The Lehmann Type II Teissier Distribution
-
V. Kumaran
and
Vishwa Prakash Jha
ABSTRACT
In this work, a two-parameter continuous distribution, namely the Lehmann type II Teissier distribution is introduced. Some important properties including the Rényi entropy, Bonferroni curves, Lorenz curves and the exact information matrix of the proposed model are derived. Seven different techniques are being used for the estimation of parameters and a simulation is carried out to observe the maximum likelihood estimates. Interval estimates of the parameters are obtained using exact information matrix and bootstrapping techniques. Finally, to show the practical significance, three datasets related to COVID-19 and rainfall are modeled using the proposed model.
Funding statement: The authors would like to thank the editor and the two anonymous reviewers for their informative and helpful comments. Also, the second author is thankful for the Institute scholarship received from the Ministry of Human Resources Development of India and NIT Tiruchirappalli.
Appendix: Algorithm for the simulation
For simulation purposes, the steps below are used to generate N number of samples of size n = n1.
Use SeedRandom[].
Using RandomReal[] generate the uniform random number substitute the uniform random sample of size n1 in the quantile function given in equation (3.11) to obtain random sample for the LII-TD. If the estimated parameters converge then only select the sample.
Continue the step 2 untill get N number of samples.
References
[1] Aarset, M. V.: How to identify bathtub hazard rate, IEEE Trans. Reliab. 36 (1987), 106–108.10.1109/TR.1987.5222310Search in Google Scholar
[2] Agu, F. I.—Eghwerido, J. T.—Nziku, C. K.: The Alpha Power Rayleigh-G family of distributions, Math. Slovaca 72 (2022), 1047–1062.10.1515/ms-2022-0073Search in Google Scholar
[3] Al-Babtain, A. A.—Elbatal, I.—Chesneau, C.—Jamal, F.: The transmuted Muth generated class of distributions with applications, Symmetry 12 (2020), Art. No. 1677.10.3390/sym12101677Search in Google Scholar
[4] Alizadeh, M.—Cordeiro, G. M.—Pinho, L. G. B.—Ghosh, I.: The Gompertz-G family of distributions, J. Stat. Theory Pract. 11 (2017), 179–207.10.1080/15598608.2016.1267668Search in Google Scholar
[5] Al-Shomrani, A.—Arif, O.—Shawky, K.—Hanif, S.—Shahbaz, M. Q.: Topp–Leone family of distributions: Some properties and application, Pak. J. Stat. Oper. Res. 12(3) (2016), 443–451.10.18187/pjsor.v12i3.1458Search in Google Scholar
[6] Anzagra, L.—Sarpong, S.—Nasiru, S.: Chen-G class of distributions, Cogent Math. Stat. 7 (2020), Art. ID 1721401.10.1080/25742558.2020.1721401Search in Google Scholar
[7] Arnold, B. C.—Balakrishnan, N.—Nagaraja, H. N.: A First Course in Order Statistics. Classics Appl. Math. 54, SIAM, 2008.10.1137/1.9780898719062Search in Google Scholar
[8] Azzalini, A.: A class of distributions which includes the normal ones, Scand. J. Stat. 12 (1985), 171–178.Search in Google Scholar
[9] Balakrishnan, N.—Barmalzan, G.—Haidari, A.: Exponentiated models preserve stochastic orderings of parallel and series systems, Comm. Stat. Theory Methods 49 (2020), 1592–1602.10.1080/03610926.2018.1532007Search in Google Scholar
[10] Bourguignon, M.—Silva, R. B.—Cordeiro, G. M.: The Weibull-G family of probability distributions, J. Data Sci. 12 (2014), 53–68.10.6339/JDS.201401_12(1).0004Search in Google Scholar
[11] Cheng, R.—Amin, N.: Maximum product of spacings estimation with application to the lognormal distribution. Mathematical Report 79-1, Cardiff: University of Wales IST, 1979.Search in Google Scholar
[12] Cordeiro, G. M.—Lemonte, A. J.: The exponentiated generalized Birnbaum–Saunders distribution, Appl. Math. Comput. 247 (2014), 762–779.10.1016/j.amc.2014.09.054Search in Google Scholar
[13] Corless, R. M.—Gonnet, G. H.—Hare, D. E.—Jeffrey, D. J.—Knuth, D. E.: On the Lambert W function, Adv. Comput. Math. 5 (1996), 329–359.10.1007/BF02124750Search in Google Scholar
[14] Efron, B.—Tibshirani, R. J.: An Introduction to the Bootstrap. Monogr. Statist. Appl. Probab. 57, Chapman and Hall, 1994.10.1201/9780429246593Search in Google Scholar
