C. Order Statistics

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In the previous unit, we studied the distributions of sums (and functions of sums) of i.i.d. random variables: \[ \hat\theta = g\left(\sum_{i=1}^n X_i\right). \] This covered the sample mean \(\bar X\) and many other common estimators.

However, we have also encountered estimators that are not simply functions of the sum. For example, in the German Tank Problem, the MLE of the number of tanks \(N\), was \(\hat N = \max(X_1, \dots, X_n)\). In Example 30.4, the MLE of \(\theta\) was the sample median. Estimators like these, which depend on the ordered observations, are called order statistics. This part of the book will develop tools to characterize the distributions of order statistics.