A simulation study of the robustness of prediction intervals for an independent observation obtained from a random sample from an assumed location-scale family of distributions

Abstract

Suppose that based on data consisting of independent repetitions of an experiment a researcher wants to predict the outcome of the next independent outcome of the experiment. The researcher models the data as being realizations of independent, identically distributed random variables { Xi, i=1,2,..n} having density f() and the next outcome as the value of an independent random variable Y , also having density f() . We assume that the density f() lies in one of three location-scale families: standard normal (symmetric); Cauchy (symmetric, heavy-tailed); extreme value (asymmetric.). The researcher does not know the values of the location and scale parameters. For f() = f0() lying in one of these families, an exact prediction interval for Y can be constructed using equivariant estimators of the location and scale parameters to form a pivotal quantity based on { Xi, i=1,2,..n} and Y. This report investigates via a simulation study the performance of these prediction intervals in terms of coverage rate and length when the assumption that f() = f0() is correct and when it is not. The simulation results indicate that prediction intervals based on the assumption of normality perform quite well with normal and extreme value data and reasonably well with Cauchy data when the sample sizes are large. The heavy tailed Cauchy assumption only leads to prediction intervals that perform well with Cauchy data and is not robust when the data are normal and extreme value. Similarly, the asymmetric extreme value model leads to prediction intervals that only perform well with extreme value data. Overall, this study indicates robustness with respect to a mismatch between the assumed and actual distributions in some cases and a lack of robustness in others.