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Gompertz Distribution (Exponential Growth Hazard)

Source: R/distributions.R
dfr_gompertz.Rd

Creates a DFR distribution with Gompertz hazard function. The Gompertz models exponentially increasing failure rate, often used for biological aging and wear-out processes that accelerate over time.

Usage

dfr_gompertz(a = NULL, b = NULL)

Arguments

a

Initial hazard rate at t=0. Must be positive.

b

Growth rate of the hazard. Must be positive.

Value

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

Details

The Gompertz distribution has:

  • Hazard: \(h(t) = a \cdot e^{bt}\)

  • Cumulative hazard: \(H(t) = (a/b)(e^{bt} - 1)\)

  • Survival: \(S(t) = \exp(-(a/b)(e^{bt} - 1))\)

Reliability Interpretation

Use Gompertz for:

  • Aging systems where failure rate grows exponentially

  • Biological mortality (human lifespans)

  • Corrosion/degradation with accelerating kinetics

When b is small, Gompertz approximates exponential early in life. As b increases, wear-out acceleration becomes more pronounced.

Examples

# Aging system: initial hazard 0.001, doubling every 1000 hours
# b = log(2)/1000 gives doubling time of 1000
system <- dfr_gompertz(a = 0.001, b = log(2)/1000)

# Hazard at various ages
h <- hazard(system)
h(0)      # 0.001 (initial)
#> [1] 0.001
h(1000)   # 0.002 (doubled)
#> [1] 0.002
h(2000)   # 0.004 (quadrupled)
#> [1] 0.004

# Survival probability
S <- surv(system)
S(5000)   # probability of surviving 5000 hours
#> [1] 3.774076e-20