Abstract
The Box-Cox (BC) transformation is widely used in data analysis for achieving approximate normality in the transformed scale. The transformation is only possible for non-negative data. This positiveness requirement implies a truncation to the distribution on the transformed scale and the distribution in the transformed scale is truncated normal. This fact has consequences for the estimation of the parameters specially if the truncated probability is high. In the seminal paper Box and Cox proposed to estimate parameters using the normal distribution which in practice means to ignore any consequences of the truncation on the estimation process. In this work we present the framework for exact likelihood estimation on the PN distribution to which we call method m1 and how to calculate the parameters estimates using consistent estimators. We also present a pseudo-Likelihood function for the same model not taking into account truncation and allowing to replace parameters µ and s for their estimates. We call m2 to this estimation method. We conclude that for cases where the truncated probability is low both methods give good estimation results. However for larger values of the truncated probability the m2 method does not present the same efficiency. © 2024 American Institute of Physics Inc.. All rights reserved.

Abstract
In an earlier work we described and applied a methodology to find an adequate distribution for Nearly Gaussian (NG) random variables. In this work, we compare two different methods, m1 and m2 to estimate a power transform parameter for NG random variables. The m1 method is heuristic and based on sample kurtosis. Herein, we describe and apply it using a new reduced data set. The second method m2 is based on the maximization of a pseudo-log-likelihood function. As an application, we compare the performance of each method using high power statistical tests for the null hypothesis of normality. The data we use are the daily errors in the forecasts of maximum and minimum temperatures in the city of Porto. We show that the high kurtosis of the original data is due to high correlation among data. We also found that although consistent with normality the data is better fitted by distributions of the power normal (PN) family than by the normal distribution. Regarding the comparison of the two parameter estimation methods we found that the m1 provides higher p-values for the observed statistics tests except for the Shapiro-Wilk test.

Abstract
The Power Normal (PN) family of distributions is obtained by inverting the Box-Cox (BC) transformation over a truncated normal (TN) (or for some cases normal) random variable. In this paper we explore the PN distribution. We give a formula for the ordinary moments and considering the bivariate PN (BPN) distribution we calculate the marginal and conditional probability density functions (pdf). We prove that they are not univariate PN distributed. We also calculate the correlation curve and we fit a power law model.

Abstract

Abstract