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Forecasting the proportion of stored energy using the unit Burr XII quantile autoregressive moving average model

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Abstract

This paper defines the unit Burr XII autoregressive moving average (UBXII-ARMA) model for continuous random variables in the unit interval, where any quantile can be modeled by a dynamic structure including autoregressive and moving average terms, time-varying regressors, and a link function. Our main motivation is to analyze the time series of the proportion of stored hydroelectric energy in Southeast Brazil and even identify a crisis period with lower water levels. We consider the conditional maximum likelihood method for parameter estimation, obtain closed-form expressions for the conditional score function, and conduct simulation studies to evaluate the accuracy of the estimators and estimated coverage rates of the parameters’ asymptotic confidence intervals. We discuss the goodness-of-fit assessment and forecasting for the new model. Our forecasts of the proportion of the stored energy outperformed those obtained from the Kumaraswamy autoregressive moving average and beta autoregressive moving average models. Furthermore, only the UBXII-ARMA detected a significant effect of lower water levels before 2002 and after 2013.

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Data Availability

The data supporting this research is publicly available and can be accessed at http://www.ons.org.br/. It is also provided in the following repository https://github.com/tatianefribeiro/ubxiiarma, with all the computer codes used in the application.

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Acknowledgements

We thank the three referees and Associate Editor for their valuable comments and suggestions. We also gratefully acknowledge partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). The author Renata Rojas Guerra acknowledges the support of Serrapilheira Institute/Serra - 2211-41692; FAPERGS/23/2551-0001595-1, FAPERGS/23/2551-0000851-3; and CNPq/306274/2022-1. The author Airlane P. Alencar acknowledges FAPESP/23/02538-0.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil

    Tatiane Fontana Ribeiro & Airlane P. Alencar

  2. Departamento de Estadística, Universidad Nacional de Colombia, Bogotá, Colombia

    Fernando A. Peña-Ramírez

  3. Departamento de Estatística, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil

    Tatiane Fontana Ribeiro & Renata Rojas Guerra

  4. Departamento de Estatística, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Gauss M. Cordeiro

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  1. Tatiane Fontana Ribeiro
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  2. Fernando A. Peña-Ramírez
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  3. Renata Rojas Guerra
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  4. Airlane P. Alencar
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  5. Gauss M. Cordeiro
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Correspondence to Tatiane Fontana Ribeiro.

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Communicated by Kelly Cristina Poldi.

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Appendix A Simulation results for other link functions and quantiles

Appendix A Simulation results for other link functions and quantiles

Tables 10 and 11 display simulation results on point and interval estimation of the parameters that index the UBXII-ARMA(1, 1) model for \(\tau =0.5\) with probit and cloglog link functions.

Table 10 Simulation results on point estimation of the UBXII-ARMA(1, 1) model considering link functions probit and cloglog with \(\tau =0.5\)
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Table 11 Estimated coverage probability from the asymptotic confidence intervals for UBXII-ARMA(1, 1) model’s parameters with link functions probit and cloglog (\(\tau =0.5\))
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Tables 12 and 13 display simulation results on point and interval estimation of the parameters that index the UBXII-ARMA(1, 1) model for \(\tau \in \{0.1,0.9\}\) with logit link function.

Table 12 Simulation results on point estimation of the UBXII-ARMA(p, q) model for \(\tau \in \{0.1,0.9\}\)
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Table 13 Estimated coverage probability from the asymptotic confidence intervals for the parameters that index the UBXII-ARMA(1, 1) model with \(\tau \in \{0.1,0.9\}\)
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Ribeiro, T.F., Peña-Ramírez, F.A., Guerra, R.R. et al. Forecasting the proportion of stored energy using the unit Burr XII quantile autoregressive moving average model. Comp. Appl. Math. 43, 27 (2024). https://doi.org/10.1007/s40314-023-02513-5

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