Underlying Assumptions of the Power-Normal Distribution

In the power transformation (Box & Cox, 1964), parameters are usually estimated under the assumption that the transformed distribution is a normal distribution even though the transformed distribution is a truncated normal distribution. In the present paper, we evaluate the asymptotic influence of the truncation on estimation of the parameters of the power-normal distribution (Goto, Uesaka, & Inoue, 1979), which specifies original observations before the application of power transformation. Then we demonstrate that when the degree of the truncation of the transformed distribution is large, the parameter estimates based on the ordinary estimation method which ignore the truncation might have large bias through the simulation study and the case study.