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.


Introduction
The slash distribution is the result of the quotient of two independent random variables, one with a standard normal distribution and the other with a uniform distribution on the interval (0, 1), with the following stochastic representation where µ ∈ R is the location parameter and σ > 0 is the scale parameter and q is the parameter related to kurtosis. Will be denoted by Y ∼ S(µ, σ, q) and its density function has the following expression where Γ(a) = ∞ 0 t a−1 e −t dt is the gamma function and Γ(a, x) = ∞ x t a−1 e −t dt is the gamma function incomplete. This distribution presents heavier tails than the normal distribution, that is, it has more kurtosis. Properties of this family are discussed in Rogers and Tukey [1] and Mosteller and Tukey [2].
Maximum likelihood estimators for location and scale parameters are discussed in Kafadar [3]. Wang and Genton [4] described multivariate symmetrical and skewmultivariate extensions of the slash-distribution while Gómez et al. [5] (and Erratum in Gómez and Venegas, 2008) extend the slash distribution by introducing the slash-elliptical family; asymmetric version of this family is discussed in work of Arslan [6]. Genc [7] discussed a symmetric generalization of the slash distribution. More recently, Gómez et al. [8] utilize the slash-elliptical family to extend the Birnbaum-Saunders distribution.
In (1), µ = 0 and σ = 1, we retrieve the standard slash distribution. What is more q = 1 we obtain the canonical slash distribution. When q tends to infinity, the standard normal distribution is recovered.
When U ∼ exp (2), in (1), the distribution obtained is called modified slash distribution studied by Reyes et al. [9]. Whose function of density is given by and will be denoted by X ∼ MS(0, 1, q), where q is kurtosis parameter. When U ∼ B(α, β) and q = 1, in (1), the distribution obtained is called extended slash (ES) distribution studied by Rojas et al. [10]. Whose function of density is given by is denoted as Y ∼ ES(µ, σ, α, β) with µ ∈ R, σ, α, β > 0 and φ denotes the pdf of the standard normal distribution (see Johnson et al. [11]) and B(·, ·) denotes the beta function. We will say that X has a student's t distribution with ν degrees of freedom and with location parameter µ and scale parameter σ, which we will denote by X ∼ T(µ, σ, ν) and you have a stochastic representation given by and continuous probability density function is given by with support on (−∞; ∞). The moment's order r of the random variable X with student's t distribution can be explained by the function Gamma. If X ∼ T(0, 1, ν) then µ r = E[X r ] = ν r/2 a r/2 2 r/2 Γ( ν−r 2 ) Γ( ν 2 ) , ν > r, where a r/2 = ∞ −∞ x r φ(x)dx for r even, then Rui Li-Saralees Nadarajah [12] makes a review of all the generalizations of the student's t distribution published to date, where they show that the main motivation of these extensions is to model heavy tails or data with high kurtosis.
In the study of symmetric distributions with heavy tails El-Bassiouny et al. [13] present the generalized student's slash t distribution. We will say that X ∼ GLST(µ, σ, α, β, ν, q), with parameter q > 0, has pdf given by where q is kurtosis parameter and B(·, ·) denotes the beta function. Another recent extension of the slash model was proposed by El-Morshedy, A. H. et al. [14]. These authors introduced the double slash (DSL) distribution with density function given by with µ ∈ R, σ, q 1 and q 2 > 0. When U ∼ Ga(2β, β) and q = 1, in (1), the distribution generalized modified slash distribution, denoted GMS(µ, σ, β), studied by Reyes, J., Barranco-Chamorro, I., and Gómez, H. W. [15]. Whose function of density is given by where µ ∈ R, σ, β > 0 and is the confluent hypergeometric function of the second kind. Details about this function can be seen in Abramowitz and Stegun, p. 505. With the motivation of finding a distribution that is a generalization of the student's t distribution and that presents heavier tails than the distributions found so far in the literature, in this article, we introduce a new generalization of the student's t distribution (GT) whose stochastic representation is given by where W ∼ N(0, 1), V ∼ χ 2 (ν) are independent with ν > 0 and q > 0 and we will denote it as Y ∼ GT(µ, σ, ν, q).
The paper is organized as follows. In Section 2 the probability density function (pdf) is given and some properties of the GT distribution are presented and shows that the distribution student's t is a particular case of the distribution GT. Additionally, moments of order r are obtained, including the kurtosis coefficient. In Section 3 derivation of the moment and maximum likelihood estimators are discussed. A simulation study is presented to illustrate the behavior of the estimator of the parameters µ, σ, and q, for ν = 8. Section 4 results of using the proposed model in two real applications are reported. Section 5 presents quantile regression. Section 6 presents the main conclusions.

