Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution

Peter Nyandiri Mecha; Isaac Chumba Kipchirchir; George Odweso Muhua; Joseph Antony Makoteku Ottieno

doi:doi:10.11648/j.ajtas.20251402.11

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Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution

Peter Nyandiri Mecha, Isaac Chumba Kipchirchir, George Odweso Muhua, Joseph Antony Makoteku Ottieno

Published in American Journal of Theoretical and Applied Statistics (Volume 14, Issue 2)

Received: 15 July 2024 Accepted: 24 December 2024 Published: 13 March 2025

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Abstract

In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.

Published in	American Journal of Theoretical and Applied Statistics (Volume 14, Issue 2)
DOI	10.11648/j.ajtas.20251402.11
Page(s)	51-60
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), 2025. Published by Science Publishing Group

Keywords

Generators, Order Statistic, Discrete, Mixing, Distribution, Mixture

References

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[2]	Adamidis, K; Dimitrakopoulous, D. and S. Loukas (2005): “On an extension of the exponential-geometric distribution” Statistics and Probability Letters 73, 259-269.
[3]	Al-Zahrani, B. and H. Sagor (2014): “The Poisson - Lomax” Revsta Colombiana de Estadistica, Vol 37, No 1, pp 223-243.
[4]	Cancho, V. G.; Louzada-Neto, F. and G. D Barriga (2011): “The Poisson-exponential lifetime’ distribution”. Computational Statistics and Data Analysis 55, 677-686.
[5]	Eugene, N; Lee, C. and Famoye F. (2002); “Beta- Normal distribution and its application” Communication in statistics-Theory and methods, 31, 497-512.
[6]	Flores, J. D; Borges, P. Cancho, V. G and Louzada F. (2013): “The complementary exponential power series distribution” Brazilian Journal of Probability and Statistics; Vol 27, No 4, pp 565-584.
[7]	Hajebi, M.; Rezaei, S. and S. Nadarajah (2013): “An exponential-Negative binomial distribution” Revstat- Statistical Journal 11, 191-210.
[8]	Jones, M. C (2004); “Families of Distributions arising from distributions of order statistics.” Test 13, 1, 1-43.
[9]	Kus, C (2007): “A new lifetime distribution” Computational Statistics and Data Analysis 51, 4497- 4509.
[10]	Lozada, F.; Roman, M. and V. G Cancho (2011): “The complementary exponential geometric distribution: model, properties and a comparison with counter part” Computational Statistics and data Analysis, 55, 2516- 2524.
[11]	Tahmasbi, R. and S. Rezaei (2008): “a two-parameter lifetime distribution with decreasing failure rate” Computational Statistics and Data Analysis, 52, 3889- 3901.
[12]	David, H. A., & Nagaraja, H. N. (2004). Order statistics. John Wiley & Sons.
[13]	Lee, E. T., & Wang, J. (2003). Statistical methods for survival data analysis (Vol. 476). John Wiley & Sons.
[14]	Marshall and Olkin (1997). A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families? Biometrika, 641- 652.
[15]	Alice, T. and Jose, K. K. (2003) Marshall-Olkin Pareto Processes. Far East Journal of Theoretical Statistics, 9, 2, 117-132.
[16]	Ghitany, M. E., AL-Awadhi, F. A and Alkhalfan, L. A. (2007) Marshall-Olkin extended Lomax distribution and its applications to censored data, Communications in statistics-Theory and Methods, 36, 1855-1866.
[17]	Peter N. Mecha, Isaac C. Kipchirchir and J. A. M. Otieno, Generators of discrete mixtures based on order statistics with application to exponential distribution, Advances and Applications in Statistics 67 (2021), 161-178.

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APA Style

Mecha, P. N., Kipchirchir, I. C., Muhua, G. O., Ottieno, J. A. M. (2025). Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. American Journal of Theoretical and Applied Statistics, 14(2), 51-60. https://doi.org/10.11648/j.ajtas.20251402.11

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ACS Style

Mecha, P. N.; Kipchirchir, I. C.; Muhua, G. O.; Ottieno, J. A. M. Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. Am. J. Theor. Appl. Stat. 2025, 14(2), 51-60. doi: 10.11648/j.ajtas.20251402.11

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AMA Style

Mecha PN, Kipchirchir IC, Muhua GO, Ottieno JAM. Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. Am J Theor Appl Stat. 2025;14(2):51-60. doi: 10.11648/j.ajtas.20251402.11

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@article{10.11648/j.ajtas.20251402.11,
  author = {Peter Nyandiri Mecha and Isaac Chumba Kipchirchir and George Odweso Muhua and Joseph Antony Makoteku Ottieno},
  title = {Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {14},
  number = {2},
  pages = {51-60},
  doi = {10.11648/j.ajtas.20251402.11},
  url = {https://doi.org/10.11648/j.ajtas.20251402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20251402.11},
  abstract = {In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.},
 year = {2025}
}

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TY  - JOUR
T1  - Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution
AU  - Peter Nyandiri Mecha
AU  - Isaac Chumba Kipchirchir
AU  - George Odweso Muhua
AU  - Joseph Antony Makoteku Ottieno
Y1  - 2025/03/13
PY  - 2025
N1  - https://doi.org/10.11648/j.ajtas.20251402.11
DO  - 10.11648/j.ajtas.20251402.11
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  - 51
EP  - 60
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20251402.11
AB  - In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.
VL  - 14
IS  - 2
ER  -

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Author Information

Peter Nyandiri Mecha

Department of Mathematics, University of Nairobi, Nairobi, Kenya

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Isaac Chumba Kipchirchir

Department of Mathematics, University of Nairobi, Nairobi, Kenya

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George Odweso Muhua

Department of Mathematics, University of Nairobi, Nairobi, Kenya

Contact Email
Joseph Antony Makoteku Ottieno

Department of Mathematics, University of Nairobi, Nairobi, Kenya

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Plain Text BibTeX RIS

APA Style

Mecha, P. N., Kipchirchir, I. C., Muhua, G. O., Ottieno, J. A. M. (2025). Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. American Journal of Theoretical and Applied Statistics, 14(2), 51-60. https://doi.org/10.11648/j.ajtas.20251402.11

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ACS Style

Mecha, P. N.; Kipchirchir, I. C.; Muhua, G. O.; Ottieno, J. A. M. Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. Am. J. Theor. Appl. Stat. 2025, 14(2), 51-60. doi: 10.11648/j.ajtas.20251402.11

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AMA Style

Mecha PN, Kipchirchir IC, Muhua GO, Ottieno JAM. Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution. Am J Theor Appl Stat. 2025;14(2):51-60. doi: 10.11648/j.ajtas.20251402.11

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@article{10.11648/j.ajtas.20251402.11,
  author = {Peter Nyandiri Mecha and Isaac Chumba Kipchirchir and George Odweso Muhua and Joseph Antony Makoteku Ottieno},
  title = {Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {14},
  number = {2},
  pages = {51-60},
  doi = {10.11648/j.ajtas.20251402.11},
  url = {https://doi.org/10.11648/j.ajtas.20251402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20251402.11},
  abstract = {In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution
AU  - Peter Nyandiri Mecha
AU  - Isaac Chumba Kipchirchir
AU  - George Odweso Muhua
AU  - Joseph Antony Makoteku Ottieno
Y1  - 2025/03/13
PY  - 2025
N1  - https://doi.org/10.11648/j.ajtas.20251402.11
DO  - 10.11648/j.ajtas.20251402.11
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  - 51
EP  - 60
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20251402.11
AB  - In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.
VL  - 14
IS  - 2
ER  -

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