A Biweight Prediction Interval for Random Samples

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Abstract
This study takes a biweight approach to providing a prediction interval for a single future observation. This problem—equivalent to the problem of estimating quantiles—is of fundamental importance in ascertaining the nature of a process. The prediction or quantile problem is difficult for a robust estimator, since much of the desired information is in the extremes of the sample, whose influence the robust estimator is designed to resist. The situation is further complicated when the population has heavy tails. The proposed biweight prediction interval efficiently extracts this information from the extremes of the sample without breaking down. It provides accurate quantile estimates or prediction even in very demanding situations, such as when the sample size is small but the desired prediction level is large. There is clearly a need for such an interval, since predictions must often be made from a limited number of observations. For example, it may be desirable to predict the crashworthiness of an automobile. Unfortunately, the desired measurements are expensive to obtain. Further, if one allows for a heavy-tailed underlying distribution, the traditional prediction interval based on the Studentt statistic may be much too wide. In this study, however, the proposed biweight prediction interval accurately predicts for the Gaussian distribution and several heavy-tailed distributions. It does so even in the demanding situation of providing a 95% prediction interval based on only 10 observations.

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