sfa (version 1.2.0)

hscore: Hyvarinen score for a Normal--Half-Normal frontier fit

Description

Evaluates the Hyvarinen score used to choose the robustness tuning parameter of the maximum \(L_q\)-likelihood and minimum density-power divergence estimators, following Sugasawa and Yonekura (2021). Lower is better.

Usage

hscore(e, sigma_v, sigma_u, c, stable = TRUE)

Value

A single numeric value, the mean observationwise score.

Arguments

e

Numeric vector of composed residuals \(y - x'\beta\).

sigma_v, sigma_u

Positive scalars, the noise and inefficiency scale parameters.

c

Robustness tuning parameter, \(c = 1-q\) for MLqE and \(c = \alpha\) for MDPD. c = 0 is the maximum likelihood endpoint.

stable

Logical. TRUE (default) evaluates in log space; FALSE uses the natural-scale expression.

Details

The score is $$H(c) = n^{-1}\sum_i \{2 D''(e_i) + D'(e_i)^2\},\qquad D = (f^c-1)/c,$$ with derivatives taken with respect to the composed residual.

The score is evaluated in logarithms. The natural-scale expression contains the fitted density raised to the power \(c-2\); since \(c<1\) that exponent is close to \(-2\), so a single observation whose density underflows makes the score non-finite. The failure is silent and directional: the tuning candidates that remain evaluable are those closest to maximum likelihood, so a naive implementation reports too little robustness. Set stable = FALSE to reproduce that behaviour for comparison.

References

Hyvarinen, A. (2005). Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research 6, 695--709.

Sugasawa, S. and Yonekura, S. (2021). On selection criteria for the tuning parameter in robust divergence. Entropy 23(9), 1147.

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

See Also

hscore_select, calibrate_c, density_weights

Examples

set.seed(1)
e <- rnorm(500, 0, 0.3) - abs(rnorm(500, 0, 0.6))
hscore(e, sigma_v = 0.3, sigma_u = 0.6, c = 0.20)

## the two evaluations agree wherever both are computable
hscore(e, 0.3, 0.6, 0.20) - hscore(e, 0.3, 0.6, 0.20, stable = FALSE)

## but only one of them survives an observation in the far tail
e2 <- c(e, -60)
hscore(e2, 0.3, 0.6, 0.15)                  # finite
hscore(e2, 0.3, 0.6, 0.15, stable = FALSE)  # not finite