 46.93055 169.2361
> exp(norm.ci(t0 = log(air.mean), var.t0 = 1/air.n)[2:3])
[1] 61.38157 190.31782
>
> # Now a more complicated example - the ratio estimate for the city data.
> ratio <- function(d, w)
+ sum(d$x * w)/sum(d$u *w)
> city.v <- var.linear(empinf(data = city, statistic = ratio))
> norm.ci(t0 = ratio(city,rep(0.1,10)), var.t0 = city.v)
conf
[1,] 0.95 1.167046 1.873579
>
>
>
> cleanEx()
> nameEx("plot.boot")
> ### * plot.boot
>
> flush(stderr()); flush(stdout())
>
> ### Name: plot.boot
> ### Title: Plots of the Output of a Bootstrap Simulation
> ### Aliases: plot.boot
> ### Keywords: hplot nonparametric
>
> ### ** Examples
>
> # We fit an exponential model to the air-conditioning data and use
> # that for a parametric bootstrap. Then we look at plots of the
> # resampled means.
> air.rg <- function(data, mle) rexp(length(data), 1/mle)
>
> air.boot <- boot(aircondit$hours, mean, R = 999, sim = "parametric",
+ ran.gen = air.rg, mle = mean(aircondit$hours))
> plot(air.boot)
>
> # In the difference of means example for the last two series of the
> # gravity data
> grav1 <- gravity[as.numeric(gravity[, 2]) >= 7, ]
> grav.fun <- function(dat, w) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2] - mns[1], s2hat)
+ }
>
> grav.boot <- boot(grav1, grav.fun, R = 499, stype = "w", strata = grav1[, 2])
> plot(grav.boot)
> # now suppose we want to look at the studentized differences.
> grav.z <- (grav.boot$t[, 1]-grav.boot$t0[1])/sqrt(grav.boot$t[, 2])
> plot(grav.boot, t = grav.z, t0 = 0)
>
> # In this example we look at the one of the partial correlations for the
> # head dimensions in the dataset frets.
> frets.fun <- function(data, i) {
+ pcorr <- function(x) {
+ # Function to find the correlations and partial correlations between
+ # the four measurements.
+ v <- cor(x)
+ v.d <- diag(var(x))
+ iv <- solve(v)
+ iv.d <- sqrt(diag(iv))
+ iv <- - diag(1/iv.d) %*% iv %*% diag(1/iv.d)
+ q <- NULL
+ n <- nrow(v)
+ for (i in 1:(n-1))
+ q <- rbind( q, c(v[i, 1:i], iv[i,(i+1):n]) )
+ q <- rbind( q, v[n, ] )
+ diag(q) <- round(diag(q))
+ q
+ }
+ d <- data[i, ]
+ v <- pcorr(d)
+ c(v[1,], v[2,], v[3,], v[4,])
+ }
> frets.boot <- boot(log(as.matrix(frets)), frets.fun, R = 999)
> plot(frets.boot, index = 7, jack = TRUE, stinf = FALSE, useJ = FALSE)
>
>
>
> cleanEx()
> nameEx("saddle")
> ### * saddle
>
> flush(stderr()); flush(stdout())
>
> ### Name: saddle
> ### Title: Saddlepoint Approximations for Bootstrap Statistics
> ### Aliases: saddle
> ### Keywords: smooth nonparametric
>
> ### ** Examples
>
> # To evaluate the bootstrap distribution of the mean failure time of
> # air-conditioning equipment at 80 hours
> saddle(A = aircondit$hours/12, u = 80)
$spa
pdf cdf
0.01005866 0.24446677
$zeta.hat
[1] -0.02580078
>
> # Alternatively this can be done using a conditional poisson
> saddle(A = cbind(aircondit$hours/12,1), u = c(80, 12),
+ wdist = "p", type = "cond")
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.090909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.909091
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.090909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.909091
$spa
pdf cdf
0.01005943 0.24438736
$zeta.hat
A1 A2
-0.02580805 0.89261577
$zeta2.hat
[1] 0.6931472
>
> # To use the Lugananni-Rice approximation to this
> saddle(A = cbind(aircondit$hours/12,1), u = c(80, 12),
+ wdist = "p", type = "cond",
+ LR = TRUE)
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.090909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.909091
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.090909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.909091
$spa
pdf cdf
0.01005943 0.24447362
$zeta.hat
A1 A2
-0.02580805 0.89261577
$zeta2.hat
[1] 0.6931472
>
> # Example 9.16 of Davison and Hinkley (1997) calculates saddlepoint
> # approximations to the distribution of the ratio statistic for the
> # city data. Since the statistic is not in itself a linear combination
> # of random Variables, its distribution cannot be found directly.
