Demet Aydın

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In this study, a new weighted version of the inverse Rayleigh distribution based on two different weight functions is introduced. Some statistical and reliability properties of the introduced distribution including the moments, moment generating function, entropy measures (i.e., Shannon and R´enyi) and survival and hazard rate functions are derived. The maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. So, a numerical method has been required to compute maximum likelihood estimates. Finally, the daily mean wind speed data set has been analysed to show the usability of the new weighted inverse Rayleigh distribution.


New weighted inverse Rayleigh distribution; Shannon entropy; hazard rate function; Fisher information matrix; wind speed data.

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A. Ahmad , S. P. Ahmad and A. Ahmed: Transmuted inverse Rayleigh distribution: a generalization of the inverse Rayleigh distribution. Mathematical Theory and Modeling 4(7) (2014), 90–98.

S. F. Bush: Nanoscale Communication Networks. Artech House, 2010.

R. A. Fisher: The effects of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics 6 (1934), 13–25.

M. E. Ghitany , F. Alqallaf , D. K. Al-Mutairi and H. A. Husain: A two-parameter weighted Lindley distribution and its applications to survival data. Mathematics and computers in Simulation 81(6) (2011), 1190–1201.

R. E. Glaser: Bathtub and related failure rate characterizations. Journal of American Statistical Association 75 (1980), 667–672.

R. C. Gupta and J. P. Keating: Relations for reliability measures under length biased sampling. Scandinavian Journal of Statistics 13 (1986), 49–56.

R. C. Gupta and S. N. Kirmani: The role of weighted distributions in stochastic modeling. Communications in Statistics-Theory and methods 19(9) (1990), 3147–3162.

K. Fatima and S. P. Ahmad: Weighted inverse Rayleigh distribution. International Journal of Statistics and Systems 12(1) (2017), 119–137.

J. X. Kersey: Weighted inverse Weibull and beta-inverse Weibull distribution. Georgia Southern University, 2010.

M. S. Khan: Modified inverse Rayleigh distribution. International Journal of Computer Applications 87(13) (2014), 28–33.

A. Klein and G. Mélard: Computation of the Fisher information matrix for time series models. Journal of Computational and Applied Mathematics 64(1–2) (1995), 57–68.

J. Leao , H. Saulo , M. Bourguignon , R. Cintra , L. Rgo and G. Cordeiro: On some properties of the beta inverse Rayleigh distribution. Chilean Journal of Statistics 4(2) (2013), 111–131.

V. Leiva , A. Sanhueza and J. M. Angulo: A length-biased version of the Birnbaum-Saunders distribution with application in water quality. Stochastic Environmental Research and Risk Assessment 23(3) (2009), 299–307.

B. O. Oluyede: On inequalities and selection of experiments for length biased distributions. Probability in the Engineering and Informational Sciences 13(2) (1999), 169–185.

A. K. Nanda and K. Jain: Some weighted distribution results on univariate and bivariate cases. Journal of Statistical Planning and Inference 77(2) (1999), 169–180.

G. P. Patil: Encountered data, statistical ecology, environmental statistics, and weighted distribution methods. Environmetrics 2(4) (1991), 377–423.

C. R. Rao: On discrete distributions arising out of methods of ascertainment. The Indian Journal of Statistics, Series A, 27 (1965), 311–324.

A. Rényi: On measures of information and entropy. Statistics and Probability 1 (1961), 547–561.

R. Roman: Theoretical properties and estimation in weighted Weibull and related distributions. M. S. Thesis, Georgia Southern University, Georgia, 2010.

A. Saghir , S. Tazeem and I. Ahmad: The length-biased weighted exponentiated inverted Weibull distribution. Cogent Mathematics 3(1) (2016), 1267299.

M. Q. Shahbaz , S. Shahbaz and N. S. Butt: The Kumaraswamy inverse Weibull distribution. Pakistan Journal of Statistics and Operation Research 8(3) (2012), 479–489.

E. Shannon: A mathematical theory of communication. The Bell System Technical Journal 27(10) (1948), 379–423.

V. Sherina and B. O. Oluyede: Weighted inverse Weibull distribution: statistical properties and applications. Theoretical Mathematics and Applications 4(2) (2014), 1–30.

V. N. Trayer: Doklady Acad. Nauk, Belorus, 1964.

V. G. Voda: On the inverse Rayleigh random variable. Reports of Statistical Application Research 19 (1972), 13–21.



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