[15] Eghwerido, J. T.: The Marshall–Olkin Teissier generated model for lifetime data, Journal of the Belarusian State University. Mathematics and Informatics 1 (2022), 46–65.10.33581/2520-6508-2022-1-46-65Search in Google Scholar
[16] Eghwerido, J. T.: The alpha power Teissier distribution and its applications, Afr. Stat. 16 (2021), 2733–2747.10.16929/as/2021.2733.181Search in Google Scholar
[17] Eghwerido, J. T.—Agu, F.: The shifted Gompertz-G family of distributions: Properties and applications, Math. Slovaca 71 (2021), 1291–1308.10.1515/ms-2021-0053Search in Google Scholar
[18] Eghwerido J. T.—Nzei, L. C.—Agu, F. I.: The alpha power Gompertz distribution: characterization, properties, and applications, Sankhya A 83 (2021), 449–475.10.1007/s13171-020-00198-0Search in Google Scholar
[19] El-Gohary, A.—Alshamrani, A.—Al-Otabi, A. N.: The generalized Gompertz distribution, Appl. Math. Model 2013 (2013), 13–24.10.1016/j.apm.2011.05.017Search in Google Scholar
[20] Geyer, C. J.: Stat 5102 notes: Fisher information and confidence intervals using maximum likelihood, Lecture Notes, 2007; https://www.stat.umn.edu/geyer/old03/5102/notes/fish.pdf.Search in Google Scholar
[21] Ghitany, M. E.—Al-Mutairi, D. K.—Balakrishnan, N.—Al-Enezi, L. J.: Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64 (2013), 20–33.10.1016/j.csda.2013.02.026Search in Google Scholar
[22] Giorgi, G. M.—Nadarajah, S.: Bonferroni and Gini indices for various parametric families of distributions, Metron 68 (2010), 23–46.10.1007/BF03263522Search in Google Scholar
[23] Irshad, M. R.—Maya, R.—Krishna, A.: Exponentiated power Muth distribution and associated inference, J. Indian Soc. Probab. Stat. 22 (2021), 265–302.10.1007/s41096-021-00104-3Search in Google Scholar
[24] Jodrá, P.—Arshad, M.: An intermediate Muth distribution with increasing failure rate, Comm. Statist. Theory Methods 51(23) (2022), 8310–8327.10.1080/03610926.2021.1892133Search in Google Scholar
[25] Jodrrá, P.—Gómez, H. W.—Jiménez-Gamero, M. D.—Alba-Fernández, M. V.: The power Muth distribution, Math. Model. Anal. 22 (2017), 186–201.10.3846/13926292.2017.1289481Search in Google Scholar
[26] Jodra, P.—Jimenez-Gamero, M. D.—Alba-Fernandez, M. V.: On the Muth distribution, Math. Model. Anal. 20 (2015), 291–310.10.3846/13926292.2015.1048540Search in Google Scholar
[27] Karamikabir, H.—Afshari, M.—Alizadeh, M.—Hamedani G. G.: A new extended generalized Gompertz distribution with statistical properties and simulations, Comm. Statist. Theory Methods 50 (2021), 251–79.10.1080/03610926.2019.1634209Search in Google Scholar
[28] Kharazmi, O.—Saadatinik, A.—Jahangard, S.: Odd hyperbolic cosine exponential-exponential (OHC-EE) distribution, Ann. Data Sci. 6 (2019), 765–785.10.1007/s40745-019-00200-zSearch in Google Scholar
[29] Laurent, A. G.: Failure and mortality from wear and ageing. The Teissier model. In: A Modern Course on Statistical Distributions in Scientific Work, Springer, Dordrecht, 1975, pp. 301–320.10.1007/978-94-010-1845-6_22Search in Google Scholar
[30] LEE, C.—FAMOYE, F.—ALZAATREH, A. Y.: Methods for generating families of univariate continuous distributions in the recent decades, Wiley Interdiscip. Rev. Comput. Stat. 5 (2013), 219–238.10.1002/wics.1255Search in Google Scholar
[31] Lehmann, E. L.: The power of rank tests, Ann. Math. Statist. 24 (1953), 23–43.10.1214/aoms/1177729080Search in Google Scholar
[32] Ly, A.—Marsman, M.—Verhagen J.—-Grasman R. P.—Wagenmakers E. J.: A tutorial on Fisher information, J. Math. Psych. 80 (2017), 40–55.10.1016/j.jmp.2017.05.006Search in Google Scholar
[33] Marshall, A. W.—Olkin, I.: Life Distributions, Springer, New York, 2007.Search in Google Scholar
[34] Mazucheli, J.—Louzada, F.—Ghitany, M. E.: Comparison of estimation methods for the parameters of the weighted Lindley distribution, Appl. Math. Comput. 220 (2013), 463–471.10.1016/j.amc.2013.05.082Search in Google Scholar
