Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data

John Kimani; Nicholas Makumi; Kilai Mutua

doi:doi:10.11648/j.ajtas.20261504.11

Research Article |

| Peer-Reviewed

Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data

John Kimani^*

, Nicholas Makumi, Kilai Mutua

Published in American Journal of Theoretical and Applied Statistics (Volume 15, Issue 4)

Received: 26 May 2026 Accepted: 7 July 2026 Published: 7 July 2026

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Probability distribution theory is fundamental to statistical modeling, especially in survival analysis, where correct representation of time-to-event data is critical. Classical distributions such as the Weibull and exponential have been used with great success but fall behind in modeling complex datasets with heavy-tailed behavior. The Inverse Unit Teissier Distribution (IUTD) presents a good solution to the issue; however, it is one-parameter-tailed. The authors introduced a new distribution called the Exponentiated Inverse Unit Teissier Distribution (EIUTD) as a modification of the IUTD to tackle the single-parameter constraint by incorporation of a shape parameter via exponentiation of the baseline IUTD. The present work developed the cumulative distribution function (CDF) and probability density function (PDF) of the EIUTD in a systematic way, investigated its statistical properties such as moments, quantile function, order statistics, Shannon and Renyi entropy, and skewness and kurtosis, estimated parameters using Maximum Likelihood Estimation (MLE), and performed simulation studies that showed the consistency and efficiency of the estimators for different sample sizes. The modeling capability exhibited by the EIUTD model across two different public health applications confirmed its strong performance. For the Kenya DHS 2022 child mortality data set (n = 77), the EIUTD produced a substantially better statistical fit than the IUTD base model with respect to both AIC (529.19 vs. 653) and BIC (533.88 vs. 655.35 ), while demonstrating acceptable goodness-of-fit based on the Kolmogorov-Smirnov test (KS p = 0.1289). For the COVID-19 recovery times of vaccinated individuals in Kenya (n = 107), the EIUTD model provided competitive performance (AIC = 471.96, p = 0.1825) when compared with both the Lognormal and Gamma models and additionally provided a clearer hazard interpretation via the α-β parameterization than either of the other models. The overall flexible parameterization capability offered by the EIUTD model suggests that it is an appropriate survival analysis method for demographic health research and also for infectious disease epidemiology.

Published in	American Journal of Theoretical and Applied Statistics (Volume 15, Issue 4)
DOI	10.11648/j.ajtas.20261504.11
Page(s)	112-132
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Exponentiated Distribution, Inverse Unit Teissier, Survival Analysis, Heavy-tailed Data, Maximum Likelihood Estimation

