# Symmetry, Vol. 14, Pages 22: Symmetry Analysis of the Uncertain Alternative Box-Cox Regression Model

#### ByLiang Fang

Dec 24, 2021

Symmetry, Vol. 14, Pages 22: Symmetry Analysis of the Uncertain Alternative Box-Cox Regression Model

Symmetry doi: 10.3390/sym14010022

Authors:
Liang Fang
Zaiying Zhou
Yiping Hong

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 &amp;lambda; tending to negative infinity for uncertain LSE or uncertain MLE. Such symmetry of residuals about the origin is reasonable in applications of experts&amp;rsquo; 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 &amp;lsquo;alternative Box-Cox model&amp;rsquo; 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 &amp;lambda; 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.

MDPI Publishing