smooth.ADAM.rstudent

ADAM.rstudent()

Return studentised (leave-one-out) residuals.

Each residual is scaled by the distribution-specific scale estimate recomputed without that observation (leave-one-out). Compared to rstandard(), the result is more sensitive to individual outliers because no single point inflates the global scale estimate it is judged against.

The leave-one-out sums are computed in O(n) using identities such as Σ e[-i]² = Σ e² − e[i]², avoiding an explicit loop. Degrees of freedom: df = nobs - nparam - 1.

Standardisation by distribution:

  • dnorm: (e[i] - ē) / √(Σe[-i]² / df)

  • dlaplace: (e[i] - ē) / (Σ|e[-i]| / df)

  • ds: (e[i] - ē) / (Σ√|e[-i]| / (2 df))²

  • dgnorm: (e[i] - ē) / (Σ|e[-i]|^β · β/df)^(1/β)

  • dlnorm: exp(log_e[i] / √(Σlog_e[-i]² / df)), where log_e = log(e) - mean(log e) - σ²/2

  • dinvgauss / dgamma: e[i] / mean(e[-i])

Mirrors R’s rstudent.adam().

Returns:

Studentised residuals, length nobs.

Return type:

NDArray

Raises:

ValueError – If the model has not been fitted yet.

See also

rstandard

Simpler standardised residuals (faster).

outlierdummy

Outlier detection based on studentised residuals.

Examples

>>> model = ADAM(model="AAN")
>>> model.fit(y)
>>> stu_res = model.rstudent()
>>> # Studentised residuals are slightly more spread than rstandard
>>> stu_res.std() >= model.rstandard().std()
True

Parent Class: ADAM