References

[1]	Adamidis, K., & Loukas, S. (1998). A lifetime distribution with decreasing failure rate. Statistics & Probability Letters, 39(1), 35-42. https://doi.org/10.1016/S0167-7152(98)00012-4
[2]	Adamidis, K., Dimitrakopoulou, T., & Loukas, S. (2005). On an extension of the exponential-geometric distribution. Statistics & Probability Letters, 73(3), 259-269. https://doi.org/10.1016/j.spl.2005.04.034
[3]	Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716-723. https://doi.org/10.1109/TAC.1974.1100705
[4]	Al-Saiari, A. Y., Baharith, L. A., & Mousa, S. A. (2014). Marshall-Olkin extended Burr type XII distribution. International Journal of Statistics and Probability, 3(1), 78-84.
[5]	Alsadat, N., Elgarhy, M., Karakaya, K., Gemeay, A. M., Chesneau, C., & Abd El-Raouf, M. M. (2023). Inverse unit Teissier distribution: Theory and practical examples. Axioms, 12(5), Article 502. https://doi.org/10.3390/axioms12050502
[6]	Alzaatreh, A., Lee, C., & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1), 63-79. https://doi.org/10.1007/s40300-013-0007-y
[7]	Alzaatreh, A., Lee, C., & Famoye, F. (2014). T-normal family of distributions: A new approach to generalize the normal distribution. Journal of Statistical Distributions and Applications, 1(1), Article 16. https://doi.org/10.1186/2195-5832-1-16
[8]	Anderson, T. W., & Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268), 765-769. https://doi.org/10.1080/01621459.1954.10501232
[9]	Aryal, G. R., & Tsokos, C. P. (2011). Transmuted Weibull distribution: A generalization of the Weibull probability distribution. European Journal of Pure and Applied Mathematics, 4(2), 89-102.
[10]	Billingsley, P. (1995). Probability and measure (3rd ed.). Wiley. https://doi.org/10.1002/9781118122372
[11]	Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury Press.
[12]	Chahkandi, M., & Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis, 53(12), 4433-4440. https://doi.org/10.1016/j.csda.2009.06.004
[13]	Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898. https://doi.org/10.1080/00949655.2010.485193
[14]	Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society: Series B, 34(2), 187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
[15]	Dirac, P. A. M. (1930). The principles of quantum mechanics. Oxford University Press.
[16]	Embrechts, P., Kluppelberg, C., & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Springer. https://doi.org/10.1007/978-3-642-33483-2
[17]	Eugene, N., Lee, C., & Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, 31(4), 497-512. https://doi.org/10.1081/STA-120003130
[18]	Gauss, C. F. (1809). Theoria motus corporum coelestium. Perthes et Besser.
[19]	Gibbons, J. D., & Chakraborti, S. (2010). Nonparametric statistical inference (5th ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9781420072620
[20]	Gupta, R. C., Gupta, P. L., & Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics - Theory and Methods, 27(4), 887-904. https://doi.org/10.1080/03610929808832134
[21]	Gupta, A. K., & Nadarajah, S. (Eds.). (2004). Handbook of beta distribution and its applications. CRC Press. https://doi.org/10.1201/9781482276596
[22]	Haight, F. A. (1967). Handbook of the Poisson distribution. Wiley.
[23]	Hormander, L. (1983). The analysis of linear partial differential operators I. Springer. https://doi.org/10.1007/978-3-662-08224-7
[24]	Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1/2), 149-176. https://doi.org/10.2307/2332539
