There has been a growing interest in the construction of flexible parametric families of distributions that exhibit asymmetry and peakedness differing from those of symmetric distributions (1–3). Many of these methods center around overcoming the assumptions of normality found in the empirical analysis of many parametric models.

An empirical analysis in various studies suggests that the assumption of normality of real data is often untenable (4, 5), and asymmetry is commonly observed (6, 7). It is highly acknowledged that data with heavy-peaked distribution are encountered in the empirical analysis (8), as is asymmetric distribution (9, 10). In all cases, it is important to adopt a flexible distribution that can directly address asymmetry and peakedness (6, 9).

There has been a different approach to develop asymmetric counterparts of symmetric distributions. Many of these approaches centered on overcoming assumptions of normality (11, 12). In many works of literature, asymmetry is achieved via the transformation of the skewing function (9, 10), which lacks a wide range of skewness and kurtosis. Moreover, the technique of creating asymmetric counterparts of symmetric distributions has a longer history (13–15).

The approach that is commonly considered for constructing classes of asymmetric distributions from symmetric distribution is authored by Azzalini (16, 17). The initial idea appeared in O’hagan and Leonard (18) in the context of base distribution. Azzalini (16, 17) introduced asymmetric distributions called skew-normal (SN). The idea was further extended in Azzalini (13) introduced multivariate asymmetry distributions. An ideal class of distributions obtained from this methodology includes symmetric distributions, mathematical tractability, and a wide range of skewness and kurtosis. The theoretical and statistical properties of the methodology have been studied by various researchers (2, 3, 19, 20).

As noted in (5, 12) the generalized error distribution (GED) has short tails, making it unsuitable for modeling data with heavier tails. One method to solve this problem, as suggested in Azzalini (16), is to use an asymmetric pdf with flexible tails and excess kurtosis. Azzalini’s methodology generates distributions with flexible tails and excess kurtosis.

In this study, we follow Azzalini’s methodology to introduce a new distribution that is flexible enough for modeling data with heavier tails and excess kurtosis. More data with heavier tails and excess kurtosis are adequately modeled to the distribution and play an important role in this context. This new distribution is called asymmetric generalized error distribution (AGED) and is denoted by {AGED(α):αϵℝ}, where α represents the asymmetry parameter so that AGED(α=0) corresponds to the generalized error distribution. We outline some properties of the distribution, provide a graphical representation of the distribution, and discuss some inferences.

The GED is a symmetric and unimodal member of the exponential family of distribution introduced by Subbotin (22) and has been used by different authors with different parameterizations (23, 24). A random variable X have a generalized error distribution if its probability density function (pdf) is expressed by (21):

where μ∈ℝ is the location parameter, σ∈ℝ+ is the scale parameter, and β∈ℝ+ is the shape parameter. Here, Γ(.) is the Euler gamma function. We denote it by X~GED(x;μ,σ,β).

It is convenient to work with the alternative expression given in Eq. 1, which allows for mean zero and variance unity (25). The variance of the GED is a function of β (26, 27). To rescale its variance, a scaling parameter η is introduced, and a substitution is made for σ→ση1/β in Eq. 1 to get the following equivalent pdf (25):

Or

In this section, the method of generating an asymmetric distribution from a symmetric distribution is presented to develop the new asymmetric generalized error distribution (AGED). Here, the method of constructing classes of asymmetric distributions suggested by Azzalini (16, 17) is used. The authors introduced a methodology that can be used to derive an asymmetric distribution from an existing symmetric distribution. This is expressed in Proposition 1.

Graphs of probability density and cumulative distribution function of AGED are illustrated in Figure 1 for some values of parameters that give possible shapes of function. The asymmetry parameter α controls the magnitude of the asymmetry exhibited by the probability density function. The AGED can take a number of forms, including symmetric, near symmetric, and asymmetric. As α→+∞, the asymmetric generalized error distribution converges in distribution to the half asymmetric generalized error distribution, and for α→0, the distribution reduced to the generalized error distribution. However, for α=0.5, the generalized error distribution converges to the asymmetric generalized error distribution (Figure 2).

Extremely illustrated properties instantly follow from definition 1 and Figure 1 is as follows:

If X~AGED(μ,σ,β,α), then the following properties are concluded directly from theorem (1) and Figures 1, 3, 4:

Let X be a random variable from AGED with pdf defined in Eq. 5, the r′th moments of the random variable X is obtained as follows (36):

where Xg~GED(μ,σ,β). However, for a random variable X~AGED(μ,σ,β,α) with pdf given in Eq. 5, it follows the form of the binomial theorem:

Consider a random variable Z~AGED(0,1,β,α) with pdf in Eq. 5, then

where W=zβ and similarly,

and k′th moment of Z becomes:

Therefore, for a random variable X~AGED(μ,σ,β,α), the r′th moment for a random variable X can be defined as:

In particular, the first four moments of a random variable X are defined as:

The skewness and kurtosis of the asymmetric generalized error distribution are functions of μ,σ,β, and α. However, the actual equations in terms of μ,σ,β, and α are quite expansive. In compact form, we can write the variance (ω), skewness (γ3), and kurtosis (γ4) of X using the standardized moments of Z and defined as:

We perform a brief comparison illustrating that the tails of the AGED are heavier than those of the GED. Table 1 noted that the AGED has much heavier tails than the GED, as Figure 1 depicts the AGED for different values of parameters.

Similarly, the degree of asymmetry and peakedness of the AGED for different values of β and α are shown, and for small values of β, the kurtosis coefficient increases in the AGED. The ranges of both coefficients are smaller in GED. Thus, the AGED is more flexible for modeling data with larger coefficients of asymmetry and kurtosis.

In this section, we go over how to estimate the AGED parameters using the maximum likelihood approach.

To establish the performance of an estimator, we conduct a simulation study. We choose parameter values that are consistent with the graph depicted in Figure 1. The effect of various shape parameter values on the distribution is shown in Figure 1.

The simulations of the AGED are done based on the accept-reject method. Three designs are presented and used to generate random samples from AGED for a parameter considered. The designs for parameters of the AGED considered are (0,1,2,α), (0,1,1,α), and (0,1,3,α) for designs 1, 2, and 3, respectively. We use three values of the asymmetry parameter, α=0.5,1,2, to cover the cases where the distribution is asymmetric. The realization plot, histogram, and density plot are assessed.

The main finding of this study is the derivation of a new probability distribution that reveals interesting properties, especially with various asymmetry and kurtosis behaviors; we call it the asymmetric generalized error distribution (AGED). AGED is a new contribution to the field of statistical theory and provides a more flexible pdf, cdf, and hazard function than the base distribution. The mathematical and statistical features of the distribution are derived and discussed. To estimate the distribution parameters, maximum likelihood estimators are derived. A simulation study is done using the acceptance-rejection algorithm. In the applications, the datasets have high kurtosis and skewness. The criteria indicate that the AGED provides better fits to the datasets. This implies that the new distribution is a good alternative for modeling data with asymmetric and excess kurtosis behavior. We expect that the distribution can be applied to many more real datasets, and the findings of the study can be used as the basis for future research in the field.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

TA: Conceptualization, Investigation, Software, Data curation, Formal analysis, Methodology, Visualization, Writing – original draft, Writing – review & editing. AG: Conceptualization, Software, Data curation, Formal analysis, Methodology, Visualization, Writing – original draft, Writing – review & editing.