A zero–one-inflated smooth unit-Laplace distribution for boundary-inflated proportional data

Fuxiang Liu; Tingting Li; Fuxiang Liu; Tingting Li

doi:10.3934/math.20261362

AIMS Mathematics

2026, Volume 11, Issue 10: 34975-35004. doi: 10.3934/math.20261362

Previous Article Next Article

Research article

A zero–one-inflated smooth unit-Laplace distribution for boundary-inflated proportional data

Fuxiang Liu ^1,2,
Tingting Li ^{1,2
,
,}

1.
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, Hubei 443002, China
2.
Department of Mathematics, College of Mathematics and Physics, China Three Gorges University, Yichang, Hubei 443002, China

Received: 29 June 2026 Revised: 13 September 2026 Accepted: 29 September 2026 Published: 26 October 2026
MSC : 62E15, 62F12, 65C05, 65C20

Bounded observations with exact endpoint values arise in many applications involving proportions, rates, and normalized indices. Such data require a model that is theoretically able to describe the continuous variation of observations in (0, 1), while practically allowing positive probability masses at the exact boundary values 0 and 1. This paper proposes a zero–one-inflated smooth unit-Laplace distribution for boundary-inflated proportional data. The construction first transforms a Laplace random variable to a smooth one-parameter distribution on the open unit interval and then combines this continuous law with point masses at 0 and 1. The resulting model admits closed-form expressions for the distribution function, density function, quantile function, and likelihood. Consequently, in the nondegenerate case, all maximum likelihood estimators admit closed-form expressions and can be computed with linear complexity in the sample size. Several mathematical properties are derived, including endpoint behavior, density-shape characteristics, raw moments, the Lorenz curve, and the Gini coefficient. Large-sample properties are established through the Fisher information matrix. Numerical experiments are carried out to examine bias, root mean squared error, standard errors, and confidence-interval coverage. An application to school-level proportional observations from the Programme for International Student Assessment (PISA) 2022 database illustrates the usefulness of the proposed computational framework for modeling boundary inflation and distributional concentration.
- boundary-inflated data,
- smooth unit-Laplace distribution,
- zero–one-inflated model,
- computational inference,
- Monte Carlo simulation
Citation: Fuxiang Liu, Tingting Li. A zero–one-inflated smooth unit-Laplace distribution for boundary-inflated proportional data[J]. AIMS Mathematics, 2026, 11(10): 34975-35004. doi: 10.3934/math.20261362

Related Papers:

Abstract

Bounded observations with exact endpoint values arise in many applications involving proportions, rates, and normalized indices. Such data require a model that is theoretically able to describe the continuous variation of observations in (0, 1), while practically allowing positive probability masses at the exact boundary values 0 and 1. This paper proposes a zero–one-inflated smooth unit-Laplace distribution for boundary-inflated proportional data. The construction first transforms a Laplace random variable to a smooth one-parameter distribution on the open unit interval and then combines this continuous law with point masses at 0 and 1. The resulting model admits closed-form expressions for the distribution function, density function, quantile function, and likelihood. Consequently, in the nondegenerate case, all maximum likelihood estimators admit closed-form expressions and can be computed with linear complexity in the sample size. Several mathematical properties are derived, including endpoint behavior, density-shape characteristics, raw moments, the Lorenz curve, and the Gini coefficient. Large-sample properties are established through the Fisher information matrix. Numerical experiments are carried out to examine bias, root mean squared error, standard errors, and confidence-interval coverage. An application to school-level proportional observations from the Programme for International Student Assessment (PISA) 2022 database illustrates the usefulness of the proposed computational framework for modeling boundary inflation and distributional concentration.

