Jfunctions {logcondens}

Numerical Routine J and Some Derivatives

Description

J00 represents the function J(x, y, v), where for real numbers x, y and v \in [0, 1],

J(x, y, v) = \int_0^v \exp((1-t)x + ty) d t = \frac{\exp(x + v(y - x)) - \exp(x)}{y - x}.

The functions Jab give the respective derivatives J_{ab} for v = 1, i.e.

J_{ab}(x, y) = \frac{\partial^{a+b}}{\partial x^a \partial y^b} J(x, y).

Specifically,

J_{10}(x, y) = \frac{\exp(y) - \exp(x) - (y - x) \exp(x)}{(y - x)^2};

J_{11}(x, y) = \frac{(y - x)(\exp(x) + \exp(y)) + 2 (\exp(y) - \exp(x))}{(y - x)^3};

J_{20}(x, y) = 2\frac{\exp(y) - \exp(x) - (y - x)\exp(x)-(y - x)^2 \exp(x)}{(y - x)^3}.

Usage

J00(x, y, v)
J10(x, y)
J11(x, y)
J20(x, y)

Arguments

x

Vector of length d with real entries.

y

Vector of length d with real entries.

v

Number in [0, 1]^d.

Value

Value of the respective function.

Note

Taylor approximations are used if y-x is small. We refer to Duembgen et al (2011, Section 6) for details.

These functions are not intended to be invoked by the end user.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html

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 https://arxiv.org/abs/0707.4643.

Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. doi:10.18637/jss.v039.i06