MLE for the highest jersey number

Question

A league issues jerseys numbered 1, 2, \dots, N per team with N unknown, and a scout records a uniform sample of numbers from one squad: 4, 19, 11, 23, 7, 15, 22. Give the maximum likelihood estimate of N.

Accepted Answer

The likelihood L(N) = N^{-n}\,\mathbf{1}\{N \ge \maxi xi\} is zero below the sample maximum and decreasing above it, so \hat N = \maxi xi = 23. Takeaway. The MLE of an upper bound N from uniform draws is the largest value seen; more draws can only push it up.

MLE for the highest jersey number

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