Log-Logistic Distribution (Non-Monotonic Hazard)

Creates a DFR distribution with log-logistic hazard function. The log-logistic has a non-monotonic hazard that increases then decreases, useful for modeling processes with an initial risk that diminishes.

Usage

dfr_loglogistic(alpha = NULL, beta = NULL)

Arguments

alpha

Scale parameter. Median lifetime when beta > 1.

beta

Shape parameter. Controls hazard shape: beta <= 1: monotonically decreasing hazard beta > 1: hazard increases to a peak then decreases

Value

A dfr_dist object with analytical rate function. Cumulative hazard uses numerical integration.

Details

The log-logistic distribution has:

• Hazard: \(h(t) = \frac{(\beta/\alpha)(t/\alpha)^{\beta-1}}{1 + (t/\alpha)^\beta}\)

• Survival: \(S(t) = \frac{1}{1 + (t/\alpha)^\beta}\)

• Median: \(\alpha\) (when beta > 1)

Reliability Interpretation

The log-logistic is useful when:

• Initial failures decrease after screening period

• Risk peaks early then declines (therapy response)

• Hazard is not monotonic throughout lifetime

Note: The cumulative hazard has no closed form and is computed numerically. For efficiency with large datasets, consider providing cum_haz_rate using numerical integration cached appropriately.

Examples

# Component with peak hazard around t = alpha
comp <- dfr_loglogistic(alpha = 1000, beta = 2)

# Non-monotonic hazard
h <- hazard(comp)
h(500)   # increasing phase
#> [1] 8e-04
h(1000)  # near peak
#> [1] 0.001
h(2000)  # decreasing phase
#> [1] 8e-04

# Survival function
S <- surv(comp)
S(1000)  # 50% survival at median (alpha)
#> [1] 0.5