Modified slash distribution

A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function

This work presents a new distribution that allows modeling data from a random variable with non-negative values. The new family is defined by a stochastic representation of a scaled mixture of a random variable with a Gompertz distribution (G) and a random variable with a uniform distribution on the interval (0,1). The result of this gives rise to a new random variable with a Slash Gompertz (SG) distribution that is more flexible than the Gompertz distribution, that is, it better models atypical data, presenting tails heavier than the Gompertz distribution. The density and some general properties of the resulting family are studied, including its moments and kurtosis coefficient. The inference of the parameters is carried out using the method of moments and maximum likelihood. Finally, illustrations of particular cases of this family are shown, adjusting in this case two sets of real data and estimating the parameters by maximum likelihood, where it is verified that this new family of distributions fits the reliability function better than the distributions of Gompertz (G), Slash Birnbaum Saunders (SBS), Slash Weibull (SW), and Gompertz-Verhults (GV).

A New Generalization of the Student’s t Distribution with an Application in Quantile Regression

In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student’s t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student’s t distribution, student’s t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student’s t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis.

A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model

Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results.

An α-Monotone Generalized Log-Moyal Distribution with Applications to Environmental Data

Modeling environmental data plays a crucial role in explaining environmental phenomena. In some cases, well-known distributions, e.g., Weibull, inverse Weibull, and Gumbel distributions, cannot model environmental events adequately. Therefore, many authors tried to find new statistical distributions to represent environmental phenomena more accurately. In this paper, an α-monotone generalized log-Moyal (α-GlogM) distribution is introduced and some statistical properties such as cumulative distribution function, hazard rate function (hrf), scale-mixture representation, and moments are derived. The hrf of the α-GlogM distribution can form a variety of shapes including the bathtub shape. The α-GlogM distribution converges to generalized half-normal (GHN) and inverse GHN distributions. It reduces to slash GHN and α-monotone inverse GHN distributions for certain parameter settings. Environmental data sets are used to show implementations of the α-GlogM distribution and also to compare its modeling performance with its rivals. The comparisons are carried out using well-known information criteria and goodness-of-fit statistics. The comparison results show that the α-GlogM distribution is preferable over its rivals in terms of the modeling capability.

A Locally Both Leptokurtic and Fat-Tailed Distribution with Application in a Bayesian Stochastic Volatility Model

In the paper, we begin with introducing a novel scale mixture of normal distribution such that its leptokurticity and fat-tailedness are only local, with this “locality” being separately controlled by two censoring parameters. This new, locally leptokurtic and fat-tailed (LLFT) distribution makes a viable alternative for other, globally leptokurtic, fat-tailed and symmetric distributions, typically entertained in financial volatility modelling. Then, we incorporate the LLFT distribution into a basic stochastic volatility (SV) model to yield a flexible alternative for common heavy-tailed SV models. For the resulting LLFT-SV model, we develop a Bayesian statistical framework and effective MCMC methods to enable posterior sampling of the parameters and latent variables. Empirical results indicate the validity of the LLFT-SV specification for modelling both “non-standard” financial time series with repeating zero returns, as well as more “typical” data on the S&P 500 and DAX indices. For the former, the LLFT-SV model is also shown to markedly outperform a common, globally heavy-tailed, t-SV alternative in terms of density forecasting. Applications of the proposed distribution in more advanced SV models seem to be easily attainable.

Slashed Lomax Distribution and Regression Model

In this article, the slashed Lomax distribution is introduced, which is an asymmetric distribution and can be used for fitting thick-tailed datasets. Various properties are explored, such as the density function, hazard rate function, Renyi entropy, r-th moments, and the coefficients of the skewness and kurtosis. Some useful characterizations of this distribution are obtained. Furthermore, we study a slashed Lomax regression model and the expectation conditional maximization (ECM) algorithm to estimate the model parameters. Simulation studies are conducted to evaluate the performances of the proposed method. Finally, two sets of data are applied to verify the importance of the slashed Lomax distribution.

An Asymmetric Distribution with Heavy Tails and Its Expectation–Maximization (EM) Algorithm Implementation

In this paper we introduce a new distribution constructed on the basis of the quotient of two independent random variables whose distributions are the half-normal distribution and a power of the exponential distribution with parameter 2 respectively. The result is a distribution with greater kurtosis than the well known half-normal and slashed half-normal distributions. We studied the general density function of this distribution, with some of its properties, moments, and its coefficients of asymmetry and kurtosis. We developed the expectation–maximization algorithm and present a simulation study. We calculated the moment and maximum likelihood estimators and present three illustrations in real data sets to show the flexibility of the new model.

On Generalized Slash Distributions: Representation by Hypergeometric Functions

The popular concept of slash distribution is generalized by considering the quotient Z = X/Y of independent random variables X and Y, where X is any continuous random variable and Y has a general beta distribution. The density of Z can usually be expressed by means of generalized hypergeometric functions. We study the distribution of Z for various parent distributions of X and indicate a possible application in finance.

Generalized Modified Slash Birnbaum–Saunders Distribution

In this paper, a generalization of the modified slash Birnbaum–Saunders (BS) distribution is introduced. The model is defined by using the stochastic representation of the BS distribution, where the standard normal distribution is replaced by a symmetric distribution proposed by Reyes et al. It is proved that this new distribution is able to model more kurtosis than other extensions of BS previously proposed in the literature. Closed expressions are given for the pdf (probability density functio), along with their moments, skewness and kurtosis coefficients. Inference carried out is based on modified moments method and maximum likelihood (ML). To obtain ML estimates, two approaches are considered: Newton–Raphson and EM-algorithm. Applications reveal that it has potential for doing well in real problems.

Modified Slash Lindley Distribution

In this paper we introduce a new distribution, called the modified slash Lindley distribution, which can be seen as an extension of the Lindley distribution. We show that this new distribution provides more flexibility in terms of kurtosis and skewness than the Lindley distribution. We derive moments and some basic properties for the new distribution. Moment estimators and maximum likelihood estimators are calculated using numerical procedures. We carry out a simulation study for the maximum likelihood estimators. A fit of the proposed model indicates good performance when compared with other less flexible models.