Jfunctions: Numerical Routine J and Some Derivatives
Description
J00 represents the function $J(x, y, v),$ where for real numbers $x, y$ and $v \in [0, 1],$
$$J(x, y, v) = \int_0^v \exp((1-t)x + ty) d t = \frac{\exp(x + v(y - x)) - \exp(x)}{y - x}.$$
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.
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. (2011)
logcondens: Computations Related to Univariate Log-Concave Density Estimation.
Journal of Statistical Software, 39(6), 1--28. http://www.jstatsoft.org/v39/i06