# The order statistics and the uniform distribution

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 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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