Q00: Numerical Routine Q

In logcondens: Estimate a Log-Concave Probability Density from Iid Observations

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 q and/or Q.
a	Vector of length m with real entries.
u	Vector of length m with real entries.
v	Vector of length m with real entries.
gamma	Real number. Standard deviation to be used.
QFhat	Logical. Should Q be computed?

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, [e-mail],
http://www.kasparrufibach.ch

Lutz Duembgen, [e-mail],
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

logcondens documentation built on Aug. 22, 2023, 5:06 p.m.