dfr.dist

Dynamic Failure Rate Distributions for Survival Analysis

Capacitors that wear out faster than any Weibull can describe. Software systems with bathtub-shaped crash rates. Post-surgical patients whose risk drops sharply, then slowly climbs again. Standard parametric survival families cannot express these hazard patterns — but dfr.dist can.

Write the hazard function you need — any R function of time and parameters — and the package derives everything else: survival curves, CDFs, densities, quantiles, sampling, log-likelihoods, MLE fitting, and residual diagnostics.

Why dfr.dist?

Feature	dfr.dist	survival	flexsurv
Custom hazard functions	Yes	No	Limited
Built-in distributions	Exp, Weibull, Gompertz, Log-logistic	Weibull, Exp	Many
User-supplied derivatives	score + Hessian	No	No
Censoring support	Right + Left	Right	Right
Model diagnostics	Cox-Snell, Martingale, Q-Q	Limited	Limited
Likelihood model interface	Full	Basic	Partial

Features

• Flexible hazard specification: Define any hazard function h(t, par, …)
• Built-in distributions: Exponential, Weibull, Gompertz, Log-logistic with optimized implementations
• Complete distribution interface: hazard, survival, CDF, PDF, quantiles, sampling
• Likelihood model support: Log-likelihood, score, Hessian for MLE
• Custom derivatives: Supply analytical score and Hessian functions, or let the package fall back to numerical differentiation via numDeriv
• Model diagnostics: Residuals (Cox-Snell, Martingale) and Q-Q plots
• Censoring support: Handle exact, right-censored, and left-censored survival data
• Ecosystem integration: Works with algebraic.dist, likelihood.model, algebraic.mle

Installation

Install from GitHub:

# install.packages("devtools")
devtools::install_github("queelius/dfr.dist")

Quick Start

library(dfr.dist)

Built-in Distributions

Use the convenient constructors for classic survival distributions:

# Exponential: constant hazard (memoryless)
exp_dist <- dfr_exponential(lambda = 0.5)

# Weibull: power-law hazard (wear-out or infant mortality)
weib_dist <- dfr_weibull(shape = 2, scale = 3)

# Gompertz: exponentially increasing hazard (aging)
gomp_dist <- dfr_gompertz(a = 0.01, b = 0.1)

# Log-logistic: non-monotonic hazard (increases then decreases)
ll_dist <- dfr_loglogistic(alpha = 10, beta = 2)

All distribution functions are automatically available:

S <- surv(exp_dist)
S(2)  # Survival probability at t=2
#> [1] 0.3678794

h <- hazard(weib_dist)
h(1)  # Hazard at t=1
#> [1] 0.2222222

Maximum Likelihood Estimation

# Simulate failure times
set.seed(42)
times <- rexp(50, rate = 1)
df <- data.frame(t = times, delta = 1)

# Fit via MLE
solver <- fit(dfr_exponential())
result <- solver(df, par = c(0.5), method = "BFGS")
coef(result)  # Estimated rate
#> [1] 0.8808457

Custom Hazard Functions

Model complex failure patterns like bathtub curves:

# h(t) = a*exp(-b*t) + c + d*t^k
# Infant mortality + useful life + wear-out
bathtub <- dfr_dist(
  rate = function(t, par, ...) {
    par[1] * exp(-par[2] * t) + par[3] + par[4] * t^par[5]
  },
  par = c(a = 1, b = 2, c = 0.02, d = 0.001, k = 2)
)

h <- hazard(bathtub)
curve(sapply(x, h), 0, 15, xlab = "Time", ylab = "Hazard rate",
      main = "Bathtub hazard curve")

Model Diagnostics

Check model fit with residual analysis:

# Fit exponential to data
fitted_exp <- dfr_exponential(lambda = coef(result))

# Cox-Snell residuals Q-Q plot
qqplot_residuals(fitted_exp, df)

Mathematical Background

For a lifetime $T$, the hazard function is: $$h(t) = \frac{f(t)}{S(t)}$$

From the hazard, all other quantities follow:

Function	Formula	Method
Cumulative hazard	$H(t) = \int_0^t h(u) du$	cum_haz()
Survival	$S(t) = e^{-H(t)}$	surv()
CDF	$F(t) = 1 - S(t)$	cdf()
PDF	$f(t) = h(t) \cdot S(t)$	density()

Likelihood for Survival Data

For exact observations: $\log L = \log h(t) - H(t)$

For right-censored: $\log L = -H(t)$

# Mixed data with censoring
df <- data.frame(
  t = c(1, 2, 3, 4, 5),
  delta = c(1, 1, 0, 1, 0)  # 1 = exact, 0 = censored
)

ll <- loglik(dfr_exponential())
ll(df, par = c(0.5))
#> [1] -9.579442

Documentation

Start Here:

• Package Overview - Motivation, complete example, and audience guide
• Quick Start Guide - 5-minute introduction

Real-World Applications:

• Reliability Engineering - Five case studies

Going Deeper:

• Dynamic Failure Rate Distributions - Mathematical foundations
• Creating Custom Distributions - The three-level optimization paradigm
• Custom Derivatives for MLE - Analytical score and Hessian functions

Reference:

• Function Reference

Related Packages

• algebraic.dist: Generic distribution interface
• likelihood.model: Likelihood model framework
• algebraic.mle: MLE utilities