The Beta-Danish Distribution for Lifetime Data Analysis

Implements the four-parameter Beta-Danish distribution and its three-parameter Exponentiated Danish submodel for survival, reliability and lifetime data analysis, following Ahmad and Danish (2025) . Density, distribution, quantile, survival, hazard and random generation functions are evaluated so as to retain accuracy in the heavy upper tail, where the survival function is regularly varying. Estimation covers maximum likelihood for complete and right-censored samples, ridge-penalized fitting for weakly identified regimes, a grouped likelihood for times recorded on a coarse grid, and Bayesian sampling. Inference provides log-scale Wald and profile likelihood intervals, together with a reparameterization in terms of the identified composite of the two shape parameters. Structural properties include raw, incomplete and conditional moments with their existence conditions, Shannon, Renyi and Tsallis entropies, mean residual life, mean deviations, Lorenz and Bonferroni curves, probability weighted moments, order statistics, stress-strength reliability, hazard shape classification and the tail index. Regression modules cover accelerated failure time models, mixture and promotion-time cure models, and competing risks with Aalen-Johansen comparison and Gray's test. Analyses can be run directly from a delimited text file or spreadsheet.

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 accommodates monotonic, unimodal and bathtub-shaped hazards within a single parametric family.

Its upper tail is regularly varying with index -b, so the survival function decays polynomially rather than exponentially. That is what makes the family useful for heavy-tailed lifetime data, and it has two consequences the package takes seriously: E(Z^r) is finite only when b > r, and the moment generating function does not exist at all.

Distribution functions

dbetadanish(), pbetadanish(), qbetadanish(), sbetadanish(), hbetadanish() and rbetadanish(), with ded(), ped(), qed(), sed(), hed() and red() naming the ED submodel directly. All are evaluated so as to hold accuracy far into the tail: the quantile function uses the beta mirror identity rather than subtracting a near-one probability from one, and the survival function is never formed by cancellation.

Estimation and inference

• Maximum likelihood for complete and right-censored samples
• A grouped likelihood for times recorded on a coarse grid, where treating a rounded value as exact would understate every standard error
• Ridge-penalized fitting for the weakly identified regime
• Bayesian sampling by random-walk Metropolis
• Log-scale Wald and profile-likelihood intervals; an unbounded profile is reported as such rather than truncated at the grid edge
• bd_identified_coef() reports the fit through the identified composite ac, which is what the data determine when a and c cannot be separated
• Diagnostics that warn when a fit lands on the b = 1 ridge, when the information matrix is singular, or when starts were discarded as degenerate

Structural properties

Raw, incomplete and conditional moments with their existence conditions; Shannon (closed form), Renyi and Tsallis entropies; mean residual life and mean inactivity time; mean deviations; Lorenz and Bonferroni curves; probability weighted moments; order-statistic densities, distributions and moments; stress-strength reliability; hazard-shape classification via Glaser's criterion; and the tail index.

Regression

Accelerated failure time models, mixture and promotion-time cure models, and competing risks with Aalen-Johansen comparison and Gray's test, each with Cox-Snell residual diagnostics.

Working from a file

bd_analyze_csv() takes a delimited file or spreadsheet through reading, fitting, tabulation and optional figure output in one call. See the "Analysing Your Own Data from a CSV File" vignette.

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("melanoma")
melanoma$event <- ifelse(melanoma$status == 1, 1, 0)

fit_aft <- fit_bd_aft(
  survival::Surv(time, event) ~ age + thickness,
  data = melanoma
)
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

Working from a CSV File

The whole workflow can be driven from a spreadsheet, with no modelling code. Two columns are enough: time and status (1 = event, 0 = censored).

# Not sure of the layout? Write a template and fill it in.
bd_csv_template("my_data.csv", type = "covariate")

# Read a file; time and status are guessed from common column names.
dat <- read_survival_data("my_data.csv", covar_cols = "all")
attr(dat, "bd_data_report")   # what was read, dropped, and inferred

# Or run the whole analysis in one call.
res <- bd_analyze_csv(
  "my_data.csv",
  analysis   = "univariate",   # or "aft", "cure", "competing"
  model      = "both",         # Beta-Danish and the ED submodel
  output_dir = "results"       # tables as CSV, figures as PNG
)

res                       # headline summary
res$tables$estimates      # tidy parameter table
res$failures              # empty if everything succeeded

Nothing is written to disk unless output_dir is supplied. Each model is fitted independently, so one failure is recorded rather than losing the whole run.

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 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. [email protected]

Muhammad Yameen Danish — Allama Iqbal Open University, Islamabad, Pakistan. [email protected]

License

GPL-3