Interpreting a non-integer shape parameter for the gamma distribution

Question

An often-cited example of the gamma distribution is that for integer shape parameter k, and scale parameter lambda, the gamma distribution can be conceptualized as the sum of k independent (identically) exponentially-distributed random variables with termination rate lambda. My question is this: what does this shape parameter mean when it is not an integer? I am a graduate student in psychology trying to model termination rates for behavioural activities in a hidden markov model, and if the "best fitting" gamma distribution has non-integer shape parameter, I would like to know how to decompose the gamma into a sum of exponential processes which, in the non-integer case, would presumably have different termination rates. Any help would be greatly appreciated.

Answer 1

So you're saying that you want to write a gamma with non-integer shape parameter as a sum of exponentials.

This can't be done. The moment generating function of a random variable $X$ is $m_X(t) = E[e^{tX}]$; this has the property that $m_{X+Y}(t) = m_X(t) m_Y(t)$.

The moment generating function of $Gamma(k, \theta)$ is $(1-\theta t)^{-k}$. If $k$ is an integer than this is the product of $k$ factors of $(1-\theta t)^{-1}$, which are the MGFs of exponentials; if $k$ is not an integer then there is no such decomposition.

(It's possible there may be an approximate decomposition as a sum of exponentials but I'm not sure how useful that would be. It would help to know why you believe it would be useful to have such a decomposition.)