Order statistics definition
Daily Note - 10/05/2024
1. Order Statistics
In statistics, the k-th order statistic of a statistical sample is equal to its k-th smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases are the minimum and maximum value of a sample. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.
The first order statistic is always the minimum of the sample. It’s denoted by \(X_{(1)} = \min\{X_{1}, ..., X_{n}\}\).
Similarly, for a sample size n, the largest order statistic is the maximum, denoted by \(X_{(n)} = \max\{X_{1}, ..., X_{n}\}\).
The sample range is the difference between the maximum and minimum order statistics: \(R = X_{(n)} - X_{(1)}\).
Similar is the interquantile range, which is the difference between the 75th and 25th percentiles, denoted by \(IQR = X_{(0.75n)} - X_{(0.25n)}\).
The sample median may or may not be an order statistic, since there is a single middle value only when the number n of observations is odd.
A most simple an beautiful explanation:
Given a sample of \(n\) variates \(X_{1}, ..., X_{n}\), reorder them so that \(Y_{1} < Y_{2} < ... < Y_{N}\). Then \(Y_{i}\) is called the \(i^{th}\) oder statistic, sometimes also denote by \(X_{(i)}\).