Manski maximum score and Horowitz's smoothed version

Maximum score maximises \(\sum (2Y_i-1)1\{X_i'b>0\}\) under only a conditional-MEDIAN restriction, tolerating arbitrary heteroskedasticity, but converges at \(n^{-1/3}\) to a non-normal Chernoff limit – ordinary standard errors do NOT apply. Smoothing the indicator with a kernel CDF lifts the rate to \(n^{-r/(2r+1)}\) and restores normality. Mirrors morie.fn.hrzmscr and hrzsms.

Usage

morie_maximum_score(X, y, smoothed = FALSE, h = NULL, r = 2L)

Arguments

X

numeric covariate matrix.

y

binary 0/1 response.

smoothed

logical; the smoothed estimator when TRUE.

h

smoothing bandwidth.

r

assumed smoothness order.

Value

list: beta, rate_exponent, limit_distribution, standard_errors_valid.

References

Horowitz, Ch. 3.

Examples

set.seed(1)
X <- matrix(rnorm(200), ncol = 2)
morie_maximum_score(X, as.numeric(X %*% c(1, -0.8) > 0))$rate_exponent
#> [1] -0.3333333