statsmodels.genmod.families.family.NegativeBinomial.resid_anscombe¶

NegativeBinomial.resid_anscombe(endog, mu, var_weights=1.0, scale=1.0)[source]¶

The Anscombe residuals

Parameters:¶

endogndarray

The endogenous response variable

mundarray

The inverse of the link function at the linear predicted values.

var_weightsarray_like

1d array of variance (analytic) weights. The default is 1.

scalefloat, optional

An optional argument to divide the residuals by sqrt(scale). The default is 1.

Returns:¶

resid_anscombendarray

The Anscombe residuals as defined below.

Notes

Anscombe residuals for Negative Binomial are the same as for Binomial upon setting $n=-\frac{1}{\alpha }$. Due to the negative value of $-\alpha \ast Y$ the representation with the hypergeometric function $H2F1(x)=hyp2f1(2/3.,1/3.,5/3.,x)$ is advantageous

$$resid\mathrm{\_}anscomb{e}_i=\frac{3}{2}\ast (Y_i^{(}2/3)\ast H2F1(-\alpha \ast Y_i)-{\mu }_i^{(}2/3)\ast H2F1(-\alpha \ast {\mu }_i))/({\mu }_i\ast (1+\alpha \ast {\mu }_i)\ast scal{e}^3{)}^{(}1/6)\ast \sqrt{(}var\mathrm{\_}weights)$$

Note that for the (unregularized) Beta function, one has $Beta(z,a,b)=z^a/a\ast H2F1(a,1-b,a+1,z)$