quantilesLogConDens {logcondens}R Documentation

Function to compute p0-Quantile of F

Description

Function to compute p_0-quantile of

widehat F_m(t) = int_{x_1}^t widehat f_m(t) dt,

where widehat f_m is the log-concave density estimator, received via activeSetLogCon.

Usage

quantilesLogConDens(p0, x, f, F, IsKnot)

Arguments

p0 Real number where quantil should be computed.
x Sorted vector of original observations x = (x_1, ..., x_m).
f Vector (widehat f_m(x_1), ..., widehat f_m(x_m)), representing the function widehat f_m, as computed by activeSetLogCon.
F Vector (widehat F_{m,i})_{i=1}^m with entries

widehat F_{m,i} = int_{x_1}^{x_i} exp(widehat varphi_m(t)) dt,

as computed by activeSetLogCon.

IsKnot Column vector with entries
IsKnot_i = 1{widehat varphi_m has a kink at x_i}, as computed by activeSetLogCon.

Value

Returns the real number q_0 = inf_{x}{widehat F_m(x) >= p_0}.

Note

Since the log-density is piecewise linear, the corresponding distribution function can be analytically computed from x and widehat varphi_m. The function quantilesLogConDens simply inverts this analytical function. However, for extreme quantiles (that are of course still smaller than x_m) and m big, this may be numerically unstable and should be replaced by numerical search (e.g. bisection).

Author(s)

Kaspar Rufibach, kaspar.rufibach@stanford.edu,
http://www.stanford.edu/~kasparr

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.stat.unibe.ch/~duembgen

Examples

## estimate gamma density
set.seed(1977)
x <- sort(rgamma(200, 2, 1))
res <- activeSetLogCon(x, w = NA, print = FALSE)

## compute 0.95 quantile of estimated F
quantilesLogConDens(0.95, x, exp(res$phi), res$F, res$IsKnot)

[Package logcondens version 1.0 Index]