Derivatives of the First-Passage Time Density and Cumulative
Distribution Function, and Random Sampling from the (Truncated)
First-Passage Time Distribution
Description
First, we provide functions to calculate the partial derivative of the first-passage time diffusion probability density function (PDF) and cumulative
distribution function (CDF) with respect to the first-passage time t (only for PDF), the upper barrier a, the drift rate v, the relative starting point w, the
non-decision time t0, the inter-trial variability of the drift rate sv, the inter-trial variability of the rel. starting point sw, and the inter-trial variability
of the non-decision time st0. In addition the PDF and CDF themselves are also provided. Most calculations are done on the logarithmic scale to make it more stable.
Since the PDF, CDF, and their derivatives are represented as infinite series, we give the user the option to control the approximation errors with the argument
'precision'. For the numerical integration we used the C library cubature by Johnson, S. G. (2005-2013) . Numerical integration is
required whenever sv, sw, and/or st0 is not zero. Note that numerical integration reduces speed of the computation and the precision cannot be guaranteed
anymore. Therefore, whenever numerical integration is used an estimate of the approximation error is provided in the output list.
Note: The large number of contributors (ctb) is due to copying a lot of C/C++ code chunks from the GNU Scientific Library (GSL).
Second, we provide methods to sample from the first-passage time distribution with or without user-defined truncation from above. The first method is a new adaptive
rejection sampler building on the works of Gilks and Wild (1992; ) and Hartmann and Klauer (in press). The second method is a rejection
sampler provided by Drugowitsch (2016; ). The third method is an inverse transformation sampler. The fourth method is a
"pseudo" adaptive rejection sampler that builds on the first method. For more details see the corresponding help files.