References

[1]	OECD, PISA 2022 results (Volume I): The state of learning and equity in education, Paris: OECD Publishing, 2023. https://doi.org/10.1787/53f23881-en
[2]	OECD, TALIS 2018 results (Volume I): Teachers and school leaders as lifelong learners, Paris: OECD Publishing, 2019. https://doi.org/10.1787/1d0bc92a-en
[3]	S. G. Rivkin, E. A. Hanushek, J. F. Kain, Teachers, schools, and academic achievement, Econometrica, 73 (2005), 417–458. https://doi.org/10.1111/j.1468-0262.2005.00584.x doi: 10.1111/j.1468-0262.2005.00584.x
[4]	E. A. Hanushek, L. Woessmann, School resources and student achievement: A review of cross-country economic research, In: Cognitive abilities and educational outcomes, Cham: Springer, 2017,149–171. https://doi.org/10.1007/978-3-319-43473-5_8
[5]	L. E. Papke, J. M. Wooldridge, Econometric methods for fractional response variables with an application to 401(k) plan participation rates, J. Appl. Econom., 11 (1996), 619–632.
[6]	R. Ospina, S. L. P. Ferrari, Inflated beta distributions, Stat. Papers, 51 (2010), 111–126. https://doi.org/10.1007/s00362-008-0125-4 doi: 10.1007/s00362-008-0125-4
[7]	R. Ospina, S. L. P. Ferrari, A general class of zero-or-one inflated beta regression models, Comput. Stat. Data Anal., 56 (2012), 1609–1623. https://doi.org/10.1016/j.csda.2011.10.005 doi: 10.1016/j.csda.2011.10.005
[8]	K. L. Mengersen, C. P. Robert, D. M. Titterington, Mixtures: Estimation and applications, John Wiley & Sons, Ltd, 2011. https://doi.org/10.1002/9781119995678
[9]	S. Ferrari, F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat., 31 (2004), 799–815. https://doi.org/10.1080/0266476042000214501 doi: 10.1080/0266476042000214501
[10]	E. Gómez-Déniz, J. M. Sarabia, A family of generalised beta distributions: Properties and applications, Ann. Data Sci., 5 (2018), 401–420. https://doi.org/10.1007/s40745-018-0143-6 doi: 10.1007/s40745-018-0143-6
[11]	M. C. Jones, Kumaraswamy's distribution: A beta-type distribution with some tractability advantages, Stat. Methodol., 6 (2009), 70–81. https://doi.org/10.1016/j.stamet.2008.04.001 doi: 10.1016/j.stamet.2008.04.001
[12]	A. Grassia, On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions, Aust. J. Stat., 19 (1977), 108–114. https://doi.org/10.1111/j.1467-842X.1977.tb01277.x doi: 10.1111/j.1467-842X.1977.tb01277.x
[13]	J. Mazucheli, A. F. B. Menezes, S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat., 46 (2019), 700–714. https://doi.org/10.1080/02664763.2018.1511774 doi: 10.1080/02664763.2018.1511774
[14]	J. Mazucheli, A. F. B. Menezes, L. B. Fernandes, R. P. de Oliveira, M. E. Ghitany, The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates, J. Appl. Stat., 47 (2020), 954–974. https://doi.org/10.1080/02664763.2019.1657813 doi: 10.1080/02664763.2019.1657813
[15]	J. Mazucheli, A. F. Menezes, S. Dey, Unit-Gompertz distribution with applications, Statistica, 79 (2019), 25–43. https://doi.org/10.6092/issn.1973-2201/8497 doi: 10.6092/issn.1973-2201/8497
[16]	K. Modi, V. Gill, Unit Burr-Ⅲ distribution with application, J. Stat. Manag. Syst., 23 (2020), 579–592. https://doi.org/10.1080/09720510.2019.1646503 doi: 10.1080/09720510.2019.1646503
[17]	C. R. da Silva, T. L. Pereira, L. M. A. Lima-Filho, P. J. Duarte-Neto, Zero-one-inflated unit gamma distribution and its application to basic sanitation data, J. Stat. Theory Pract., 20 (2026), 53. https://doi.org/10.1007/s42519-026-00570-7 doi: 10.1007/s42519-026-00570-7
[18]	S. R. Bapat, R. Bhardwaj, On an inflated unit-Lindley distribution, J. Stat. Res., 55 (2022), 299–311. https://doi.org/10.3329/jsr.v55i2.58807 doi: 10.3329/jsr.v55i2.58807
