prep_boot {HQM}

Precomputation for wild bootstrap

Description

Implements key components for the wild bootstrap of the hqm estimator in preparation for obtaining confidence bands.

Usage

prep_boot(g_xt, alpha, Ni, Yi, size_s_grid, br_X, br_s, t, b, int_X, x, n)

Arguments

g_xt

A vector obtained by g_xt.

alpha

A vector of the marker only hazard on the marker grid obtained by get_alpha.

Ni

A matrix made by make_Ni indicating the occurence.

Yi

A matrix made by make_Yi indicating the exposure.

size_s_grid

Size of the time grid.

br_X

Vector of grid points for the marker values.

br_s

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

t

Numeric value of the time the function should be evaluated.

b

Bandwidth.

int_X

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

x

Numeric value of the last observed marker value.

n

Number of individuals.

Details

The function implements

A_B(t) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \int^{T}_0 \hat{g}_{i,t,x_*}(X_i(s)) V_i\{dN_i(s) - \hat{\alpha}_i(X_i(s))Z_i(s)ds\},

and

B_B(t) = \frac{1}{\sqrt{n}}\sum_{i = 1}^n V_i\{\hat{\Gamma}(t,x_*)^{-1}W_i(t,x_*) - \hat{h}_{x_*}(t)\},

where V \sim N(0,1),

W_i(t) =\int_0^T\hat{\alpha}_i(X_i(t+s))Z_i(t+s)Z_i(s)K_b(x_*,X_i(s))\mathrm {d}s,

and

\hat{\Gamma}(t,x) = \frac{1}{n} \sum_{i = 1}^n \int_{0}^{T-t} Z_i(t+s)Z_i(s) K_b(x,X_i(s))ds,

with Z being the exposure and X the marker.

Value

A list of 5 items. The first two are vectors for calculating A_B and the third one a vector for B_B. The 4th one is the value of the hqm estimator that can also be obtained by h_xt and the last one is the value of \Gamma.

See Also

Conf_bands

Examples

pbc2_id = to_id(pbc2)
size_s_grid <- size_X_grid <- 100
n = max(as.numeric(pbc2$id))
s = pbc2$year
X = pbc2$serBilir
XX = pbc2_id$serBilir
ss <- pbc2_id$years
delta <- pbc2_id$status2
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)

N <- make_N(pbc2, pbc2_id, br_X, br_s, ss, XX, delta)
Y <- make_Y(pbc2, pbc2_id, X_lin, br_X, br_s,
            size_s_grid, size_X_grid, int_s, int_X, event_time = 'years', n)

b = 1.7
alpha<-get_alpha(N, Y, b, br_X, K=Epan )

Yi <- make_Yi(pbc2, pbc2_id, X_lin, br_X, br_s,
              size_s_grid, size_X_grid, int_s, int_X, event_time = 'years', n)
Ni  <- make_Ni(br_s, size_s_grid, ss, delta, n)

t = 2
x = 2

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

Boot_all = prep_boot(g, alpha, Ni, Yi, size_s_grid, br_X, br_s, t, b, int_X, x, n)
Boot_all