BetaDanish

Overview

The BetaDanish package implements the four-parameter Beta-Danish distribution and its three-parameter Exponentiated Danish (ED) submodel for survival, reliability, and lifetime-data analysis. The distribution was introduced by Ahmad and Danish (2025) and offers a flexible alternative to classical lifetime models such as the Weibull, gamma, and log-normal, while accommodating monotonic, unimodal, and bathtub-shaped hazards within a single parametric family.

Beyond the core distribution, the package provides a comprehensive toolkit for modern survival modelling:

• Maximum-likelihood and Bayesian inference for complete and right-censored data
• Accelerated failure time (AFT) regression
• Mixture and promotion-time cure models
• Competing risks analysis with Aalen-Johansen comparisons and Gray’s test
• Closed-form moments, Shannon entropy, order statistics, mean residual life, hazard-shape classification, and stress-strength reliability
• A complete diagnostic toolkit: survival, hazard, density, P-P, Q-Q and Cox-Snell residual plots

Installation

You can install the development version of BetaDanish from GitHub with:

# install.packages("devtools")
devtools::install_github("bilal-aiou/BetaDanish")

Optional packages used by advanced modules (declared in Suggests):

install.packages(c("MCMCpack", "coda", "cmprsk", "flexsurv", "MASS"))

The Beta-Danish Distribution

Construction

The Beta-Danish distribution is obtained by applying the beta-generated family of Eugene, Lee, and Famoye (2002) to the Danish baseline, whose cumulative distribution function and density are

\[G(z; c, k) = \left(\frac{kz}{1+kz}\right)^c, \qquad g(z; c, k) = ck\,(kz)^{c-1}(1+kz)^{-(c+1)}, \qquad z > 0,\]

with shape parameter \(c > 0\) and scale parameter \(k > 0\).

Cumulative distribution function

For a random variable \(Z \sim \text{BetaDanish}(a, b, c, k)\) with \(a, b, c, k > 0\), the CDF is the regularised incomplete beta function evaluated at the Danish CDF:

\[F_Z(z; a, b, c, k) = I_{G(z; c, k)}(a, b) = I_{(kz/(1+kz))^c}(a, b), \qquad z > 0,\]

where \(I_x(a, b) = B_x(a, b) / B(a, b)\) and \(B(a, b)\) is the beta function.

Probability density function

Differentiating the CDF yields the density

\[f_Z(z; a, b, c, k) = \frac{ck}{B(a, b)} \cdot \frac{(kz)^{ca - 1}}{(1 + kz)^{ca + 1}} \cdot \left[1 - \left(\frac{kz}{1 + kz}\right)^c\right]^{b - 1}, \qquad z > 0.\]

The case \(a = 1\) yields the three-parameter Exponentiated Danish (ED) submodel used throughout the case studies.

Survival, hazard, and CDF

The package provides numerically stable d/p/q/r/h functions analogous to the base R distribution interface:

Quick Start

library(BetaDanish)

# 1. Evaluate the density, CDF, hazard
dbetadanish(x = 2, a = 1.5, b = 2, c = 3, k = 0.5)
pbetadanish(q = 2, a = 1.5, b = 2, c = 3, k = 0.5)
hbetadanish(x = 2, a = 1.5, b = 2, c = 3, k = 0.5)

# 2. Simulate data and fit a model
set.seed(123)
dat <- simulate_bd_data(n = 200, a = 1.5, b = 2, c = 3, k = 0.5,
                        censor_rate = 0.2)
fit <- fit_betadanish(survival::Surv(time, status) ~ 1, data = dat)
summary(fit)

# 3. Diagnostic plots
plot(fit, type = "all")

# 4. Compare against standard distributions
compare_distributions(fit)

Featured Examples

Bayesian estimation

data("remission")
fit_bayes <- bayes_betadanish(
  time     = remission$time,
  status   = remission$status,
  submodel = TRUE,
  burnin   = 2000,
  mcmc     = 5000,
  seed     = 1
)
print(fit_bayes)

See the Bayesian Estimation vignette for full details.

AFT regression

data("brain_cancer")
fit_aft <- fit_bd_aft(
  survival::Surv(Survtime, Survstatus) ~ Age + Grade + Surgery,
  data = brain_cancer
)
summary(fit_aft)
plot(fit_aft)   # Cox-Snell residual diagnostic

Cure models

Both mixture and promotion-time cure models on the Exponentiated Danish kernel:

fit_cure <- fit_bd_cure(
  formula_aft  = survival::Surv(time, status) ~ 1,
  formula_cure = ~ group,
  data         = mydata,
  type         = "mixture"
)
summary(fit_cure)

Competing risks

fit_cr <- fit_bd_competing(time = time, cause = cause)
res <- cif_compare(fit_cr, plot = TRUE)
res$gray_test

The fitted Beta-Danish cumulative incidence functions are overlaid against the nonparametric Aalen-Johansen estimator, with Gray’s test reported for cause equality.

Structural Properties

Built-in Datasets

Vignettes

The package ships with five comprehensive tutorials:

1. Introduction to BetaDanish — Overview, parameterization, first-time-user walkthrough
2. Brain Cancer Case Study — Complete real-world analysis
3. Bayesian Estimation — Posterior inference with bayes_betadanish()
4. Competing Risks — Joint cause-specific modelling with CIF comparison
5. Cure Models — Mixture and promotion-time formulations

Browse them with:

browseVignettes("BetaDanish")

Citation

If you use BetaDanish in published work, please cite:

Ahmad, B., & Danish, M. Y. (2025). The Beta-Danish distribution for lifetime data analysis. Journal of Applied Mathematics, Statistics and Informatics, 21(1). https://doi.org/10.2478/jamsi-2025-0010

A BibTeX entry is available via:

citation("BetaDanish")

References

Ahmad, B., & Danish, M. Y. (2025). The Beta-Danish distribution for lifetime data analysis. Journal of Applied Mathematics, Statistics and Informatics, 21(1). https://doi.org/10.2478/jamsi-2025-0010

Eugene, N., Lee, C., & Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics — Theory and Methods, 31(4), 497–512.

Getting Help

• Bug reports and feature requests: please open an issue at https://github.com/bilal-aiou/BetaDanish/issues
• Function reference: see the Reference tab on this site, or type ?fit_betadanish in R
• Vignettes: see the Articles tab on this site, or vignette(package = "BetaDanish")

Authors

Bilal Ahmad (maintainer) — Allama Iqbal Open University, Islamabad, Pakistan. bilalahmad.imcbh9@gmail.com

Muhammad Yameen Danish — Allama Iqbal Open University, Islamabad, Pakistan. yameen.danish@aiou.edu.pk

License

GPL-3