> # Instead the statistic is expressed as the solution to a linear
> # estimating equation and hence its distribution can be found. We
> # get the saddlepoint approximation to the pdf and cdf evaluated at
> # t = 1.25 as follows.
> jacobian <- function(dat,t,zeta)
+ {
+ p <- exp(zeta*(dat$x-t*dat$u))
+ abs(sum(dat$u*p)/sum(p))
+ }
> city.sp1 <- saddle(A = city$x-1.25*city$u, u = 0)
> city.sp1$spa[1] <- jacobian(city, 1.25, city.sp1$zeta.hat) * city.sp1$spa[1]
> city.sp1
$spa
pdf cdf
0.05565040 0.02436306
$zeta.hat
[1] -0.02435547
>
>
>
> cleanEx()
> nameEx("saddle.distn")
> ### * saddle.distn
>
> flush(stderr()); flush(stdout())
>
> ### Name: saddle.distn
> ### Title: Saddlepoint Distribution Approximations for Bootstrap Statistics
> ### Aliases: saddle.distn
> ### Keywords: nonparametric smooth dplot
>
> ### ** Examples
>
> # The bootstrap distribution of the mean of the air-conditioning
> # failure data: fails to find value on R (and probably on S too)
> air.t0 <- c(mean(aircondit$hours), sqrt(var(aircondit$hours)/12))
> ## Not run: saddle.distn(A = aircondit$hours/12, t0 = air.t0)
>
> # alternatively using the conditional poisson
> saddle.distn(A = cbind(aircondit$hours/12, 1), u = 12, wdist = "p",
+ type = "cond", t0 = air.t0)
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.344718
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.655282
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.344718
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.655282
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.832240
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.167760
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.832240
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.167760
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.444538
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.555462
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.444538
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.555462
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.494449
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.505551
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.494449
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.505551
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.594269
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.405731
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.594269
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.405731
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.544359
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.455641
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.544359
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.455641
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.519404
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.480596
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.519404
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.480596
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.561394
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.438606
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.561394
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.438606
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.290549
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.709451
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.290549
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.709451
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.019703
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.980297
Warning in dpois(y, mu, log = TRUE) : non-integer x = 11.019703
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.980297
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.748857
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.251143
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.748857
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.251143
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.478011
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.521989
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.478011
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.521989
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.207165
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.792835
Warning in dpois(y, mu, log = TRUE) : non-integer x = 10.207165
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.792835
Warning in dpois(y, mu, log = TRUE) : non-integer x = 9.936320
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.063680
Warning in dpois(y, mu, log = TRUE) : non-integer x = 9.936320
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.063680
Warning in dpois(y, mu, log = TRUE) : non-integer x = 9.665474
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.334526
Warning in dpois(y, mu, log = TRUE) : non-integer x = 9.665474
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.334526
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.785225
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.214775
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.785225
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.214775
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.175822
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.824178
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.175822
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.824178
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.566419
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.433581
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.566419
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.433581
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.957016
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.042984
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.957016
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.042984
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.347613
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.652387
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.347613
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.652387
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.738210
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.261790
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.738210
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.261790
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.128807
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.871193
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.128807
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.871193
Saddlepoint Distribution Approximations
Call :
saddle.distn(A = cbind(aircondit$hours/12, 1), u = 12, wdist = "p",
type = "cond", t0 = air.t0)
Quantiles of the Distribution
0.1% 27.4
0.5% 35.4
1.0% 39.7
2.5% 46.7
5.0% 53.5
10.0% 62.5
20.0% 75.3
50.0% 104.5
80.0% 139.0
90.0% 158.8
95.0% 175.9
97.5% 191.2
99.0% 209.6
99.5% 222.4
99.9% 249.5
Smoothing spline used 20 points in the range 9.8 to 304.7.
