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Sains Malaysiana 52(2)(2023): 669-682
http://doi.org/10.17576/jsm-2023-5202-26
Statistical
Properties and Estimation of the Three-Parameter Lindley Distribution with
Application to COVID-19 Data
(Sifat Statistik dan Anggaran Taburan Lindley Tiga Parameter dengan
Aplikasi pada Data COVID-19)
MATHIL
KAMIL THAMER1,2,* & RAOUDHA ZINE1
1Laboratory of Probability and Statistics, Faculty of
Sciences of Sfax, Sfax,
Tunisia
2Department of Economics, College of Administration and
Economics, University of Anbar, Iraq
Diserahkan: 14 Julai 2021/Diterima: 9 Mei 2022
Abstract
In 2017, the three-parameter
Lindley distribution has been studied. The present paper is a continuation of
the investigation of the properties of this distribution because of its high
flexibility for modeling lifetime data. We studied some statistical properties
of this distribution as central tendency measures, dispersion measures, and
shape measures. In addition, the mode and the quantile function of the
distribution were derived by the authors. The three parameters were estimated
by the Maximum Product of Spacing Method (MPS) due
to its fame in estimating parameters. A simulation study is carried out to
examine the consistency of estimators using mean square error (MSE). The
estimators showed that they have the property of consistency because MSEs
decrease with increasing the size of the sample. On the practical side, the MPS
estimates were used to obtain statistical properties, probability density function
(p.d.f), cumulative distribution function (c.d.f), survival function, and hazard function for real
data which represents COVID-19 Data in Iraq/Al-Anbar Province. We found the
flexibility of the distribution in representing life data and the possibility
of getting the patients' probability of death and probability of survival for
the time.
Keywords: COVID-19
data; mathematical model; maximum product of spacing method; three-parameter
Lindley distribution
Abstrak
Pada tahun 2017, taburan Lindley tiga parameter telah dikaji. Makalah ini adalah kesinambungan daripada penyelidikan sifat pengedaran ini kerana kefleksibelannya yang tinggi untuk memodelkan data sepanjang hayat. Kami mengkaji beberapa sifat statistik taburan ini sebagai ukuran kecenderungan pusat, ukuran penyebarandan ukuran bentuk. Di samping itu, mod dan fungsi kuantil taburan diperoleh oleh penulis. Ketiga-tiga parameter tersebut dianggarkan menggunakan Kaedah Maksimum Jarak Jauh (MPS) kerana kemasyhurannya dalam menganggar parameter. Suatu kajian simulasi dijalankan untuk mengkaji ketekalan penganggar menggunakan min ralat kuasa dua (MSE). Penganggar menunjukkan bahawa mereka memiliki sifat ketekalan kerana MSE menurun dengan peningkatan ukuran sampel. Dari segi praktikal, anggaran MPS digunakan untuk memperoleh sifat statistik, fungsi ketumpatan kebarangkalian (p.d.f), fungsi taburan kumulatif (c.d.f), fungsi kemandirian dan fungsi bahaya untuk data sebenar yang mewakili Data COVID-19 di Wilayah Iraq/Al-Anbar. Kami mendapati kefleksibelan penyebaran dalam mewakili data kehidupan dan kemungkinan mendapat kebarangkalian kematian pesakit dan kebarangkalian bertahan untuk masa ini.
Kata kunci: Data COVID-19; model matematik; produk maksimum kaedah jarak; taburan Lindley tiga parameter
RUJUKAN
Abd El-Monsef,
M.M.E. 2015. A new Lindley distribution with location parameter. Communications in Statistics-Theory and
Methods 45(17): 5204-5219.
Afify, A.Z., Nassar,
M., Cordeiro, G.M. & Kumar, D. 2020. The Weibull Marshall-Olkin Lindley distribution: Properties and estimation. Journal of Taibah University for Science 14(1): 192-204.
Al-Bayati, R.S.S. 2018. Some estimation methods for lindley distribution. A Master Thesis, College of Science, Mustansiriyah University, Baghdad, Iraq (Unpublished).
Al-Mofleh, H., Afify, A.Z. &
Ibrahim, N.A. 2020. A new extended two-parameter distribution: Properties,
estimation methods, and applications in medicine and geology. Mathematics 8(9): 1578.
Arslan, T., Acitas, S. & Senoglu, B.
2017. Generalized Lindley and power Lindley distributions for modeling the wind
speed data. Energy Conversion and
Management 152: 300-311.
Bakouch, H.S., Al-Zahrani, B.M., Al-Shomrani, A.A., Marchi, V.A. & Louzada,
F. 2012. An extended Lindley distribution. Journal
of the Korean Statistical Society 41: 75-85.
