MLE for the largest serial number

Question

A competitor stamps its products with serial numbers $1, 2, \dots, N$ , each equally likely to be the one you happen to buy, so each observation is uniform on the integers $\{1, \dots, N\}$ with unknown $N$ . You purchase six items and read off the serials 120, 47, 205, 88, 173, 62.

Derive the maximum likelihood estimator of $N$ and give its value here.

Accepted Answer

Each observation has probability 1/N if it lies in \{1, \dots, N\} and 0 otherwise. For n independent draws the likelihood is L(N) = \prod{i=1}^n \frac{1}{N}\,\mathbf{1}\{xi \le N\} = \frac{1}{N^n}\,\mathbf{1}\{N \ge \maxi xi\}. Two things matter. First, N must be at least the largest number you saw, or the likelihood is zero. Second, among the allowed values, 1/N^n is decreasing in N, so you want N as small as the constraint permits. Both point to the same place: \hat N = \maxi xi. The largest observed serial is 205, so \hat N = 205. Takeaway. For a discrete uniform on \{1, \dots, N\}, the M…

MLE for the largest serial number

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