bcapar {bcaboot}  R Documentation 
Compute parametric bootstrap confidence intervals
Description
bcapar computes parametric bootstrap confidence intervals for a realvalued parameter theta in a pparameter exponential family. It is described in Section 4 of the reference below.
Usage
bcapar(
t0,
tt,
bb,
alpha = c(0.025, 0.05, 0.1, 0.16),
J = 10,
K = 6,
trun = 0.001,
pct = 0.333,
cd = 0,
func
)
Arguments
t0 
Observed estimate of theta, usually by maximum likelihood. 
tt 
A vector of parametric bootstrap replications of theta of
length 
bb 
A 
alpha 
percentiles desired for the bca confidence limits. One
only needs to provide 
J , K 
Parameters controlling the jackknife estimates of Monte
Carlo error: 
trun 
Truncation parameter used in the calculation of the
acceleration 
pct 
Proportion of "nearby" b vectors used in the calculation
of 
cd 
If cd is 1 the bca confidence density is also returned; see Section 11.6 in reference Efron and Hastie (2016) below 
func 
Function 
Value
a named list of several items:

lims : Bca confidence limits (first column) and the standard limits (fourth column). Also the abc limits (fifth column) if
func
is provided. The second column,jacksd
, are the jackknife estimates of Monte Carlo error;pct
, the third column are the proportion of the replicatestt
less than eachbcalim
value 
stats : Estimates and their jackknife Monte Carlo errors:
theta
=\hat{\theta}
;sd
, the bootstrap standard deviation for\hat{\theta}
;a
the acceleration estimate;az
another acceleration estimate that depends less on extreme values oftt
;z0
the biascorrection estimate;A
the bigA measure of raw acceleration;sdd
delta method estimate for standard deviation of\hat{\theta}
;mean
the average oftt

abcstats : The abc estimates of
a
andz0
, returned iffunc
was provided 
ustats : The biascorrected estimator
2 * t0  mean(tt)
.ustats
givesustat
, an estimatesdu
of its sampling error, and jackknife estimates of monte carlo error for bothustat
andsdu
. Also given isB
, the number of bootstrap replications 
seed : The random number state for reproducibility
References
DiCiccio T and Efron B (1996). Bootstrap confidence intervals. Statistical Science 11, 189228
T. DiCiccio and B. Efron. More accurate confidence intervals in exponential families. Biometrika (1992) p231245.
Efron B (1987). Better bootstrap confidence intervals. JASA 82, 171200
B. Efron and T. Hastie. Computer Age Statistical Inference. Cambridge University Press, 2016.
B. Efron and B. Narasimhan. Automatic Construction of Bootstrap Confidence Intervals, 2018.
Examples
data(diabetes, package = "bcaboot")
X < diabetes$x
y < scale(diabetes$y, center = TRUE, scale = FALSE)
lm.model < lm(y ~ X  1)
mu.hat < lm.model$fitted.values
sigma.hat < stats::sd(lm.model$residuals)
t0 < summary(lm.model)$adj.r.squared
y.star < sapply(mu.hat, rnorm, n = 1000, sd = sigma.hat)
tt < apply(y.star, 1, function(y) summary(lm(y ~ X  1))$adj.r.squared)
b.star < y.star %*% X
set.seed(1234)
bcapar(t0 = t0, tt = tt, bb = b.star)