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2022, DOAJ (DOAJ: Directory of Open Access Journals)
This paper reviews works on Skellam distribution, its extensions and areas of applications. Available literature shows that this distribution is flexible for modelling integer data where they appear as count data or paired count data in the field of finance, medicine, sports, and science. Bivariate Skellam distribution, dynamic Skellam model and other extensions are also discussed and additional literature are provided.
Communications in Statistics - Theory and Methods, 2015
In this paper, we introduce a new distribution on Z 2 , which can be viewed as a natural bivariate extension of the Skellam distribution. The main feature of this distribution a possible dependence of the univariate components, both following univariate Skellam distributions. We explore various properties of the distribution and investigate the estimation of the unknown parameters via the method of moments and maximum likelihood. In the experimental section, we illustrate our theory. First, we compare the performance of the estimators by means of a simulation study. In the second part, we present two applications to a real data set and show how an improved fit can be achieved by estimating mixture distributions.
Symmetry, 2019
In this paper, we consider a stochastic representation of the epsilon–skew–Cauchy distribution, viewed as a member of the family of skewed distributions discussed in Arellano-Valle et al. (2005). The stochastic representation facilitates derivation of distributional properties of the model. In addition, we introduce symmetric and asymmetric extensions of the Cauchy distribution, together with an extension of the epsilon–skew–Cauchy distribution. Multivariate versions of these distributions can be envisioned. Bivariate examples are discussed in some detail.
Pakistan Journal of Statistics, 2012
A skew logistic distribution is proposed by considering a new skew function where the skew function is not a cumulative distribution function (cdf). Some of its distributional properties are derived. Its suitability in empirical modeling is investigated by comparative fitting of two real life data sets.
Brazilian Journal of Probability and Statistics, 2016
Following the methodology of Azzalini, researchers have developed skew logistic distribution and studied its properties. The cumulative distribution function in their case is not explicit and therefore numerical methods are employed for estimation of parameters. In this paper, we develop a new skew logistic distribution based on the methodology of Fernández and Steel and derive its cumulative distribution function and also the characteristic function. For estimating the parameters, Method of Moments, Modified Method of Moment and Maximum likelihood estimation are used. With the help of simulation study, for different sample sizes, the parameters are estimated and their consistency was verified through Box Plot. We also proposed a regression model in which probability of occurrence of an event is derived from our proposed new skew logistic distribution. Further, proposed model fitted to a well studied lean body mass of Australian athlete data and compared with other available competing distributions.
2013
• In this paper, we introduce a generalized skew logistic distribution that contains the usual skew logistic distribution as a special case. Several mathematical properties of the distribution are discussed like the cumulative distribution function and moments. Furthermore, estimation using the method of maximum likelihood and the Fisher information matrix are investigated. Two real data applications illustrate the performance of the distribution.
2015
In this paper we study a new class of skew-Cauchy distributions inspired on the family extended two-piece skew normal distribution. The new family of distributions encompasses three well known families of distributions, the normal, the two-piece skew-normal and the skew-normal-Cauchy distributions. Some properties of the new distribution are investigated, inference via maximum likelihood estimation is implemented and results of a real data application, which reveal good performance of the new model, are reported
Pakistan Journalof Statistics, 2020
We propose a new family of continuous distributions with two extra parameters named Transmuted Exponential-G family of distributions. We provide a special member for the new family of distributions. An explicit expression for some of its mathematical and structural properties such as reliability function, failure rate, ordinary moments, incomplete moments, generating function, Renyi entropy and order statistics were derived and presented. The method of maximum likelihood is used to estimate the parameters of the developed family of distributions. A simulation study is carried out to assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors. Real-life data are used to validate the robustness of the developed family of distribution.
2019
In this paper, we introduce a new distribution for positively skewed data by combining the Birnbaum-Saunders and centered skew-normal distributions. Several of its properties are developed. Our model accommodates both positively and negatively skewed data. Also, we show that our proposal circumvents some problems related to another Birnbaum-Saunders distribution based on the usual skew-normal model, previously presented in the literature. We derive both maximum likelihood and Bayesian inference, comparing them through a suitable simulation study. The convergence of the expectation conditional maximization (for maximum likelihood inference) and MCMC algorithms (for Bayesian inference) are verified and several factors of interest are compared. In general, as the sample size increases, the results indicate that the Bayesian approach provided the most accurate estimates. Our model accommodates the asymmetry of the data more properly than the usual Birnbaum-Saunders distribution, which is illustrated through real data analysis.
Mathematics and Statistics, 2023
We have introduced a novel continuous distribution known as the Klongdee distribution, which is a combination of the exponential distribution with parameter (θ α) and the gamma distribution with parameters (2, θ α). We thoroughly examined various statistical properties that provide insights into probability distributions. These properties encompass measures such as the cumulative distribution function, moments about the origin, and the moment-generating function. Additionally, we explored other important measures including skewness, kurtosis, C.V., and reliability measures. Furthermore, we explore parameter estimation using nonlinear least squares methods. The numerical results presented compare the unweighted and weighted least squares (UWLS and WLS) methods, maximum likelihood estimation (MLE), and method of moments (MOM). Based on our findings, the MLE demonstrates superior performance compared to other parameter estimation methods. Moreover, we demonstrate the application of this distribution within an actuarial context, specifically in the analysis of collective risk models using a mixed Poisson framework. By incorporating the proposed distribution into the mixed Poisson model and analyzing a real-life dataset, it has been determined that the Poisson-Klongdee model outperforms alternative models in terms of performance. Highlighting its capability to mitigate the problem of overcharges, the Poisson-Klongdee model has been proven to be a valuable tool.
Journal of Applied Statistics, 2020
A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.
