The Alpha-Beta Skew Generalized 𝒕 Distribution: Properties and Applications

In this paper, we propose a new distribution, namely alpha-beta-skew generalized t distribution. The proposed distribution is really flexible and includes as special models some important distributions like Normal, t-student, Cauchy and etc as its marginal component distributions. It features a probability density function with up to three modes. The moment generating function as well as the main moments are provided. Inference is based on a usual maximum-likelihood estimation approach and a small Monte Carlo simulation is conducted for studying the asymptotic properties of the maximum-likelihood estimate. The usefulness of the new model is illustrated in a real data.


Introduction
Traditionally, the normality assumption is one of the conditions in statistical procedures.However, in most cases of real-life problems, the normality assumption has not established and non-normal distributions for modeling data sets having skewness and/or kurtosis are more prevalent, see for example (Tiku and eatl,2011) and (Celik and eatl , 2015).Therefore, the construction of the non-normal distributions has been an enormous interest and attracted the attention of researchers.The Generalized t distribution was defined by (McDonald and Newey,1998) to develop a partially adaptive M-regression procedure.The procedure includes many other estimation methods such as least squares, least absolute deviation and   .It has been followed up more recently by (Theodossiou,1998) and (Arslan and Genç,2003).Suppose the random variable X have the generalized t distribution, then the probability density function (pdf) is,   (, , , ) =  2 1/2 (1/,) ( where , ,  are distributional parameters,  corresponds to the standard deviation,  and  are parameters corresponding to the shape of distribution, (⋅,⋅)is the Beta function and denoted by  ∼ (, , ).The shape parameters  and  control the tails of the distribution.Larger values of  and  are associated with thinner tails of the distribution.
Similarly, smaller values of the shape parameters correspond to thicker tails.Thus, the GT distribution is useful in accommodating both leptokurtic and platykurtic symmetric unimodal distributions.
The GT distribution has several important subdistributions as special or limiting cases of the shape parameters.For example, for  = 2, we get the usual t distribution with 2 degrees of freedom, and for  → ∞ and  → ∞, we get the uniform and power exponential distributions, respectively.Figure 0 shows the distribution tree of the GT distribution family.
Figure 1: Distribution tree of the GT distribution family Skew symmetric distributions is a rapidly growing and they have attracted significant attention in climatology, economics, finance and other areas of the sciences.A random variable  is said to have the skew symmetric distribution if its pdf is given by where −∞ <  < ∞ and  and  are valid pdf and cdf (respectively) of a distribution symmetric around 0 (see, for example, (Gupta and eatl,2002) ).If one takes  in Equation (2) to be given as Equation (1) then one would have the skew generalized t distribution.The calculation of the mathematical properties of this distribution and their applications could be find in (Theodossiou,1998) and (Nadarajah,2008).Recently (Acitas,2015), introduced a new class of skew generalized t distributions called alpha-skew generalized t distribution, with the pdf,   (; , , ) = (1−) ) () The aim of this paper is to introduce a new family of distributions as an extension of alpha skew generalized t distribution with the pdf (3).The rest of this paper is organized as follows: In Section 2, the new family of distributions is defined and studied in details.In Section 3, the random variable generating method has been investigated.
In the following, without loss of generality, we assume  and  equals 0 and 1 respectively.When  tends to ±∞ in equation ( 4), the pdf becomes It is easy to see that the function given in equation ( 5) is also a pdf.Thus, this limiting case of ABSGT is called a bimodal generalized t (BGT) distribution.BGT is symmetric and bimodal distribution.It can be seen as an extension of Elal-Olivero's bimodal normal distribution.Proposition 2.4 expresses the cdf of Eqution 4 in terms of the incomplete beta function ratio.The cdf of generalized t distribution is obtained by (Nadarajah,2008) and (Acitas,2015) by following the lines of (Nadarajah,2008), obtained the cdf of ASGT distribution.In the proposition 2.4 we get the cdf of ABSGT by following the similar lines.Proposition 2.4 Let  ∼ (, , , ) then the cdf of random variable Z is given as, and, where   (, ) denotes the incomplete beta function.
Proof.The cdf of ABSGT is obtained as shown below.
Proof.We have,   We will refer to a maximum-likelihood estimate as any point in the parameter space at which the likelihood function has the global maximum.Taking the derivatives of equation ( 8) with respect to the parameters and setting them to zero yield the following likelihood equations: Note that the likelihood equations are not in a simple closed form.Therefore, the maximumlikelihood estimators have to be calculated through some numerical procedures, for example, the Newton-Raphson and the L-BFGS-B (limited memory Broyden-Fletcher-Goldfarb-Shanno method to solve the bound constrained optimization problem) methods.
The programs such as R provide computing routines for solving such nonlinear optimization problems.A few words can be said about the initial guesses of the parameters by visual inspection as follows.We first draw the histogram of the data set under study and then draw some pdfs over it.We then search the best, that is, the most likelihood, superimposed fit since it is important to choose near values to the true values.The parameter values in the best fit can be chosen as the initial guesses.
In the rest of this section, a small Monte Carlo simulation experiment is conducted to evaluate the maximum likelihood All results are obtained from 5000 Monte Carlo replications and the simulations were carried out using the package 'HI' and 'stats' in statistical software R.

