logcondens (version 2.1.5)

activeSetLogCon: 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 \(x_1, \ldots, x_n\).

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 http://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. http://www.jstatsoft.org/v39/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

# NOT RUN {
## 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)
# }