test_that("kappa_hat equals the explicit projection-matrix formula", {
  set.seed(1)
  n <- 20
  xy <- expand.grid(x = 1:5, y = 1:4)
  D <- as.matrix(dist(xy)); Wm <- 1 / D; diag(Wm) <- 0; Wm <- Wm / sum(Wm)
  X <- cbind(1, rnorm(n), rbinom(n, 1, 0.5))
  y <- rnorm(n)

  M  <- X %*% solve(t(X) %*% X) %*% t(X)
  In <- diag(n)
  num <- as.numeric(t(y) %*% (In - M) %*% Wm %*% y)
  den <- as.numeric(t(y) %*% Wm %*% (In - M) %*% Wm %*% y)
  ref <- num / den

  expect_equal(kappa_hat(y, Wm, X), ref, tolerance = 1e-10)
})

test_that("kappa_hat returns a single finite number", {
  set.seed(2)
  n <- 12; xy <- expand.grid(x = 1:4, y = 1:3)
  D <- as.matrix(dist(xy)); Wm <- 1 / D; diag(Wm) <- 0; Wm <- Wm / sum(Wm)
  X <- cbind(1, rep(0:1, length.out = n))
  k <- kappa_hat(rnorm(n), Wm, X)
  expect_length(k, 1L)
  expect_true(is.finite(k))
})