Home Mathematics An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
Article
Licensed
Unlicensed Requires Authentication

An extended type I half-logistic family of distributions: Properties, applications and different method of estimations

  • Majid Hashempour
Published/Copyright: June 11, 2022
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 3

Abstract

We propose a new family of continuous distributions with two shape parameters called the Extended type I(ET1HL-G). We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics. Then we study half-logistic distribution as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two data sets are illustration the purposes.

MSC 2010: Primary 60E05; 62G05; Secondary 62N05; 62P30
Keywords: half-logistic distribution; maximum likelihood; moment; simulation; quantile function

  1. (Communicated by Gejza Wimmer)

References

[1] Afify, A. Z.—Altun, E.—Alizadeh, M.—Ozel, G.—Hamedani, G. G.: The odd exponentiated half-logistic-G Family: Properties, characterizations and applications, Chil. J. Stat. 8(2) (2017), 65–91.Search in Google Scholar

[2] Alizadeh, M.—Emadi, M.—Doostparast, M.: A new two-parameter lifetime distribution: Properties, applications and different method of estimations, Stat. Optim. Inf. Comput. 7(2) (2019), 291–310.10.19139/soic.v7i2.653Search in Google Scholar

[3] Alizadeh, M.—Nematollahi, A.—Altun, E.—Rasekhi, M.: A study on a new type 1 half-logistic family of distributions and its applications, Stat. Optim. Inf. Comput. 8(4) (2020), 934–949.10.19139/soic-2310-5070-674Search in Google Scholar

[4] Al-Mofleh, H.—Elgarhy, M.—Afify, A.—Zannon, M.: Type II exponentiated half logistic generated family of distributions with applications, Electron. J. Appl. Stat. Anal. 13(2) (2020), 536–561.Search in Google Scholar

[5] Alzaatreh, A.—Lee, C.—Famoye, F.: A new method for generating families of continuous distributions, Metron 71(1) (2013), 63–79.10.1007/s40300-013-0007-ySearch in Google Scholar

[6] Anderson, T. W.—Darling, D. A: Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes, The Annals of Mathematical Statistics (1952), 193–212.10.1214/aoms/1177729437Search in Google Scholar

[7] Balakrishnan, N.: Order statistics from the half logistic distribution, J. Stat. Comput. Simul. 20(4) (1985), 287–309.10.1080/00949658508810784Search in Google Scholar

[8] Chen, Z.: A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statist. Probab. Lett. 49(2) (2000), 155–161.10.1016/S0167-7152(00)00044-4Search in Google Scholar

[9] Cheng, RCH.—Amin, NAK.: Maximum Product-of-Spacings Estimation with Applications to the Lognormal Distribution, Technical Report, Department of Mathematics, University of Wales, 1979.Search in Google Scholar

[10] Cheng, RCH.—Amin, NAK.: Estimating parameters in continuous univariate distributions with a shifted origin, J. R. Stat. Soc. B 3 (1983), 394–403.10.1111/j.2517-6161.1983.tb01268.xSearch in Google Scholar

[11] Choi, K.—Bulgren, W.: An estimation procedure for mixtures of distributions, J. R. Stat. Soc. Ser. B. Stat. Methodol. (1968), 444–460.10.1111/j.2517-6161.1968.tb00743.xSearch in Google Scholar

[12] Cordeiro, G. M.—de Castro, M.: A new family of generalized distributions, J. Stat. Comput. Simul. 81(7) (2011), 883–898.10.1080/00949650903530745Search in Google Scholar

[13] Cordeiro, G. M.—Alizadeh, M.—Ortega, E. M. : The exponentiated half-logistic family of distributions: Properties and applications, J. Probab. Stat. 2014 (2014).10.1155/2014/864396Search in Google Scholar

[14] Cordeiro, G. M.—Alizadeh, M.—Diniz Marinho, P. R.: The type I half-logistic family of distributions, J. Stat. Comput. Simul. 86(4) (2016), 707–728.10.1080/00949655.2015.1031233Search in Google Scholar

[15] Dey, S.—Mazucheli, J.—Nadarajah, S.: Kumaraswamy distribution: different methods of estimation, J. Comput. Appl. Math. (2017), 1–18.10.1007/s40314-017-0441-1Search in Google Scholar

[16] Elgarhy, M.—ul Haq, M. A.—Perveen, I.: Type II half logistic exponential distribution with applications, Annals of Data Science 6(2) (2019), 245–257.10.1007/s40745-018-0175-ySearch in Google Scholar

[17] Ghitany, M. E.—Atieh, B.—Nadarajah, S.: Lindley distribution and its application, J. Stat. Comput. Simul. 78(4) (2008), 493–506.10.1016/j.matcom.2007.06.007Search in Google Scholar

[18] 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

[19] Gupta, R. D.—Kundu, D.: Theory & methods: Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), 173–188.10.1111/1467-842X.00072Search in Google Scholar

[20] Hashempour, M.: A new two-parameter lifetime distribution with flexible hazard rate function: properties, applications and different method of estimations, Math. Slovaca 71(4) (2021), 983–1004.10.1515/ms-2021-0034Search in Google Scholar

