How to fit parameters in Exponential Function Kernel The data that I have is conversion X,and t, time for conversion.  Which means, t is the time from start up until the event X=1 (a conversion)
I have the following kernel function, which I'm trying to fit.
$$
g(t) =  \beta \omega e^{( -\omega t)}
$$
However, how should I fit the two parameters $\beta $  and $\omega$?
For more background, this is the paper I'm working through.
 A: In the linked paper, the form
$$
g_i(t) =  \beta_i \omega_i e^{( -\omega_i t)}
$$
is proposed as the general form of a hazard function. The paper models "conversion" (e.g., purchase on the web) in terms of multiple online advertising channels. Each channel $i$ is assumed to have a specific initial "impact" $\beta_i \ge 0$ in terms of influencing conversion when a user connects to it at time $t=0$, but decaying over time as determined by its specific $\omega_i \ge 0$. The ideas is to identify particular channels that are most effective in terms of "conversion."
Data were collected on the times that users connected to each channel, and the time of (first) conversion or last data time without conversion. For each user, the hazard function was modeled as a sum of hazards of the above form, given the known times that the user connected to a channel. Users only differed in terms of the times at which they connected to a channel, with additive hazards from all channels of the above form each starting from the corresponding connection time.
This is not the standard type of survival problem that assumes an ultimate event probability of 1. In that standard survival problem, the integral of the hazard function over $t = (0,\infty)$ must diverge.
That integral for the above form of $g_i(t)$, in contrast, is the finite value $\beta_i$. An assumption of ultimate event probability of 1 situation would not make sense in the situation examined by the paper, however, with almost 380 million users, 2605 advertising channels, and only 4281 conversions.
The paper shows how to estimate the $\beta_i$ and $\omega_i$ by a maximum likelihood approach that takes advantage of the above parametric form in a way that includes likelihoods both from those who purchased/converted and those who had not yet done so at last follow-up (censored). As noted in comments on the question, it would not be correct to restrict analysis only to users who "converted," as that ignores the large number of censored cases.
