Prediction intervals for the original response when using Manly’s exponential transformations

ABSTRACT

Motivated by A-10 single engine aircraft climb experiments, in this article we consider prediction interval estimation in the original units of observation after fitting a linear model using Manly’s exponential transformations. We assume that the residuals obtained from fitting the model in the transformed space are iid zero-mean normal random variables, at least approximately. We discuss the bias in the retransformed mean and derive a reduced bias estimator for the kth moment of the original response, given settings of the design variables. This is then used to compute reduced-bias estimates for the mean and variance of the untransformed response at various locations in design space. We then exploit Chebychev’s inequality, along with our proposed moment estimator, to construct an approximate 100(1 − α)% prediction interval on the original response, given settings of the design factors. We used Monte Carlo simulation to evaluate the performance of our proposed prediction interval estimator relative to a more commonly used alternative in practice. Our results suggest the proposed method is often the better alternative when the sample size is small and/or when the underlying model is misspecified.

KEYWORDS:

• data transformations
• linear models
• Manly’s exponential transformations
• prediction intervals
• retransformation bias

Additional information

Notes on contributors

Marcus B. Perry

Marcus B. Perry is an Associate Professor of Applied Statistics in the Department of Information Systems, Statistics and Management Science in the Culverhouse College of Commerce at the University of Alabama. His research interests include statistical process control, design and analysis of experiments, regression analysis, and predictive modeling.