Skip to main content
SpringerLink
Log in

Multivariate frequency polygon for stationary random fields

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

The purpose of this paper is to investigate the multivariate frequency polygon as a density estimator for stationary random fields indexed by multidimensional lattice points space. Optimal cell widths that asymptotically minimize integrated mean square error (IMSE) are derived. Under weak conditions, the IMSE of frequency polygons achieves the same rate of convergence to zero as that of kernel estimators. The frequency polygon can also attain the optimal uniform rate of convergence and the almost sure convergence under general conditions. Finally, a result of \(L^1\) convergence is given. Frequency polygons thus appear to be very good density estimators with respect to the criteria of IMSE, of uniform convergence, of almost sure convergence and of \(L^1\) convergence. We apply our results to simulated data and real data.

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
Fig. 3

Similar content being viewed by others

Data availability statement

The dataset analyzed in Section 3.2 is publicly available. It was extracted on 26 July 2023 from the Environment and Climate Change Canada Historical Hydrometric Data web site, https://wateroffice.ec.gc.ca/mainmenu/historical_data_index_e.html.

References

  • Biau, G. (2003). Spatial kernel density estimation. Mathematical Methods of Statistics, 12, 371–390.

    MathSciNet  Google Scholar 

  • Bolthausen, E. (1982). On the central limit theorem for stationary random fields. Annals of Probability, 10, 1047–1050.

    Article  MathSciNet  Google Scholar 

  • Carbon, M. (2006). Polygone des fréquences pour des champs aléatoires. Note aux Comptes Rendus de l’Académie des Sciences de Paris, Série I, 342, 693–696.

    MathSciNet  Google Scholar 

  • Carbon, M. (2014). Histograms for stationary linear random files. Statistical Inference for Stochastic Processes 17, 245–266.

    Article  MathSciNet  Google Scholar 

  • Carbon, M., Hallin, M., Tran, L.T. (1996). Kernel density estimation for random fields: The \(L_1\) theory. Journal of Non Parametric Statistics, 6, 157–170.

    Article  Google Scholar 

  • Carbon, M., Garel, B., Tran, L.T. (1997a). Frequency polygons for weakly dependent processes. Statistics and Probability Letters, 33, 1–13.

    Article  MathSciNet  Google Scholar 

  • Carbon, M., Tran, L.T., Wu, B. (1997b). Kernel density estimation for random fields (Density estimation for random fields). Statistics and Probability Letters, 36, 115–125.

    Article  MathSciNet  Google Scholar 

  • Davydov, Y.A. (1970). The invariant principle for stationary processes. Theory of Probability and Its Applications, 14, 487–498.

    Article  Google Scholar 

  • Deo, C.M. (1973). A note on empirical processes of strong mixing sequences. Annals of Probability, 1, 870–875.

    MathSciNet  Google Scholar 

  • Doukhan, P. (1994). Mixing: properties and examples. Springer.

    Book  Google Scholar 

  • Francq, C., Tran, L.T. (2002) Nonparametric estimation of density, regression and dependence coefficients. Nonparametric Statistics, 14, 729–747.

    Article  MathSciNet  Google Scholar 

  • Friedman, D., Diaconis, P. (1981). On the histogram as a density estimator: L 2 theory. Zeitschrift für Wahrschenlichtkeitstheory und Verwandte Gebiete, 67, 453–476.

    Article  MathSciNet  Google Scholar 

  • Guyon, X. (1987). Estimation d’un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas Markovien. Proceedings of the 6th Franco-Belgian Meeting of Statisticians.

  • Guyon, X., Richardson, S. (1984). Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants. Zeitschrift für Wahrschenlichtkeitstheory und Verwandte Gebiete, 66, 297–314.

    Article  Google Scholar 

  • Hall, P., Heyde, C.C. (1980). Martingale limit theory and its application. Academic Press.

    Google Scholar 

  • Hallin, M., Lu, Z., Tran, L.T. (2001). Density estimation for spatial processes. Bernouilli, 7, 657–688.

