LogConcDEAD (version 1.0-3)

lcd.mle: Compute the maximum likelihood estimator of a log-concave density

Description

This function uses Shor's $r$-algorithm to compute the maximum likelihood estimator of a log-concave density based on an i.i.d. sample. The estimator is determined by its value at data points.

Usage

lcd.mle (X, initial=NULL, verbose=-1, alpha=5, c=1, sigmatol=10^-8,
 integraltol=10^-4, ytol=10^-4, stepscale=5.1,
stepscale2=2, stepscale3=1.5, stepscale4=1.05, desiredsize=3.3)

Arguments

X

Data in $R^d$, in the form of an $n \times d$ numeric matrix

initial

Starting point for the $r$-algorithm. If NULL (the default), a kernel estimate is used.

verbose

ll{ -1 (default) prints nothing 0 prints warning messages $n>0$ prints summary information every $n$ iterations}

alpha

Scaling parameter for SolvOpt

c

Starting step size

sigmatol

Stopping criterion: relative change in $\sigma$ (see Details for details)

ytol

Stopping criterion: relative change in $y$ (see Details for details)

integraltol

Stopping criterion: difference from 1 of integral of $\exp(\bar{h}_y)$ (see Details for details)

stepscale,stepscale2,stepscale3,stepscale4

Parameters for SolvOpt

desiredsize

Another parameter for SolvOpt (changing this is not recommended)

Value

• x
  Data (reordered)
• logMLE
  Vector of the log of the MLE at the observation points
• NumberOfEvaluations
  Number of steps, number of function evaluations, number of subgradient evaluations. If the SolvOpt algorithm fails, the first component will be an error message.
• MinSigma
  Minimum value of the objective function
• b
  see Details
• beta
  see Details
• chull
  Vector containing final triangulation of convex hull of data
• verts
  Details of triangulation for use in lcd.eval
• vertsoffset
  Details of triangulation for use in lcd.eval

Details

The MLE of a log-concave density is of the form $$\bar{h}_y(x) = \inf \lbrace h(x) \colon h \textrm{ concave }, h(X_i) \geq y_i \textrm{ for } i = 1, \ldots, n \rbrace$$ for some $y \in R^n$.

Functions of this form are piecewise linear on the convex hull of $X_1, \ldots, X_n$, and of the form $$\min \lbrace \langle b_j, x \rangle - \beta_j \rbrace$$ for some vectors $b_j$ and constants $\beta_j$. This function uses Shor's $r$-algorithm (an iterative subgradient-based procedure) to minimise over vectors $y$ in $R^n$ the function $$\sigma(y) = -\frac{1}{n} \sum_{i=1}^n y_i + \int \exp(\bar{h}_y(x)) \, dx.$$ This is equivalent to finding the MLE. An implementation of Shor's $r$-algorithm based on SolvOpt is used.

Computing $\sigma$ makes use of the qhull (http:\www.qhull.org) library, adapted from the Rimplementation in geometry.

References

Cule, M. L., Samworth, R. J., and Stewart, M. I. (2007) Computing the nonparametric maximum likelihood estimator of a log-concave density In preparation

Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T. (1996) The Quickhull algorithm for convex hulls ACM Trans. on Mathematical Software, 22(4) p.469-483 http://www.qhull.org

Kappel, F. and Kuntsevich, A. V. (2000) An implementation of Shor's r-algorithm Computational Optimization and Applications 15(2) p.193-205

http://www.uni-graz.at/imawww/kuntsevich/solvopt/

N. Z. Shor (1985) Minimization methods for nondifferentiable functions Springer-Verlag

See Also

geometry

Examples

## some simple normal data
set.seed(101)
x <- matrix(rnorm(200), ncol=2)
out <- lcd.mle(x)