An Expository Note on Unit - Gompertz Distribution with Applications


  • Mohammed Zafar Anis Indian Statistical Institute
  • Debsurya De Indian Statistical Institute



Log concave, Reliability functions, Stochastic orders


In a recent paper, Mazucheli et al. (2019) introduced the unit-Gompertz (UG) distribution and studied some of its properties. It is a continuous distribution with bounded support, and hence may be useful for modelling life-time phenomena. We present counter-examples to point out some subtle errors in their work, and subsequently correct them. We also look at some other interesting properties of this new distribution. Further, we also study some important reliability measures and consider some stochastic orderings associated with this new distribution.


R. AABERGE (2000). Characterizations of Lorenz curves and income distributions. Social Choice and Welfare, 17, pp. 639–653.

A. ALZAATREH, F. FAMOYE, C. LEE (2013). A new method for generating families of continuous distributions. Metron, 71, pp. 63–79.

P. K. ANDERSEN, O. BORGAN, R. D. GILL, N. KEIDING (1993). Statistical Methods Based on Counting Processes. Springer Verlag, New York.

N. BALAKRISHNAN, C. RAO (1998). Order Statistics: Applications. Elsevier, New York.

N. BÄUERLE, E. BAYRAKTAR (2014). A note on applications of stochastic ordering to control problems in insurance and finance. Stochastics: An International Journal of Probability and Stochastic Processes, 86, no. 2, pp. 330–340.

E. BEADLE, J. SCHROEDER, B. MORAN, S. SUVOROVA (2008). An overview of Rényi Entropy and some potential applications. Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers, 42, pp. 1698–1704.

H. BLOCK, T. H. SAVITS, H. SINGH (1998). The reversed hazard rate function. Probability in the Engineering and Informational Sciences, 12, pp. 69–90.

E. BONFERRONI (1930). Elementi di Statistica Generale. Libreria Seber, Firenze.

W. BRYC (1996). Conditional moment representations for dependent random variables. Electronic Journal of Probability, 1, no. 7, p. 14.

M. CHANDRA, N. SINGPURWALLA (1981). Relationships between some notions which are common to reliability theory and economics. Mathematics of Operations Research, 5, pp. 113–121.

N. K. CHANDRA, D. ROY (2001). The reversed hazard rate function. Probability in the Engineering and Informational Sciences, 15, pp. 95–102.

M. A. DOMÍNGUEZ, I. LOBATO (2004). Consistent estimation of models defined by conditional moment restrictions. Econometrica, 72, pp. 1601–1615.

L. EECKHOUDT, C. GOLLIER (1995). Demand for risky assets and the monotone probability ratio order. Journal of Risk and Uncertainty, 11, pp. 113–122.

N. EUGENE, C. LEE, F. FAMOYE (2002). Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31, no. 4, pp. 497–512.

M. FINKELSTEIN (2002). On the reversed hazard rate. Reliability Engineering & System Safety, 78, pp. 71–75.

S. FUHRMAN, M. J. CUNNINGHAM, X.WEN, G. ZWEIGER, J. J. SEILHAMER, R. SOMOGYI (2000). The application of Shannon entropy in the identification of putative drug targets. Biosystems, 55, pp. 5–14.

T. GAŁKA (2015). On the application of Shannon entropy and continuous entropy in the evaluation of diagnostic symptoms. International Journal of Condition Monitoring, 5, pp. 12–17.

M. GHITANY (2004). The monotonicity of the reliability measures of the beta distribution. Applied Mathematics Letters, 17, pp. 1277–1283.

G. M. GIORGI, M. CRESCENZI (2001). A look at the Bonferroni inequality measure in a reliability framework. Statistica, 61, no. 4, pp. 571–583.

R. E. GLASER (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75, no. 371, pp. 667–672.

C. GROVES-KIRKBY, A. DENMAN, P. PHILLIPS (2009). Lorenz curve and Gini coefficient: Novel tools for analysing seasonal variation of environmental radon gas. Journal of Environmental Management, 90, pp. 2480–2487.

F. GUESS, F. PROSCHAN (1988). Mean residual life: Theory and applications. Handbook of Statistics, 7, pp. 215–224.

M. S. HUGHES, J. N. MARSH, J. M. ARBEIT, R. G. NEUMANN, R. W. FUHRHOP, K. D. WALLACE, L. THOMAS, J. SMITH, K. AGYEM, G. M. LANZA, S. A. WICKLINE, J. E. MCCARTHY (2009). Application of Rényi entropy for ultrasonic molecular imaging. Journal of the Acoustical Society of America, 125, pp. 3141–3145.

A. JACOBSON, A.MILMAN, D. KAMMEN (2005). Letting the (energy) Gini out of the bottle: Lorenz curves of cumulative electricity consumption and Gini coefficients as metrics of energy distribution and equity. Energy Policy, 33, pp. 1825–1832.

C. L. JAYASINGHE, P. ZEEPHONGSEKUL (2013). Non-parametric smooth estimation of the expected inactivity time function. Journal of Statistical Planning and Inference, 143, no. 5, pp. 911–928.

