Chen's (2002) rank estimator of T

\(T_n(y) = \arg\max_{t}\ [n(n-1)]^{-1}\sum_i\sum_{j \ne i} (d_{iy} - d_{jy_0}) 1\{X_i'b_n - X_j'b_n \ge t\}\) (6.67), with \(d_{iy} = 1\{Y_i \ge y\}\).

Usage

morie_chen_transform(
  x,
  y,
  bandwidth = NULL,
  beta_hat = NULL,
  y0 = NULL,
  y_grid = NULL,
  t_grid = NULL
)

Arguments

x

numeric matrix of covariates.

y

numeric response.

bandwidth

ignored; (6.67) uses no kernel.

beta_hat

beta, rescaled to |b_1| = 1; first canonical direction when NULL.

y0

location-normalisation point; median of y when NULL.

y_grid

points at which to return T_hat.

t_grid

the compact interval M searched over.

Value

list: y_grid, T_hat, objective_max, y0, beta, t_grid, uses_kernel, rate_exponent, faster_than_horowitz, normalisation, n, d, method.

Details

The construction rests on a sign, not a smoothing: \(E(d_{iy} - d_{jy_0} | X_i, X_j) \ge 0\) exactly when \(X_i'\beta - X_j'\beta \ge T(y)\), so maximising a pairwise sum over t locates \(T(y)\). It is a U-statistic of order two – hence the \(n(n-1)\) normalisation, and hence the quadratic cost.

This is not faster than Horowitz's estimator. Both are \(n^{-1/2}\) (Theorem 6.6). The book compares them and finds neither dominates: Horowitz's has smaller mean-square error near the centre of the range of y, Chen's further out, and no known estimator is efficient uniformly over y. bandwidth is accepted for interface symmetry and NOT used – (6.67) contains no kernel. Mirrors morie.fn.hrzchet.

References

Horowitz, Sec. 6.3.3, eq. (6.67), CT1-CT6, Theorem 6.6; Chen (2002).

Examples

set.seed(1)
n <- 80
x <- cbind(rnorm(n), rnorm(n))
y <- exp(x %*% c(1, -0.5) + rlogis(n))
morie_chen_transform(x, y, beta_hat = c(1, -0.5))$uses_kernel
#> [1] FALSE