Defective Generalized Gompertz Distribution

• DGGD is a cure-rate survival model where a negative α induces defectiveness, resulting in a nonzero long-term survival (cure) fraction.
• It offers dual parameterizations for quantile regression and direct cure probability estimation, blending classical and generalized Gompertz methods.
• Empirical applications and simulation studies demonstrate DGGD’s effectiveness and advantages over traditional mixture cure models in survival analysis.

Searching arXiv for papers on defective generalized Gompertz distributions and related defective Gompertz cure models. Search query: "defective generalized Gompertz distribution cure rate survival arXiv" The defective generalized Gompertz distribution (DGGD) is a cure-rate survival model obtained by placing a generalized Gompertz law in a parameter regime where the survival function converges to a positive constant rather than to zero; that nonzero limit is interpreted as the cured, immune, or long-term survivor fraction. In recent arXiv work, DGGD appears in two closely related but notationally different formulations: a defective generalized Gompertz model used for cure-rate quantile regression, and a reparametrized defective generalized Gompertz model in which the limiting survival probability is itself treated as an explicit regression target (Rodrigues et al., 2021, Neto et al., 15 Jul 2025). Across these formulations, the defining structural feature is the same: defectiveness is intrinsic to the distribution, not added through a separate mixture parameter.

1. Definitions and parameterizations

Recent arXiv papers use at least two generalized Gompertz parameterizations for DGGD. In the quantile-regression formulation, the baseline generalized Gompertz distribution is written with parameters $(\lambda,\alpha,\theta)$ and survival

$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$

with the proper model defined for $\lambda>0$, $\alpha>0$, and $\theta>0$; the distribution becomes defective when $\alpha<0$ (Rodrigues et al., 2021). In the reparametrized cure-regression formulation, the generalized Gompertz cdf is expressed as

$$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$

with survival

$$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$

and the DGGD is defined by $\alpha<0$, $\beta>0$, $$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$0 (Neto et al., 15 Jul 2025).

Formulation	Survival representation	Defective regime
Quantile-regression DGGD (Rodrigues et al., 2021)	$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$1	$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$2
Reparametrized DGGD (Neto et al., 15 Jul 2025)	$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$3	$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$4

These formulations share the same defect mechanism: the sign change in the Gompertz-type parameter $$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$5 moves the model from a proper survival law to an improper one. In the notation of the uterine-cancer paper, the DGGD nests the defective Gompertz when $$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$6 (Neto et al., 15 Jul 2025).

2. Defectiveness and cure fraction

The defining feature of a defective survival distribution is

$$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$7

so that the cdf does not converge to $$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$8. In survival-analysis terms, the missing mass is interpreted as $$S(t\mid \lambda,\alpha,\theta)=1-\left(1-\exp\left\{-\frac{\lambda}{\alpha}\left(\exp\{\alpha t\}-1\right)\right\}\right)^\theta,$$9, the probability of never experiencing the event of interest. The defective Gompertz foundation for this mechanism is explicit in recent work: for the standard Gompertz parameterization

$\lambda>0$0

the model becomes defective when $\lambda>0$1, and then

$\lambda>0$2

so the continuous part integrates to $\lambda>0$3 and the remaining mass is at $\lambda>0$4 (Neto et al., 31 Jul 2025). An alternative defective Gompertz parameterization used in a bivariate construction writes

$\lambda>0$5

again with a positive survival limit representing cure (Peres et al., 2020).

For DGGD, the generalized parameter modifies that defective tail. In the quantile-regression model,

$\lambda>0$6

and in the reparametrized cure-regression model,

$\lambda>0$7

Using the defective Gompertz baseline cure fraction $\lambda>0$8, the latter may be written as

$\lambda>0$9

so the generalized Gompertz power parameter transforms the baseline defective Gompertz cure probability into the long-term survival probability (Rodrigues et al., 2021, Neto et al., 15 Jul 2025).

This cure mechanism differs from standard mixture cure models of the form

$\alpha>0$0

because the cure fraction is induced by the survival tail itself rather than introduced as a separate free parameter at model definition. Recent defective Gompertz work explicitly presents this as a parsimonious alternative to mixture cure models, with the same family covering cured and non-cured scenarios depending on parameter values (Neto et al., 31 Jul 2025, Peres et al., 2020).

