Tolerance Limits for Distributions

Definition

Let $X_1,\dots ,X_n$ be Random Sample from a distribution with cdf $F(x)$, and $Y_1<\cdots <Y_n$ be Order Statistic, then If $P[F(Y_i)-F(Y_j)\ge p]=\gamma$, then $(Y_i,Y_j)$ is $100\gamma \%$ tolerance limits for $100p\%$ of the probability for the distribution of $X$

Properties

Joint PDF

$h(z_1,z_2,\dots ,z_n)=\{\begin{array}{ll}n!& 0<z_1<z_2<\cdots <z_n<1\\ 0& \text{otherwise}\end{array}$

Computation of $\gamma$

$P(Z_i-Z_j\ge p)=P(Z_{j-i}\ge p)=\gamma ={\int }_p^1{h}_{j-i}(v)dv$ where $h_k(v)=\frac{n!}{(k-1)!(n-k)!}{v}^{k-1}(1-v{)}^{n-k}$

A probability that the tolerance interval $(Z_j,Z_i)$ contains $100p\%$ of the probability