class_docstr: Cauchy Distribution
  ----
  This class contains methods relating to the Cauchy distribution for time series.
docstr: GAS Cauchy Update term potentially using second-order information - native Python function
  
  Parameters
  ----------
  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
  ----------
  - Adjusted score of the Cauchy family
docstr_tok: ["GAS","Cauchy","Update","term","potentially","using","second-order","information","-","native","Python","function","Parameters","--","--","--","--","--","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","--","--","--","--","--","-","Adjusted","score","of","the","Cauchy","family"]
code: def second_order_score(y, mean, scale, shape, skewness):
      """GAS Cauchy Update term potentially using second-order information - native Python function
  
      Parameters
      ----------
      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
      ----------
      - Adjusted score of the Cauchy family
      """
      return (
          2.0
          * (y - mean)
          / (np.power(scale, 2) + (np.power(y - mean, 2)))
          / (
              2.0
              * ((np.power(scale, 2)) - np.power(y - mean, 2))
              / np.power(
                  (np.power(scale, 2)) + np.power(y - mean, 2), 2
              )
          )
      )
  
imports: import numpy
  import scipy.stats
  import scipy.special
  from .family import Family
  from .flat import Flat
  from .normal import Normal
  from .gas_recursions import gas_recursion_cauchy_orderone, gas_recursion_cauchy_ordertwo
  from .gas_recursions import gasx_recursion_cauchy_orderone, gasx_recursion_cauchy_ordertwo
  from .gas_recursions import gas_llev_recursion_cauchy_orderone, gas_llev_recursion_cauchy_ordertwo
  from .gas_recursions import gas_llt_recursion_cauchy_orderone, gas_llt_recursion_cauchy_ordertwo
  from .gas_recursions import gas_reg_recursion_cauchy_orderone, gas_reg_recursion_cauchy_ordertwo
label_desc: Use image node_id 14 for calling the Cauchy obj's underlying member method code with example usage: obj.second_order_score(y, mean, scale, shape, skewness) and returns: unknown