The Generalized Student's t Distribution
We present the generalized student's t distribution with heavier tails compared to similar distributions. Initially we will present its density function.

Density Function
We will use the stochastic representation where W is distributed standard normal, V is distributed chi square, with ν degrees of freedom, W and V are independent random variables, µ, σ are location and scale parameters, respectively, ν degrees of freedom and q > 0 is the parameter related to the distribution kurtosis. We use the notation Y ∼ GT(µ, σ, ν, q), and for the standard case, we denote X ∼ GT(0, 1, ν, q). Proposition 1. Let Y ∼ GT(µ, σ, ν, q). Then, the pdf of Y is given by Proof. Since W and V are two independent random variables, such that W ∼ N(0, 1) and V ∼ χ 2 (ν) , then the joint pdf of (Y, T) where y ∈ R and t > 0. By marginalizing the result follows immediately para y = µ. Doing y = µ the other expression is obtained. (14), then la fdp de Y is called the canonical generalized student's t distribution.
Making a = ν/2 and b = ( y−µ σ ) 2 ν 2 and making the change of variables w = t 4a and applying the result obtained in Reyes et al. [9] ∞ 0 t a e − x 2 where x = 2 y−µ σ the result is obtained. Figure 1 on the left shows the PDFs of the generalized student's t distribution for q = 1 compared to the Student's t for ν = 5, the normal distribution, the generalized bar t distribution and the double bar distribution. In which, it can be seen that as the variable tends to ∞ to the right (or to the left), the new model captures more data than the other comparative distributions. Furthermore, it is observed that to the extent that q is smaller, the distribution has greater kurtosis.

Tails Comparison of GT and Student's t Distributions
In this part, we perform a comparison of the upper tails between the GT distribution and student's t distribution. For this, we consider the canonical version (q = 1) of GT distribution considering student's t distribution with ν = 5 degrees of freedom. Table 1 shows P(Y > y) for different values of y in the mentioned distributions. The GT distribution has tails much heavier than the student's t distribution. Table 1. Tails comparison GT distributions and student's t distribution.
0.0301 0.0103 0.0041 0.0002 Remark 1. Table 1 illustrates the fact that the generalized student's t distributions have heavier tails than the tails of the student's t distribution. Figure 2 shows the quantile function of the generalized student's t distribution compared to quantile function of student's t for different values of q and ν = 5.

Compared GT Quantiles with T Quantiles
. Then an approximation of quantile p of Y is where t p and j p denotes the quantiles p of student's t and chi-square distribution whit ν degrees of freedom.
. Figure 3 shows the quantiles of the generalized student's t distribution compared to quantile of proposition 2 for values q = 1 and ν = 5.
If q = 2 then y p = t p ; 2.
if ν → ∞ then y p = z p where z p is the quantile p of standard normal distribution.
In Table 2 we present quantiles generalized student's t for n degrees of freedom and q = 1.

Properties of the Generalized Student's t Distribution
In this section, we present some properties of the generalized student's t distribution. Proof.

1.
Making q tend to infinity in representation (13), the result is immediately obtained; where f V es la fdp chi-square distribution with ν degrees of freedom. The result follows using transformation t = v 1/q and direct integral computations; 3.
Making q = 2 we obtain the density student's with ν degrees of freedom.
Remark 2. Proposition 3 shows first that the generalized student's t distribution contains the normal distribution as a special case (q → ∞). Moreover, it also shows that the generalized student's t distribution is a scale mixture between the normal and the chi-square distribution with ν degrees of freedom. The third property shows that for q = 2, the density function for the generalized student's t coincides with the density function of the student's t distribution with ν degrees of freedom.