    Article  MathSciNet  Google Scholar 

  • Hallin, M., Lu, Z., Tran, L.T. (2004a). Local linear spatial regression. Annals of Statistics, 32, 2469–2500.

    Article  MathSciNet  Google Scholar 

  • Hallin, M., Lu, Z., Tran, L.T. (2004b). Kernel density estimation for spatail processes : The \(L_1\) theory. Journal of Multivariate Analysis, 88, 61–75.

    Article  MathSciNet  Google Scholar 

  • Harel, M., Lenain, J.-F., Ngatchou-Wandji, J. (2016). Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields. Journal of Nonparametric Statistics, 28(2), 296–321.

    Article  MathSciNet  Google Scholar 

  • Hjort, N.L. (1986). On frequency polygons and averaged shifted histograms in higher dimensions. Technical Report, No. 22, Stanford University.

  • Nahapetian, B.S. (1980). The central limit theorem for random fields with mixing conditions. Advances in Probability, 6, 531–548.

    MathSciNet  Google Scholar 

  • Nahapetian, B.S. (1987). An approach to proving limit theorems for dependent random variables. Theory of Probability and Its Applications, 32, 535–539.

    Google Scholar 

  • Neaderhouser, C.C. (1980) Convergence of block spins defined on random fields. Journal of Statistical Physics, 22, 673–684.

    Article  ADS  MathSciNet  Google Scholar 

  • Politis, D.N., Romano, J.P. (1993) Nonparametric resampling for homogeneous strong mixing random fields. Journal of Multivariate Analysis, 47, 301–328.

    Article  MathSciNet  Google Scholar 

  • Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Annals of Probability, 23, 1188–1203.

    Article  MathSciNet  Google Scholar 

  • Robinson, P.M. (1983). Nonparametric estimators for time series. Journal of Time Series Analysis, 4, 185–207.

    Article  MathSciNet  Google Scholar 

  • Robinson, P.M. (2011). Asymptotic theory for non parametric regression with spatial data. Journal of Econometrics, 165, 5–19.

    Article  MathSciNet  Google Scholar 

  • Rosenblatt, M. (1985). Stationary sequences and random fields. Birkhauser.

    Book  Google Scholar 

  • Scott, D.W. (1985) Frequency polygons, theory and applications. Journal of the American Statistical Association, 80, 348–354.

    Article  MathSciNet  Google Scholar 

  • Scott, D.W. (2015). Multivariate density estimation: Theory, practice and visualization. Wiley.

    Book  Google Scholar 

  • Stone, C.J. (1983). Optimal uniform rate of convergence for non parametric estimators of a density function and its derivative. In Revzi, M.H., Rustagi, J.S., Siegmund D. (Eds.) Recent advances in statistics: Papers in Honor of H. Chernoff. (pp. 393–406). Academic Press.

    Chapter  Google Scholar 

  • Takahata, H. (1983). On the rates in the central limit theorem for weakly dependent random fields. Zeitschrift für Wahrschenlichtkeitstheory und Verwandte Gebiete, 62, 477–480.

    MathSciNet  Google Scholar 

  • Terrell, G.R. (1983). The multilinear frequency spline. Technical Report, Department of Math Sciences, Rice University, Houston.

  • Tran, L.T. (1990). Kernel density estimation on random fields. Journal of Multivariate Analysis, 34, 37–53.

    Article  MathSciNet  Google Scholar 

  • Tran, L.T., Yakowitz, S. (1993). Nearest neighbor estimators for random fields. Journal of Multivariate Analysis, 44, 23–46.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the Editor and two anonymous reviewers whose comments have lead to an improved version of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Université Laval, 1056, avenue de la Médecine, Québec, QC, G1V 0A6, Canada

    Michel Carbon & Thierry Duchesne

Authors
  1. Michel Carbon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thierry Duchesne
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thierry Duchesne.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 929 KB)

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carbon, M., Duchesne, T. Multivariate frequency polygon for stationary random fields. Ann Inst Stat Math 76, 263–287 (2024). https://doi.org/10.1007/s10463-023-00883-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-023-00883-5

Keywords

Search

Navigation