logcondens (version 2.0.2)

Jfunctions: Numerical Routine J and Some Derivatives

Description

J00 represents the function $J(x, y, v),$ where for real numbers $x, y$ and $v \in [0, 1],$ $$J(x,y,v)={\int }_0^v\exp ((1-t)x+ty)dt=\frac{\exp (x+v(y-x))-\exp (x)}{y-x}.$$ 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.

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. (2010) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, to appear.