Discrete uniform MLE from the sample maximum alone

Question

Tokens are numbered 1, 2, \dots, N with N unknown, and n = 10 tokens are drawn uniformly from \{1, \dots, N\}. The individual draws were discarded, but the log records that the largest value observed was 90. What is the maximum likelihood estimate of N, and why is the largest value all you need?

Accepted Answer

The likelihood is L(N) = N^{-n}\,\mathbf{1}\{N \ge \maxi xi\}. It depends on the data only through \maxi xi: that quantity is a sufficient statistic for N. Two draws with the same maximum give the identical likelihood function no matter what the other values were. Since L is decreasing in N above the maximum, \hat N = \maxi xi = 90. Takeaway. The sample maximum is sufficient for the discrete uniform upper bound, so knowing only the maximum lets you compute the MLE.

Discrete uniform MLE from the sample maximum alone

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