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!

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

  1. Casella fan Says:

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

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