Q00 {logcondens} | R Documentation |
Numerical Routine Q
Description
This function is used in the computation of \widehat f^*
and \widehat F^*
.
Usage
Q00(x, a, u, v, gamma, QFhat = FALSE)
Arguments
x |
Number at which to compute |
a |
Vector of length |
u |
Vector of length |
v |
Vector of length |
gamma |
Real number. Standard deviation to be used. |
QFhat |
Logical. Should |
Value
The vector(s) q
and/or Q
.
Note
Taylor approximation is used if a
is small. In addition, as described in Duembgen and Rufibach (2011) at
the end of Appendix C, in extreme situations, e.g. when data sets contain extreme spacings, numerical problems may
occur in the computation of the function q_\gamma
(eq. (7) in Duembgen and Rufibach, 2011).
For it may happen that the exponent is rather large while the difference of Gaussian CDFs is very
small. To moderate these problems, we are using the following bounds:
\exp(- m^2/2) \bigl( \Phi(\delta) - \Phi(-\delta) \bigr) \ \le \ \Phi(b) - \Phi(a) \ \le \ \exp(- m^2/2) \cosh(m\delta) \bigl( \Phi(\delta) - \Phi(-\delta) \bigr)
for arbitrary numbers a < b
and m := (a + b) / 2
, \delta := (b - a) / 2
.
However, the function Q00
is 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,
https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html
References
Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1–28. https://www.jstatsoft.org/v39/i06