Earthline Journal of Mathematical Sciences
A new mixed Poisson model is proposed as a better alternative for modelling count data in the presence of overdispersion and/or heavy-tail. The mathematical properties of the model were derived. The maximum likelihood estimation method is employed to estimate the model’s parameters and its applications to the three real data sets discussed. The model is used to model sets of frequencies that have been used in different literature on the subject. The results of the new model were compared with Poisson, Negative Binomial and Generalized Poisson-Sujatha distributions (POD, NBD and GPSD, respectively). The parameter estimates expected frequencies and the goodness-of-fit statistics under each model are computed using R software. The results show that the proposed PSD fits better than POD, NBD and GPSD for all the data sets considered. Hence, PSD is a better alternative provided to model count data exhibiting overdispersion property.
Statistics, 2014
A new one-parameter discrete distribution is introduced. Its mathematical properties and estimation procedures are derived. Four real data sets are used to show that the new model performs at least as well as the traditional one-parameter discrete models and other newly proposed two-parameter discrete models.
Complexity
There is an increasing interest in expanding the one-parameter Lindley distribution to two-parameter, three-parameter, and five-parameter. The univariate one-parameter Lindley distribution is still one of the most applicable distributions in data analysis especially in lifetime data. Modeling dependent random quantities required bivariate parametric probability distributions. This study presents a new bivariate three-parameter probability distribution called bivariate modified Lindley distribution. The one-parameter modified Lindley distribution is used as a base line to construct the new model. Its statistical properties including cumulative function, density function, marginals, moments, conditional distributions, and copula are discussed. Simulation is constructed to declare theoretical properties, to show the flexibility of the new model and to investigate the goodness of fit. Two sets of real data, financial data and UEFA Champion’s League data, are used to show the applicabili...
Computational Statistics & Data Analysis, 2011
In this article, we perform statistical inference on a skew model that belongs to a class of distributions proposed by . Specifically, we introduce two ways to represent this model by means of which moments and generation of random numbers can be obtained. In addition, we carry out estimation of the model parameters by moment and maximum likelihood methods. Asymptotic inference based on both of these methods is also produced. We analyze the expected Fisher information matrix associated with the model and highlight the fact that this does not have the singularity problem, as occurs with the corresponding information matrix of the skew-normal model introduced by . Furthermore, we conduct a simulation study to compare the performance of the moment and maximum likelihood estimators. Finally, an application based on real data is carried out.
Journal of Statistical Research of Iran, 2010
In this paper, we discuss a new generalization of univariate skew-Cauchy distribution with two parameters, we denoted this by GSC (λ 1 , λ 2 ), that it has more flexible than the skew-Cauchy distribution (denoted by SC (λ)), introduced by Behboodian et al. (2006). Furthermore, we establish some useful properties of this distribution and by two numerical example, show that GSC (λ 1 , λ 2 ) can fits the data better than SC (λ).
Journal of Statistical Theory and Practice, 2019
This paper introduced a five-parameter skewed Kotz (SK) distribution that may be viewed as a generalized skewed T distribution. Its mathematical properties are investigated, and parameters are estimated using the maximum likelihood method. The usefulness of this new distribution has been illustrated by deriving explicit formulae for the value-at-risk (VaR) and the average value-at-risk (AVaR). The obtained results are clearly generalizations of those that were established earlier by Dokov et al. (J Appl Funct Anal 3(1):189-208, 2008). On the other hand, simulation studies have been conducted and showed the accuracy of the VaR and AVaR computations. Furthermore, an application on financial returns of the Universal Health Services stock provided evidence that the SK distribution better fits the empirical distribution than both normal and skewed T distributions. The empirical study revealed the suitability of the SK distribution, specially for modelling data that fall within a small range, with a high excess kurtosis.
African Journal of Mathematics and Statistics Studies, 2024
In this paper, the record is set straight on the technique for the development of classical distributions, where a new model called the Sky-Log distribution is proposed as an illustrative example of the methodical approach. The statistical properties of the proposed distribution were derived, and the very many known generating functions exist for the distribution. Lionel Messi's football record data were analyzed to validate the essence of the proposed model. Finally, it was discovered that the proposed distribution sub-model, termed Sky-X distribution, and the exponential distribution, are exact model fit alternatives.
Statistics in Transition, 2024
This paper introduces a novel three-parameter skew-log-logistic distribution. The research involves the development of a new random variable based on Azzalini and Capitanio's (2013) proposition. Additionally, various statistical properties of this distribution are explored. The paper presents a maximum likelihood method for estimating the distribution's parameters. The density function exhibits unimodality with heavy right tails, while the hazard function exhibits rapid increase, unimodality, and slow decrease, resulting in a right-skewed curve. Furthermore, four real datasets are utilized to assess the applicability of this new distribution. The AIC and BIC criteria are employed to assess the goodness of fit, revealing that the new distribution offers greater flexibility compared to the baseline distribution.
Densities and Sampling for the Skellam Distribution.
Communications in Statistics: Case Studies, Data Analysis and Applications, 2019
In environmental studies, many data are typically skewed and it is desired to have a flexible statistical model for this kind of data. In this paper, we study a class of skewed distributions by invoking arguments as described by Ferreira and Steel (2006, Journal of the American Statistical Association, 101: 823-829). In particular, we consider using the logistic kernel to derive a class of univariate distribution called the truncated-logistic skew symmetric (TLSS) distribution. We provide some structural properties of the proposed distribution and develop the statistical inference for the TLSS distribution. A simulation study is conducted to investigate the efficacy of the maximum likelihood method. For illustrative purposes, two real data sets from environmental studies are used to exhibit the applicability of such a model.
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