1
$\begingroup$

I'm trying to find the expected value of a random variable $t_i$ which is the solution of

$$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$

where $t_1,..., t_{i-1} , \alpha, \beta, \mu$ are all known constants and where $\epsilon_i$ is a random variable with exponential(1) distribution. The text I'm reading says that this can be done using simulations, but I have no idea of how to do it in R. I have tried finding $t_i$ as a function of $\epsilon_i$ but I only got this far

$$ \log(t_i)-\beta t_i=\log\left(\epsilon_i+\mu t_{i-1}+(i-1) \frac{\alpha}{\beta}\right)-\log\sum^{i-1}_{k=1}e^{\beta t_k} -\log\mu- \log\left(\frac{\alpha}{\beta}\right)$$

Any ideas of what to do next are welcome!

$\endgroup$
1
$\begingroup$

Assuming that the algebraic expression you have is correct, do the following:

Suppose that you want the time $t_i$ simulation and you have data up to time $t_{i-1}$:

  1. Take a random draw for $\varepsilon_i \sim Exp(1)$
  2. Plug in values to the expression you have to create $x_i = \log(t_i) - \beta t_i$
  3. Use the Lambert W function to get $t_i$. $$t_i = -\frac{W(-e^{x_i} \beta)}{\beta}$$ Here's a link to how to use the Lambert W package in R: http://www.inside-r.org/packages/cran/LambertW/docs/W

After you have a large number of simulations, take the average.

$\endgroup$
  • $\begingroup$ If you just want the solution to W, it may be faster to use the lamWpackage, which is used as the Lambert W function calculation engine in LambertW. $\endgroup$ – Avraham Dec 6 '18 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.