Likelihood-Based Inference for the Asymmetric Beta-Skew Alpha-Power Distribution

Abstract

This paper introduces a new family of asymmetric distributions that allows to fit unimodal as well as bimodal and trimodal data sets. The model extends the normal model by introducing two parameters that control the shape and the asymmetry of the distribution. Basic properties of this new distribution are studied in detail. The problem of estimating parameters is addressed by considering the maximum likelihood method and Fisher information matrix is derived. A small Monte Carlo simulation study is conducted to examine the performance of the obtained estimators. Finally, two data set are considered to illustrate the developed methodology.

The Censored Beta-Skew Alpha-Power Distribution

“…For β = 0 y α = 1 the CBSAP model is reduced to the tobit model. In this case, following Martínez-Flórez et al [22] it can be seen that expected information matrix of the CBSAP model, say I CP (θ) is non-singular.…”

“…This model is joined to the bimodal models studied by Kim [18], Arnold et al [19], Gómez et al [9], and Bolfarine et al [10], among others. Other proposals involve models of multimodal type, among these, the flexible class of skew-symmetric distributions by Ma and Genton [20], the alpha-beta-skew-normal (ABSN) model by Shafiei et al [21] and, the asymmetric beta-skew alpha-power model by Martínez-Flórez et al [22]. In particular, the ABSN model by Shafiei et al [21] has pdf given by…”

“…Based on the alpha-power family of distribution and the BSN model, Martínez-Flórez et al [22] introduced a new asymmetric distribution useful for fitting a multimodal data set with high/low asymmetry. The new model, which is named as the beta-skew alpha-power (BSAP) distribution has non-singular information matrix and pdf given by…”

“…Thus, the maximum likelihood estimator for θ can be obtained numerically via iterative algorithms such as, the Newton-Raphson or quasi-Newton. The elements of the observed and expected Fisher information matrix are obtained directly from the respective elements of the BSAP model, see Martínez-Flórez et al [22], by using z i = (ln(y i ) − µ)/σ instead of z i = (y i − µ)/σ, hence, the matrix I l (θ) of the LBSAP(θ) is non-singular. Thus, the covariances matrix of the estimators vector of the LBSAP distribution is given by V(θ) = I −1 l (θ).…”

“…The model in ( 32) is denoted by Y ∼ CLBSAP(θ), with θ = (µ, σ, β, α) . Estimation for the parameters vector is carried out in a similar way as for the censored BSAP(θ) model by Martínez-Flórez et al [22], where the score functions and the information matrices have the same structure and it is only necessary to change z i = (ln(y i ) − µ)/σ instead of z i = (y i − µ)/σ and z c = (ln(c) − µ)/σ instead of z c = (c − µ)/σ.…”