3. Reparameterization and regression structures

A major DGGD development is reparameterization around interpretable estimands. The quantile-regression paper first rewrites the defective survival in mixture form,

$\alpha>0$1

where $\alpha>0$2 is the proper susceptible survival distribution. This reformulation is used to define the $\alpha>0$3-th quantile of susceptible survival, $\alpha>0$4, and to link it to covariates through

$\alpha>0$5

Under this model, covariates act directly on a chosen susceptible quantile, while the cure fraction depends on the same DGGD parameters rather than on a separate logistic cure component (Rodrigues et al., 2021).

The uterine-cancer paper instead makes cure itself the explicit estimand. Starting from

$\alpha>0$6

it solves for the original scale parameter as

$\alpha>0$7

With

$\alpha>0$8

the reparametrized survival becomes

$\alpha>0$9

Covariates are then placed directly on the cure fraction through a logistic link: $\theta>0$0 The paper explicitly notes that this model is neither a proportional hazards model nor an accelerated failure time model in standard form, because covariates act through the cure probability and thereby alter the entire survival shape nonlinearly (Neto et al., 15 Jul 2025).

A related but distinct development appears in defective Gompertz joint modeling. There, the subject-specific cure fraction under a shared-parameter longitudinal–survival model is

$\theta>0$1

with $\theta>0$2 containing baseline covariates and shared random effects. That paper does not define a generalized Gompertz extension, but it explicitly provides a blueprint for future DGGD work: defectiveness via limiting survival, regression through the hazard scale, and subject-specific cure fractions derived from the defective tail (Neto et al., 31 Jul 2025).

4. Likelihood-based and Bayesian inference

For right-censored data, the quantile-regression DGGD uses the standard likelihood

$\theta>0$3

with parameter vector $\theta>0$4. The posterior is

$\theta>0$5

with a truncated normal prior on $\theta>0$6, a gamma prior on $\theta>0$7, and normal priors on the regression coefficients. Posterior computation is performed by the Adaptive Metropolis algorithm with multivariate normal proposal, implemented in the LaplacesDemon package in R (Rodrigues et al., 2021).

The reparametrized DGGD cure-regression model uses

$\theta>0$8

where $\theta>0$9. Independent priors are specified as normal for $\alpha<0$0, normal for $\alpha<0$1, and gamma for $\alpha<0$2, and posterior computation is carried out with Hamiltonian Monte Carlo via rstan, using the No-U-Turn Sampler (Neto et al., 15 Jul 2025).

Simulation evidence supports the practical estimability of DGGD parameters under moderate to large samples. In the quantile-regression paper, Monte Carlo experiments with $\alpha<0$3 show decreasing bias and mean squared error with increasing sample size, and cure fraction estimates with low bias even in small samples (Rodrigues et al., 2021). In the reparametrized DGGD paper, 1000 Monte Carlo replicates were run for $\alpha<0$4; by $\alpha<0$5, the reported posterior means were $\alpha<0$6, $\alpha<0$7, $\alpha<0$8, $\alpha<0$9, and $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$0, with coverage between $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$1 and $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$2 (Neto et al., 15 Jul 2025).

Methodologically adjacent work suggests broader inferential routes. Defective Gompertz joint models have been rewritten as latent Gaussian models and estimated by INLA through INLAjoint and R-INLA; that paper does not define DGGD, but it explicitly argues that the same latent-Gaussian strategy should carry over if the generalized defective likelihood remains compatible with INLA machinery (Neto et al., 31 Jul 2025). Likewise, the copula-based bivariate defective Gompertz paper develops censored likelihoods, maximum likelihood estimation via maxLik, and Bayesian estimation via R2jags, all of which are structurally transferable once DGGD marginals are specified (Peres et al., 2020).

5. Empirical uses

The DGGD quantile-regression model has been applied to male breast cancer data from São Paulo, Brazil, comprising 872 men diagnosed from 2000 to 2019, with follow-up until February 2020 and at least two months of follow-up; $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$3 of observations were censored. Covariates were age categories and clinical stage, and the model was fitted for $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$4. The main reported findings were that men older than 65 showed no evidence of different susceptible survival quantiles relative to those under 55, stage II differed from stage I, and stages III and IV had negative effects across all quantiles, with the effects becoming less negative at higher quantiles. Reported cure fractions declined monotonically from stage I to stage IV; for example, stage I, age $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$5 had posterior mean cure probability $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$6 with interval $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$7, while stage IV, age $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$8 had $$F(t;\alpha,\beta,\psi)=\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$9 with interval $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$0 (Rodrigues et al., 2021).

