Process for a Gamma distribution with non integer shape parameter

2010-09-15T13:56:49Z

I am sampling the distribution of lifetimes of computers participating in massive volunteer computing initiatives (BOINC projects). While a phenomenological Weibull distribution makes a good description, I find that a better fit happens with a Gamma distribution with a shape parameter $k < 1$

Now, while a Gamma distribution with integer shape parameter is produced as the waiting time for the $k$-th event of a Poisson process, I do not know of a similar, process-based, description of the meaning of a Gamma distribution for a non integer parameter, nor to say for non integer and less than unity. Is there one?

Answer by Pietro Majer for Process for a Gamma distribution with non integer shape parameter

2010-09-15T14:10:50Z

This is just a hint (actually yours is a question I planned to ask here myself sooner or later): What's the model for the gamma distribution with non-integer parameter $\lambda$?

Since possibly the most relevant aspect is the semigroup law w.r.to convolution, my feeling is that it should describe properly waiting times for reaching given amount $\lambda$ of mass, arriving continuously and randomly under independence assumptions, like e.g. rain, falls of powder on the soil. I'm very interested in further more precise answers.

Answer by Shai Covo for Process for a Gamma distribution with non integer shape parameter

2011-01-02T23:43:50Z

Here is some interesting, though natural, view of the problem (including a small but quite elegant result at the end). Denote by ${\rm Gamma}(k,\lambda)$ the gamma distribution with density function $f(x;k,\lambda) = \lambda ^k x^{k - 1} e^{ - \lambda x}/ \Gamma (k)$ , $x>0$ . Here $k$ and $\lambda$ are arbitrary positive constants. Clearly, the problem reduces to the case $0<k<1$ . Indeed, if $k$ is non-integer and greater than $1$ , then a ${\rm Gamma}(k,\lambda)$ random variable can be viewed as a sum of independent ${\rm Gamma}(\left\lfloor k \right\rfloor,\lambda)$ and ${\rm Gamma}(k - \left\lfloor k \right\rfloor,\lambda)$ random variables. Now, let's consider a gamma process $X = \lbrace X_t : t \geq 0\rbrace$ such that $X_1 \sim {\rm exponential}(\lambda)$ . That is, $X$ is a process with stationary independent increments, starting at $0$ and having right-continuous with left limits sample paths (i.e., a L\'evy process), such that $X_t - X_s \sim {\rm Gamma}(t-s,\lambda)$ , for any $0 \leq s < t$ . The process $X$ is a pure jump process, having positive jumps only, and it holds $X_t = \sum\nolimits_{0 < s \le t} {\Delta X_s } $ , where $\Delta X_s = X_s - X_{s-}$ . The number of jumps is a.s. infinite countable in any time interval, no matter how small (yet, $\Delta X_t =0$ a.s., for any fixed $t>0$ ). The process $X$ is characterized by its jump measure (L\'evy measure) given by $\nu({\rm d}x)=x^{-1}e^{-\lambda x}\,{\rm d}x$ : given a time interval $[t_1,t_2]$ , the number of jumps of $X$ whose sizes lie in $[a,b] \subset (0,\infty)$ is Poisson distributed with mean $(t_2-t_1)\int_a^b {x^{ - 1} e^{ - \lambda x } \,{\rm d}x}$ (the key concept here is Poisson random measure). Hence, roughly speaking, the process usually increases by tiny jumps (whose sum is negligible in practice).

Now, consider the process $X$ on the time interval $[0,1]$ . On the one hand, $X_1$ is the (infinite) sum of jumps of $X$ up to time $t=1$ . On the other hand, $X_1$ is distributed as the waiting time until the first occurrence in a Poisson process with rate $\lambda$ . This is already quite interesting. Returning to the original problem, we can view a ${\rm Gamma}(k,\lambda)$ random variable, $0<k<1$ fixed, as the random variable $X_k$ , i.e., the sum of jumps of $X$ up to time $k$ (note that $\lim _{k \uparrow 1} X_k = X_1 $ , a.s.). So, in some respect, this already gives a process-based description of the meaning of a Gamma distribution for a non-integer parameter. But much more can be said, using the fact that if $X$ and $Y$ are independent ${\rm Gamma}(\alpha,\lambda)$ and ${\rm Gamma}(\beta,\lambda)$ rv's, respectively, then the ratio $X/(X+Y)$ has a ${\rm Beta}(\alpha,\beta)$ distribution. Namely, the ratio $R = X_k / X_1$ , $0<k<1$ , is distributed as a ${\rm Beta}(k,1-k)$ random variable, implying that $R$ has density function $f_{R} (x;k) = x^{k-1}(1-x)^{-k}\sin(\pi k)/\pi$ , $0<x<1$ , where we have used $\Gamma(k)\Gamma(1-k) = \pi/\sin(\pi k)$ . In particular, $f_R (x;1/2) = 1/(\pi \sqrt {x(1 - x)} )$ , $0<x<1$ .

Process for a Gamma distribution with non integer shape parameter - MathOverflow

Process for a Gamma distribution with non integer shape parameter

Answer by Pietro Majer for Process for a Gamma distribution with non integer shape parameter

Answer by Shai Covo for Process for a Gamma distribution with non integer shape parameter