M. K. JHA, S. DEY, R. M. ALOTAIBI, Y. TRIPATHI (2020). Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring. Quality & Reliability Engineering International, 36, pp. 965–987.

M. K. JHA, S. DEY, Y. TRIPATHI (2019). Reliability estimation in a multicomponent stress–strength based on unit-Gompertz distribution. International Journal of Quality & Reliability Management, 37, pp. 428–450.

J. D. KALBFLEISCH, J. LAWLESS (1989). Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS. Journal of the American Statistical Association, 84, pp. 360–372.

M. KAYID, H. AL-NAHAWATI, I. A. AHMAD (2011). Testing behavior of the reversed hazard rate. Applied Mathematical Modelling, 35, no. 5, pp. 2508–2515.

B. KLEFSJÖ (1984). Reliability interpretations of some concepts from economics. Naval Research Logistics Quarterly, 31, pp. 301–308.

S. KOLTCOV (2018). Application of Rényi and Tsallis entropies to topic modeling optimization. Physica A: Statistical Mechanics and its Applications, 512, pp. 1192–1204.

D. KUMAR, S. DEY, E. ORMOZ, S. M. T. K. MIRMOSTAFAEE (2020). Inference for the unit-Gompertz model based on record values and inter-record times with an application. Rendiconti del Circolo Matematico di Palermo Series 2, 69, p. 1295–1319.

C. KUNDU, A. NANDA (2010). Some reliability properties of the inactivity time. Communications in Statistics - Theory and Methods, 39, no. 5, pp. 899–911.

K. R. M. NAIR, N. SREELAKSHMI (2016). The new Zenga curve in the context of reliability analysis. Communications in Statistics - Theory & Methods, 45, no. 22, pp. 6540–6552.

A. NANDA, S. CHOWDHURY (2020). Shannon’s entropy and its generalisations towards statistical inference in last seven decades. International Statistical Review. Early view.

A. K. NANDA, M. SHAKED (2001). The hazard rate and the reversed hazard rate orders, with application to order statistics. Annals of the Institute of Statistical Mathematics, 53, no. 4, pp. 853–864.

T. G. PHAM,N. TURKKAN (1994). The Lorenz and the scaled total-time-on-test transformcurves: a unified approach. IEEE Transactions on Reliability, 43, pp. 76–84.

P. L. RAMOS, D. C. NASCIMENTO, P. H. FERREIRA, K. T. WEBER, T. E. SANTOS, F. LOUZADA (2019). Modeling traumatic brain injury lifetime data: Improved estimators for the generalized gamma distribution under small samples. PLoS One, 14, p. 8.

A. RÉNYI (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, University of California Press, Berkeley, CA, pp. 547–561.

M. M. RISTI´C, N. BALAKRISHNAN (2012). The gamma exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, pp. 1191–1206.

M. SHAKED, J. SHANTHIKUMAR (2007). Stochastic Orders. Springer, New York.

C. E. SHANNON (1948). A mathematical theory of communication. Bell System Technical Journal, 27, pp. 379–423.

J. G. SHANTHIKUMAR, G. YAMAZAKI, H. SAHASEGAWA (1991). Characterization of optimal order of servers in a tandem queue with blocking. Operations Research Letters, 10, pp. 17–22.

M. SHERAZ, S. DEDU, V. PREDA (2015). Entropy measures for assessing volatile markets. Procedia Economics and Finance, 22, pp. 655–662.

K. S. SONG (2001). Rényi information, loglikelihood and an intrinsic distribution measure. Journal of Statistical Planning and Inference, 93, pp. 51–69.

L. TARKO (2011). A new manner to use application of Shannon entropy in similarity computation. Journal of Mathematical Chemistry, 49, pp. 2330–2344.

C. TEPEDELENLIOGLU, A. RAJAN, Y. ZHANG (2011). Applications of stochastic ordering to wireless communications. IEEE Transactions on Wireless Communications, 10, no. 12, pp. 4249–4257.

M. LORENZ (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9, no. 70, pp. 209–219.

J. MAZUCHELI, A. F. MENEZES, S. DEY (2019). Unit-Gompertz Distribution with applications. Statistica, 79, no. 1, pp. 25–43.

J.MI (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Transactions on Reliability, 44, no. 3, pp. 388–391.

K.MOSLER, M. SCARSINI (1993). Stochastic Orders and Applications: A Classified Bibliography. Springer-Verlag, Berlin Heidelberg.

E. J. VERES-FERRER, J. PAVÍA (2014). On the relationship between the reversed hazard rate and elasticity. Statistical Papers, 55, pp. 275–284.

M. ZENGA (2007). Inequality curve and inequality index based on the ratios between lower and upper arithmetic means. Statistica & Applicazioni, 5, pp. 3–27.

K. ZOGRAFOS,N. BALAKRISHNAN (2009). On families of beta and generalized gamma generated distributions and associated inference. Statistical Methodology, 6, pp. 344–362.




How to Cite

Anis, M. Z., & De, D. (2020). An Expository Note on Unit - Gompertz Distribution with Applications. Statistica, 80(4), 469–490.