[21] Hassan, A. S.—Elgarhy, M. A. E.—Shakil, M.: Type II half logistic family of distributions with applications, Pak. J. Stat. Oper. Res. (2017), 245–264.10.18187/pjsor.v13i2.1560Search in Google Scholar

[22] Jones, M. C.: Families of distributions arising from distributions of order statistics, Test 13(1) (2004), 1–43.10.1007/BF02602999Search in Google Scholar

[23] Kang, S. B.—Seo, J. I.: Estimation in an exponentiated half logistic distribution under progressively type-II censoring, Communications for Statistical Applications and Methods 18(5) (2011), 657–666.10.5351/CKSS.2011.18.5.657Search in Google Scholar

[24] Leadbetter, M. R.—Lindgren, G.—Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes, Springer Science & Business Media, 2012.Search in Google Scholar

[25] Lehmann, E. L.: The power of rank tests, The Annals of Mathematical Statistics (1953), 23–43.10.1007/978-1-4614-1412-4_33Search in Google Scholar

[26] Murthy, D. P.—Xie, M.—Jiang, R.: Weibull Models, Vol. 505, John Wiley & Sons, 2004.Search in Google Scholar

[27] Nichols, M. D.—Padgett, W. J.: A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International 22(2) (2006), 141–151.10.1002/qre.691Search in Google Scholar

[28] Nadarajah, S.—Haghighi, F.: An extension of the exponential distribution, Statistics 45(6) (2011), 543–558.10.1080/02331881003678678Search in Google Scholar

[29] Oliveira, J.—Santos, J.—Xavier, C.—Trindade, D.—Cordeiro, G. M.: The McDonald half-logistic distribution: Theory and practice, Commun. Stat. – Theory Methods 45(7) (2016), 2005–2022.10.1080/03610926.2013.873131Search in Google Scholar

[30] 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

[31] ZeinEldin, R. A.—Hashmi, S.—Elsehety, M.—Elgarhy, M.: Type II Half Logistic Kumaraswamy Distribution with Applications, J. Stat. Comput. Simul. 2020 (2020).10.1155/2020/1343596Search in Google Scholar

Appendix A: PDF of competitive models in application section

f E S H L ( x ; α ) = 2 α e − x ( 1 − e − x ) α − 1 ( 1 + e − x ) α + 1 x > 0 , α > 0 , f N O L L − S H L ( x ; α , β ) = 2 e − x ( 1 + e − x ) 2 1 − e − x 1 + e − x α − 1 1 − 1 − e − x 1 + e − x β − 1 α + ( β − α ) 1 − e − x 1 + e − x 1 − e − x 1 + e − x α + 1 − 1 − e − x 1 + e − x β 2 = x > 0 , α > 0 , β > 0 , f K w S H L ( x ; α , β ) = 2 α β e − x ( 1 − e − x ) α − 1 ( 1 + e − x ) α + 1 1 − ( 1 − e − x 1 + e − x ) α β − 1 x > 0 , α > 0 , β > 0 , f B S H L ( x ; α , β ) = 2 β e − β x ( 1 − e − x ) α − 1 B ( α , β ) ( 1 + e − x ) α + β x > 0 , α > 0 , β > 0 , f M c S H L ( x ; α , β , c ) = 2 c e − x ( 1 − e − x ) α c − 1 B ( α , β ) ( 1 + e − x ) α c + 1 1 − ( 1 − e − x 1 + e − x ) c β − 1 x > 0 , α > 0 , β > 0 , c > 0 , f L i ( x ; α ) = α 2 1 + α ( 1 + x ) e − α x x > 0 , α > 0 , f P L ( x ; α , β ) = α 2 β 1 + α x β − 1 ( 1 + x β ) e − α x β x > 0 , α > 0 , β > 0 , f G E ( x ; α , β ) = α β e − α x ( 1 − e − α x ) β − 1 x > 0 , α > 0 , β > 0 , f N H ( x ; α , β ) = α β ( 1 + α x ) β − 1 e 1 − ( 1 + α x ) β x > 0 , α > 0 , β > 0 , f L N ( x ; α , β ) = 1 x β 2 π e − ( log ⁡ ( x ) − α ) 2 2 β 2 x > 0 , α ∈ R , β > 0 , f G a ( x ; α , β ) = α β ( 1 + α x ) β − 1 e 1 − ( 1 + α x ) β x > 0 , α > 0 , β > 0 , f W ( x ; α , β ) = α β ( x β − 1 e − α x β x > 0 , α > 0 , β > 0

Appendix B: R codes for obtaining start value and finding ML estimators

library(”GenSA”)

library(”AdequacyModel”)

x<-scan()

1.578 1.582 1.858 2.595 2.710 2.899 2.940 3.087 3.669 3.848

4.740 4.848 5.170 5.783 5.866 5.872 6.152 6.406 6.839 7.042

7.187 7.262 7.466 7.479 7.481 8.292 8.443 8.475 8.587 9.053

9.172 9.229 9.352 10.046 11.182 11.270 11.490 11.623 11.848

12.695 14.369 14.812 15.662 16.296 16.410 17.181 17.675

19.742 29.022 29.047

hist(x, prob = T)

