Great post about order statistics and their importance in non-parametric methods.

A Blog on Probability and Statistics

In this post, we show that the order statistics of the uniform distribution on the unit interval are distributed according to the beta distributions. This leads to a discussion on estimation of percentiles using order statistics. We also present an example of using order statistics to construct confidence intervals of population percentiles. For a discussion on the distributions of order statistics of random samples drawn from a continuous distribution, see the previous postĀ The distributions of the order statistics.

Suppose that we have a random sample $latex X_1,X_2,\cdots,X_n$ of size $latex n$ from a continuous distribution with common distribution function $latex F_X(x)=F(x)&s=-1$ and common density function $latex f_X(x)=f(x)&s=-1$. The order statistics $latex Y_1<Y_2< \cdots <Y_n&s=-1$ are obtained by ordering the sample $latex X_1,X_2,\cdots,X_n$ in ascending order. In other words, $latex Y_1&s=-1$ is the smallest item in the sample and $latex Y_2&s=-1$ is the second smallest item in the sampleā¦

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