IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i8p1213-d789052.html
   My bibliography  Save this article

Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution

Author

Listed:
  • Agustín G. Nogales

    (Departamento de Matemáticas, IMUEx, Universidad de Extremadura, 06006 Badajoz, Spain)

Abstract

In this paper, several related estimation problems are addressed from a Bayesian point of view, and optimal estimators are obtained for each of them when some natural loss functions are considered. The problems considered are the estimation of a regression curve, a conditional distribution function, a conditional density, and even the conditional distribution itself. These problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, and parametric and nonparametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function commonly used in estimating a real function of the unknown parameter. The cornerstone of these Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate the results.

Suggested Citation

  • Agustín G. Nogales, 2022. "Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution," Mathematics, MDPI, vol. 10(8), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1213-:d:789052
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/8/1213/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/8/1213/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265, October.
    2. Agustín G. Nogales, 2022. "On Bayesian estimation of densities and sampling distributions: The posterior predictive distribution as the Bayes estimator," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(2), pages 236-250, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Qianwen Tan & Subhashis Ghosal, 2021. "Bayesian Analysis of Mixed-effect Regression Models Driven by Ordinary Differential Equations," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 3-29, May.
    2. Laura Liu & Hyungsik Roger Moon & Frank Schorfheide, 2023. "Forecasting with a panel Tobit model," Quantitative Economics, Econometric Society, vol. 14(1), pages 117-159, January.
    3. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    4. A Stefano Caria & Grant Gordon & Maximilian Kasy & Simon Quinn & Soha Osman Shami & Alexander Teytelboym, 2024. "An Adaptive Targeted Field Experiment: Job Search Assistance for Refugees in Jordan," Journal of the European Economic Association, European Economic Association, vol. 22(2), pages 781-836.
    5. Reiß, Markus & Schmidt-Hieber, Johannes, 2020. "Posterior contraction rates for support boundary recovery," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6638-6656.
    6. Dmitry B. Rokhlin, 2021. "Relative utility bounds for empirically optimal portfolios," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 437-462, June.
    7. Suzanne Sniekers & Aad Vaart, 2020. "Adaptive Bayesian credible bands in regression with a Gaussian process prior," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 386-425, August.
    8. Shota Gugushvili & Frank van der Meulen & Moritz Schauer & Peter Spreij, 2018. "Nonparametric Bayesian volatility estimation," Papers 1801.09956, arXiv.org, revised Mar 2019.
    9. Martin Burda & Remi Daviet, 2023. "Hamiltonian sequential Monte Carlo with application to consumer choice behavior," Econometric Reviews, Taylor & Francis Journals, vol. 42(1), pages 54-77, January.
    10. Agustín G. Nogales, 2022. "On Consistency of the Bayes Estimator of the Density," Mathematics, MDPI, vol. 10(4), pages 1-6, February.
    11. Minerva Mukhopadhyay & Didong Li & David B. Dunson, 2020. "Estimating densities with non‐linear support by using Fisher–Gaussian kernels," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1249-1271, December.
    12. Meier, Alexander & Kirch, Claudia & Meyer, Renate, 2020. "Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    13. I. Votsi & G. Gayraud & V. S. Barbu & N. Limnios, 2021. "Hypotheses testing and posterior concentration rates for semi-Markov processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 707-732, October.
    14. Zach Dietz & William Lippitt & Sunder Sethuraman, 2023. "Stick-Breaking processes, Clumping, and Markov Chain Occupation Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 129-171, February.
    15. Francesco Denti & Michele Guindani & Fabrizio Leisen & Antonio Lijoi & William Duncan Wadsworth & Marina Vannucci, 2021. "Two‐group Poisson‐Dirichlet mixtures for multiple testing," Biometrics, The International Biometric Society, vol. 77(2), pages 622-633, June.
    16. Bhattacharya, Indrabati & Ghosal, Subhashis, 2021. "Bayesian multivariate quantile regression using Dependent Dirichlet Process prior," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    17. Dmitry B. Rokhlin, 2020. "Relative utility bounds for empirically optimal portfolios," Papers 2006.05204, arXiv.org.
    18. Hatjispyros, Spyridon J. & Merkatas, Christos & Walker, Stephen G., 2023. "Mixture models with decreasing weights," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    19. David M. Kaplan, 2015. "Bayesian and frequentist tests of sign equality and other nonlinear inequalities," Working Papers 1516, Department of Economics, University of Missouri.
    20. Sergei Seleznev, 2019. "Truncated priors for tempered hierarchical Dirichlet process vector autoregression," Bank of Russia Working Paper Series wps47, Bank of Russia.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1213-:d:789052. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.