activeSetLogCon {logcondens}

Computes a Log-Concave Probability Density Estimate via an Active Set Algorithm

Description

Given a vector of observations {\bold{x}_n} = (x_1, \ldots, x_n) with not necessarily equal entries, activeSetLogCon first computes vectors {\bold{x}_m} = (x_1, \ldots, x_m) and {\bold{w}} = (w_1, \ldots, w_m) where w_i is the weight of each x_i s.t. \sum_{i=1}^m w_i = 1. Then, activeSetLogCon computes a concave, piecewise linear function \widehat \phi_m on [x_1, x_m] with knots only in \{x_1, \ldots, x_m\} such that

L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^\infty \exp(\phi(t)) dt

is maximal. To accomplish this, an active set algorithm is used.

Usage

activeSetLogCon(x, xgrid = NULL, print = FALSE, w = NA)

Arguments

x

Vector of independent and identically distributed numbers, not necessarily unique.

xgrid

Governs the generation of weights for observations. See preProcess for details.

print

print = TRUE outputs the log-likelihood in every loop, print = FALSE does not. Make sure to tell R to output (press CTRL+W).

w

Optional vector of weights. If weights are provided, i.e. if w != NA, then xgrid is ignored.

Value

xn

Vector with initial observations x_1, \ldots, x_n.

x

Vector of observations x_1, \ldots, x_m that was used to estimate the density.

w

The vector of weights that had been used. Depends on the chosen setting for xgrid.

phi

Vector with entries \widehat \phi_m(x_i).

IsKnot

Vector with entries IsKnot_i = 1\{\widehat \phi_m has a kink at x_i\}.

L

The value L(\widehat {\bold{\phi}}_m) of the log-likelihood-function L at the maximum \widehat {\bold{\phi}}_m.

Fhat

A vector (\widehat F_{m,i})_{i=1}^m of the same size as {\bold{x}} with entries

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

H

Vector (H_1, \ldots, H_m)' where H_i is the derivative of

t \to L(\phi + t\Delta_i)

at zero and \Delta_i(x) = \min(x - x_i, 0).

n

Number of initial observations.

m

Number of unique observations.

knots

Observations that correspond to the knots.

mode

Mode of the estimated density \hat f_m.

sig

The standard deviation of the initial sample x_1, \ldots, x_n.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

References

Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at https://arxiv.org/abs/0707.4643.

Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log–concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40–68.

Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. doi:10.18637/jss.v039.i06

See Also

activeSetLogCon can be used to estimate a log-concave density. However, to generate an object of class dlc that allows application of summary and plot we recommend to use logConDens.

The following functions are used by activeSetLogCon:

J00, J10, J11, J20, Local_LL, Local_LL_all, LocalCoarsen, LocalConvexity, LocalExtend, LocalF, LocalMLE, LocalNormalize, MLE

Log concave density estimation via an iterative convex minorant algorithm can be performed using icmaLogCon.

Examples

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

## plot resulting functions
par(mfrow = c(2, 2), mar = c(3, 2, 1, 2))
plot(res$x, exp(res$phi), type = 'l'); rug(x)
plot(res$x, res$phi, type = 'l'); rug(x)
plot(res$x, res$Fhat, type = 'l'); rug(x)
plot(res$x, res$H, type = 'l'); rug(x)

## compute and plot function values at an arbitrary point
x0 <- (res$x[100] + res$x[101]) / 2
Fx0 <- evaluateLogConDens(x0, res, which = 3)[, "CDF"]
plot(res$x, res$Fhat, type = 'l'); rug(res$x)
abline(v = x0, lty = 3); abline(h = Fx0, lty = 3)

## compute and plot 0.9-quantile of Fhat
q <- quantilesLogConDens(0.9, res)[2]
plot(res$x, res$Fhat, type = 'l'); rug(res$x)
abline(h = 0.9, lty = 3); abline(v = q, lty = 3)