>
> # Distribution of the ratio of a sample of size 10 from the bigcity
> # data, taken from Example 9.16 of Davison and Hinkley (1997).
> ratio <- function(d, w) sum(d$x *w)/sum(d$u * w)
> city.v <- var.linear(empinf(data = city, statistic = ratio))
> bigcity.t0 <- c(mean(bigcity$x)/mean(bigcity$u), sqrt(city.v))
> Afn <- function(t, data) cbind(data$x - t*data$u, 1)
> ufn <- function(t, data) c(0,10)
> saddle.distn(A = Afn, u = ufn, wdist = "b", type = "cond",
+ t0 = bigcity.t0, data = bigcity)
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Warning in eval(family$initialize) :
non-integer counts in a binomial glm!
Saddlepoint Distribution Approximations
Call :
saddle.distn(A = Afn, u = ufn, wdist = "b", type = "cond", t0 = bigcity.t0,
data = bigcity)
Quantiles of the Distribution
0.1% 1.070
0.5% 1.092
1.0% 1.104
2.5% 1.122
5.0% 1.139
10.0% 1.158
20.0% 1.184
50.0% 1.237
80.0% 1.304
90.0% 1.348
95.0% 1.392
97.5% 1.436
99.0% 1.494
99.5% 1.537
99.9% 1.636
Smoothing spline used 20 points in the range 1.014 to 1.96.
>
> # From Example 9.16 of Davison and Hinkley (1997) again, we find the
> # conditional distribution of the ratio given the sum of city$u.
> Afn <- function(t, data) cbind(data$x-t*data$u, data$u, 1)
> ufn <- function(t, data) c(0, sum(data$u), 10)
> city.t0 <- c(mean(city$x)/mean(city$u), sqrt(city.v))
> saddle.distn(A = Afn, u = ufn, wdist = "p", type = "cond", t0 = city.t0,
+ data = city)
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.866400
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.511350
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.622251
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.866400
Warning in dpois(y, mu, log = TRUE) : non-integer x = 8.511350
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.622251
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.210844
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.107208
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.681949
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.210844
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.107208
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.681949
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.038622
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.809279
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.152100
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.038622
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.809279
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.152100
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.452511
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.160314
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.387175
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.452511
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.160314
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.387175
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.159455
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.835832
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.004713
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.159455
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.835832
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.004713
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.155629
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.115412
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.728960
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.155629
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.115412
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.728960
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.205022
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.133273
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.661705
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.205022
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.133273
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.661705
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.722845
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.033105
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.244049
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.722845
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.033105
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.244049
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.787431
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.388294
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.824275
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.787431
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.388294
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.824275
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.415407
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.940756
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.643837
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.415407
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.940756
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.643837
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.043383
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.493218
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.463399
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.043383
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.493218
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.463399
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.671359
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.045680
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.282961
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.671359
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.045680
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.282961
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.299336
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.598142
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.102523
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.299336
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.598142
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.102523
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.433935
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.409277
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.156788
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.433935
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.409277
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.156788
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.360158
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.271008
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.368834
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.360158
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.271008
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.368834
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.319252
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.639889
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.040859
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.319252
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.639889
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.040859
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.278346
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.008770
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.712884
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.278346
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.008770
Warning in dpois(y, mu, log = TRUE) : non-integer x = 4.712884
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.237440
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.377650
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.384909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.237440
Warning in dpois(y, mu, log = TRUE) : non-integer x = 1.377650
Warning in dpois(y, mu, log = TRUE) : non-integer x = 5.384909
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.196534
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.746531
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.056935
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.196534
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.746531
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.056935
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.155629
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.115412
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.728960
Warning in dpois(y, mu, log = TRUE) : non-integer x = 3.155629
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.115412
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.728960
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.734728
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.299661
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.965612
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.734728
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.299661
Warning in dpois(y, mu, log = TRUE) : non-integer x = 6.965612
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.228786
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.666383
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.104831
Warning in dpois(y, mu, log = TRUE) : non-integer x = 2.228786
Warning in dpois(y, mu, log = TRUE) : non-integer x = 0.666383
Warning in dpois(y, mu, log = TRUE) : non-integer x = 7.104831
Saddlepoint Distribution Approximations
Call :
saddle.distn(A = Afn, u = ufn, wdist = "p", type = "cond", t0 = city.t0,
data = city)
Quantiles of the Distribution
0.1% 1.216
0.5% 1.236
1.0% 1.248
2.5% 1.272
5.0% 1.301
10.0% 1.340
20.0% 1.393
50.0% 1.502
80.0% 1.618
90.0% 1.680
95.0% 1.732
97.5% 1.777
99.0% 1.830
99.5% 1.866
99.9% 1.938
Smoothing spline used 20 points in the range 1.182 to 2.061.