[25]	Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Continuous univariate distributions (2nd ed., Vol. 1). Wiley. https://doi.org/10.1002/9781119011679
[26]	Jones, M. C. (2004). Families of distributions arising from distributions of order statistics. TEST, 13(1), 1-43. https://doi.org/10.1007/BF02595726
[27]	Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481. https://doi.org/10.1080/01621459.1958.10501452
[28]	Kendall, M. G., & Stuart, A. (1958). The advanced theory of statistics (Vol. 1). Griffin.
[29]	Kenya National Bureau of Statistics. (2022). Kenya Demographic and Health Survey 2022. KNBS.
[30]	Klein, J. P., & Moeschberger, M. L. (2006). Survival analysis (2nd ed.). Springer. https://doi.org/10.1007/0-387-21611-8
[31]	Kolmogorov, A. (1933). Sulla determinazione empirica delle leggi di distribuzione. Giornale dell'Istituto Italiano degli Attuari, 4(1), 83-91.
[32]	Lawless, J. F. (2003). Statistical models and methods for lifetime data (2nd ed.). Wiley. https://doi.org/10.1002/9781118033005
[33]	Lee, C., Famoye, F., & Alzaatreh, A. (2013). Methods for generating families of univariate continuous distributions. Wiley Interdisciplinary Reviews: Computational Statistics, 5(3), 219-238. https://doi.org/10.1002/wics.1255
[34]	Marshall, A. W., & Olkin, I. (1997). A new method for adding a parameter to a family of distributions. Biometrika, 84(3), 641-652. https://doi.org/10.1093/biomet/84.3.641
[35]	Merovci, F. (2013). Transmuted exponentiated exponential distribution. Mathematical Sciences and Applications E-Notes, 1(2), 112-122.
[36]	Mudholkar, G. S., & Srivastava, D. K. (1995). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 44(3), 503-509. https://doi.org/10.1109/24.406588
[37]	Nadarajah, S., & Kotz, S. (2006). The exponentiated type distributions. Acta Applicandae Mathematicae, 92(2), 97-111. https://doi.org/10.1007/s10440-006-9055-1
[38]	Nadarajah, S., & Kotz, S. (2011). A new family of distributions based on the exponential distribution. Journal of Statistical Computation and Simulation, 81(7), 883-898.
[39]	Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. M. (2012). The Kumaraswamy G distribution. Journal of Statistical Computation and Simulation, 82(9), 1341-1365. https://doi.org/10.1080/00949655.2011.561443
[40]	Parzen, E. (1979). Nonparametric statistical data modeling. Journal of the American Statistical Association, 74(365), 105-131. https://doi.org/10.1080/01621459.1979.10481618
[41]	Pearson, K. (1895). Contributions to the mathematical theory of evolution. II. Philosophical Transactions of the Royal Society of London. Series A, 186, 343-414. https://doi.org/10.1098/rsta.1895.0010
[42]	Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461-464. https://doi.org/10.1214/aos/1176344136
[43]	Shaw, W. T., & Buckley, I. R. C. (2007). The alchemy of probability distributions. arXiv:0901.0434.
[44]	Smirnov, N. V. (1939). On the estimation of the discrepancy between empirical curves. Bulletin Moscow University, 2(2), 3-16.
[45]	Stigler, S. M. (1986). The history of statistics. Harvard University Press. https://doi.org/10.2307/j.ctvjghwv2
[46]	Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18(3), 293-297. https://doi.org/10.1115/1.4010337
[47]	Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83. https://doi.org/10.2307/3001968
[48]	Zografos, K., & Balakrishnan, N. (2009). On families of beta- and generalized gamma-generated distributions. Statistical Methodology, 6(4), 344-362. https://doi.org/10.1016/j.stamet.2009.01.003