Real Data
In this section, we consider two real data sets to model with the ABSGT distribution.The first data set is an example of unimodal data called Roller data.Faithful geyser data is the second one which is bimodal.All data analyses were conducted in R program.The maximum-likelihood estimators for all the parameters were calculated using the optim function with the L-BFGS-B method in R program.Then, we find the inverse of the resulting Hessian matrix using the solve function.The square roots of the diagonal elements of this inverse matrix give the standard errors of the estimates.We compared the proposed distribution  with alpha-skew generalized t distribution of (Wahed and ali,2001) and  of (Elal-Olivero,2010).Akaike Information Criterion (AIC) is used for model comparison.Further, since are nested models the likelihood ratio (LR) test is used to discriminate between them.The LR test is carried out to test the following hypothesis:  0 :  = 0, that is the sample is drawn from (, , , , ) against the alternative  1 :  ≠ 0, that is the sample is drawn from (, , , , , ).Aplication 5.1 In this study, we model roller data by using ABSGT distribution.This data set has 1,150 observations which are available at http://lib.stat.cmu.edu/jasadata/laslettwebsite, see also (Gómez,2011) and (Acitas,2015) who analyzed the same data.ML estimates of the parameters and AIC values are given in Table 2.By comparing the AICs, one can see the ABSLG distribution is the best among the fitted models.AIC values suggest that ABSGT distribution is more reliable than ASN and ASGT distributions for roller data.It is also obvious from Figure 3 that ASGT provides a substantially good fitting than ASGT distribution.Results obtained from AIC are also supported by the following likelihood ratio test (LRT) given below.The value of LR test statistics is 42.1 which exceed the 95% critical value.Thus there is evidence in favors of the alternative hypothesis that the sampled data comes from  not from .Note that (Acitas,2015) conclude that the ASGT distribution yields a better than the ASN.Aplication 5.2 Faithful geyser data includes 272 observations which denote the waitingtime between eruptions and the duration-time of these eruptions for Old Faithful geyser in Yellow National Park, Wyoming, USA.This popular data is available at Rsystem and see also (Arellano-Valle,2010) in which it is indicated that data is negatively skewed and bimodal.In this study, we use ABSGT distribution to model this popular data and compare results with ASGT distribution.ML estimates of the parameters and AIC values for ABSGT, ASGT and ASN distributions are given in Table 3.The histogram and the fitted density for Faithful geyser data are given in Figure 4.The value of LR test statistics is 13.61 which exceed the 95% critical value.Thus, there is evidence in favors of the alternative hypothesis that the sampled data comes from  not from .

Figure 2 : 1 2+𝑑
Figure 2: The pdf of the (, ) distribution for some selected values of the parameters.MomentsIn this subsection, moments of ABSGT distribution are derived.As a result, the skewness and the kurtosis measures are given.Here, it should be realized that the even and the odd moments of ABSGT distribution can be obtained by using the moments of GT distribution.This is because of the fact that the pdf of ABSGT distribution includes the pdf of GT distribution.The following proposition gives for both even and odd n values of (  ) based on this phenomena where  ∼ (, , , ).Proposition 3.1 Let  ∼ (, , , ) then for  ∈ ,  2 = ( 2 ) =

Figure 3 :
Figure 3: The kurtosis function of  distribution function with p = 2 and q = 10.

Figure 4 :
Figure 4: The skewness function of  distribution function with p = 4 and q = 25.

Figure 5 :
Figure 5: The mean square error of  ̂ and  ̂ vesus .Figures 5 represents the MSE of the parameters  and  for fixed values of another parameters in the model.As can be seen in the Figures 5 , by increasing the sample size, the MSE of estimated parameters decreases.

Figure 6 :
Figure 6: The histogram and the fitted  and  densities for Roller data.

Figure 7 :
Figure 7: The histogram and the fitted  and  densities for Eruption data.

Table 1 ,
we give the skewness and the kurtosis values of ABSGT distribution for some selected values of the shape parameters.It is obvious that Table1provides extra information for modeling performance of ABSGT distribution.

Table 1 : The skewness and the kurtosis values of the ABSGT distribution based on some selected values of the shape parameters
, ,  and .

Table 3 : Estimates of the parameters and AIC for fitted model for real data.
.9143 -3.2509 3.2541 1.3427 12.2041 1.9124 509.87 531.50