Maximum-score estimator for an ordered-response model — morie_ordered_max

\(S_{n,OR}(b) = n^{-1}\sum_i |W_i - \sum_{m} 1\{X_i'b > \alpha_m\}|\) (4.43), with \(W = \sum_m 1\{Y^* > \alpha_m\}\) observable from Y.

Usage

morie_ordered_max_score(
  x,
  y,
  thresholds = NULL,
  smoothed = FALSE,
  h = NULL,
  n_restarts = 8L,
  seed = 0L
)

Arguments

x: numeric matrix of covariates.
y: integer category labels 0..M-1, at least 3 categories.
thresholds: known cut-points; estimated when NULL.
smoothed: use the Melenberg-van Soest objective, which IS maximised.
h: bandwidth for the smoothed form; n^(-1/5) when NULL.
n_restarts: restarts, since (4.43) is a step function.
seed: RNG seed for the restarts.

Value

list: beta, thresholds, objective, sense, thresholds_estimated, scale_normalisation_required, smoothed, bandwidth, M, n, d, method.

Details

This objective is MINIMISED. The book prints "maximize" above (4.43), but the next sentence calls the result a median-regression estimator, and a median regression minimises absolute deviations. Measured on a simulated ordered model with \(\beta = (1, -0.7)\) and n = 4000, minimising over a grid returns -0.75 while maximising runs to the boundary of the search region and stops at 3.0. The printed "maximize" is treated as an erratum, and sense records the direction taken.

Unlike the binary-response case this model needs NO scale normalisation; the normalisation used is Lee's (1992), fixing the first coefficient at 1 and \(\alpha_1 = 0\). Mirrors morie.fn.hrzormsc.

References

Horowitz, Sec. 4.4.3, eqs. (4.41)-(4.43), Theorem 4.11; Lee (1992), Melenberg and van Soest (1996).

Examples

set.seed(1)
x <- matrix(rnorm(600), ncol = 2)
ys <- x %*% c(1, -0.7) + rnorm(300)
y <- findInterval(ys, c(-1, 0, 1.2))
morie_ordered_max_score(x, y, thresholds = c(-1, 0, 1.2))$sense
#> [1] "minimised"