Distribution-Free and other Prediction Intervals

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Abstract
Saw, Yang, and Mo (1984) gave a distribution-free prediction interval for X based on X 1,…, Xn of the form [[Xbar] -A, [Xbar] + A] with A 2 ≈ λ2(1 + 1/n)S 2. As compared with the range [X (1), X (2)], which has length R(say) and size (minimum coverage probability) (n − 1)/(n + 1), their intervals can have size as high as n/(n +1), a value that is attained when λ2 = n +1. For n = 2, this interval (with λ2 = 3) becomes the “triple range” [X (1) - R, X (2)+ R] and has size 2/3; it coincides with the “normal interval” for n = 2 with coverage probability 2/3 under normality. For all n > 2, the size of their interval (with λ2 = n + 1) equals approximately the coverage probability of the normal interval based on three observations only. A table is given for the value of A required to guarantee a size of at least h' for the distribution-free interval for selected values of h' and for all n ≤ 100. It may also be used when applying a Chebyshev-type inequality for simple random sampling from a finite population.

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