Distribution-free and other prediction intervals

  • Konijn H
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Abstract

Saw, Yang, and Mo (1984) gave a distribution-free prediction interval for X based on X1,..., Xn, of the form [X̄ - A, X̄ + A] with A2 = λ2(1 + 1/n)S2. 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 λ 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.

Author-supplied keywords

  • Chebyshev’s inequality
  • Distribution-free inference
  • Finite population sampling
  • Tolerance interval
  • Triple range

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Authors

  • Hendrik S. Konijn

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