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Jackknife for Hájek-dominated generalized U-statistics (overview)

Source: vignettes/jackknife-gen-ustats.Rmd
jackknife-gen-ustats.Rmd

Overview

RdistNN includes two delete-1 jackknife utilities for rank-weighted functionals:

Both return:

  • theta0: baseline aggregate
  • theta_loo: all leave-one-out aggregates
  • info: metadata (s, number of targets, output dimension, normalization)

Important distinction:

  • jackknife_fast_rank() and jackknife_slow_rank() are generic utilities for compact-support rank-weighted L-stat rules;
  • the user-level exact DNN wrappers do not truncate to the first s observations. They keep the full rank matrix and reevaluate the DNN rank-weight rule after delete-one updates.

Example Setup 

n <- 40
p <- 2
s <- 3L

X <- matrix(rnorm(n * p), n, p)
Y <- rnorm(n)

# Toy compact-support example: build only the first s+1 ranked observations
# for each target point. This illustrates the generic L-stat utilities below;
# it is not the internal exact-DNN jackknife path used by np_dnn()/cate_dnn()/plqr_dnn().
wt <- weights_dnn(
  X = X, X0 = X,
  method = "exact",
  distance = "l2",
  k = s, s = s,
  return = "ord"
)
ranks <- t(wt$ord[seq_len(s + 1L), , drop = FALSE])  # T x (s+1)

# L-stat coefficients
a <- rep(1, s)

# Scalar signal represented as n x 1 matrix
phi <- cbind(Y)

Fast vs Slow Jackknife 

jk_fast <- jackknife_fast_rank(ranks = ranks, a = a, phi = phi, normalize = TRUE)
jk_slow <- jackknife_slow_rank(ranks = ranks, a = a, phi = phi, normalize = TRUE)

all.equal(jk_fast$theta0, jk_slow$theta0, tolerance = 1e-12)
all.equal(jk_fast$theta_loo, jk_slow$theta_loo, tolerance = 1e-12)

Turning LOO Aggregates into Standard Errors 

theta_loo <- jk_fast$theta_loo[, 1]
theta_bar <- mean(theta_loo)
n_ref <- length(theta_loo)

var_jk <- ((n_ref - 1) / n_ref) * sum((theta_loo - theta_bar)^2)
se_jk <- sqrt(var_jk)
se_jk

Notes

  • jackknife_fast_rank() supports both matrix phi and function-valued phi(j, t).
  • The slow version is useful for debugging and parity checks in tests.
  • For user-level wrappers (np_dnn(), cate_dnn(), plqr_dnn()), these steps are handled internally when se = TRUE.