# Modeling by means of a negative binomial process

The negative binomial distribution with parameters $p\in(0,1)$ and $t>0$ is sometimes defined as the distribution of the number of failures before the $t$th success. This is supported on the set $\{0,1,2,3,\ldots\}.$ Another convention defines it as the number of trials needed to get $t$ successes, supported on the set $\{t, t+1, t+2, \ldots\}.$ But if the first convention is used, then the process is infinitely divisible. Therefore we can define a stochastic processes $N_t,\,t\ge0,$ so that for $0\le s<t,$ $N_t-N_s$ has a negative binomial distribution with parameters $p$ and $t-s,$ and such increments on pairwise disjoint intervals are independent.

My question is: What are the uses of that continuous-time process in statistical modeling?

• Modeling of item demand for inventory control purposes comes to mind, although not statistical (strictly speaking). – jbowman Jun 12 '18 at 20:18
• @jbowman : Why do you call it "not statistical"? – Michael Hardy Jun 13 '18 at 3:15
• I was thinking of it in terms of its use in optimization, as in something out of Production & Operations Management in an IE/OR department, but it is still a stochastic model and is treated as such, so I guess it really is statistical. Another example might be equipment failures over time that generate demand for replacement / repair parts, I've done some work along those lines in the past. – jbowman Jun 13 '18 at 4:32
• @jbowman : Have you, or has anyone, published anything about this? – Michael Hardy Jun 13 '18 at 4:40
• Craig Sherbrooke, "Optimal Inventory Modeling of Systems", p. 62-64 (in my edition) discusses why he uses the N.B. distribution from a "mathematical properties matching real world observations" point of view. His METRIC family of algorithms, with of course many variants and extensions by others form one of the two great branches of multi-echelon inventory control algorithms; I've implemented variants myself for several companies, and they were widely used by the Department of Defense (at one time, I'm not up to date on it now.) – jbowman Jun 13 '18 at 17:57