The process below yields a distribution of "response times" (RT), and I'd like to know if there is an analytic solution to obtain the density function of this distribution.
An RT is recorded at the earliest time ($t_1$) at which a function, $f(t)$, reaches a threshold.
The threshold varies across iterations of the process according to a random uniform distribution with bounds specified by the parameters $P_1$ and $P_2$.
$f(t)$ begins at $t=0$ and has a form whose derivative is defined by the function $g(t)$:
$$g(t) = P_3\left (\frac{P_4}{P_4+P_5}\right )\left (1-\exp\left [-(P_4+P_5)\int_{0}^{t}h(s)ds\right ]\right )$$
where $h(t)$ is defined as:
$$h(t) = P_6\Gamma(t,P_8,n)[1 - \Gamma(t-P_7,P_9,n)]$$
and the gamma function defined as:
$$\Gamma(t,\beta,n) = 1- \left ( e^{-\beta t} \right )\left ( \sum_{n-1}^{j=0} \frac{(\beta t)^{j}}{j!} \right )$$
So, there are in total 10 parameters in the process: $P_1$ through $P_9$, and $n$. I could of course perform a Euler simulation to generate a distribution of RT values for given a particular set of 10 parameter values, but it would be nice to avoid this because I'm hoping to use a Nelder–Mead search algorithm to find the set of 10 parameter values that maximizes the likelihood of a set of observed RT data points, and it really slows down the search if I have to compute estimates of the density function for every candidate set of parameter values.
Thanks for any help you're able to provide!
Pursuant to comments, here is a further description of the process:
The process is an adaptation of the model presented by Smith & Racliff (PDF) which models perception, attention and decision in the context of a speeded detection task. Function $h(t)$ models perception: accumulation of information about a brief visual stimulus. Function $g(t)$ models the transfer of this information to memory via attention, and function $h(t)$ is a linear ballistic decision process in the vein of Brown & Heathcote (PDF). I thought that supplementing an LBA as the decision mechanism in place of Smith & Ratcliff's diffusion process might simplify the derivation of the final density function.
$P_1$ and $P_2$ are the upper and lower bounds of the random uniform threshold. $P_3$ is the asymptote of $g(t)$. $P_4$ is the intensity of the stimulus. $P_5$ is the intensity of the background. $P_6$ is the amplitude of the stimulus. $P_7$ is the duration of the stimulus. $P_8$ is the rate at which the stimulus comes on. $P_9$ is the rate at which the stimulus turns off.
Smith & Ratcliff (2009) note that $n$ might be reasonably fixed to a value of 3, so possibly that can further simplify things. Additionally, while Smith & Ratcliff sought to fit their model to different stimulus intensities, I don't need this particular feature, so I'm not sure if I need parameters $P_4$ and $P_5$ (though I'm not sure how to properly remove them either).