[19]	F. Cribari-Neto, J. Santos, Inflated Kumaraswamy distributions, An. Acad. Bras. Ciênc., 91 (2019), e20180955. https://doi.org/10.1590/0001-3765201920180955 doi: 10.1590/0001-3765201920180955
[20]	G. Martínez-Flórez, R. Tovar-Falón, C. Barrera-Causil, Inflated unit-Birnbaum-Saunders distribution, Mathematics, 10 (2022), 667. https://doi.org/10.3390/math10040667 doi: 10.3390/math10040667
[21]	S. Kotz, T. J. Kozubowski, K. Podgórski, The Laplace distribution and generalizations: A revisit with applications to communications, economics, engineering, and finance, Birkhäuser Boston, 2001. https://doi.org/10.1007/978-1-4612-0173-1
[22]	V. S. Stojanović, T. J. Spasojević, M. Jovanović, Laplace-logistic unit distribution with application in dynamic and regression analysis, Mathematics, 12 (2024), 2282. https://doi.org/10.3390/math12142282 doi: 10.3390/math12142282
[23]	I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic Press, 2014. https://doi.org/10.1016/C2010-0-64839-5
[24]	F. W. Olver, D. W. Lozier, R. Boisvert, C. W. Clark, The NIST handbook of mathematical functions, New York: Cambridge University Press, 2010.
[25]	B. C. Arnold, J. M. Sarabia, Majorization and the Lorenz order with applications in applied mathematics and economics, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-93773-1
[26]	M. Shaked, J. G. Shanthikumar, Stochastic orders, New York: Springer, 2007. https://doi.org/10.1007/978-0-387-34675-5
[27]	J. L. Gastwirth, The estimation of the Lorenz curve and Gini index, Rev. Econ. Stat., 54 (1972), 306–316. https://doi.org/10.2307/1937992 doi: 10.2307/1937992
[28]	G. Casella, R. Berger, Statistical inference, 2 Eds., New York: Chapman and Hall/CRC, 2024. https://doi.org/10.1201/9781003456285
[29]	A. Wald, Tests of statistical hypotheses concerning several parameters when the number of observations is large, Trans. Amer. Math. Soc., 54 (1943), 426–482. https://doi.org/10.1090/S0002-9947-1943-0012401-3 doi: 10.1090/S0002-9947-1943-0012401-3
[30]	T. P. Morris, I. R. White, M. J. Crowther, Using simulation studies to evaluate statistical methods, Stat. Med., 38 (2019), 2074–2102. https://doi.org/10.1002/sim.8086 doi: 10.1002/sim.8086
[31]	J. Antonelli, M. Cefalu, N. Palmer, D. Agniel, Doubly robust matching estimators for high dimensional confounding adjustment, Biometrics, 74 (2018), 1171–1179. https://doi.org/10.1111/biom.12887 doi: 10.1111/biom.12887
[32]	H. Ma, T. Nguyen, J. Jiang, Developing a practical measure: An asymmetric mean squared prediction error for small area estimation, J. Amer. Stat. Assoc., 121 (2026), 1647–1660. https://doi.org/10.1080/01621459.2025.2555056 doi: 10.1080/01621459.2025.2555056
[33]	Organisation for Economic Co-operation and Development (OECD), PISA 2022 database, 2023. Available from: https://www.oecd.org/en/data/datasets/pisa-2022-database.html.
[34]	E. F. Schuster, Incorporating support constraints into nonparametric estimators of densities, Commun. Stat. Theory Methods, 14 (1985), 1123–1136. https://doi.org/10.1080/03610928508828965 doi: 10.1080/03610928508828965
[35]	B. W. Silverman, Density estimation for statistics and data analysis, New York: Routledge, 1998. https://doi.org/10.1201/9781315140919
[36]	D. Rodriguez-Segura, C. Campton, L. Crouchn, T. S. Slade, Looking beyond changes in averages in evaluating foundational learning: Some inequality measures, Int. J. Educ. Dev., 87 (2021), 102411. https://doi.org/10.1016/j.ijedudev.2021.102411 doi: 10.1016/j.ijedudev.2021.102411
[37]	J. Petersson, Using the Gini coefficient for assessing heterogeneity within classes and schools, SN Soc. Sci., 3 (2023), 186. https://doi.org/10.1007/s43545-023-00772-x doi: 10.1007/s43545-023-00772-x

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)