logcondens (version 2.1.5)

icmaLogCon: Computes a Log-Concave Probability Density Estimate via an Iterative Convex Minorant 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. In order to be able to apply the pool - adjacent - violaters algorithm, computations are performed in the parametrization

$${\bold{\eta}}({\bold{\phi}}) = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^i (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m \Bigr).$$

To find the maximum of \(L\), a variant of the iterative convex minorant using the pool - adjacent - violaters algorithm is used.

Usage

icmaLogCon(x, xgrid = NULL, eps = 10^-8, T1 = 2000, 
    robustif = TRUE, print = FALSE)

Arguments

x

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

xgrid

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

eps

An arbitrary real number, typically small. Iterations are halted if the directional derivative of \({\bold{\eta}} \to L({\bold{\eta}})\) in the direction of the new candidate is \(\le \varepsilon\).

T1

Maximal number of iterations to perform.

robustif

robustif = TRUE performs the robustification and Hermite interpolation procedure detailed in Rufibach (2006, 2007), robustif = FALSE does not. In the latter case, convergence of the algorithm is no longer guaranteed.

print

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

Value

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.

f

Vector with entries \(\widehat f_m(x_i).\)

xn

Vector with initial observations \(x_1, \ldots, x_n\).

Loglik

The value \(L(\widehat \phi_m)\) of the log-likelihood-function \(L\) at the maximum \(\widehat \phi_m.\)

Iterations

Number of iterations performed.

sig

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

References

Rufibach K. (2006) Log-concave Density Estimation and Bump Hunting for i.i.d. Observations. PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. Available at http://www.zb.unibe.ch/download/eldiss/06rufibach_k.pdf.

Rufibach, K. (2007) Computing maximum likelihood estimators of a log-concave density function. J. Stat. Comput. Simul. 77, 561--574.

See Also

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

The following functions are used by icmaLogCon:

phieta, etaphi, Lhat_eta, quadDeriv, robust, isoMean.

Examples

# NOT RUN {
set.seed(1977)
x <- rgamma(200, 2, 1)
# }
# NOT RUN {
res <- icmaLogCon(x, T1 = 2000, robustif = TRUE, print = TRUE)

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