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This paper explores bimodal skew-symmetric distributions, a versatile family of distributions characterized by parameters that control asymmetry and kurtosis. These distributions encapsulate both symmetrical and well-known asymmetrical behaviors. A simulation study evaluates the model's estimation accuracy, detailing the score function and the robustness of the observed information matrix, which is proven to be non-singular under specific conditions. We apply the bimodal skew-normal model to protein data from cancer cells, comparing its performance against four established distributions supported on the entire real line. Results indicate superior performance by the proposed model, underscoring its potential for enhancing analytical precision in biological research. [ABSTRACT FROM AUTHOR]

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. Therefore, it presents an alternative for modeling this type of data. Several mathematical and statistical properties of this distribution have been derived, such as moments, hazard function, the Bonferroni curve, and entropy. Moreover, we investigate the characterizations of the proposed distribution based on its hazard function. Parameter estimation has been performed using both the maximum likelihood method and method of the moments. Because of this, we were able to determine the best critical region and the information matrix, facilitating the calculation of asymptotic confidence intervals. A simulation study is presented to analyze the behavior of the obtained estimators for different sample sizes. To demonstrate the suitability of the proposed distribution, applications and goodness-of-fit tests have been performed on two practical data sets. [ABSTRACT FROM AUTHOR]

Recent years have shown growth in the potential applications of extensions of skew and half-normal distributions. In this paper, we provide and study, in detail, an extended class of such distributions. The presented distribution has a scale parameter and two shape parameters. The motivations behind the development of this distribution are: 1- the probability distribution function has skewed, unimodal and bathtub shapes with different styles and the hazard rate function is increasing with various shapes, which makes this distribution flexible enough to analyze reliability data; 2- it can be expressed as a mixture of the Generalized Half-Normal distribution and the new weighted Generalized Half-Normal distribution. A number of statistical results are derived. Monte Carlo simulation analysis is carried out withMaximum Likelihood Estimation to assess the performance of this technique for a variety of the distribution parameter values. The power of the distribution is demonstrated with real applications using two reliability datasets from the industry. The results reveal that the proposed distribution outperformed the Generalized Lindley, Exponentiated Exponential, Gamma, Alpha-Skew-Normal, the Half-Alpha-Skew-Normal, and the Modified Generalized Half-Normal distributions. [ABSTRACT FROM AUTHOR]

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005) and Arnold et al. (2009) to the exponentiated normal distribution studied in Durrans (1992), 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 byBPN(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. [ABSTRACT FROM AUTHOR]

This paper focuses on the development of a new extension of the generalized skew-normal distribution introduced in Gómez et al. [Generalized skew-normal models: properties and inference. Statistics. 2006;40(6):495–505]. To produce the generalization a new parameter is introduced, the signal of which has the flexibility of yielding unimodal as well as bimodal distributions. We study its properties, derive a stochastic representation and state some expressions that facilitate moments derivation. Maximum likelihood is implemented via the EM algorithm which is based on the stochastic representation derived. We show that the Fisher information matrix is singular and discuss ways of getting round this problem. An illustration using real data reveals that the model can capture well special data features such as bimodality and asymmetry. [ABSTRACT FROM PUBLISHER]

We consider a distribution obtained by combining two well-known mechanisms for generating skewed distributions. In this manner we arrive at a flexible model which subsumes and extends several skew distributions which have been discussed in the literature. One approach to the problem of generating skewed distributions was first popularized by Azzalini [A class of distributions which includes the normal ones. Scand J Stat. 1985;12:171–178]. The single constraint skew normal distribution that was studied by Azzalini is of the formwhere φ and Φ denote, respectively, the standard normal density and distribution function and α∈ℝ is a skewing parameter. Multiple constraint variations of this distribution have also been considered. The second skewing approach that we will consider was proposed by Mudholkar and Hutson [The epsilon-skew-normal distribution for analyzing near-normal data. J Statist Plann Inference. 2000;83:291–309] and called an epsilon-skew-normal distribution. The combination of an Azzalini mechanism with that of Mudholkar and Hutson is investigated in this paper with special focus on the distributions obtained using the standard normal as the base distribution. The resulting flexible model includes both unimodal and bimodal cases and can be expected to fit a wider variety of data configurations than either of the models involving a single skewing mechanism. Distributional and inferential properties of the doubly skewed model are discussed and the model is used to obtain improved fits to two well-known data sets. [ABSTRACT FROM AUTHOR]

The Fisher information matrix for Generalized skew-normal (GSN) distribution is derived. The expressions for the elements of the matrices require of integrals that are solved numerically using a suitable software. [ABSTRACT FROM AUTHOR]