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,
$$J_{10}(x, y) = \frac{\exp(y) - \exp(x) - (y - x) \exp(x)}{(y - x)^2};$$
$$J_{11}(x, y) = \frac{(y - x)(\exp(x) + \exp(y)) + 2 (\exp(y) - \exp(x))}{(y - x)^3};$$
$$J_{20}(x, y) = 2\frac{\exp(y) - \exp(x) - (y - x)\exp(x)-(y - x)^2 \exp(x)}{(y - x)^3}.$$
J00(x, y, v)
J10(x, y)
J11(x, y)
J20(x, y)Value of the respective function.
Vector of length \(d\) with real entries.
Vector of length \(d\) with real entries.
Number in \([0, 1]^d\).
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Lutz Duembgen, duembgen@stat.unibe.ch,
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html
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. tools:::Rd_expr_doi("https://doi.org/10.18637/jss.v039.i06")