signed mixtures [X’ed]

Following a question on X validated, the hypoexponential distribution, I came across (for the second time) a realistic example of a mixture (of exponentials) whose density wrote as a signed mixture, i.e. involving both negative and positive weights (with sum still equal to one). Namely,

$\displaystyle f(x)=\sum_i^d \lambda_i e^{-\lambda_ix}\prod_{j=1,i\neq j}^{d}\frac{\lambda_j}{\lambda_j-\lambda_i}\quad x,\lambda_j>0$

representing the density of a sum of d Exponential variates. The above is only well-defined when all rates differ, while a more generic definition involving matrix exponentiation exists. But the case when (only) two rates are equal can rather straightforwardly be derived by a direct application of L’Hospital rule, which my friend George considered as the number one calculus rule!

This entry was posted on March 26, 2023 at 12:23 am and is filed under Books, Kids, Statistics with tags convolution, cross validated, finite mixtures, George Casella, Hospital's rule, hypoexponential distributions, matrix exponentiation, Michel de L'Hôpital, probability distribution. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “signed mixtures [X’ed]”

Casella fan Says:
March 26, 2023 at 6:59 am

Are there any of George Casella’s lectures / talks available online? I haven’t been able to find any.

Reply
- xi'an Says:
  March 26, 2023 at 4:22 pm
  
  Not that I know of, alas.
  
  Reply

Xi'an's Og