#————————–

pdf_ET1HLHL<- function(x, alpha, beta){

A<-(1-exp(-x))/(1+exp(-x))

B<-2*exp(-x)/(1+exp(-x))^2

C<-(1-(1-A)^alpha)/(1+(1-A)^beta)

D<-(alpha*B*(1-A)^(alpha-1)+beta*B*(1-A)^(beta-1)+(alpha-beta)*B

*(1-A)^(alpha+beta-1))/(1+(1-A)^beta)^2

return(D)

}

fit.ET1HLHL<- function(data, density) {

minusllike<- function(x) -sum(log(density(data, x[1], x[2])))

lower<- c(0.001, 0.001) #may need some changes here

upper<- c(1000, 1000)

out<- GenSA(lower = lower, upper = upper,

fn = minusllike, control=list(verbose=TRUE, max.time = 2))

return(out[c(”value”, ”par”, ”counts”)])

}

fit.ET1HLHL(x, pdf_ET1HLHL)

#———————–

library(”AdequacyModel”)

cdf_ET1HLHL<-function(par, x){

 alpha=par[1]

 beta=par[2]

A<-(1-exp(-x))/(1+exp(-x))

B<-2*exp(-x)/(1+exp(-x))^2

C<-(1-(1-A)^alpha)/(1+(1-A)^beta)

D<-(alpha*B*(1-A)^(alpha-1)+beta*B*(1-A)^(beta-1)+(alpha-beta)*B

*(1-A)^(alpha+beta-1))/(1+(1-A)^beta)^2

return(C)

 }

pdf_ET1HLHL<-function(par, x){

 alpha=par[1]

 beta=par[2]

A<-(1-exp(-x))/(1+exp(-x))

B<-2*exp(-x)/(1+exp(-x))^2

C<-(1-(1-A)^alpha)/(1+(1-A)^beta)

D<-(alpha*B*(1-A)^(alpha-1)+beta*B*(1-A)^(beta-1)+(alpha-beta)*B

*(1-A)^(alpha+beta-1))/(1+(1-A)^beta)^2

return(D)

}

goodness.fit(pdf=pdf_ET1HLHL, cdf=cdf_ET1HLHL, starts=c(0.1368680, 0.2407834), data=x, method=”B”, domain=c(0, Inf), mle=NULL)
Received: 2020-10-12
Accepted: 2021-02-12
Published Online: 2022-06-11
Published in Print: 2022-06-27

© 2022 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular Papers
  2. The poset of morphism-extension classes of countable graphs
  3. Characterization of monadic BL-algebras by state operators
  4. Comments on efficient batch verification test for digital signatures based on elliptic curves
  5. On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
  6. An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers
  7. Some results on certain types of Putcha semigroups
  8. Inner functions in QK spaces and multipliers
  9. Certain subclasses of meromorphic multivalent q-starlike and q-convex functions
  10. Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes
  11. Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type
  12. Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group
  13. Intuitionistic fuzzy Tribonacci I-convergent sequence spaces
  14. Lipschitz class functions and their general Fourier coefficients
  15. Spectra and fine spectra of the generalized upper difference operator with triple repetition Δ3ab on the Hahn sequence space
  16. A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere
  17. An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
  18. On the Unit-Chen distribution with associated quantile regression and applications
  19. A note on Lévy subordinators in cones of fuzzy sets in Banach spaces
  20. Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors
  21. Upper bound for variance of finite mixtures of power exponential distributions
  22. The dimension Dind of finite topological T0-spaces
https://doi.org/10.1515/ms-2022-0051
Keywords for this article
half-logistic distribution; maximum likelihood; moment; simulation; quantile function

Articles in the same Issue

  1. Regular Papers
  2. The poset of morphism-extension classes of countable graphs
  3. Characterization of monadic BL-algebras by state operators
  4. Comments on efficient batch verification test for digital signatures based on elliptic curves
  5. On necessary and sufficient conditions for the monogeneity of a certain class of polynomials
  6. An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers
  7. Some results on certain types of Putcha semigroups
  8. Inner functions in QK spaces and multipliers
  9. Certain subclasses of meromorphic multivalent q-starlike and q-convex functions
  10. Algebraic dependences of meromorphic mappings into a projective space sharing few hyperplanes
  11. Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type
  12. Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group
  13. Intuitionistic fuzzy Tribonacci I-convergent sequence spaces
  14. Lipschitz class functions and their general Fourier coefficients
  15. Spectra and fine spectra of the generalized upper difference operator with triple repetition Δ3ab on the Hahn sequence space
  16. A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere
  17. An extended type I half-logistic family of distributions: Properties, applications and different method of estimations
  18. On the Unit-Chen distribution with associated quantile regression and applications
  19. A note on Lévy subordinators in cones of fuzzy sets in Banach spaces
  20. Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors
  21. Upper bound for variance of finite mixtures of power exponential distributions
  22. The dimension Dind of finite topological T0-spaces
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Winner of the OpenAthens UX Award 2026
Winner of the OpenAthens UX Award 2026
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 7.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2022-0051/html
Scroll to top button