logConCI: Compute pointwise confidence interval for a density assuming...

In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

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. Two main approaches are implemented: In the first, 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). In the second, the confidence interval is estimated via the boostrap.

Usage

logConCI(res, xx0, conf.level = c(0.8, 0.9, 0.95, 0.99)[3], 
    type = c("DR", "ks", "nrd", "ECDFboot", "NPMLboot")[2], 
    htype = c("hscv", "hlscv", "hpi", "hns")[4], BB = 500)

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). The default is 95%.
type	Vector of strings indicating type of confidence interval to compute. When type = ks is chosen, then htype should also be specified. The default is type = ks.
htype	Vector of strings indicating bandwidth selection method if type = ks. The default is htype = hns.
BB	number of iterations in the bootstrap if type = NPMLboot or type = ECDFboot. The default is BB = 500.

Details

In Balabdaoui et al. (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.

Three of 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.

Two of the confidence intervals are based on the bootstrap. For type = "ECDFboot" confidence intervals based on re-sampling from the empirical cumulative distribution function are computed. For type = "NPMLboot" confidence intervals based on re-sampling from the nonparametric maximum likelihood estimate of log-concave density are computed. Bootstrap confidence intervals take a few minutes to compute!

The default option is type = "ks" with htype = "hns". Currently available confidence levels are 80%, 90%, 95% and 99%, with a default of 95%.

Azadbakhsh et al. (2014) provides an empirical study of the relative performance of the various approaches available in this function.

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.
up_npml	Upper confidence interval limit when type = NPMLboot.
lo_npml	Lower confidence interval limit when boot = NPMLboot.
up_ecdf	Upper confidence interval limit when boot = ECDFboot.
lo_ecdf	Lower confidence interval limit when boot = ECDFboot.

Author(s)

Mahdis Azadbakhsh

Hanna Jankowski, [e-mail]

References

Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. Comput. Statist. Data Anal., 75, 248–264.

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. https://CRAN.R-project.org/package=ks

Examples

## Not run: 
## ===================================================
## Confidence intervals at a fixed point for the density
## ===================================================
data(reliability)
x.rel <- sort(reliability)

# calculate 95
grid <- seq(min(x.rel), max(x.rel), length.out = 200)
res <- logConDens(x.rel)
ci  <- logConCI(res, grid, type = c("nrd", "ECDFboot"))	

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$fhat, col = "black", lwd = 2)
lines(grid, ci$lo_nrd, col = "red", lwd = 2, lty = 2)
lines(grid, ci$up_nrd, col = "red", lwd = 2, lty = 2)
lines(grid, ci$lo_ecdf, col = "blue", lwd = 2, lty = 2)
lines(grid, ci$up_ecdf, col = "blue", lwd = 2, lty = 2)
legend("topleft", col = c("black", "blue", "red"), lwd = 2, lty = c(1, 2, 2), legend = 
c("log-concave NPMLE", "CI for type = nrd", "CI for type = ECDFboot"), bty = "n")

## End(Not run)

logcondens documentation built on Aug. 22, 2023, 5:06 p.m.