R: Compute pointwise confidence interval for a density assuming...
logConCI
R Documentation
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.
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) ar{C}(0).
The nuisance parameter c_2(x) depends on the true density f and the second derivative of its logarithm. The limiting process ar{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 ar{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.
Azadbakhsh, M., Jankowski, H. and Gao, X. (2014).
Computing confidence intervals for log-concave densities.
Comput. Statist. Data Anal., to appear.
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.