[35] Morales, D.—Pardo, L.—Vajda, I.: Some new statistics for testing hypotheses in parametric models, J. Multivariate Anal. 62 (1997), 137–168.10.1006/jmva.1997.1680Search in Google Scholar
[36] Muth, E. J.: Reliability models with positive memory derived from the mean residual life function, Theory and Applications of Reliability 2 (1977), 401–435.Search in Google Scholar
[37] Nadarajah, S.—Cordeiro, G. M.—Ortega, E. M.: General results for the Kumaraswamy-G distribution, J. Stat. Comput. Simul. 82 (2012), 951–979.10.1080/00949655.2011.562504Search in Google Scholar
[38] Nekoukhou, V.—Khalifeh, A.—Bidram, H.: Univariate and bivariate extensions of the generalized exponential distributions, Math. Slovaca 71 (2021), 1581–1598.10.1515/ms-2021-0073Search in Google Scholar
[39] Nielsen, F.: Cramér-Rao lower bound and information geometry. In: Connected at Infinity II, Hindustan Book Agency, 2013, pp. 18–37.10.1007/978-93-86279-56-9_2Search in Google Scholar
[40] Poonia, N.—Azad, S.: Alpha power exponentiated Teissier distribution with application to climate datasets, Theor. Appl. Climatol. 149 (2022), 339–353.10.1007/s00704-022-04039-ySearch in Google Scholar
[41] Raya, M. A.: A new extension of the Lomax distribution with properties and applications to failure times data, Pak. J. Stat. Oper. Res. 15 (2019), 461–479.10.18187/pjsor.v15i2.2657Search in Google Scholar
[42] Rinne, H.: Estimating the lifetime distribution of private motor-cars using prices of used cars: the Teissier model. In: Statistiks Zwischen Theorie und Praxis, 1981, pp. 172–184.Search in Google Scholar
[43] Sharma, V. K.—SINGH, S. V.—Shekhawat, K.: Exponentiated Teissier distribution with increasing, decreasing and bathtub hazard functions, J. Appl. Stat. 49 (2022), 371–93.10.1080/02664763.2020.1813694Search in Google Scholar PubMed PubMed Central
[44] Stute, W.—Manteiga, W. G.—Quindimil, M. P.: Bootstrap based goodness-of-fit-tests, Metrika 40 (1993), 243–256.10.1007/BF02613687Search in Google Scholar
[45] Swain, J. J.—Venkatraman, S.—Wilson J. R.: Least-squares estimation of distribution functions in Johnson’s translation system, J. Stat. Comput. Simul. 29 (1988), 271–297.10.1080/00949658808811068Search in Google Scholar
[46] Tahir, M. H.—Cordeiro, G. M.—Mansoor, M.—Zubair, M.: The Weibull–Lomax distribution: properties and applications, Hacet. J. Math. Stat. 44 (2015), 455–474.10.15672/HJMS.2014147465Search in Google Scholar
[47] Teissier G.: Recherches sur le vieillissement et sur les lois de mortalite, Annales de physiologie et de physicochimie biologique 10 (1934), 237–284.Search in Google Scholar
[48] Tomazella, V. L.—Ramos, P. L.—Ferreira, P. H.—Mota, A. L.—Louzada, F.: The Lehmann type II inverse Weibull distribution in the presence of censored data, Comm. Statist. Simulation Comput. 51(12) (2022), 7057–7073.10.1080/03610918.2020.1823000Search in Google Scholar
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Articles in the same Issue
- Remembering Professor Štefan Znám, 9.2.1936–17.7.1993
- Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures
- Enlargements of Quantales
- Multiplicative Dependent Pairs in the Sequence of Padovan Numbers
- Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions
- On the Rational Parametric Solution of Diagonal Quartic Varieties
- Theory of Certain Non-Univalent Analytic Functions
- Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)
- A Conjecture on H3(1) for Certain Starlike Functions
- Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces
- Complete Monotonicity and Inequalities Involving the k-Gamma and k-Polygamma Functions
- Study of Oscillation Criteria of Odd-Order Differential Equations with Mixed Neutral Terms
- On a System of Difference Equations Defined by the Product of Separable Homogeneous Functions
- On the Existence of Bi-Lipschitz Equivalent Metrics in Semimetric Spaces
- The Lehmann Type II Teissier Distribution
- Asymptotic Predictive Inference of Negative Lower Tail Index Distributions
- On Numerical Problems in Computing Life Annuities Based on the Makeham–Beard Law
- The Rational Zero-Divisor Cup-Length of Oriented Partial Flag Manifolds