Cite This Article

Plain Text BibTeX RIS

APA Style

Kimani, J., Makumi, N., Mutua, K. (2026). Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. American Journal of Theoretical and Applied Statistics, 15(4), 112-132. https://doi.org/10.11648/j.ajtas.20261504.11

Copy | Download

ACS Style

Kimani, J.; Makumi, N.; Mutua, K. Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. Am. J. Theor. Appl. Stat. 2026, 15(4), 112-132. doi: 10.11648/j.ajtas.20261504.11

Copy | Download

AMA Style

Kimani J, Makumi N, Mutua K. Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. Am J Theor Appl Stat. 2026;15(4):112-132. doi: 10.11648/j.ajtas.20261504.11

Copy | Download

@article{10.11648/j.ajtas.20261504.11,
  author = {John Kimani and Nicholas Makumi and Kilai Mutua},
  title = {Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {4},
  pages = {112-132},
  doi = {10.11648/j.ajtas.20261504.11},
  url = {https://doi.org/10.11648/j.ajtas.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261504.11},
  abstract = {Probability distribution theory is fundamental to statistical modeling, especially in survival analysis, where correct representation of time-to-event data is critical. Classical distributions such as the Weibull and exponential have been used with great success but fall behind in modeling complex datasets with heavy-tailed behavior. The Inverse Unit Teissier Distribution (IUTD) presents a good solution to the issue; however, it is one-parameter-tailed. The authors introduced a new distribution called the Exponentiated Inverse Unit Teissier Distribution (EIUTD) as a modification of the IUTD to tackle the single-parameter constraint by incorporation of a shape parameter via exponentiation of the baseline IUTD. The present work developed the cumulative distribution function (CDF) and probability density function (PDF) of the EIUTD in a systematic way, investigated its statistical properties such as moments, quantile function, order statistics, Shannon and Renyi entropy, and skewness and kurtosis, estimated parameters using Maximum Likelihood Estimation (MLE), and performed simulation studies that showed the consistency and efficiency of the estimators for different sample sizes. The modeling capability exhibited by the EIUTD model across two different public health applications confirmed its strong performance. For the Kenya DHS 2022 child mortality data set (n = 77), the EIUTD produced a substantially better statistical fit than the IUTD base model with respect to both AIC (529.19 vs. 653) and BIC (533.88 vs. 655.35 ), while demonstrating acceptable goodness-of-fit based on the Kolmogorov-Smirnov test (KS p = 0.1289). For the COVID-19 recovery times of vaccinated individuals in Kenya (n = 107), the EIUTD model provided competitive performance (AIC = 471.96, p = 0.1825) when compared with both the Lognormal and Gamma models and additionally provided a clearer hazard interpretation via the α-β parameterization than either of the other models. The overall flexible parameterization capability offered by the EIUTD model suggests that it is an appropriate survival analysis method for demographic health research and also for infectious disease epidemiology.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data
AU  - John Kimani
AU  - Nicholas Makumi
AU  - Kilai Mutua
Y1  - 2026/07/07
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261504.11
DO  - 10.11648/j.ajtas.20261504.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 112
EP  - 132
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261504.11
AB  - Probability distribution theory is fundamental to statistical modeling, especially in survival analysis, where correct representation of time-to-event data is critical. Classical distributions such as the Weibull and exponential have been used with great success but fall behind in modeling complex datasets with heavy-tailed behavior. The Inverse Unit Teissier Distribution (IUTD) presents a good solution to the issue; however, it is one-parameter-tailed. The authors introduced a new distribution called the Exponentiated Inverse Unit Teissier Distribution (EIUTD) as a modification of the IUTD to tackle the single-parameter constraint by incorporation of a shape parameter via exponentiation of the baseline IUTD. The present work developed the cumulative distribution function (CDF) and probability density function (PDF) of the EIUTD in a systematic way, investigated its statistical properties such as moments, quantile function, order statistics, Shannon and Renyi entropy, and skewness and kurtosis, estimated parameters using Maximum Likelihood Estimation (MLE), and performed simulation studies that showed the consistency and efficiency of the estimators for different sample sizes. The modeling capability exhibited by the EIUTD model across two different public health applications confirmed its strong performance. For the Kenya DHS 2022 child mortality data set (n = 77), the EIUTD produced a substantially better statistical fit than the IUTD base model with respect to both AIC (529.19 vs. 653) and BIC (533.88 vs. 655.35 ), while demonstrating acceptable goodness-of-fit based on the Kolmogorov-Smirnov test (KS p = 0.1289). For the COVID-19 recovery times of vaccinated individuals in Kenya (n = 107), the EIUTD model provided competitive performance (AIC = 471.96, p = 0.1825) when compared with both the Lognormal and Gamma models and additionally provided a clearer hazard interpretation via the α-β parameterization than either of the other models. The overall flexible parameterization capability offered by the EIUTD model suggests that it is an appropriate survival analysis method for demographic health research and also for infectious disease epidemiology.
VL  - 15
IS  - 4
ER  -

Copy | Download

Author Information

John Kimani

Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0003-3116-6046
Nicholas Makumi

Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email
Kilai Mutua

Department of Pure and Applied Sciences, Kirinyaga University, Kerugoya, Kenya

Contact Email

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Kimani, J., Makumi, N., Mutua, K. (2026). Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. American Journal of Theoretical and Applied Statistics, 15(4), 112-132. https://doi.org/10.11648/j.ajtas.20261504.11

Copy | Download

ACS Style

Kimani, J.; Makumi, N.; Mutua, K. Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. Am. J. Theor. Appl. Stat. 2026, 15(4), 112-132. doi: 10.11648/j.ajtas.20261504.11

Copy | Download

AMA Style

Kimani J, Makumi N, Mutua K. Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data. Am J Theor Appl Stat. 2026;15(4):112-132. doi: 10.11648/j.ajtas.20261504.11

