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