Solution
Moments for the Tank Numbers
Show the problem again
Tanks are numbered 1, 2, ..., N with N unknown. You observe five serial numbers: 7, 31, 12, 50 and 25. Use the method of moments, matching the sample mean to the mean of the discrete uniform distribution on {1, ..., N}, to estimate N. What is the estimate?
Worked solution
The mean of the discrete uniform on {1, ..., N} is (N+1)/2. Setting this equal to the sample mean (7 + 31 + 12 + 50 + 25)/5 = 25 gives N̂ = 2·25 − 1 = 49. Notice that this is smaller than the observed serial number 50, so the estimate is impossible as a value of N. This is the standard weakness of the method of moments for models whose support depends on the parameter: it ignores the hard constraint N ≥ max.
Source: Variation on the German tank problem. Statement written for AxiomIQ.