Basu, S., Singh, S.K. & Singh, U.
2017. Parameter estimation of inverse Lindley distribution for Type-I censored
data. Computational Statistics 32(1):
367-385.
Cheng,
R.C.H. & Amin, N.A.K. 1979. Maximum product-of-spacings estimation with application to the lognormal distribution. University of Wales
IST, Math. Report. p. 79-1.
Déniz, G.E. & Ojeda, C.E. 2011. The
discrete Lindley distribution: Properties and applications. Journal of Statistical Computation and
Simulation 81(11): 1405-1416.
Déniz, G.E., Sordo,
M.A. & Ojeda, C.E. 2014. The Log-Lindley distribution as an alternative to
the beta regression model with applications in insurance. Insurance: Mathematics and Economics 54: 49-57.
Do Espirito Santo, A.P.J. & Mazucheli,
J. 2015. Comparison of estimation methods for the Marshall–Olkin extended Lindley distribution. Journal of
Statistical Computation and Simulation 85(17): 3437-3450.
Ghitany, M.E., Atieh, B. & Nadarajah, S.
2008. Lindley distribution and its application. Mathematics Computing and Simulation 78: 493-506.
Ghitany, M.E., Alqallaf, F., Al-Mutairi, D.K.
& Husain, H.A. 2011. A two-parameter weighted Lindley distribution and its
applications to survival data. Mathematics
and Computers in Simulation 81(6): 1190-1201.
Jodrá, P. 2010. Computer generation of
random variables with Lindley or Poisson-Lindley distribution via the Lambert W
function. Mathematics and Computers in
Simulation 81(4): 851-859.
Kantar,
Y.M. & Şenoğlu, B. 2008. A comparative
study for the location and scale parameters of the Weibull distribution with
given shape parameter. Computers &
Geosciences 34(12): 1900-1909.
Krishna,
H. & Kumar, K. 2011. Reliability estimation in Lindley distribution with
progressively type II right censored sample. Mathematics and Computers in Simulation 82(2): 281-294.
Kumar,
S.C. & Jose, R. 2019. On double Lindley distribution and some of its
properties. American Journal of
Mathematical and Management Sciences 38(1): 23-43.
Lindley,
D.V. 1958. Fiducial distribution and Bayes theorem. Journal of the Royal Statistical Society, Series B 20(1): 102-107.
Maiti, S.S. & Mukherjee, I. 2018. On
estimation of the PDF and CDF of the Lindley distribution. Communications in Statistics-Simulation and Computation 47(5):
1370-1381.
Mazucheli, J., Menezes,
A.F.B. & Chakraborty, S. 2018. On the one parameter unit-Lindley
distribution and its associated regression model for proportion data. Journal of Applied Statistics 46(4):
700-714.
Nassar, M., Afify, A.Z., Dey, S. & Kumar,
D. 2018. A new extension of Weibull distribution: Properties and different
methods of estimation. Journal of
Computational and Applied Mathematics 336: 439-457.
Shanker, R. & Mishra,
A. 2013a. A quasi Lindley distribution. African
Journal of Mathematics and Computer Science Research 6(4): 64-71.
Shanker, R. & Mishra,
A. 2013b. A two-parameter Lindley distribution. Statistics in Transition-New Series 14(1): 45-56.
Shanker, R., Kamlesh, K.K. & Fesshaye, H.
2017. A two parameter Lindley distribution: Its properties and applications. Biostatistics and Biometrics Open Access
Journal 1(4): 85-90.
Shanker, R., Fesshaye, H. & Sharma, S. 2016. On two parameter
Lindley distribution and its applications to model lifetime data. Biom. Biostat. Int. J.
3(1): 9-15.
Shanker, R., Sharma, S.
& Shanker, R. 2013. A two-parameter Lindley
distribution for modeling waiting and survival times data. Applied Mathematics 4: 363-368.
Shanker, R., Shukla,
K.K., Shanker, R. & Tekie,
A.L. 2017. A three-parameter Lindley distribution. American Journal of Mathematics and Statistics 7(1): 15-26.
Sharma,
V.K., Singh, S.K., Singh, U. & Agiwal, V. 2015.
The inverse Lindley distribution: A stress-strength reliability model with
application to head and neck cancer data. Journal
of Industrial and Production Engineering 32(3): 162-173.
Singh,
S.K., Singh, U. & Sharma, V.K. 2014. The truncated Lindley distribution:
Inference and application. Journal of
Statistics Applications and Probability 3(2): 219-228.
Zakerzadeh, H. & Dolati, A. 2009. Generalized Lindley distribution. Journal of Mathematical Extension 3(2):
13-25.
*Pengarang untuk surat-menyurat; email: mathil.thamir@uoanbar.edu.iq
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