>
>
>
> cleanEx()
> nameEx("simplex")
> ### * simplex
>
> flush(stderr()); flush(stdout())
>
> ### Name: simplex
> ### Title: Simplex Method for Linear Programming Problems
> ### Aliases: simplex
> ### Keywords: optimize
>
> ### ** Examples
>
> # This example is taken from Exercise 7.5 of Gill, Murray and Wright (1991).
> enj <- c(200, 6000, 3000, -200)
> fat <- c(800, 6000, 1000, 400)
> vitx <- c(50, 3, 150, 100)
> vity <- c(10, 10, 75, 100)
> vitz <- c(150, 35, 75, 5)
> simplex(a = enj, A1 = fat, b1 = 13800, A2 = rbind(vitx, vity, vitz),
+ b2 = c(600, 300, 550), maxi = TRUE)
Linear Programming Results
Call : simplex(a = enj, A1 = fat, b1 = 13800, A2 = rbind(vitx, vity,
vitz), b2 = c(600, 300, 550), maxi = TRUE)
Maximization Problem with Objective Function Coefficients
x1 x2 x3 x4
200 6000 3000 -200
Optimal solution has the following values
x1 x2 x3 x4
0.0 0.0 13.8 0.0
The optimal value of the objective function is 41400.
>
>
>
> cleanEx()
> nameEx("smooth.f")
> ### * smooth.f
>
> flush(stderr()); flush(stdout())
>
> ### Name: smooth.f
> ### Title: Smooth Distributions on Data Points
> ### Aliases: smooth.f
> ### Keywords: smooth nonparametric
>
> ### ** Examples
>
> # Example 9.8 of Davison and Hinkley (1997) requires tilting the resampling
> # distribution of the studentized statistic to be centred at the observed
> # value of the test statistic 1.84. In the book exponential tilting was used
> # but it is also possible to use smooth.f.
> grav1 <- gravity[as.numeric(gravity[, 2]) >= 7, ]
> grav.fun <- function(dat, w, orig) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2] - mns[1], s2hat, (mns[2]-mns[1]-orig)/sqrt(s2hat))
+ }
> grav.z0 <- grav.fun(grav1, rep(1, 26), 0)
> grav.boot <- boot(grav1, grav.fun, R = 499, stype = "w",
+ strata = grav1[, 2], orig = grav.z0[1])
> grav.sm <- smooth.f(grav.z0[3], grav.boot, index = 3)
>
> # Now we can run another bootstrap using these weights
> grav.boot2 <- boot(grav1, grav.fun, R = 499, stype = "w",
+ strata = grav1[, 2], orig = grav.z0[1],
+ weights = grav.sm)
>
> # Estimated p-values can be found from these as follows
> mean(grav.boot$t[, 3] >= grav.z0[3])
[1] 0.0240481
> imp.prob(grav.boot2, t0 = -grav.z0[3], t = -grav.boot2$t[, 3])
$t0
[1] -1.840118
$raw
[1] 0.02254379
$rat
[1] 0.02409135
$reg
[1] 0.02224027
>
>
> # Note that for the importance sampling probability we must
> # multiply everything by -1 to ensure that we find the correct
> # probability. Raw resampling is not reliable for probabilities
> # greater than 0.5. Thus
> 1 - imp.prob(grav.boot2, index = 3, t0 = grav.z0[3])$raw
[1] 0.08678105
> # can give very strange results (negative probabilities).
