Random number generation and estimation with the bimodal asymmetric power-normal distribution

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Recently, Bolfarine et al. [Bimodal symmetric-asymmetric power-normal families. Commun Statist Theory Methods. Forthcoming. doi:10.1080/03610926.2013.765475] introduced a bimodal asymmetric model having the normal and skew normal as special cases. Here, we prove a stochastic representation for their bimodal asymmetric model and use it to generate random numbers from that model. It is shown how the resulting algorithm can be seen as an improvement over the rejection method. We also discuss practical and numerical aspects regarding the estimation of the model parameters by maximum likelihood under simple random sampling. We show that a unique stationary point of the likelihood equations exists except when all observations have the same sign. However, the location-scale extension of the model usually presents two or more roots and this fact is illustrated here. The standard maximization routines available in the R system (Broyden–Fletcher–Goldfarb–Shanno (BFGS), Trust, Nelder–Mead) were considered in our implementations but exhibited similar performance. We show the usefulness of inspecting profile loglikelihoods as a method to obtain starting values for maximization and illustrate data analysis with the location-scale model in the presence of multiple roots. A simple Bayesian model is discussed in the context of a data set which presents a flat likelihood in the direction of the skewness parameter.