Skip to main content
SpringerLink
Log in

On multivariate quantile regression analysis

  • Original Paper
  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. For example, Hallin et al. (2010) proposed a directional quantile estimator involving hyperplanes that define halfspace Tukey-depth level sets.

  2. Note that multivariate unconditional quantiles are obtained as a special case when \(k=1\) and \(x=1\).

  3. An extreme point in a convex set is a point that cannot be expressed as a convex combination of any other points in the set.

  4. In turn, the solution of the dual problem (26b) evaluated at \(k^{*}\) can be used to conduct Lagrange multiplier/Rank tests on the parameters \(\beta \). See Koenker (2005, pp. 81–92).

  5. With (\(v_L,v_M)\) as bounds on simulated \(v_2 \), choosing \(y_2 =\frac{v_2-v_L }{v_M-v_L }\in [0,1]\) means that the distribution of \(Y_2 \) is a truncated version of the distribution of \(V_2 \), with truncation effects that vary with the simulated sample. As a result, we do not have a closed-form expression for the joint distribution of \((Y_1,Y_2)\). In this context, the quantile estimates reported below have the simple objective of illustrating that our proposed estimator is empirically tractable.

References

  • Billingsley P (1986) Probability and measure. Wiley, New York

    MATH  Google Scholar 

  • Cai Y (2010) Multivariate quantile function models. Stat Sin 20:481–496

    MathSciNet  MATH  Google Scholar 

  • Chaudhuri P (1996) On a geometric notation of quantiles for multivariate data. J Am Stat Assoc 91:862–872

    Article  MATH  Google Scholar 

  • Chernozhukov V, Galichon A, Hallin M, Henry N (2017) Monge–Kantorovich depth, quantiles, ranks and signs. Ann Stat 45:223–256

    Article  MathSciNet  MATH  Google Scholar 

  • El Bantli F, Hallin M (1999) L\(_{1}\) estimation in linear models with heterogeneous white noise. Stat Probab Lett 45:305–315

    Article  MathSciNet  MATH  Google Scholar 

  • Hallin M, Paindaveine D, Šiman M (2010) Multivariate quantiles and multi-output regression quantiles: from L\(_{1}\) optimization to halfspace depth. Ann Stat 38:635–668

    Article  MATH  Google Scholar 

  • Huber PJ (1964) Robust estimation of a location parameter. Ann Math Stat 35:73–101

    Article  MathSciNet  MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Koenker R, Bassett G (1978) Regression quantile. Econometrica 46:33–50

    Article  MathSciNet  MATH  Google Scholar 

  • Kong L, Mizera I (2012) Quantile tomography: using quantiles with multivariate data. Stat Sin 22:1589–1610

    MathSciNet  MATH  Google Scholar 

  • Knight K (1998) Limiting distributions for L\(_{1}\) regression estimators under general conditions. Ann Stat 96:755–770

    MathSciNet  MATH  Google Scholar 

  • Manski CF (1988) Analog estimation methods in econometrics. Chapman and Hall, New York

    MATH  Google Scholar 

  • Mizera I, Wellner JA (1998) Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables. Ann Stat 26:672–691

    Article  MathSciNet  MATH  Google Scholar 

  • Serfling R (2010) Equivariance and invariance properties of multivariate quantile and related functions and the role of standardization. J Nonparametr Stat 22:915–936

    Article  MathSciNet  MATH  Google Scholar 

  • Serfling R, Zuo Y (2010) Discussion. Ann Stat 38:676–684

    Article  MATH  Google Scholar 

  • Tukey JW (1974) Mathematics and the picturing of of data. In: Proceedings of the International Congress of Mathematicians, vol 2, Vancouver, BC, pp 523–531

  • Wei Y (2008) An approach to multivariate covariate-dependent quantile contours with application to bivariate conditional growth charts. J Am Stat Assoc 103:397–409

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Agricultural and Applied Economics, Taylor Hall, University of Wisconsin, Madison, WI, 53706, USA

    Jean-Paul Chavas

Authors
  1. Jean-Paul Chavas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Chavas.

Additional information

The author would like to thank two reviewers for useful comments on an earlier draft of the paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chavas, JP. On multivariate quantile regression analysis. Stat Methods Appl 27, 365–384 (2018). https://doi.org/10.1007/s10260-017-0407-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-017-0407-x

Keywords

JEL Classification

Search

Navigation