HQM (version 0.1.1)

g_xt: Computation of a key component for wild bootstrap

Description

Implements a key part for the wild bootstrap of the hqm estimator.

Usage

g_xt(br_X, br_s, size_s_grid, int_X, x, t, b, Yi, Y, n)

Value

A vector of \(\hat{g}_{t,x}(z)\) for all values \(z\) on the marker grid.

Arguments

br_X

Marker value grid points that will be used in the evaluatiuon.

br_s

Time value grid points that will be used in the evaluatiuon.

size_s_grid

Size of the time grid.

int_X

Position of the linear interpolated marker values on the marker grid.

x

Numeric value of the last observed marker value.

t

Numeric value of the time the function should be evaluated.

b

Bandwidth.

Yi

A matrix made by make_Yi indicating the exposure.

Y

A matrix made by make_Y indicating the exposure.

n

Number of individuals.

Details

The function implements $$\hat{g}_{t,x}(z) = \frac{1}{n} \sum_{ j = 1}^n \int^{T-t}_0 \hat{E}(X_j(t+s))^{-1} K_b(z,X_j(t+s)) Z_j(t+s)Z_j(s)K_b(x,X_j(s))ds,$$ for every value \(z\) on the marker grid, where \(\hat{E}(x) = \frac{1}{n} \sum_{j=1}^n \int_0^T K_b(x,X_j(s))Z_j(s)ds\), \(Z\) the exposure and \(X\) the marker.

Examples

pbc2_id = to_id(pbc2)
size_s_grid <- size_X_grid <- 100
n = max(as.numeric(pbc2$id))
X = pbc2$serBilir
s = pbc2$year
br_s = seq(0, max(s), max(s)/( size_s_grid-1))
br_X = seq(min(X), max(X), (max(X)-min(X))/( size_X_grid-1))

X_lin = lin_interpolate(br_s, pbc2_id$id, pbc2$id, X, s)

int_X <- findInterval(X_lin, br_X)
int_s = rep(1:length(br_s), n)

Yi<-make_Yi(pbc2, pbc2_id, X_lin, br_X, br_s, size_s_grid, size_X_grid, int_s, int_X, 'years', n)
Y<-make_Y(pbc2, pbc2_id, X_lin, br_X, br_s,size_s_grid,size_X_grid, int_s,int_X, 'years', n)

t = 2
x = 2
b = 10

g_xt(br_X, br_s, size_s_grid, int_X, x, t, b, Yi, Y, n)