logcondens (version 2.0.8)

logConCI: Compute pointwise confidence interval for a density assuming log-concavity

Description

Compute approximate confidence interval for the true log-concave density, on a grid of points. The confidence interval at a fixed point is based on the pointwise asymptotic theory for the log-concave maximum likelihood estimator (MLE) developed in Balabdaoui, Rufibach, and Wellner (2009).

Usage

logConCI(res, xx0, conf.level = c(0.8, 0.9, 0.95, 0.99)[3], 
type = c("DR", "ks", "nrd")[2], htype = c("hscv", "hlscv", "hpi", "hns")[4])

Arguments

res

An object of class dlc, usually a result of a call to logConDens.

xx0

Vector of grid points at which to calculate the confidence interval.

conf.level

Confidence level for the confidence interval(s).

type

Vector of strings indicating estimating method for nuisance parameters.

htype

Vector of strings indicating bandwidth selection method if type = ks.

Value

• The function returns a list containing the following elements:
• fhat
  MLE evaluated at grid points.
• up_DR
  Upper confidence interval limit when type = DR.
• lo_DR
  Lower confidence interval limit when type = DR.
• up_ks_hscv
  Upper confidence interval limit when type = ks and htype = hscv.
• lo_ks_hscv
  Lower confidence interval limit when type = ks and htype = hscv.
• up_ks_hlscv
  Upper confidence interval limit when type = ks and htype = hlscv.
• lo_ks_hlscv
  Lower confidence interval limit when type = ks and htype = hlscv.
• up_ks_hpi
  Upper confidence interval limit when type = ks and htype = hpi.
• lo_ks_hpi
  Lower confidence interval limit when type = ks and htype = hpi.
• up_ks_hns
  Upper confidence interval limit when type = ks and htype = hns.
• lo_ks_hns
  Lower confidence interval limit when type = ks and htype = hns.
• up_nrd
  Upper confidence interval limit when type = nrd.
• lo_nrd
  Lower confidence interval limit when type = nrd.

Details

In Balabdaoui, Rufibach, and Wellner (2009) it is shown that (if the true density is strictly log-concave) the limiting distribution of the MLE of a log-concave density $\widehat f_n$ at a point $x$ is $$n^{2/5}(\widehat f_n(x)-f(x)) \to c_2(x) \bar{C}(0).$$ The nuisance parameter $c_2(x)$ depends on the true density $f$ and the second derivative of its logarithm. The limiting process $\bar{C}(0)$ is found as the second derivative at zero of a particular operator (called the "envelope") of an integrated Brownian motion plus $t^4$. The confidence intervals are based on inverting the above limit using estimated quantiles of $\bar{C}(0)$, and estimating the nuisance parameter $c_2(x)$. The options for the function logConCI provide different ways to estimate this nuisance parameter. If type="DR", $c_2(x)$ is estimated using derivatives of the smoothed MLE as calculated by the function logConDens (this method does not perform well in simulations and is therefore not recommended). If type="ks", $c_2(x)$ is estimated using kernel density estimates of the true density and its first and second derivatives. This is done using the R package ks, and, with this option, a bandwidth selection method htype must also be chosen. The choices in htype correspond to the various options for bandwidth selection available in ks. If type="nrd", the second derivative of the logarithm of the true density in $c_2(x)$ is estimated assuming a normal reference distribution. The default option is type="ks" with htype="hns". Currently available confidence levels are 80%, 90%, 95% and 99%, with a default of 95%.

References

Azadbakhsh, M., Jankowski, H. and Gao, X. (2012) Computing confidence intervals for log-concave densities. Preprint. Baladbaoui, F., Rufibach, K. and Wellner, J. (2009) Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist., 37(3), 1299--1331. Tarn Duong (2012). ks: Kernel smoothing. R package version 1.8.10. http://CRAN.R-project.org/package=ks

Examples

## ===================================================
## Confidence intervals at a fixed point for the density
## ===================================================
data(reliability)
x.rel <- sort(reliability)

# calculate 95% confidence interval and plot the result
grid <- seq(min(x.rel), max(x.rel), length.out = 200)
res <- logConDens(x.rel)
ci  <- logConCI(res, grid, type = "nrd")	

par(las = 1, mar = c(2.5, 3.5, 0.5, 0.5))
hist(x.rel, n = 25, col = gray(0.9), main = "", freq = FALSE, 
    xlab = "", ylab = "", ylim = c(0, 0.0065), border = gray(0.5))
lines(grid, ci$fhat, col = "red", lwd = 2)
lines(grid, ci$lo_nrd, col = "red", lwd = 2, lty = 2)
lines(grid, ci$up_nrd, col = "red", lwd = 2, lty = 2)