>
>
>
> cleanEx()
> nameEx("tilt.boot")
> ### * tilt.boot
>
> flush(stderr()); flush(stdout())
>
> ### Name: tilt.boot
> ### Title: Non-parametric Tilted Bootstrap
> ### Aliases: tilt.boot
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # Note that these examples can take a while to run.
>
> # Example 9.9 of Davison and Hinkley (1997).
> grav1 <- gravity[as.numeric(gravity[,2]) >= 7, ]
> grav.fun <- function(dat, w, orig) {
+ strata <- tapply(dat[, 2], as.numeric(dat[, 2]))
+ d <- dat[, 1]
+ ns <- tabulate(strata)
+ w <- w/tapply(w, strata, sum)[strata]
+ mns <- as.vector(tapply(d * w, strata, sum)) # drop names
+ mn2 <- tapply(d * d * w, strata, sum)
+ s2hat <- sum((mn2 - mns^2)/ns)
+ c(mns[2]-mns[1],s2hat,(mns[2]-mns[1]-orig)/sqrt(s2hat))
+ }
> grav.z0 <- grav.fun(grav1, rep(1, 26), 0)
> tilt.boot(grav1, grav.fun, R = c(249, 375, 375), stype = "w",
+ strata = grav1[,2], tilt = TRUE, index = 3, orig = grav.z0[1])
TILTED BOOTSTRAP
Exponential tilting used
First 249 replicates untilted,
Next 375 replicates tilted to -2.648,
Next 375 replicates tilted to 1.879.
Call:
tilt.boot(data = grav1, statistic = grav.fun, R = c(249, 375,
375), stype = "w", strata = grav1[, 2], tilt = TRUE, index = 3,
orig = grav.z0[1])
Bootstrap Statistics :
original bias std. error
t1* 2.846154 -0.2728113 2.513218
t2* 2.392353 -0.2450416 1.187570
t3* 0.000000 -0.7305799 2.152173
>
>
> # Example 9.10 of Davison and Hinkley (1997) requires a balanced
> # importance resampling bootstrap to be run. In this example we
> # show how this might be run.
> acme.fun <- function(data, i, bhat) {
+ d <- data[i,]
+ n <- nrow(d)
+ d.lm <- glm(d$acme~d$market)
+ beta.b <- coef(d.lm)[2]
+ d.diag <- boot::glm.diag(d.lm)
+ SSx <- (n-1)*var(d$market)
+ tmp <- (d$market-mean(d$market))*d.diag$res*d.diag$sd
+ sr <- sqrt(sum(tmp^2))/SSx
+ c(beta.b, sr, (beta.b-bhat)/sr)
+ }
> acme.b <- acme.fun(acme, 1:nrow(acme), 0)
> acme.boot1 <- tilt.boot(acme, acme.fun, R = c(499, 250, 250),
+ stype = "i", sim = "balanced", alpha = c(0.05, 0.95),
+ tilt = TRUE, index = 3, bhat = acme.b[1])
>
>
>
> cleanEx()
> nameEx("tsboot")
> ### * tsboot
>
> flush(stderr()); flush(stdout())
>
> ### Name: tsboot
> ### Title: Bootstrapping of Time Series
> ### Aliases: tsboot ts.return
> ### Keywords: nonparametric ts
>
> ### ** Examples
>
> lynx.fun <- function(tsb) {
+ ar.fit <- ar(tsb, order.max = 25)
+ c(ar.fit$order, mean(tsb), tsb)
+ }
>
> # the stationary bootstrap with mean block length 20
> lynx.1 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "geom")
>
> # the fixed block bootstrap with length 20
> lynx.2 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "fixed")
>
> # Now for model based resampling we need the original model
> # Note that for all of the bootstraps which use the residuals as their
> # data, we set orig.t to FALSE since the function applied to the residual
> # time series will be meaningless.
