Process for a Gamma distribution with non integer shape parameter - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:26:48Z http://mathoverflow.net/feeds/question/38821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38821/process-for-a-gamma-distribution-with-non-integer-shape-parameter Process for a Gamma distribution with non integer shape parameter arivero 2010-09-15T13:56:49Z 2011-01-02T23:57:00Z <p>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 $k &lt; 1$</p> <p>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?</p> http://mathoverflow.net/questions/38821/process-for-a-gamma-distribution-with-non-integer-shape-parameter/38822#38822 Answer by Pietro Majer for Process for a Gamma distribution with non integer shape parameter Pietro Majer 2010-09-15T14:10:50Z 2010-09-15T15:03:05Z <p>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 $\lambda$? </p> <p>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. </p> http://mathoverflow.net/questions/38821/process-for-a-gamma-distribution-with-non-integer-shape-parameter/50969#50969 Answer by Shai Covo for Process for a Gamma distribution with non integer shape parameter Shai Covo 2011-01-02T23:43:50Z 2011-01-02T23:57:00Z <p>Here is some interesting, though natural, view of the problem (including a small but quite elegant result at the end). Denote by <code>${\rm Gamma}(k,\lambda)$</code> the gamma distribution with density function <code>$f(x;k,\lambda) = \lambda ^k x^{k - 1} e^{ - \lambda x}/ \Gamma (k)$</code>, <code>$x&gt;0$</code>. Here <code>$k$</code> and <code>$\lambda$</code> are arbitrary positive constants. Clearly, the problem reduces to the case <code>$0&lt;k&lt;1$</code>. Indeed, if <code>$k$</code> is non-integer and greater than <code>$1$</code>, then a <code>${\rm Gamma}(k,\lambda)$</code> random variable can be viewed as a sum of independent <code>${\rm Gamma}(\left\lfloor k \right\rfloor,\lambda)$</code> and <code>${\rm Gamma}(k - \left\lfloor k \right\rfloor,\lambda)$</code> random variables. Now, let's consider a gamma process <code>$X = \lbrace X_t : t \geq 0\rbrace$</code> such that <code>$X_1 \sim {\rm exponential}(\lambda)$</code>. That is, <code>$X$</code> is a process with stationary independent increments, starting at <code>$0$</code> and having right-continuous with left limits sample paths (i.e., a L\'evy process), such that <code>$X_t - X_s \sim {\rm Gamma}(t-s,\lambda)$</code>, for any <code>$0 \leq s &lt; t$</code>. The process <code>$X$</code> is a pure jump process, having positive jumps only, and it holds <code>$X_t = \sum\nolimits_{0 &lt; s \le t} {\Delta X_s }$</code>, where <code>$\Delta X_s = X_s - X_{s-}$</code>. The number of jumps is a.s. infinite countable in any time interval, no matter how small (yet, <code>$\Delta X_t =0$</code> a.s., for any fixed <code>$t&gt;0$</code>). The process <code>$X$</code> is characterized by its jump measure (L\'evy measure) given by <code>$\nu({\rm d}x)=x^{-1}e^{-\lambda x}\,{\rm d}x$</code>: given a time interval <code>$[t_1,t_2]$</code>, the number of jumps of <code>$X$</code> whose sizes lie in <code>$[a,b] \subset (0,\infty)$</code> is Poisson distributed with mean <code>$(t_2-t_1)\int_a^b {x^{ - 1} e^{ - \lambda x } \,{\rm d}x}$</code> (the key concept here is Poisson random measure). Hence, roughly speaking, the process usually increases by tiny jumps (whose sum is negligible in practice). </p> <p>Now, consider the process <code>$X$</code> on the time interval <code>$[0,1]$</code>. On the one hand, <code>$X_1$</code> is the (infinite) sum of jumps of <code>$X$</code> up to time <code>$t=1$</code>. On the other hand, <code>$X_1$</code> is distributed as the waiting time until the first occurrence in a Poisson process with rate <code>$\lambda$</code>. This is already quite interesting. Returning to the original problem, we can view a <code>${\rm Gamma}(k,\lambda)$</code> random variable, <code>$0&lt;k&lt;1$</code> fixed, as the random variable <code>$X_k$</code>, i.e., the sum of jumps of <code>$X$</code> up to time <code>$k$</code> (note that <code>$\lim _{k \uparrow 1} X_k = X_1$</code>, 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 <code>$X$</code> and <code>$Y$</code> are independent <code>${\rm Gamma}(\alpha,\lambda)$</code> and <code>${\rm Gamma}(\beta,\lambda)$</code> rv's, respectively, then the ratio <code>$X/(X+Y)$</code> has a <code>${\rm Beta}(\alpha,\beta)$</code> distribution. Namely, the ratio <code>$R = X_k / X_1$</code>, <code>$0&lt;k&lt;1$</code>, is distributed as a <code>${\rm Beta}(k,1-k)$</code> random variable, implying that <code>$R$</code> has density function <code>$f_{R} (x;k) = x^{k-1}(1-x)^{-k}\sin(\pi k)/\pi$</code>, <code>$0&lt;x&lt;1$</code>, where we have used <code>$\Gamma(k)\Gamma(1-k) = \pi/\sin(\pi k)$</code>. In particular, <code>$f_R (x;1/2) = 1/(\pi \sqrt {x(1 - x)} )$</code>, <code>$0&lt;x&lt;1$</code>.</p>