Copy | Download

@article{10.11648/j.ajtas.20261504.11,
  author = {John Kimani and Nicholas Makumi and Kilai Mutua},
  title = {Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {15},
  number = {4},
  pages = {112-132},
  doi = {10.11648/j.ajtas.20261504.11},
  url = {https://doi.org/10.11648/j.ajtas.20261504.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20261504.11},
  abstract = {Probability distribution theory is fundamental to statistical modeling, especially in survival analysis, where correct representation of time-to-event data is critical. Classical distributions such as the Weibull and exponential have been used with great success but fall behind in modeling complex datasets with heavy-tailed behavior. The Inverse Unit Teissier Distribution (IUTD) presents a good solution to the issue; however, it is one-parameter-tailed. The authors introduced a new distribution called the Exponentiated Inverse Unit Teissier Distribution (EIUTD) as a modification of the IUTD to tackle the single-parameter constraint by incorporation of a shape parameter via exponentiation of the baseline IUTD. The present work developed the cumulative distribution function (CDF) and probability density function (PDF) of the EIUTD in a systematic way, investigated its statistical properties such as moments, quantile function, order statistics, Shannon and Renyi entropy, and skewness and kurtosis, estimated parameters using Maximum Likelihood Estimation (MLE), and performed simulation studies that showed the consistency and efficiency of the estimators for different sample sizes. The modeling capability exhibited by the EIUTD model across two different public health applications confirmed its strong performance. For the Kenya DHS 2022 child mortality data set (n = 77), the EIUTD produced a substantially better statistical fit than the IUTD base model with respect to both AIC (529.19 vs. 653) and BIC (533.88 vs. 655.35 ), while demonstrating acceptable goodness-of-fit based on the Kolmogorov-Smirnov test (KS p = 0.1289). For the COVID-19 recovery times of vaccinated individuals in Kenya (n = 107), the EIUTD model provided competitive performance (AIC = 471.96, p = 0.1825) when compared with both the Lognormal and Gamma models and additionally provided a clearer hazard interpretation via the α-β parameterization than either of the other models. The overall flexible parameterization capability offered by the EIUTD model suggests that it is an appropriate survival analysis method for demographic health research and also for infectious disease epidemiology.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Exponentiated Inverse Unit Teissier Distribution and Its Application to Survival Data
AU  - John Kimani
AU  - Nicholas Makumi
AU  - Kilai Mutua
Y1  - 2026/07/07
PY  - 2026
N1  - https://doi.org/10.11648/j.ajtas.20261504.11
DO  - 10.11648/j.ajtas.20261504.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 112
EP  - 132
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20261504.11
AB  - Probability distribution theory is fundamental to statistical modeling, especially in survival analysis, where correct representation of time-to-event data is critical. Classical distributions such as the Weibull and exponential have been used with great success but fall behind in modeling complex datasets with heavy-tailed behavior. The Inverse Unit Teissier Distribution (IUTD) presents a good solution to the issue; however, it is one-parameter-tailed. The authors introduced a new distribution called the Exponentiated Inverse Unit Teissier Distribution (EIUTD) as a modification of the IUTD to tackle the single-parameter constraint by incorporation of a shape parameter via exponentiation of the baseline IUTD. The present work developed the cumulative distribution function (CDF) and probability density function (PDF) of the EIUTD in a systematic way, investigated its statistical properties such as moments, quantile function, order statistics, Shannon and Renyi entropy, and skewness and kurtosis, estimated parameters using Maximum Likelihood Estimation (MLE), and performed simulation studies that showed the consistency and efficiency of the estimators for different sample sizes. The modeling capability exhibited by the EIUTD model across two different public health applications confirmed its strong performance. For the Kenya DHS 2022 child mortality data set (n = 77), the EIUTD produced a substantially better statistical fit than the IUTD base model with respect to both AIC (529.19 vs. 653) and BIC (533.88 vs. 655.35 ), while demonstrating acceptable goodness-of-fit based on the Kolmogorov-Smirnov test (KS p = 0.1289). For the COVID-19 recovery times of vaccinated individuals in Kenya (n = 107), the EIUTD model provided competitive performance (AIC = 471.96, p = 0.1825) when compared with both the Lognormal and Gamma models and additionally provided a clearer hazard interpretation via the α-β parameterization than either of the other models. The overall flexible parameterization capability offered by the EIUTD model suggests that it is an appropriate survival analysis method for demographic health research and also for infectious disease epidemiology.
VL  - 15
IS  - 4
ER  -

Copy | Download