The reparametrized DGGD cure-regression model has been applied to uterine cancer cases from the Hospital Cancer Registry organized by the Oncocenter Foundation of São Paulo, covering women residing in São Paulo state between 2012 and 2020. The event was death due exclusively to uterine cancer, and the Kaplan–Meier curve reportedly showed a plateau after about 10 years. The covariates in the cure regression were age over 50, distant recurrence or metastasis, surgery, chemotherapy, and hormone therapy. Posterior means were

$$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$1

with the interval for $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$2 entirely negative, confirming the defective regime. The paper interprets surgery as a statistically significant protective factor, and age over 50, metastatic stage, and chemotherapy as associated with lower cure probability. Model comparison against the defective Gompertz regression favored DGGD on all reported criteria: $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$3 for generalized Gompertz, versus

$$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$4

for Gompertz (Neto et al., 15 Jul 2025).

These applications illustrate two distinct roles for DGGD. In one, it acts as a parametric engine for quantile regression on the susceptible subpopulation; in the other, it is reparametrized so that cure itself is the direct regression target. In both cases, the model is used specifically where the empirical survival curve suggests a nonzero plateau (Rodrigues et al., 2021, Neto et al., 15 Jul 2025).

6. Relation to defective Gompertz, copula models, and generalized extensions

DGGD belongs to a broader family of defective Gompertz-type survival models. The simplest member is the defective Gompertz itself, where cure is induced by $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$5 without an extra generalized parameter; recent joint-model work emphasizes four stated advantages over standard mixture cure models: parsimony, unified specification, numerical stability, and regression simplicity (Neto et al., 31 Jul 2025). The bivariate defective Gompertz distribution based on a Clayton copula extends this idea to paired event times, using defective Gompertz marginals and a copula survival

$$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$6

That paper does not define DGGD, but it is directly relevant because it supplies the copula machinery, censored likelihood structure, and medical applications for defective Gompertz-type marginals (Peres et al., 2020).

Proper generalized Gompertz extensions provide a contrasting reference class. The Marshall–Olkin extended generalized Gompertz (MOEGG) distribution is a four-parameter proper model with survival tending to zero for all parameter values discussed in the paper; it is therefore not defective, but it demonstrates how generalized Gompertz hazard shapes can be constant, increasing, decreasing, upside-down bathtub, or bathtub-shaped depending on parameters (Benkhelifa, 2016). Likewise, the generalized Gompertz–power series (GGPS) family is proper under its stated assumptions, with hazard functions that can be increasing, decreasing, and bathtub-shaped, and with an EM algorithm based on a latent count representation (Tahmasebi et al., 2015). These proper models are relevant because they show that hazard-shape flexibility and defectiveness are distinct properties: a generalized Gompertz family may be flexible yet still satisfy $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$7 (Benkhelifa, 2016, Tahmasebi et al., 2015).

Another close relative is the defective Marshall–Olkin Gompertz model. It is not identified as the same family as the Martinez–Achcar defective generalized Gompertz distribution, but it is explicitly a defective Marshall–Olkin extension of Gompertz, becoming improper when $$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$8. Its cure fraction is

$$S(t;\alpha,\beta,\psi)=1-\left\{1-\exp\!\left[-\frac{\beta}{\alpha}\left(e^{\alpha t}-1\right)\right]\right\}^{\psi},$$9

and in colon cancer data it outperformed ordinary defective Gompertz under both frequentist and Bayesian criteria (Neto et al., 2024). This suggests that additional shape parameters beyond the basic defective Gompertz can materially improve fit when cure is present, which is one of the central motivations for DGGD and related generalized defective Gompertz constructions.

A persistent feature of the literature is that DGGD is not represented by a single universal notation. Recent papers use different generalized Gompertz parameterizations, different regression targets, and different computational strategies, but they share a common survival-theoretic core: a generalized Gompertz law is extended into a defective regime, the nonzero limiting survival is interpreted as cure, and covariate effects are studied either through susceptible survival structure, direct cure modeling, or both (Rodrigues et al., 2021, Neto et al., 15 Jul 2025).