output:

github_document:

toc: true

```{r, include = FALSE}

[](https://CRAN.R-project.org/package=flexhaz)

[](https://github.com/queelius/flexhaz/actions/workflows/R-CMD-check.yaml)

**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 `flexhaz` 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.

Use `flexhaz` when your starting point is a hazard function and you want to

reuse it for simulation, likelihood evaluation, fitting, and composition into

reliability systems. Analytical cumulative hazards and derivatives are optional.

For regression and established survival workflows, also consider

and [`flexsurv`](https://chjackson.github.io/flexsurv/reference/flexsurvreg.html).

Both support multiple censoring types; `flexsurv` also accepts custom

distributions. `flexhaz` currently handles exact, right-censored, and

left-censored observations, using a function-based likelihood interface.

- **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`

Install from CRAN:

install.packages("flexhaz")

Or the development version from r-universe:

install.packages("flexhaz", repos = "https://queelius.r-universe.dev")

```{r setup, message=FALSE, warning=FALSE}

Use the convenient constructors for classic survival distributions:

```{r builtin}

All distribution functions are automatically available:

```{r methods}

```{r mle}

Model complex failure patterns like bathtub curves:

```{r bathtub, fig.height=4}

Check model fit with residual analysis:

```{r diagnostics}

For a lifetime $T$, the hazard function is:

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()` |

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

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

```{r likelihood}

**Start Here:**

- [Package Overview & Quick Start](https://queelius.github.io/flexhaz/articles/flexhaz-package.html) - Motivation, complete example, and quick start guide

**Real-World Applications:**

- [Reliability Engineering](https://queelius.github.io/flexhaz/articles/reliability_engineering.html) - Five case studies

**Going Deeper:**

- [Dynamic Failure Rate Distributions](https://queelius.github.io/flexhaz/articles/failure_rate.html) - Mathematical foundations

- [Creating Custom Distributions](https://queelius.github.io/flexhaz/articles/custom_distributions.html) - The three-level optimization paradigm

- [Custom Derivatives for MLE](https://queelius.github.io/flexhaz/articles/custom_derivatives.html) - Analytical score and Hessian functions

**Reference:**

- [Function Reference](https://queelius.github.io/flexhaz/reference/)

- [`algebraic.dist`](https://github.com/queelius/algebraic.dist): Generic distribution interface

- [`likelihood.model`](https://github.com/queelius/likelihood.model): Likelihood model framework

- [`algebraic.mle`](https://github.com/queelius/algebraic.mle): MLE utilities