Absolute Bounds on the Mean and Standard Deviation of Transformed Data for Constant-Derivative Transformations

摘要

We investigate absolute bounds (or inequalities) on the mean and standard deviation of transformed data values, given only a few statistics on the original set of data values. Our work applies primarily to transformation functions whose derivatives are constant-sign for a positive range (e.g. logarithm, antilog, square root, and reciprocal). With such functions we can often get reasonably tight absolute bounds, so that distributional assumptions about the data needed for confidence intervals can be eliminated. We investigate a variety of methods of obtaining such bounds, first examining bounding curves which are straight lines, them those that are quadratic polynomials. While the problem of finding the best quadratic bound is an optimization problem with no closed-form solution, we display a variety of closed-form quadratic bounds which can come close to the optimal solution. We emphasize what can be done with prior knowledge of the mean and standard deviation of the untransformed data values, but do address some other statistics too. (Author)