bcapar {bcaboot}

Compute parametric bootstrap confidence intervals

Description

bcapar computes parametric bootstrap confidence intervals for a real-valued parameter theta in a p-parameter 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 B, usually large, say B = 2000

bb

A B by p matrix of natural sufficient vectors, where p is the dimension of the exponential family.

alpha

percentiles desired for the bca confidence limits. One only needs to provide alpha values below 0.5; the upper limits are automatically computed

J, K

Parameters controlling the jackknife estimates of Monte Carlo error: J jackknife folds, with the jackknife standard errors averaged over K random divisions of bb

trun

Truncation parameter used in the calculation of the acceleration a.

pct

Proportion of "nearby" b vectors used in the calculation of t., the gradient vector of theta.

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 \hat{\theta} = func(b). If this is not missing then output includes abc estimates; see reference DiCiccio and Efron (1992) below

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 replicates tt less than each bcalim 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 of tt; z0 the bias-correction estimate; A the big-A measure of raw acceleration; sdd delta method estimate for standard deviation of \hat{\theta}; mean the average of tt

• abcstats : The abc estimates of a and z0, returned if func was provided

• ustats : The bias-corrected estimator 2 * t0 - mean(tt). ustats gives ustat, an estimate sdu of its sampling error, and jackknife estimates of monte carlo error for both ustat and sdu. Also given is B, 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, 189-228

T. DiCiccio and B. Efron. More accurate confidence intervals in exponential families. Biometrika (1992) p231-245.

Efron B (1987). Better bootstrap confidence intervals. JASA 82, 171-200

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)