GAS t Update term potentially using second-order information - native Python function

        Parameters
        ----------
        y : float
            datapoint for the time series

        mean : float
            location parameter for the t distribution

        scale : float
            scale parameter for the t distribution

        shape : float
            tail thickness parameter for the t distribution

        skewness : float
            skewness parameter for the t distribution

        Returns
        ----------
        - Adjusted score of the t family

def second_order_score(y, mean, scale, shape, skewness):
        """ GAS t Update term potentially using second-order information - native Python function

        Parameters
        ----------
        y : float
            datapoint for the time series

        mean : float
            location parameter for the t distribution

        scale : float
            scale parameter for the t distribution

        shape : float
            tail thickness parameter for the t distribution

        skewness : float
            skewness parameter for the t distribution

        Returns
        ----------
        - Adjusted score of the t family
        """
        return ((shape+1)/shape)*(y-mean)/(np.power(scale,2) + (np.power(y-mean,2)/shape))/((shape+1)*((np.power(scale,2)*shape) - np.power(y-mean,2))/np.power((np.power(scale,2)*shape) + np.power(y-mean,2),2))