Moments
In this subsection the moments of the generalized student's t distribution are deduced.
Proposition 5. Let Y ∼ GT(µ, σ, ν, q), so that the coefficient of skewness and kurtosis are: and Proof. The standardized coefficient of skewness and kurtosis are (µ 2 − µ 2 1 ) 2 and the result follows after replacing the even moments derived in Proposition 4. Figure 4 shows the kurtosis the GT distribution compared with T distribution for different values of q and ν = 8.
It can be seen that the generalized student's distribution has a greater kurtosis than the student's distribution for q less than 2, then for data with high kurtosis, it would be recommended to use the generalized student's distribution.

Moment Estimators
In the following proposition we present the moment estimators of µ, σ, and q for ν = 8. Proposition 6. Where Y 1 , . . . , Y n a random sample from the distribution of the random variable Y ∼ GT(µ, σ, ν, q), so that the moment estimators of θ = (µ, σ, ν, q) for q > 1 are given by where Y, S and γ 2 are the mean, standard deviation, and sample kurtosis coefficient.
Proof. Using (17) it follows that replacing γ 2 in (19) one obtains the numerical equation and solving (21) for q and ν one obtains q M and ν M . Further, replacing in (20) q by q M , ν by ν M , E(Y) byȲ and Var(Y) by the sample variance S 2 , we obtain the moment estimators ( µ M , σ M , ν M , q M ) for (µ, σ, ν, q).

Maximum Likelihood Estimation
Given a random sample Y i ∼ GT(µ, σ, ν, q), for i = 1, .., n, the log-likelihood function can be written as q +v] dv and hence the maximum likelihood equations are given by where Using numerical procedures Equations (27)-(30) can be solved.

Proposition 7.
Let Y 1 , . . . , Y n a random sample from the distribution of random variable Y ∼ GT(µ, σ, ν, q). Then, Proof. The random variable Z and T replacing the result is obtained. Proposition 8. Let Y 1 , . . . , Y n a random sample from the distribution of random variable Y ∼ GT(µ, σ, ν, q). Then, a level (1 − α) confidence interval for the population mean is where t 1−α/2 is the percentile of order 1 − α 2 of GT distribution.
Proof. The result is obtained from the previous proposition.

Simulation Study
To generate random numbers from the GT(µ, σ, 8, q) distribution we will use the stochastic representation given in (13) and the following algorithm: 1.

Two Illustrative Datasets
Illustrative Datasets 1 We consider the data that were first presented in Jander [16], from an entomology experiment. with respect to ants. A total of n = 730 ants were individually placed in the center of an arena. The measurements correspond to the initial direction in which they moved relative to a visual stimulus in a 180 degree angle from zero direction, rounded to the nearest 10 grades. Figure 5 depicts the histogram of these data, including estimated densities under a T, ES, MS, SGT, DSL and GT model, using maximum likelihood. Figure 6 shows the qqplots for T, ES, MS and GT models. We use the AIC (Akaike Information Criterion), which penalizes the maximized likelihood function by the excess of model parameters (AIC = −2log(lik) + 2k, where k is the number of unknown parameters being estimated, see Akaike [17]). Table 4 shows the descriptive statistics of the database, while Table 5 presents the Kolmogorov -Smirnov (KSS) statistic, corresponding values for the four given models, which also indicates that the best fit is presented by the GT model. Table 6 shows a 95% confidence interval for the population mean using generalized Student's t-quantiles. Moreover, Figure 7 depicts the empirical cumulative distribution function (cdf) and the estimated cdfs for T, ES, MS and GT models.
The estimators of moments for the dataset are: which will be used as starting points in obtaining the EMVs.  Figure 8 depicts the histogram of these data, including estimated densities under a SGT, DSL and GT model, using maximum likelihood. We use the Akaike information criterion (AIC) and Bayesian Information Criterion (BIC), see Schwarz [18], which is defined as (BIC = −2log(lik) + klog(n), where k is the number of estimated parameters and n is the sample size. Table 7 shows these results. Table 6. The 95 percent confidence interval for the mean of dataset using T and GT quantiles T.

