Maximum Likelihood Estimation of Parameters in the Power Normal Distribution

The Power-Normal (PN) distribution was first introduced by Goto et al. in the context of modeling original observations following the application of the inverse Box–Cox (BC) transformation. This family includes the normal and log-normal distributions as special cases. In this paper, we present results for the exact Maximum Likelihood Estimation (MLE) of the PN distribution and compare them with those obtained using the truncated MLE method originally proposed by Box and Cox, implemented here by jointly maximizing all three parameters. The algorithm we propose focuses on the interval [0,1] and employs a partitioning strategy to generate initial values for the Newton–Raphson (N-R). For the exact MLE, we consider two scenarios: one in which all parameters are estimated simultaneously, and another in which only λ is optimized. We find the first approach to outperform the second overall, although both yield similar accuracy for the estimation of λ. In contrast, the truncated MLE method performs worse than the exact MLE in estimating λ, but better in the average Mean Square Error (MSE) statistic than the exact maximizing in λ alone.