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Symmetry Analysis of the Uncertain Alternative Box-Cox Regression Model

The asymmetry of residuals about the origin is a severe issue in estimating a Box-Cox transformed model. In the framework of uncertainty theory, there are such theoretical issues regarding the least-squares estimation (LSE) and maximum likelihood estimation (MLE) of the linear models after the Box-Cox transformation on the response variables. Heretofore, only weighting methods for least-squares analysis have been available. This article proposes an uncertain alternative Box-Cox model to alleviate the asymmetry of residuals and avoid λ tending to negative infinity for uncertain LSE or uncertain MLE. Such symmetry of residuals about the origin is reasonable in applications of experts’ experimental data. The parameter estimation method was given via a theorem, and the performance of our model was supported via numerical simulations. According to the numerical simulations, our proposed ‘alternative Box-Cox model’ can overcome the problems of a grossly underestimated lambda and the asymmetry of residuals. The estimated residuals neither deviated from zero nor changed unevenly, in clear contrast to the LSE and MLE for the uncertain Box-Cox model downward biased residuals. Thus, though the LSE and MLE are not applicable on the uncertain Box-Cox model, they fit the uncertain alternative Box-Cox model. Compared with the uncertain Box-Cox model, the issue of a systematically underestimated λ is not likely to occur in our uncertain alternative Box-Cox model. Both the LSE and MLE can be used directly without constructing a weighted estimation method, offering better performance in the asymmetry of residuals.