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 + t y) d t = (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) = (partial ^ {a + b}) / (partial x ^ a partial y ^ b) J(x, y).

Specifically,

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

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

J_{20}(x, y) = 2(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

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

Lutz Duembgen, duembgen@stat.unibe.ch,
http://www.stat.unibe.ch/content/staff/personalhomepages/duembgen/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 http://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. http://www.jstatsoft.org/v39/i06

Results