A New Lifetime Distribution with Decreasing and Upside-Down Bathtub-Shaped Hazard Rate Function

Authors

  • Sanku Dey St. Anthony's College
  • Mazen Nassar King Abdulaziz University
  • Devendra Kumar Central University of Haryana
  • Ayman Alzaatreh Nazarbayev University
  • Muhammad Hussain Tahir The Islamia University of Bahawalpur

DOI:

https://doi.org/10.6092/issn.1973-2201/8442

Keywords:

Lomax distribution, Hazard rate function, Maximum likelihood estimation, Survival function

Abstract

We introduce a new lifetime distribution, called the alpha-power transformed Lomax (APTL) distribution which generalizes the Lomax distribution to provide better fits than the Lomax distribution and some of its known generalizations. Various properties of the proposed distribution, including explicit expressions for the quantiles, mode, moments, conditional moments, mean residual lifetime, stochastic ordering, Bonferroni and Lorenz curve, stress-strength reliability and order statistics are derived. The new distribution can have a decreasing and upside-down bathtub failure rate function depending on its parameters. The maximum likelihood estimators of the three unknown parameters of APTL are obtained. A simulation study is carried out to examine the performances of the maximum likelihood estimates in terms of their mean squared error using simulated samples. Finally, the potentiality of the distribution is analyzed by means of two real data sets. For the real data sets, this distribution is found to be superior in its ability to sufficiently model both the data sets as compared to the Lomax (L) distribution, exponentiated-Lomax (EL) distribution, gamma-Lomax (GL) distribution, beta-Lomax (BL) distribution and Kumaraswamy-Lomax (KuL) distribution.

References

I. B. ABDUL-MONIEM, H. F. ABDEL-HAMEED (2012). On exponentiated Lomax distribution. International Journal of Mathematical Archive, 3, pp. 2144–2150.

B. AL-ZAHRANI (2015). An extended Poisson-Lomax distribution. Advances in Mathematics: Scientific Journal, 4, pp. 79–89.

B. AL-ZAHRANI, H. SAGOR (2014a). The Poisson-Lomax distribution. Colombian Journal of Statistics, 37, pp. 223–243.

B. AL-ZAHRANI, H. SAGOR (2014b). Statistical analysis of the Lomax-logarithmic distribution. Journal of Statistical Computation and Simulation, 85, pp. 1883–1901.

L. BAIN, M. ENGELHARDT (1992). Introduction to Probability and Mathematical Statistics. Duxbury Press, London.

S. BENNETTE (1983). Log-logistic regression models for survival data. Applied Statistics, 32, pp. 165–171.

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

M. C. BRYSON (1974). Heavy-tailed distributions: Properties and tests. Technometrics, 16, pp. 61–68.

G. CASELLA, R. L. BERGER (1990). Statistical Inference. Duxbury Press, Belmont, CA.

G. M. CORDEIRO, E. M. M. ORTEGA, C. C. DANIEL (2013). The exponentiated generalized class of distributions. Journal of Data Science, 11, pp. 1–27.

S. DEY, A. ALZAATREH, C. ZHANG, D. KUMAR (2017a). A new extension of generalized exponential distribution with application to ozone data. OZONE: Science and Engineering, 39, pp. 273–285.

S. DEY, M. NASSAR, D. KUMAR (2017b). Alpha logarithmic transformed family of distributions with application. Annals of Data Sciences, 4, no. 4, pp. 457–482.

S. DEY, M. NASSAR, D. KUMAR (2019). Alpha power transformed inverse Lindley distribution: A distribution with an upside-down bathtub-shaped hazard function. Journal of Computational and Applied Mathematics, 348, pp. 130–145.

B. EFRON (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. Journal of the American Statistical Association, 83, pp. 414–425.

M. E. GHITANY, F. A. AL-AWADHI, L. A. ALKHALFAN (2007). Marshall-Olkin extended Lomax distribution and its application to censored data. Communication in Statistics-Theory and Methods, 36, pp. 1855–1866.

I. S. GRADSHTEYN, I. M. RYZHIK (2014). Table of Integrals, Series, and Products. Academic Press, San Diego, CA, sixth ed.

