from tramway to Panzer (or back!)…

Although it is usually presented as the tramway problem, namely estimating the number of tram or bus lines in a city given observing one line number, including The Bayesian Choice by yours truly, the original version of the problem is about German tanks, Panzer V tanks to be precise, which total number M was to be estimated by the Allies from their observation of serial numbers of a number k of tanks. The Riddler is restating the problem when the only available information is made of the smallest, 22, and largest, 144, numbers, with no information about the number k itself. I am unsure what the Riddler means by “best” estimate, but a posterior distribution on M (and k) can be certainly be constructed for a prior like 1/k x 1/M² on (k,M). (Using M² to make sure the posterior mean does exist.) The joint distribution of the order statistics is

\frac{k!}{(k-2)!} M^{-k} (144-22)^{k-2}\, \Bbb I_{2\le k\le M\ge 144}

which makes the computation of the posterior distribution rather straightforward. Here is the posterior surface (with an unfortunate rendering of an artefactual horizontal line at 237!), showing a concentration near the lower bound M=144. The posterior mode is actually achieved for M=144 and k=7, while the posterior means are (rounded as) M=169 and k=9.