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Weibull Distribution (Power-Law Hazard)

Source: R/distributions.R
dfr_weibull.Rd

Creates a DFR distribution with Weibull hazard function. The Weibull is extremely versatile: it can model increasing (wear-out), decreasing (infant mortality), or constant (exponential) failure rates.

Usage

dfr_weibull(shape = NULL, scale = NULL)

Arguments

shape

Shape parameter (k). Controls hazard behavior: k < 1: decreasing hazard (infant mortality) k = 1: constant hazard (exponential) k > 1: increasing hazard (wear-out)

scale

Scale parameter (sigma). Controls time scale.

Value

A dfr_dist object with analytical rate, cumulative hazard, and score function.

Details

The Weibull distribution has:

  • Hazard: \(h(t) = (k/\sigma)(t/\sigma)^{k-1}\)

  • Cumulative hazard: \(H(t) = (t/\sigma)^k\)

  • Survival: \(S(t) = e^{-(t/\sigma)^k}\)

  • Characteristic life (63.2% failure): \(\sigma\)

Reliability Interpretation

  • Shape < 1: Infant mortality (burn-in failures, defects)

  • Shape = 1: Random failures (reduces to exponential)

  • Shape = 2: Rayleigh distribution (linear hazard increase)

  • Shape > 2: Accelerating wear-out (fatigue, corrosion)

B-Life Calculation

The B10 life (10% failure quantile) is commonly used in reliability: \(B10 = \sigma \cdot (-\log(0.9))^{1/k}\)

Examples

# Bearing with wear-out failure (shape > 1)
bearing <- dfr_weibull(shape = 2.5, scale = 50000)

# Hazard increases with time
h <- hazard(bearing)
h(10000)  # hazard at 10k hours
#> [1] 4.472136e-06
h(40000)  # much higher at 40k hours
#> [1] 3.577709e-05

# B10 life calculation
Q <- inv_cdf(bearing)
B10 <- Q(0.10)  # 10% failure quantile

# Fit to test data with right-censoring
set.seed(123)
test_data <- data.frame(
  t = pmin(rweibull(100, shape = 2.5, scale = 50000), 30000),
  delta = as.integer(rweibull(100, shape = 2.5, scale = 50000) <= 30000)
)
solver <- fit(dfr_weibull())
result <- solver(test_data, par = c(2, 40000))
coef(result)
#> [1]     6.091528 39999.995178