R. C. GUPTA, J. KEATING (1985). Relations for reliability measures under length biased sampling. Scandinavian Journal of Statistics, 13, pp. 49–56.

R. C. GUPTA, S. KIRMANI (1990). The role of weighted distribution in stochastic modeling. Communication in Statistics-Theory and Methods, 19, pp. 3147–3162.

R. D. GUPTA, D. KUNDU (2009). A new class of weighted exponential distributions. Statistics: A Journal of Theoretical and Applied Statistics, 43, pp. 621–634.

O.HOLLAND, A.GOLAUP, A. H. AGHVAMI (2006). Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration. IEE Proceedings - Communications,

, pp. 683–690.

M. C. JONES (2015). On families of distributions with shape parameters. International Statistical Review, 83, pp. 175–192.

S. KOTZ, S.NADARAJAH (2000). Extreme Value Distributions: Theory and Applications. Imperial College Press, London.

C. KUS (2007). A new lifetime distribution. Computational Statistics and Data Analysis, 51, pp. 4497–4509.

A. LANGLANDS, S. POCOCK, G. KERR, S.GORE (1997). Long-termsurvival of patients with breast cancer: A study of the curability of the disease. British Medical Journal, 2, pp. 1247–1251.

C. LEE, F. FAMOYE, A. ALZAATREH (2013). Methods for generating families of continuous distribution in the recent decades.Wiley InterdisciplinaryReviews: Computational Statistics, 5, pp. 219–238.

E. LEE, J.W.WANG (2003). Statistical Methods for Survival Data Analysis. John Wiley, New York, 3rd ed.

A. J. LEMONTE, G. M. CORDEIRO (2013). An extended Lomax distribution. Statistics: A Journal of Theoretical and Applied Statistics, 47, pp. 800–816.

K. C. LOMAX (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49, pp. 847–852.

A.MAHDAVI, D. KUNDU (2017). A new method for generating distributions with an application to exponential distribution. Communications in Statistics-Theory and Methods, 46, pp. 6543–6557.

D. N. P. MURTHY, M. XIE, R. JIANG (2004). Weibull Models. JohnWiley, New York.

M.NASSAR, A.ALZAATREH, M.MEAD, O.ABO-KASEM (2017). Alpha power Weibull distribution: Properties and applications. Communications in Statistics-Theory and Methods, 46, pp. 10236–10252.

Z. M. NOFAL, A. Z. AFIFY, H. YOUSOF, G. M. CORDEIRO (2017). The generalized transmuted-G family of distributions. Communications in Statistics-Theory and Methods, 46, no. 8, pp. 4119–4136.

B. OLUYEDE (1999). On inequalities and selection of experiments for length-biased distributions. Probability in the Engineering and Informational Sciences, 13, pp. 169–185.

G. PATIL (2002). Weighted distributions. Encyclopedia of Environmetics, 4, pp. 2369–2377.

G. PATIL, C. RAO (1978). Weighted distributions and size-biased sampling with application to wildlife populations and human families. Biometrics, 34, pp. 179–189.

G. PATIL, C. RAO, M. RATNAPARKHI (1986). On discrete weighted distributions and their use in model choice for observed data. Communications in Statistics-Theory and Methods, 15, pp. 907–918.

F. PROSCHAN (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 5, pp. 375–383.

C. R. RAO (1965). On discrete distributions arising out of methods of ascertainment. In G. PATIL (ed.), Classical and Contagious Discrete Distributions, Statistical Publishing Society, Calcutta, pp. 320–333.

M. H. TAHIR, G. M. CORDEIRO, M. MANSOOR, M. ZUBAIR (2015). The Weibull-Lomax distribution: Properties and applications. Hacettepe Journal of Mathematics and Statistics, 44, p. 461 – 480.

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Published

2020-03-12

How to Cite

Dey, S., Nassar, M., Kumar, D., Alzaatreh, A., & Tahir, M. H. (2019). A New Lifetime Distribution with Decreasing and Upside-Down Bathtub-Shaped Hazard Rate Function. Statistica, 79(4), 399–426. https://doi.org/10.6092/issn.1973-2201/8442

Issue

Vol. 79 No. 4 (2019)

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