A new class of unimodal asymmetric distributions is introduced to the unit interval and these distributions are useful for modeling data of percentages, proportions and fractions. Therefore, we propose the unit-half-normal distribution as a contribution to the earlier path and investigate some of its mathematical properties. The maximum likelihood estimator is obtained with a comprehensive inference. This new class of distributions belongs to the exponential family, hence the uniformly minimum variance unbiased estimator of the distribution parameter is obtained. The distribution represents a power alternative to the unit interval distributions, namely the beta, Kumaraswamy and other recent ones. We investigate a small simulation study to analyze the behavior of the obtained estimators for different sample sizes. Moreover, we illustrate the goodness of fit of the proposed model for image data. Lastly, we describe a procedure of incorporating covariates into regression analysis of the proposed distribution.

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim ( 2005 Kim, H.J. (2005). On a class of two-piece skew-normal distributions. Statist.: J. Theoret. Appl. Statist. 39:537–553. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) and Arnold et al. ( 2009 Arnold, B.C. , H.W. Gómez, and H.S. Salinas. (2009). On multiple constraint skewed models. Statist. J. Theoret. Appl. Statist. 43: 279–293. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) to the exponentiated normal distribution studied in Durrans ( 1992 Durrans, S.R. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655. [Crossref], [Web of Science ®] [Google Scholar] ), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.

Many lifetime distribution models have successfully served as population models for risk analysis and reliability mechanisms. We propose a novel estimation procedure of stress–strength reliability in the case of two independent unit-half-normal distributions can fit asymmetrical data with either positive or negative skew, with different shape parameters. We obtain the maximum likelihood estimator of the reliability, its asymptotic distribution, and exact and asymptotic confidence intervals. In addition, confidence intervals of model parameters are constructed by using bootstrap techniques. We study the performance of the estimators based on Monte Carlo simulations, the mean squared error, average bias and length, and coverage probabilities. Finally, we apply the proposed reliability model in data analysis of burr measurements on the iron sheets.

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.