Unit-Gompertz Distribution with Applications


  • Josmar Mazucheli Universidade Estadual de Maringá
  • André Felipe Menezes Universidade Estadual de Maringá
  • Sanku Dey St. Anthony's College




Gompertz distribution, Maximum likelihood estimators, Monte Carlo simulation


The transformed family of distributions are sometimes very useful to explore additional properties of the phenomenons which non-transformed (baseline) family of distributions cannot. In this paper, we introduce a new transformed model, called the unit-Gompertz (UG) distribution which exhibit right-skewed (unimodal) and reversed-J shaped density while the hazard rate has constant, increasing, upside-down bathtub and then bathtub shaped hazard rate. Some statistical properties of this new distribution are presented and discussed. Maximum likelihood estimation for the parameters that index UG distribution are derived along with their corresponding asymptotic standard errors. Monte Carlo simulations are conducted to investigate the bias, root mean squared error of the maximum likelihood estimators as well as the coverage probability. Finally, the potentiality of the model is presented and compared with three others distributions using two real data sets.


A. C. BEMMAOR (1994). Modeling the diffusion of new durable goods: Word-of-mouth effect versus consumer heterogeneity. In G. LAURENT, G. L. LILIEN, B. PRAS (eds.), Research Traditions in Marketing, Springer Netherlands, Dordrecht, pp. 201–229.

L. BENKHELIFA (2017). The beta generalized Gompertz distribution. Applied Mathematical Modelling, 52, pp. 341–357.

Z. CHEN (1997). Parameter estimation of the Gompertz population. Biometrical Journal, 39, no. 1, pp. 117–124.

R. DUMONCEAUX, C. E. ANTLE (1973). Discrimination between the log-normal and the Weibull distributions. Technometrics, 15, no. 4, pp. 923–926.

M. EL-DAMCESE, A. MUSTAFA, B. EL-DESOUKY, M. MUSTAFA (2015). The odd generalized exponential Gompertz distribution. Applied Mathematics, 6, no. 14, pp. 2340–2353.

A. EL-GOHARY, A.ALSHAMRANI, A.N.AL-OTAIBI (2013). The generalized Gompertz distribution. Applied Mathematical Modelling, 37, no. 1–2, pp. 13 – 24.

P. H. FRANSES (1994). Fitting a Gompertz curve. The Journal of the Operational Research Society, 45, no. 1, pp. 109–113.

E. GÓMEZ-DÉNIZ, M. A. SORDO, E. CALDERÍN-OJEDA (2014). The log-Lindley distribution as an alternative to the Beta regression model with applications in insurance. Insurance: Mathematics and Economics, 54, pp. 49 – 57.

A. GRASSIA (1977). On a family of distributions with argument between 0 and 1 obtained by transformation of the Gamma distribution and derived compound distributions. Australian

Journal of Statistics, 19, no. 2, pp. 108–114.

A. A. JAFARI, S. TAHMASEBI (2016). Gompertz-power series distributions. Communications in Statistics - Theory and Methods, 45, no. 13, pp. 3761–3781.

A. A. JAFARI, S. TAHMASEBI, M. ALIZADEH (2014). The Beta-Gompertz distribution. Revista Colombiana de Estadística, 37, no. 1, pp. 141–158.

P. KUMARASWAMY (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46, no. 1, pp. 79 – 88.

E. J. LEHMANN, G. CASELLA (1998). Theory of Point Estimation. Springer Verlag, New York.

B. G. LINDSAY, B. LI (1997). On second-order optimality of the observed Fisher information. The Annals of Statistics, 25, no. 5, pp. 2172–2199.

R. MAKANY (1991). A theoretical basis for Gompertz's curve. Biometrical Journal, 33, no. 1, pp. 121–128.

J. B. MCDONALD (1984). Some generalized functions for the size distribution of income. Econometrica, 52, no. 3, pp. 647–665.

C. QUESENBERRY, C. HALES (1980). Concentration bands for uniformity plots. Journal of Statistical Computation and Simulation, 11, no. 1, pp. 41–53.

B. R. RAO, C. V. DAMARAJU (1992). New better than used and other concepts for a class of life distributions. Biometrical Journal, 34, no. 8, pp. 919–935.

C. B. READ (1983). Gompertz distribution. In S. KOTZ, N. JOHNSON (eds.), Encyclopedia of Statistical Sciences, JohnWiley & Sons, Inc., New York, p. 446.

S. ROY, M. A. S. ADNAN (2012). Wrapped generalized Gompertz distribution: An application to ornithology. Journal of Biometrics & Biostatistics, 3, no. 6, pp. 153–156.

M. D. STASINOPOULOS, R. A. RIGBY, G. Z. HELLER, V. VOUDOURIS, F. DE BASTIANI (2017). Flexible Regression and Smoothing: Using GAMLSS in R. Chapman & Hall/CRC, Boca Raton.

F. WILLEKENS (2001). Gompertz in context: The Gompertz and related distributions. In E. TABEAU, A. VAN DEN BERG JETHS, C.HEATHCOTE (eds.), Forecasting Mortality in Developed Countries: Insights from a Statistical, Demographic and Epidemiological Perspective, Springer Netherlands, Dordrecht, pp. 105–126.

J.-W. WU, W.-C. LEE (1999). Characterization of the mixtures of Gompertz distributions by conditional expectation of order statistics. Biometrical Journal, 41, no. 3, pp. 371–381.




How to Cite

Mazucheli, J., Menezes, A. F., & Dey, S. (2019). Unit-Gompertz Distribution with Applications. Statistica, 79(1), 25–43. https://doi.org/10.6092/issn.1973-2201/8497