def resid_anscombe(self, endog, mu):
'''
The Anscombe residuals
Parameters
----------
endog : array-like
Endogenous response variable
mu : array-like
Fitted mean response variable
Returns
-------
resid_anscombe : array
The Anscombe residuals as defined below.
Notes
-----
sqrt(n)*(cox_snell(endog)-cox_snell(mu))/(mu**(1/6.)*(1-mu)**(1/6.))
where cox_snell is defined as
cox_snell(x) = betainc(2/3., 2/3., x)*betainc(2/3.,2/3.)
where betainc is the incomplete beta function
The name 'cox_snell' is idiosyncratic and is simply used for
convenience following the approach suggested in Cox and Snell (1968).
Further note that
cox_snell(x) = x**(2/3.)/(2/3.)*hyp2f1(2/3.,1/3.,5/3.,x)
where hyp2f1 is the hypergeometric 2f1 function. The Anscombe
residuals are sometimes defined in the literature using the
hyp2f1 formulation. Both betainc and hyp2f1 can be found in scipy.
References
----------
Anscombe, FJ. (1953) "Contribution to the discussion of H. Hotelling's
paper." Journal of the Royal Statistical Society B. 15, 229-30.
Cox, DR and Snell, EJ. (1968) "A General Definition of Residuals."
Journal of the Royal Statistical Society B. 30, 248-75.
'''
cox_snell = lambda x: (special.betainc(2/3., 2/3., x)
* special.beta(2/3., 2/3.))
return np.sqrt(self.n) * ((cox_snell(endog) - cox_snell(mu)) /
(mu**(1/6.) * (1 - mu)**(1/6.)))
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