ak.tau = function(X, max.second.derivative, Y = NULL, weights = rep(1, length(X)), threshold = 0, sigma.sq = NULL, change.derivative = TRUE, alpha = 0.95, max.window = max(abs(X - threshold)), num.bucket = 200) {
# Naive initialization for sigma.sq if needed
if (is.null(sigma.sq)) {
if (is.null(Y)) {
warning("Setting noise level to 1 as default...")
sigma.sq = 1
} else {
Y.bar = sum(Y * weights) / sum(weights)
sigma.sq = sum((Y - Y.bar)^2 * weights^2) / sum(weights)
}
}
# We compute our estimator based on a histogram summary of the data,
# shifted such that the threshold is at 0. The breaks vector defines
# the boundaries between buckets.
xx = seq(-max.window, max.window, length.out = num.bucket)
bin.width = xx[2] - xx[1]
breaks = c(xx - bin.width/2, max(xx) + bin.width/2)
# Construct a (weighted) histogram representing the X-values.
inrange = which(abs(X - threshold) / max.window <= 1)
bucket = cut(X[inrange] - threshold, breaks = breaks)
bucket.map = Matrix::sparse.model.matrix(~bucket + 0, transpose = TRUE)
X.counts = as.numeric(bucket.map %*% weights[inrange])
# only consider counts that occurs inside the bandwidth
realized.idx = which(X.counts > 0)
num.realized = length(realized.idx)
# This optimizer learns bucket-wise gammas. Let k denote
# the bucket index, n[k] the number of observations in
# bucket k, and x[k] is the center of bucket k.
#
# We solve the following. Note that gamma[k] must be 0
# if n[k] is 0, so we only optimize gamma over realized indices.
#
# argmin sum_k (gamma_+[k] - gamma_-[k])^2 * n[k] + B^2 I^2
# subject to:
# sum_k n[k] (gamma_+[k] - gamma_-[k]) = 0
# sum_k n[k] (gamma_+[k] - gamma_-[k]) (2 W[k] - 1) = 2
# sum_k n[k] (gamma_+[k] - gamma_-[k]) x[k] = 0
# sum_k n[k] (gamma_+[k] - gamma_-[k]) (x[k])_+ = 0
# sum_k n[k] (gamma_+[k] + gamma_-[k]) x[k]^2 / 2 = I
# gamma_+, gamma_- >= 0
Dmat =diag(c(sigma.sq * X.counts[realized.idx],
sigma.sq * X.counts[realized.idx],
max.second.derivative^2))
dvec = rep(0, 2 * num.realized + 1)
if(!change.derivative) {
stop("Only implemented with derivate change at threshold.")
}
Amat = cbind(c(X.counts[realized.idx], -X.counts[realized.idx], 0),
c(X.counts[realized.idx] * sign(xx[realized.idx]),
-X.counts[realized.idx] * sign(xx[realized.idx]), 0),
c(X.counts[realized.idx] * xx[realized.idx],
-X.counts[realized.idx] * xx[realized.idx], 0),
c(X.counts[realized.idx] * pmax(xx[realized.idx], 0),
-X.counts[realized.idx] * pmax(xx[realized.idx], 0), 0),
c(X.counts[realized.idx] * xx[realized.idx]^2/2,
X.counts[realized.idx] * xx[realized.idx]^2/2, -1),
diag(rep(1, 2 * num.realized + 1)))
bvec = c(0, 2, 0, 0, 0, rep(rep(0, 2 * num.realized + 1)))
meq = 5
soln = quadprog::solve.QP(Dmat, dvec, Amat, bvec, meq)$solution
gamma.xx = rep(0, num.bucket)
gamma.xx[realized.idx] = soln[1:num.realized] - soln[num.realized + (1:num.realized)]
# Now map this x-wise function into a weight for each observation
gamma = rep(0, length(X))
gamma[inrange] = weights[inrange] * as.numeric(Matrix::t(bucket.map) %*% gamma.xx)
max.bias = max.second.derivative * soln[1 + 2 * num.realized]
# If outcomes are provided, also compute confidence intervals for tau.
if (!is.null(Y)) {
# The point estimate
tau.hat = sum(gamma * Y)
# A heteroskedaticity-robust variance estimate
regr.df = data.frame(X=X, W=X>=threshold, Y=Y)
Y.fit = lm(Y ~ X * W, data = regr.df[inrange,], weights=weights[inrange])
Y.resid.sq = rep(0, length(Y))
Y.resid.sq[inrange] = (Y[inrange] - predict(Y.fit))^2 * sum(weights[inrange]) / (sum(weights[inrange]) - 4)
se.hat.tau = sqrt(sum(Y.resid.sq * gamma^2))
# Confidence intervals that account for both bias and variance
tau.plusminus = get.plusminus(max.bias, se.hat.tau, alpha)
} else {
tau.hat = NULL
se.hat.tau = sqrt(sigma.sq * sum(gamma^2))
tau.plusminus = get.plusminus(max.bias, se.hat.tau, alpha)
}
ret = list(tau.hat=tau.hat,
tau.plusminus=tau.plusminus,
alpha=alpha,
max.bias = max.bias,
sampling.se=se.hat.tau,
gamma=gamma,
gamma.fun = data.frame(xx=xx[realized.idx] + threshold,
gamma=gamma.xx[realized.idx]))
class(ret) = "optrdd"
return(ret)
}
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