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Asymmetric probability distributions have been widely studied by various authors in recent decades. Special interest has been had families of flexible distributions with the capability to have into account degree of skewness and kurtosis greater than the classical distributions widely known in statistical theory. While, most of the new distributions _t unimodal data, and a few fit bimodal data, in the bimodal proposals, singularity problems have been found in the information matrices. Therefore, in this paper, extensions of the alpha-power family of distributions are developed, which have non-singular information matrix. The new proposals are based on the bimodal-normal and bimodal elliptical skew-normal distributions. These new extensions allow modeling asymmetric bimodal data, which are commonly found in several areas of scientific interest. The properties of these new distributions of probability are also studied in detail, and the statistical inference process is carried out to estimate the parameters of the proposed models. The stochastic convergence for the maximum likelihood estimator (MLE) vector can be found due to the non-singularity of the expected information matrix in the corresponding support. We also introduced extensions of the asymmetric bimodal normal and bimodal elliptical skew-normal models for the situations in which the data present censorship. A small simulation study to evaluate the properties of the MLE is also presented and, finally, two applications to real data set are presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

Assumption of normality in statistical analysis had been a common practice in many literature, but in the event where small sample is obtainable, then normality assumption will lead to erroneous conclusion in the statistical analysis. Taking a large sample had been a serious concern in practice due to various factors. In this paper, we further derived some inferential properties for log student's t-distribution (simply log-t distribution) which makes it more suitable as substitute to log-normal when carrying out analysis on right-skewed small sample data. Mathematical and Statistical properties such as the moments, cumulative distribution function, survival function, hazard function and log-concavity are derived. We further extend the results to case of multivariate log-t distribution; we obtained the marginal and conditional distributions. The parameters estimation was done via maximum likelihood estimation method, consequently its best critical region and information matrix were derived in order to obtain the asymptotic confidence interval. The applications of log-t distribution and goodness-of-fit test was carried out on two dataset from literature to show when the model is most appropriate. [ABSTRACT FROM AUTHOR]

The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing ap- proach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the gen- eralized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also ob- tained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution. [ABSTRACT FROM AUTHOR]