Distribution
Lower Limit Upper Limit

Quantile Regression
The quantile regression is used when the study objective focuses on the estimation of the different percentiles (such as the median) of a population of interest. An advantage of using quantile regression to estimate the median, rather than ordinary least squares regression current file (to estimate the mean), is that the quantile regression will be more robust in the presence of outliers. Quantile regression can be seen as a natural analogue in regression analysis when using different measures of central tendency and dispersion, in order to obtain a more complete and robust analysis of the data. Another advantage of this type of regression lies in the possibility of estimating any quantile, thus being able to assess what happens with extreme values of the population.

Quantile Regression Uni-Dimensional
Translating this concept of quantile to the regression line, we obtain the linear quantile regression.
If we assume that ∀i (1, ..., n) with τ (0, 1) and that the conditional expected value is not necessarily zero, but the τ-ésimo quantile of the error with respect to the regressive variable is zero (Q τ ( i,τ /X) = 0), then the τ-ésimo quantile of Y i with respect to X can be written as The estimates of β 0,τ y β 1,τ are found bŷ being To estimate the parameters, the function described in the equation should be minimized. For this, there is a way to approach the minimization problem as a linear programming problem. This allows us to obtain the regression line for the value of a certain quantile. Therefore, the first of the limitations will be solved raised at the end of the previous section, for simple linear regression. Furthermore, since the quartiles have robust properties, it is also possible to solve the second of the limitations that arose with the classical regression line.

Quantile Regression Student's t
In this case, in the regression equation ∀i (1, ..., n) the response variable Y ∼ T(µ, σ, ν), it is possible to generate random numbers for the T(µ, σ, ν) distribution, which the parameters µ, σ and ν they are estimated using maximum likelihood for the data. Then, one way to obtain the quantiles of Y is using the stochastic representation.

Quantile Regression Slash Logistic
In this case, in the regression equation ∀i (1, ..., n) the response variable Y ∼ GSLOG(µ, σ, q), it is possible to generate random numbers for the SLOG(µ, σ, q) distribution, which the parameters µ, σ, and q they are estimated using maximum likelihood for the data. Then, one way to obtain the quantiles of Y is using the stochastic representation.
Compute Y 2 = T U 1/q . Using this new variable Y 2 quantile regression is applied to the data (X, Y 2 ).

Quantile Regression Generalized Student's t
In this case, in the regression equation ∀i (1, ..., n) the response variable Y ∼ GT(µ, σ, ν, q), it is possible to generate random numbers for the GT(µ, σ, ν, q) distribution, which the parameters µ, σ, ν, and q they are estimated using maximum likelihood for the data. Then, one way to obtain the quantiles of Y is using the stochastic representation given in (13) 1.

Application 2
We consider now data concerning the body mass index and Lean Body Mass of 202 Australian athletes. The data are available for download at http://azzalini.stat.unipd.it/ SN/index.html (accessed on 15/10/2021). Table 8 shows statistics for these data for which the maximum likelihood estimators of (β 0 , β 1 ) and its corresponding coefficients AIC and BIC fit models for data. are shown in Tables 9 and 10, respectively. In Figure 9 the quantile regression of the data is shown using the T, SLOG and GT models. Quantile Regression SLOG lbm bmi q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q . Quantile regression for BMI and LBM data with student's t distribution (left), slash logistic distribution (center) and generalized student's t distribution (right).

Discussion
We have introduced a new distribution called the generalized student's t distribution (GT). The main idea is to replace the exponent 1/2 of the chi-square distribution by a exponent 1/q where q > 0 is the kurtosis parameter. We consider the density function of the distribution and study some of its properties, as well as its moments. The parameter estimation was analyzed using the method of moments and maximum likelihood estimation. We present two illustrations, in the first a set of real data are studied where we show that the GT distribution fits the data better than the T, ES, MS, SGT, and DSL distributions. In the other application, we use quantile regression to fit a linear model to a paired dataset where the response variable shows high kurtosis where it is shown that the GT distribution fits better than the T and SLOG distributions to model the residuals.

Acknowledgments:
The authors would like to thank the referee for his/her constructive suggestions that improved the final version of this paper.

Conflicts of Interest:
The authors declare no conflict of interest.