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"rqProcess" <-
function (formula, data, taus, nullH = "location", ...)
{
z <- summary(f <- lm(formula, data = data, x = TRUE))
xbar <- apply(f$x,2,mean)
vars <- names(z$coef[-1, 1])
p <- length(z$coef[, 1])
n <- nrow(z$x)
Jinv <- z$cov.unscaled
pivot <- any(taus < 0) || any(taus > 1)
if(!pivot){ #grid method
if(abs(diff(range(diff(taus)))) > sqrt(.Machine$double.eps))
stop("rqProcess must be evaluated on equally spaced taus")
ntaus <- length(taus)
coefs <- matrix(0, ntaus, p)
Cov <- array(0, c(p, p, ntaus))
for (i in 1:ntaus) {
z <- summary(rq(formula, data = data, tau = taus[i], method = "fn"),
covariance = TRUE, ...)
coefs[i, ] <- z$coef[, 1]
Cov[, , i] <- z$cov/(taus[i]*(1 - taus[i]))
}
qtaus <- coefs %*% xbar
Vhat <- t(coefs)[-1,,drop=FALSE]
vhat <- t(coefs)[-1,,drop=FALSE]
J <- solve(Jinv)
p <- nrow(J)
if (nullH == "location-scale") {
f <- lm(coefs[,-1] ~ coefs[,1])
b <- matrix(f$coef,2,p-1)[2,]
R <- matrix(f$resid,ntaus,p-1)
for (j in 1:length(taus)) {
V <- Cov[, , j]
v <- V[-1, -1] + V[1, 1] * outer(b,b) -
outer(V[-1, 1], b) - t(outer(V[-1, 1], b))
v <- solve(v)
v <- chol(0.5 * (v + t(v)))
Vhat[,j] <- v %*% R[j,]
for (i in 2:p) {
v <- V[i, i] + V[1, 1] * b[i-1]^2 - 2 * V[i, 1] * b[i-1]
vhat[i-1,j] <- R[j, i-1]/sqrt(v)
}
}
}
else if (nullH == "location") {
b <- apply(coefs, 2, mean)
R <- t(coefs) - b
for (j in 1:length(taus)) {
V <- Cov[, , j]
A <- solve(V[-1, -1,drop=FALSE])
B <- chol(0.5 * (A + t(A)))
Vhat[,j] <- B %*% R[-1, j,drop=FALSE]
vhat[,j] <- R[-1, j,drop=FALSE] / (sqrt(diag(V))[-1])
}
}
}
else{
z <- rq(formula,data = data, tau = -1)
taus <- z$sol[1,]
ntaus <- length(taus)
qtaus <- z$sol[2,]
qden <- qdensity(taus, qtaus)
A <- solve(Jinv[-1,-1,drop=FALSE])
B <- z$sol[-(1:3),,drop=FALSE]
if(nullH == "location")
R <- B[-1,,drop=FALSE] - c(((B[-1,-1] + B[-1,-ntaus])/2) %*% diff(taus))
else if(nullH == "location-scale")
R <- t(lm(t(B[-1,,drop=FALSE]) ~ B[1,])$resid)
Vhat <- (chol(A) %*% t(t(R) * qden$s) )[,!qden$trim,drop=FALSE]
vhat <- (t(t(R) * qden$s)/(sqrt(diag(Jinv))[-1]))[,!qden$trim,drop=FALSE]
taus <- taus[!qden$trim]
qtaus <- qtaus[!qden$trim]
}
dimnames(Vhat) <- list(vars, NULL)
dimnames(vhat) <- list(vars, NULL)
x <- list(taus = taus, qtaus = qtaus, Vhat = Vhat, vhat = vhat)
class(x) <- "rqProcess"
x
}
"qdensity" <- function(u,q,alpha = .05) {
#Computes Siddiqui estimate of quantile density function, fhat(F^{-1}(tau))
#Based on quantile regression process, using trimming proportion alpha
#linear interpolation calls approx() april, 2006
#h <- 0.6 * bandwidth.rq(u, length(u),hs=FALSE) #local bandwidth
h <- bandwidth.rq(u, length(u)) #local bandwidth
trim <- ((u - h) < alpha) | ((u + h) > 1 - alpha)
qlo <- approx(u, q, u - h)$y
qup <- approx(u, q, u + h)$y
s <- (2 * h)/(qup - qlo)
list(s=s, trim = trim)
}
"KhmaladzeTest" <-
function (formula, data = NULL, taus = -1, nullH = "location",
trim = c(0.05, 0.95), ...)
{
f <- rqProcess(formula, data = data, taus=taus, nullH = nullH, ...)
Vtilde <- khmaladzize(f$taus, f$qtaus, f$Vhat, nullH)
vtilde <- khmaladzize(f$taus, f$qtaus, f$vhat, nullH)
trim <- (f$taus >= trim[1] & f$taus <= trim[2])
Tvtilde <- (vtilde - vtilde[, 2])/sqrt(max(f$taus) - min(f$taus))
TVtilde <- apply(abs(Vtilde - Vtilde[, 2])/
sqrt(max(f$taus) - min(f$taus)), 2, "sum")[trim]
Tn <- max(TVtilde)
THn <- apply(abs(Tvtilde[, trim,drop = FALSE]), 1, max)
x <- list(nullH = nullH, Tn = Tn, THn = THn)
class(x) <- "KhmaladzeTest"
x
}
"khmaladzize" <-
function (taus, qtaus, Z, nullH) {
dtaus <- diff(taus)
dtaus <- c(dtaus[1], dtaus)
score <- akj(qtaus, qtaus, dtaus)
L <- length(taus)
gdot2 <- -score$psi
gdot <- cbind(rep(1, L), gdot2)
if (nullH == "location-scale") {
gdot3 <- gdot2 * qtaus
gdot <- cbind(gdot, gdot3)
}
kmin <- 0
p <- nrow(Z)
for (i in 1:p) {
v <- Z[i, ]
dv <- c(0, diff(v))
x1 <- gdot * sqrt(dtaus)
x1 <- x1[L:1, ]
y1 <- rev(dv/sqrt(dtaus))
bhat <- lm.fit.recursive(x1, y1, int = FALSE)
bhat <- bhat[, L:1]
dvhat <- diag(gdot %*% bhat) * dtaus
vhat <- cumsum(dvhat)
v <- v[kmin:L - kmin]
vhat <- vhat[kmin:L - kmin]
Z[i, ] <- v - vhat
}
return(Z)
}
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