logcondens (version 2.0.2)

logcon-package: Estimate a Log-Concave Probability Density from iid Observations

Description

The main function of this package is logConDens: A function to compute the maximum likelihood estimator (MLE) of a log-concave density from one-dimensional i.i.d. observations as well as the kernel smoothed version derived from it. A list of additional functions that can be used to compute quantities relevant in that context can be found below. Two algorithms are offered to compute the estimate: An active set (logConDens) and an iterative algorithm based on the pool-adjacent-violaters algorithm (icmaLogCon). The latter of these functions is only part of this package for historical reasons: it was the first method that was proposed to estimate a log-concave density, see Rufibach (2007). The more efficient way of computing the estimate is via an active set algorithm. The smoothed versions of the log-concave density and distribution function estimates discussed in Section 3 of Duembgen and Rufibach (2009) are available in the function evaluateLogConDens. For the computation of a log-concave estimate in any dimension the package LogConDEAD can be used.

Details

ll{ Package: logcondens Type: Package Version: 2.0.2 Date: 2011-02-23 License: GPL (>=2) } The following additional functions are provided in the package: evaluateLogConDens Computes the value of the estimated log-density, density, and distribution function of the MLE and the smoothed estimator at a given x0. quantilesLogConDens Computes quantiles of the estimated distribution function at a given x0. intECDF Compute the integrated empirical distribution function of the observations at a given x0. intF Compute the integral of the distribution function corresponding to the log-concave density estimator at a given x0. logconTwoSample Compute a permutation test for the difference between two distribution functions. logConROC Compute ROC curve based on log-concave density estimates within cases and controls. isoMean Compute the weighted least squares regression under a monotonicity constraint (the Grenander estimator). This function is used as part of icmaLogCon but is also of independent interest. The following datasets have been used in several publications to illustrate log-concave density estimation and are therefore included in this package: reliability Dataset that contains the data analyzed in Duembgen and Rufibach (2009, Figure 2). See the help file for logConDens for the analysis of this data. brightstar Dataset that contains the data analyzed in Mizera and Koenker (2009, Section 5). The sample consists of measurements of radial and rotational velocities for the stars from the Bright Star Catalog, see Hoffleit and Warren (1991). pancreas Data from pancreatic cancer serum biomarker study, first published by Wieand et al (1989). Contains data on serum measurements for a cancer antigen (CA-125) and a carbohydrate antigen (CA19.9) both of which are measured on a continuous non-negative scale. The measurements were taken within a case-control study on 90 cases with pancreatic cancer and 51 controls who did not have cancer but pancreatitis. A print and summary for objects of class dlc, generated by logConDens, are available as well.

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. (2010) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, to appear. Hoffleit, D., Warren, W.H. (1991). The Bright Star Catalog. Yale University Observatory, New Heaven. Mizera, I., Koenker, R. (2010). Quasi-concave density estimation. Ann. Statist., 38(5), 2998--3027. 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.stub.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. Rufibach, K. (2011). A smooth ROC curve estimator based on log-concave density estimates. Preprint.

Examples

## estimate gamma density
set.seed(1977)
x <- rgamma(100, 2, 1)
res <- logConDens(x, smoothed = FALSE, print = TRUE)
summary(res)

## compare performance to ICMA
res2 <- icmaLogCon(x, T1 = 2000, robustif = TRUE, print = TRUE)

res$L
res2$L

## plot resulting functions
par(mfrow = c(2, 2), mar = c(3, 2, 1, 2))
plot(res, which = "density")
plot(res, which = "log-density")
plot(res, which = "CDF")
plot(res$x, res$H, type = 'l'); rug(res$x)

## compute function values at an arbitrary point
x0 <- (res$x[50] + res$x[51]) / 2
evaluateLogConDens(x0, res)

## compute 0.5 quantile of Fhat
quantilesLogConDens(0.5, res)