sfa (version 1.2.0)

influence_sfa: Which observations are driving this frontier fit?

Description

Empirical influence function and case-influence measures for a fitted stochastic frontier. Answers which observations move the estimates, and whether the specification lets any single observation move them without bound.

Usage

influence_sfa(object, scale = TRUE)

Value

An object of class "sfa_influence":

influence

\(n \times p\) matrix, the empirical influence function.

d

Length-\(n\) case influence \(n\,s_i'Vs_i\), the same quadratic form as Cook's distance and invariant to reparameterisation.

sensitivity_std

\(\sup_i \sqrt{d_i}\), the self-standardised sensitivity -- the one to compare across models.

sensitivity

\(\sup_i \|IF_i\|\), the raw sup-norm.

max_abs_by_parameter

Largest \(|IF|\) in each coordinate.

Arguments

object

An "sfareg" fit. It must retain its likelihood, so fit with keep_objective = TRUE; the score matrix cannot otherwise be built.

scale

Multiply the influence function by \(n\), so that influence[i, ] approximates the effect on the estimate of deleting observation \(i\) and is therefore on the same scale as the coefficients. Default TRUE.

Details

The standard outlier rules do not transfer to this model. Least-median-of-squares and its relatives ignore the asymmetry of the composed error, so a genuinely large \(u_i\) -- an inefficient firm, the thing being measured -- reads as an outlier and is discarded, which defeats the analysis.

The influence function is the right object instead. Maximum likelihood is an M-estimator, so its influence function is a linear transformation of the score, \(IF_i \propto I(\theta)^{-1}\psi(y_i,\theta)\), and the estimator is B-robust exactly when that is bounded. Both pieces are already available: estfun supplies the per-observation scores and vcov the inverse information.

Which sensitivity to read. The raw sup-norm depends on how the parameters happen to be scaled, so it is not comparable between models with different parameter vectors. On a clean sample of 200 it reads 72.6 for "NHN" and 1322.5 for "tHN", which appears to say the robust specification is far worse; all it reflects is tHN's extra \(\nu\) on its own scale. sensitivity_std measures the same influence function in the information metric and is invariant to reparameterisation. On that scale, contaminating a single response takes "NHN" from 13.5 to 32.3 and "tHN" from 13.1 to 17.2 -- Stead, Wheat and Greene's finding, and invisible in the raw number.

Neither number has an absolute threshold. The comparison to make is between specifications on the same data, not either number against a cut-off.

What to do about a large value. Nothing, by deletion. A large sensitivity is a property of the specification: under an unbounded influence function the next most extreme point simply replaces the one removed. The remedies are a specification with bounded influence -- Stead, Wheat and Greene's Student's \(t\) noise term, model_name = "tHN" -- or a robust divergence criterion, sfm(robust = ), whose tuning is chosen by hscore_select or calibrate_c.

What this cannot see. Like density_weights, it is computed at the fitted surface. An observation mis-recorded in a regressor can bend that surface toward itself and so appear unremarkable here. Pair this with a leverage check before concluding that an observation is not driving a result.

References

Stead, A.D., Wheat, P. and Greene, W.H. (2023). Robust maximum likelihood estimation of stochastic frontier models. European Journal of Operational Research, 309(1), 188--201.

Bernstein, D.H., Parmeter, C.F. and Wright, I.A. (2026). On Robust Estimation of the Stochastic Frontier Model. Working paper.

See Also

density_weights, hscore_select, sfm, sfareg-methods

Examples

# \donttest{
set.seed(3)
n <- 120
x <- runif(n, 1, 10)
d <- data.frame(y = 1 + 0.5 * log(x) + rnorm(n, 0, 0.3) - abs(rnorm(n, 0, 0.6)),
                x = log(x))

fit <- sfm(y ~ x, data = d, model_name = "NHN", keep_objective = TRUE)
influence_sfa(fit)

## Contaminate one response and compare specifications on the
## self-standardised scale.
d2 <- d; d2$y[1] <- d2$y[1] - 5
a <- sfm(y ~ x, data = d2, model_name = "NHN", keep_objective = TRUE)
b <- sfm(y ~ x, data = d2, model_name = "tHN", keep_objective = TRUE)
c(NHN = influence_sfa(a)$sensitivity_std,
  tHN = influence_sfa(b)$sensitivity_std)
# }