MLE for the number of warehouse bins

Question

A warehouse numbers its storage bins 1, 2, \dots, N with N unknown, and an audit samples bins uniformly at random from \{1, \dots, N\}. Six sampled labels are 500, 213, 488, 655, 302, 197. Estimate N by maximum likelihood.

Accepted Answer

For a discrete uniform the likelihood L(N) = N^{-n}\,\mathbf{1}\{N \ge \maxi xi\} is largest at the smallest feasible N, which is the largest observed label: \hat N = \maxi xi = 655. Takeaway. Whenever data are uniform on \{1, \dots, N\}, maximum likelihood returns the sample maximum as the estimate of N.

MLE for the number of warehouse bins

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