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Kumaraswamy regression model with Aranda-Ordaz link function

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In this work, we introduce a regression model for double-bounded variables in the interval (0, 1) following a Kumaraswamy distribution. The model resembles a generalized linear model, in which the response’s median is modeled by a regression structure through the asymmetric Aranda-Ordaz parametric link function. We consider the maximum likelihood approach to estimate the regression and the link function parameters altogether. We study large sample properties of the proposed maximum likelihood approach, presenting closed-form expressions for the score vector as well as the observed and Fisher information matrices. We briefly present and discuss some diagnostic tools. We provide numeric evaluation of the finite sample inferences to show the performance of the estimators. Finally, to exemplify the usefulness of the methodology, we present and explore an empirical application.

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  1. Lemma 3, in particular, is presented and proved in the supplementary material that accompanies the paper.


  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723

    Article  MathSciNet  MATH  Google Scholar 

  • Aranda-Ordaz FJ (1981) On two families of transformations to additivity for binary response data. Biometrika 68(2):357–363

    Article  MathSciNet  MATH  Google Scholar 

  • Arundel AV, Sterling EM, Biggin JH, Sterling TD (1986) Indirect health effects of relative humidity in indoor environments. Environ Health Perspect 65:351–361

    Google Scholar 

  • Bayer FM, Bayer DM, Pumi G (2017) Kumaraswamy autoregressive moving average models for double bounded environmental data. J Hydrol 555:385–396

    Article  Google Scholar 

  • Benjamin M, Rigby R, Stasinopoulos D (1998) Fitting non-Gaussian time series models. In: COMPSTAT proceedings in computational statistics Heidelburg: Physica-Verlag, pp 191–196

  • Cai Z, Xiao Z (2012) Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. J Econom 167(2):413–425

    Article  MathSciNet  MATH  Google Scholar 

  • Canterle DR, Bayer FM (2019) Variable dispersion beta regressions with parametric link functions. Stat Pap 60(5):1541–1567

    Article  MathSciNet  MATH  Google Scholar 

  • Czado C (1997) On selecting parametric link transformation families in generalized linear models. J Stat Plan Inference 61(1):125–140

    Article  MATH  Google Scholar 

  • Czado C, Munk A (2000) Noncanonical links in generalized linear models: when is the effort justified? J Stat Plan Inference 87:317–345

    Article  MathSciNet  MATH  Google Scholar 

  • Dehbi H-M, Cortina-Borja M, Geraci M (2016) Aranda-Ordaz quantile regression for student performance assessment. J Appl Stat 43(1):58–71

    Article  MathSciNet  MATH  Google Scholar 

  • Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5(3):236–244

    Google Scholar 

  • Fahrmeir L (1987) Asymptotic testing theory for generalized linear models. Statistics 18(1):65–76

    Article  MathSciNet  MATH  Google Scholar 

  • Fahrmeir L, Kaufmann H (1985) Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann Stat 1(13):342–368

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher S, Ponnambalam K (1996) Estimation of reservoir yield and storage distribution using moments analysis. J Hydrol 182(1–4):259–275

    Article  Google Scholar 

  • Fokianos K, Kedem B (1998) Prediction and classification of non-stationary categorical time series. J Multivar Anal 67:277–296

    Article  MathSciNet  MATH  Google Scholar 

  • Gomes GSDS, Ludermir TB (2013) Optimization of the weights and asymmetric activation function family of neural network for time series forecasting. Expert Syst Appl 40(16):6438–6446

    Article  Google Scholar 

  • Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Academic Press, Cambridge

    MATH  Google Scholar 

  • Gunawardhana LN, Al-Rawas GA, Kazama S (2017) An alternative method for predicting relative humidity for climate change studies. Meteorol Appl 24(4):551–559

    Article  Google Scholar 

  • Gupta AK, Nadarajah S (2004) Handbook of beta distribution and its applications. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  • Jones M (2009) Kumaraswamy distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6(1):70–81

    Article  MathSciNet  MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Econometric society monograph series, Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Kumaraswamy P (1980) A generalized probability density function for double-bounded random processes. J Hydrol 46(1):79–88

    Article  Google Scholar 

  • Mathieu J (1981) Tests of \(\chi ^2\) in the generalized linear model. Ser Stat 12(4):509–527

    Article  MathSciNet  Google Scholar 

  • McCullough P, Nelder JA (1989) Generalized linear models. Chapman & Hall, London

    Book  Google Scholar 

  • Mitnik PA (2013) New properties of the Kumaraswamy distribution. Commun Stat Theory Methods 42(5):741–755

    Article  MathSciNet  MATH  Google Scholar 

  • Mitnik PA, Baek S (2013) The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Stat Pap 54(1):177–192

    Article  MathSciNet  MATH  Google Scholar 

  • Muggeo V, Ferrara G (2008) Fitting generalized linear models with unspecified link function: a p-spline approach. Comput Stat Data Anal 52:2

    Article  MathSciNet  MATH  Google Scholar 

  • Nadarajah S (2008) On the distribution of Kumaraswamy. J Hydrol 348(3):568–569

    Article  Google Scholar 

  • Nagelkerke NJ (1991) A note on a general definition of the coefficient of determination. Biometrika 78(3):691–692

    Article  MathSciNet  MATH  Google Scholar 

  • Papker L, Wooldridge J (1996) Econometric methods for fractional response variables with an application to 401 (K) plan participation rates. J Appl Econom 11:619–632

    Article  Google Scholar 

  • Pereira GH (2017) On quantile residuals in beta regression. Commun Stat Simul Comput 46:1–15

    Article  Google Scholar 

  • Pereira TL, Cribari-Neto F (2014) Detecting model misspecification in inflated beta regressions. Commun Stat Simul Comput 43(3):631–656

    Article  MathSciNet  MATH  Google Scholar 

  • Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1988) Numerical recipes in C, vol 1. Cambridge University Press, Cambridge, p 3

    MATH  Google Scholar 

  • Ramsey JB (1969) Tests for specification errors in classical linear least-squares regression analysis. J R Stat Soc Ser B 31:350–371

    MathSciNet  MATH  Google Scholar 

  • Sánchez S, Ancheyta J, McCaffrey WC (2007) Comparison of probability distribution functions for fitting distillation curves of petroleum. Energy Fuels 21(5):2955–2963

    Article  Google Scholar 

  • Schwarz G et al (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464

    Article  MathSciNet  MATH  Google Scholar 

  • Sundar V, Subbiah K (1989) Application of double bounded probability density function for analysis of ocean waves. Ocean Eng 16(2):193–200

    Article  Google Scholar 

  • Team RC (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  • Tutz G, Petry S (2012) Nonparametric estimation of the link function including variable selection. Stat Comput 22(2):545–561

    Article  MathSciNet  MATH  Google Scholar 

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We would like to thank two anonymous referees for valuable comments and suggestions that helped to improve the quality of the paper.

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Correspondence to Guilherme Pumi.

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We gratefully acknowledge partial financial support from Capes and CNPq, Brazil.

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Pumi, G., Rauber, C. & Bayer, F.M. Kumaraswamy regression model with Aranda-Ordaz link function. TEST 29, 1051–1071 (2020).

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