Compute w_j(t) = f_j(t) * prod_i != j F_i(t) for exponential components

For component j with rate \(\lambda_j\) and other rates \(\lambda_{-j}\): $$w_j(t) = \lambda_j e^{-\lambda_j t} \prod_{i \neq j} (1 - e^{-\lambda_i t})$$

Usage

w_j_exact(t, par, j)

Arguments

t

Scalar time point (positive numeric).

par

Numeric vector of rates (length m), one per component.

j

Component index (integer, 1-based).

Value

Scalar value of \(w_j(t)\).

Details

Uses the inclusion-exclusion expansion to express this as a finite sum of exponentials.

See also

w_j_integral() for the closed-form integral of \(w_j\).

Examples

# Component 1 contribution at t = 0.5, rates = c(1, 2, 3)
w_j_exact(0.5, c(1, 2, 3), j = 1)
#> [1] 0.2978523