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A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications

Author

Listed:
  • Luis Sánchez

    (Institute of Statistics, Universidad Austral de Chile, Valdivia 5091000, Chile)

  • Víctor Leiva

    (School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Valparaíso 2362807, Chile)

  • Helton Saulo

    (Department of Statistics, Universidade de Brasília, Brasília 70910-900, Brazil)

  • Carolina Marchant

    (Faculty of Basic Sciences, Universidad Católica del Maule, Talca 3480112, Chile
    ANID-Millennium Science Initiative Program-Millennium Nucleus Center for the Discovery of Structures in Complex Data, Santiago 7820244, Chile)

  • José M. Sarabia

    (Department of Quantitative Methods, Universidad CUNEF, 28040 Madrid, Spain)

Abstract

Standard regression models focus on the mean response based on covariates. Quantile regression describes the quantile for a response conditioned to values of covariates. The relevance of quantile regression is even greater when the response follows an asymmetrical distribution. This relevance is because the mean is not a good centrality measure to resume asymmetrically distributed data. In such a scenario, the median is a better measure of the central tendency. Quantile regression, which includes median modeling, is a better alternative to describe asymmetrically distributed data. The Weibull distribution is asymmetrical, has positive support, and has been extensively studied. In this work, we propose a new approach to quantile regression based on the Weibull distribution parameterized by its quantiles. We estimate the model parameters using the maximum likelihood method, discuss their asymptotic properties, and develop hypothesis tests. Two types of residuals are presented to evaluate the model fitting to data. We conduct Monte Carlo simulations to assess the performance of the maximum likelihood estimators and residuals. Local influence techniques are also derived to analyze the impact of perturbations on the estimated parameters, allowing us to detect potentially influential observations. We apply the obtained results to a real-world data set to show how helpful this type of quantile regression model is.

Suggested Citation

  • Luis Sánchez & Víctor Leiva & Helton Saulo & Carolina Marchant & José M. Sarabia, 2021. "A New Quantile Regression Model and Its Diagnostic Analytics for a Weibull Distributed Response with Applications," Mathematics, MDPI, vol. 9(21), pages 1-21, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2768-:d:669958
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    References listed on IDEAS

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    1. Víctor Leiva & Helton Saulo & Rubens Souza & Robert G. Aykroyd & Roberto Vila, 2021. "A new BISARMA time series model for forecasting mortality using weather and particulate matter data," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(2), pages 346-364, March.
    2. Marcelo Ventura & Helton Saulo & Victor Leiva & Sandro Monsueto, 2019. "Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 35(4), pages 963-977, July.
    3. Barry C. Arnold & Enrique Castillo & José María Sarabia, 1996. "Modeling the fatigue life of longitudinal elements," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(6), pages 885-895, September.
    4. Luis Sánchez & Víctor Leiva & Manuel Galea & Helton Saulo, 2021. "Birnbaum‐Saunders quantile regression and its diagnostics with application to economic data," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 37(1), pages 53-73, January.
    5. Helton Saulo & Jeremias Leão & Víctor Leiva & Robert G. Aykroyd, 2019. "Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data," Statistical Papers, Springer, vol. 60(5), pages 1605-1629, October.
    6. Angela Noufaily & M. C. Jones, 2013. "Parametric quantile regression based on the generalized gamma distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(5), pages 723-740, November.
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    Cited by:

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    2. Josmar Mazucheli & Bruna Alves & Mustafa Ç. Korkmaz & Víctor Leiva, 2022. "Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications," Mathematics, MDPI, vol. 10(9), pages 1-23, April.
    3. Gabriela M. Rodrigues & Edwin M. M. Ortega & Gauss M. Cordeiro & Roberto Vila, 2022. "An Extended Weibull Regression for Censored Data: Application for COVID-19 in Campinas, Brazil," Mathematics, MDPI, vol. 10(19), pages 1-17, October.
    4. Helton Saulo & Roberto Vila & Giovanna V. Borges & Marcelo Bourguignon, 2022. "Parametric quantile regression for income data," Papers 2207.06558, arXiv.org.
    5. Helton Saulo & Roberto Vila & Giovanna V. Borges & Marcelo Bourguignon & Víctor Leiva & Carolina Marchant, 2023. "Modeling Income Data via New Parametric Quantile Regressions: Formulation, Computational Statistics, and Application," Mathematics, MDPI, vol. 11(2), pages 1-25, January.

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