> lynx.ar <- ar(log(lynx))
> lynx.model <- list(order = c(lynx.ar$order, 0, 0), ar = lynx.ar$ar)
> lynx.res <- lynx.ar$resid[!is.na(lynx.ar$resid)]
> lynx.res <- lynx.res - mean(lynx.res)
>
> lynx.sim <- function(res,n.sim, ran.args) {
+ # random generation of replicate series using arima.sim
+ rg1 <- function(n, res) sample(res, n, replace = TRUE)
+ ts.orig <- ran.args$ts
+ ts.mod <- ran.args$model
+ mean(ts.orig)+ts(arima.sim(model = ts.mod, n = n.sim,
+ rand.gen = rg1, res = as.vector(res)))
+ }
>
> lynx.3 <- tsboot(lynx.res, lynx.fun, R = 99, sim = "model", n.sim = 114,
+ orig.t = FALSE, ran.gen = lynx.sim,
+ ran.args = list(ts = log(lynx), model = lynx.model))
>
> # For "post-blackening" we need to define another function
> lynx.black <- function(res, n.sim, ran.args) {
+ ts.orig <- ran.args$ts
+ ts.mod <- ran.args$model
+ mean(ts.orig) + ts(arima.sim(model = ts.mod,n = n.sim,innov = res))
+ }
>
> # Now we can run apply the two types of block resampling again but this
> # time applying post-blackening.
> lynx.1b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "fixed",
+ n.sim = 114, orig.t = FALSE, ran.gen = lynx.black,
+ ran.args = list(ts = log(lynx), model = lynx.model))
>
> lynx.2b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "geom",
+ n.sim = 114, orig.t = FALSE, ran.gen = lynx.black,
+ ran.args = list(ts = log(lynx), model = lynx.model))
>
> # To compare the observed order of the bootstrap replicates we
> # proceed as follows.
> table(lynx.1$t[, 1])
2 3 4 5 6 7 8 10 11 13 14
16 21 41 4 3 5 3 1 3 1 1
> table(lynx.1b$t[, 1])
2 3 4 5 6 7 9 10 11 12 13 15 16
2 5 24 3 2 8 2 3 38 8 2 1 1
> table(lynx.2$t[, 1])
2 3 4 5 6 7 9 11 12 13 15 17 24
18 20 41 3 4 1 1 4 1 3 1 1 1
> table(lynx.2b$t[, 1])
2 3 4 5 6 7 8 10 11 12 14 15
1 3 31 7 1 6 4 3 35 4 3 1
> table(lynx.3$t[, 1])
2 3 4 5 6 7 8 9 10 11 12 13 14 16
4 2 11 1 3 6 3 2 2 50 7 6 1 1
> # Notice that the post-blackened and model-based bootstraps preserve
> # the true order of the model (11) in many more cases than the others.
>
>
>
> cleanEx()
> nameEx("var.linear")
> ### * var.linear
>
> flush(stderr()); flush(stdout())
>
> ### Name: var.linear
> ### Title: Linear Variance Estimate
> ### Aliases: var.linear
> ### Keywords: nonparametric
>
> ### ** Examples
>
> # To estimate the variance of the ratio of means for the city data.
> ratio <- function(d,w) sum(d$x * w)/sum(d$u * w)
> var.linear(empinf(data = city, statistic = ratio))
[1] 0.03248701
>
>
>
> ### * <FOOTER>
> ###
> cleanEx()
> options(digits = 7L)
> base::cat("Time elapsed: ", proc.time() - base::get("ptime", pos = 'CheckExEnv'),"\n")
Time elapsed: 44.104 0.665 45.086 0 0
> grDevices::dev.off()
null device
1
> ###
> ### Local variables: ***
> ### mode: outline-minor ***
> ### outline-regexp: "\\(> \\)?### [*]+" ***
> ### End: ***
> quit('no')
Reference in New Issue Copy Permalink