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.