Vectorized truncated log-moment of the Weibull distribution

Computes $E[\log T | T < t_i]$ for $T \sim \text{Weibull}(\alpha, \beta)$ simultaneously for a vector of truncation points.

Usage

trunc_log_moment_vec(v_max_vec, alpha, beta)

Arguments

v_max_vec: Numeric vector. Precomputed values of $(t_i/\beta)^\alpha$.
alpha: Numeric scalar. Weibull shape parameter.
beta: Numeric scalar. Weibull scale parameter.

Value

Numeric vector of $E[\log T | T < t_i]$ values, same length as v_max_vec.

Details

Uses the identity: $$E[\log T | T < t] = \log \beta + \frac{1}{\alpha} E[\log U | U < v_{\max}]$$ where $U \sim \text{Exp}(1)$ and $v_{\max} = (t/\beta)^\alpha$.

The inner expectation $E[\log U | U < v_{\max}]$ is computed via central finite difference of the lower incomplete gamma function at $s = 1$.

For very small $v_{\max}$ (below 1e-10), the asymptotic approximation $E[\log T | T < t] \approx \log t - 1/\alpha$ is used. For large $v_{\max}$ where numerical issues arise, the untruncated moment $E[\log T] = \log \beta + \psi(1)/\alpha$ is used as fallback, where $\psi$ is the digamma function.

Examples

# E[log T | T < t] for Weibull(shape=2, scale=1) at t = 0.5, 1.0, 2.0
v_max <- (c(0.5, 1.0, 2.0) / 1)^2
trunc_log_moment_vec(v_max_vec = v_max, alpha = 2, beta = 1)
#> [1] -1.2248035 -0.6301010 -0.3088497