Order Statistics

Order Statistics

Consider a lifetest in which n like items are tested until they fail. The failure times may be denoted by

$$\begin{array}{l@{\,}l} {x}_{(1)} \leq {x}_{(2)} \leq \ldots \leq {x}_{(r)} \leq \ldots \leq {x}_{(n)}.\,\end{array}$$

Here x _(r) is the r-th order statistic, also denoted by x _r: n if the sample size needs to be emphasized. The duration of the test, x _(n), may be unduly long, so that it becomes desirable to censor the test after, say, the r-th failure.

More commonly, the observations come to us unordered as x ₁, x ₂, …, x _n , say the diameters of n mass-produced items. Here n is usually small, e.g., n = 5, but such samples are taken frequently and charts of both sample means and sample rangesx _(n) − x ₍₁₎ plotted. The range is more convenient than the standard deviation and almost as efficient in small samples from the underlying normal populations generally assumed.

Basic Distribution Theory

Let X ₁, X ₂, …, X _n be independent random variables drawn from a...