Vectorized truncated power moment of the Weibull distribution

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

Usage

trunc_pow_moment_vec(k, v_max_vec, alpha, beta)

Arguments

k: Numeric scalar. Power parameter (can be non-integer).
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 $E[T^k | T < t_i]$ values, same length as v_max_vec.

Details

Uses the substitution $U = (T/\beta)^\alpha \sim \text{Exp}(1)$, so that $T^k = \beta^k U^{k/\alpha}$ and the truncation $T < t$ becomes $U < v_{\max}$. The result is: $$E[T^k | T < t] = \beta^k \frac{\gamma(k/\alpha + 1, v_{\max})}{1 - e^{-v_{\max}}}$$ where $\gamma(s, x)$ is the lower incomplete gamma function.

Computation is done in log-space for numerical stability.

For very small $v_{\max}$ (below 1e-12), the asymptotic approximation $E[T^k | T < t] \approx t^k / (k/\alpha + 1)$ is used to avoid numerical issues.

Examples

# E[T^2 | 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_pow_moment_vec(k = 2, v_max_vec = v_max, alpha = 2, beta = 1)
#> [1] 0.1197971 0.4180233 0.9253706