2.4. Parameter Estimation
The gpower-normal models have three parameters to be estimated. They are related to the corresponding TN model parameters as it has been detailed in Section 2.1. We propose a combined profile likelihood and maximum likelihood approach to estimate the parameters. The five steps of the proposed estimation approach are described below: Given a data set represented by vector y=(y1, y2, …, yn), to obtain a profile likelihood for the power p, we consider a grid of values p0, p1, …, pk.
For each pj, 1≤j≤k the transformed data xpj are calculated as xpj=gpowery,pj.
Then, for each pj, the corresponding μpj and σpj are estimated, maximizing the likelihood function of the truncated normal variable.
Then, pj, μpj and σpj are used to obtain the log-likelihood function of y whose density was given by (2): lnfyy;μ,σ2,p=nln1K2πσpj2+∑i=1nlnpyi+yi2+1−lnyi2+1−12σ2∑i=1ndpyi,μpj.
Finally, p is chosen as the one that maximizes the log-likelihood in the grid: p^=max1≤j≤klnfyy;μpj,σpj2,pj
The method described above is applied in Section 3.2 and an implementation has been written in R language [15]. The codes are available from the authors upon request.