logcondens: icmaLogCon – R documentation

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icmaLogCon

Computes a Log-Concave Probability Density Estimate via an Iterative Convex Minorant Algorithm

Description

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

L(φ) = ∑_{i=1}^m w_i φ(x_i) - \int_{-∞}^∞ exp(φ(t)) dt

is maximal. In order to be able to apply the pool - adjacent - violaters algorithm, computations are performed in the parametrization

η(φ) = (φ_1, (η_1 + ∑_{j=2}^i (x_i-x_{i-1})η_i)_{i=2}^m ).

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{η}} \to L({\bold{η}}) in the direction of the new candidate is ≤ \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, …, 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, …, x_n.
`Loglik`	The value L(\widehat φ_m) of the log-likelihood-function L at the maximum \widehat φ_m.
`Iterations`	Number of iterations performed.
`sig`	The standard deviation of the initial sample x_1, …, x_n.

Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

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.

Examples

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)

## End(Not run)

logcondens

Estimate a Log-Concave Probability Density from Iid Observations

v2.1.5

GPL (>= 2)

Authors

Kaspar Rufibach <kaspar.rufibach@gmail.com> and Lutz Duembgen <duembgen@stat.unibe.ch>

Initial release

2016-07-11