code: def reg_score_function(X, y, mean, scale, shape, skewness):
          
          return 2.0*((y-mean)*X)/(np.power(scale,2)+np.power((y-mean),2))
docs: GAS Cauchy Regression Update term using gradient only - native Python function
  
          Parameters
          ----------
          X : float
              datapoint for the right hand side variable
      
          y : float
              datapoint for the time series
  
          mean : float
              location parameter for the Cauchy distribution
  
          scale : float
              scale parameter for the Cauchy distribution
  
          shape : float
              tail thickness parameter for the Cauchy distribution
  
          skewness : float
              skewness parameter for the Cauchy distribution
  
          Returns
          ----------
          - Score of the Cauchy family