The Annals of Statistics

Quadratic distances on probabilities: A unified foundation

Bruce G. Lindsay, Marianthi Markatou, Surajit Ray, Ke Yang, and Shu-Chuan Chen

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Abstract

This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing the goodness of fit of models. Many important procedures have this structure, but the theory for these methods is dispersed and incomplete. Central to the statistical analysis of these distances is the spectral decomposition of the kernel that generates the distance. We show how this determines the limiting distribution of natural goodness-of-fit tests. Additionally, we develop a new notion, the spectral degrees of freedom of the test, based on this decomposition. The degrees of freedom are easy to compute and estimate, and can be used as a guide in the construction of useful procedures in this class.

Article information

Source
Ann. Statist. Volume 36, Number 2 (2008), 983-1006.

Dates
First available in Project Euclid: 13 March 2008

Permanent link to this document
http://projecteuclid.org/euclid.aos/1205420526

Digital Object Identifier
doi:10.1214/009053607000000956

Mathematical Reviews number (MathSciNet)
MR2396822

Zentralblatt MATH identifier
1133.62001

Subjects
Primary: 62A01: Foundations and philosophical topics 62E20: Asymptotic distribution theory
Secondary: 62H10: Distribution of statistics

Keywords
Degrees of freedom diffusion kernel goodness of fit high dimensions model assessment quadratic distance spectral decomposition

Citation

Lindsay, Bruce G.; Markatou, Marianthi; Ray, Surajit; Yang, Ke; Chen, Shu-Chuan. Quadratic distances on probabilities: A unified foundation. The Annals of Statistics 36 (2008), no. 2, 983--1006. doi:10.1214/009053607000000956. http://projecteuclid.org/euclid.aos/1205420526.


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References

  • Axler, S., Bourdon, P. and Ramey, W. (2001). Harmonic Function Theory, 2nd ed. Springer, New York.
  • Bhatia, R. (2003). Fourier Series. Reprint of the 1993 edition [Hindustan Book Agency, New Delhi]. Classroom Resource Materials Series. Mathematical Association of America, Washington, DC.
  • Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193.
  • Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 91 674–688.
  • Fan, Y. (1997). Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function. J. Multivariate Anal. 62 36–63.
  • Fan, Y. (1998). Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters. Econometric Theory 14 604–621.
  • Hastie, T., Tibshirani, R. and Friedman, J. H. (2001). The Elements of Statistical Learning: Data Mining, Inference and Prediction. Springer, New York.
  • Lévy, P. (1939). L’addition des variable aleatoires defines sur une circonference. Bull. Soc. Math. France 67 1–41.
  • Lindsay, B. G., Kettenring, J. and Siegmund, D. O. (2004). A report on the future of statistics (with discussion). Statist. Sci. 19 387–413.
  • Lindsay, B. G. and Markatou, M. (2002). Statistical Distances: A Global Framework to Inference. Springer, New York. To appear.
  • Liu, Z. J. and Rao, C. R. (1995). Asymptotic distribution of statistics based on quadratic entropy and bootstrapping. J. Statist. Plann. Inference 43 1–18.
  • Satterthwaite, F. W. (1946). An approximate distribution of estimates of variance components. Biometrics Bull. 2 110–114.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • Spitzner, D. J. (2006). Use of goodness-of-fit procedures in high dimensional testing. J. Statist. Comput. Simul. 76 447–457.
  • Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions. Princeton Univ. Press.
  • Wintner, A. (1947). On the shape of the angular case of Cauchy’s distribution curves. Ann. Math. Statist. 18 589–593.
  • Yang, K. (2004). Using the Poisson kernel in model building and selection. Ph.D. dissertation, Pennsylvania State Univ.
  • Yosida, K. (1980). Functional Analysis, 6th ed. Springer, Berlin.
  • Zuo, Y. and He, X. (2006). On the limiting distributions of multivariate depth-based rank sum statistics and related tests. Ann. Statist. 34 2879–2896.