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:

Function	Purpose
dbetadanish()	Density (PDF)
pbetadanish()	Cumulative distribution (CDF) and survival
qbetadanish()	Quantile function
rbetadanish()	Random number generation
hbetadanish()	Hazard rate

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

Function	Property
bd_entropy_shannon()	Shannon (differential) entropy
bd_order_stat_pdf()	r-th order statistic density

Built-in Datasets

Dataset	n	Description
remission	128	Bladder cancer remission times
carbon_fibres	100	Breaking stress of carbon fibres (Gba)
transplant	91	Bone marrow transplant survival
aarset	50	Aarset device failure times (bathtub hazard)
leukemia	23	Acute myelogenous leukemia survival
melanoma	205	Malignant melanoma post-surgery
brain_cancer	varies	Brain cancer survival with comorbidities

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 the underlying article. The package implements the three-parameter Exponentiated Danish (ED) submodel introduced in the published article, as well as the four-parameter Beta-Danish distribution developed in the accompanying doctoral thesis.

Article (introduces the three-parameter ED submodel):

Ahmad, B., & Danish, M. Y. (2025). Development and characterization of a flexible three-parameter lifetime distribution: theoretical properties and real-world applications. Journal of Applied Mathematics, Statistics and Informatics, 21(1). https://doi.org/10.2478/jamsi-2025-0010

Thesis (introduces the four-parameter Beta-Danish extension):

Ahmad, B. (2026). Modeling Diverse Survival Patterns: The Development and Characterization of a New Four-Parameter Lifetime Distribution (Ph.D. thesis). Allama Iqbal Open University, Islamabad, Pakistan. Supervised by Dr. Muhammad Yameen Danish.

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