The Annals of Statistics

Mixed-rates asymptotics

Peter Radchenko

Full-text: Open access


A general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function. The asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions. The new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator. The asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings. The method is explained by examples from Lasso-type estimation, k-means clustering, Shorth estimation and partial linear models.

Article information

Ann. Statist., Volume 36, Number 1 (2008), 287-309.

First available in Project Euclid: 1 February 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 62F12: Asymptotic properties of estimators

Nonstandard asymptotics rates of convergence limiting distribution empirical processes M-estimators singular quadratic approximation Lasso k-means Shorth partial splines


Radchenko, Peter. Mixed-rates asymptotics. Ann. Statist. 36 (2008), no. 1, 287--309. doi:10.1214/009053607000000668.

Export citation


  • Andrews, D. W. K. (1999). Estimation when a parameter is on a boundary. Econometrica 67 1341–1383.
  • Dudley, R. M. (1985). An extended Wichura theorem, definitions of Donsker classes, and weighted empirical distributions. Probability in Banach Spaces V. Lecture Notes in Math. 1153 141–178. Springer, Berlin.
  • Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
  • Frank, I. and Friedman, J. (1993). A statistical view of some chemometrics regression tools (with discussion). Technometrics 35 109–148.
  • Grübel, R. (1988). The length of the shorth. Ann. Statist. 16 619–628.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • Knight, K. and Fu, W. (2000). Asymptotics for lasso-type estimators. Ann. Statist. 28 1356–1378.
  • Kosorok, M. R. (2006). Introduction to empirical processes and semiparametric inference. Springer Series in Statistics. To appear. Parts available at
  • Pollard, D. (1981). Strong consistency of k-means clustering. Ann. Statist. 9 135–140.
  • Pollard, D. (1982). A central limit theorem for k-means clustering. Ann. Probab. 10 919–926.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
  • Pollard, D. and Radchenko, P. (2006). Nonlinear least-squares estimation. J. Multivariate Anal. 97 548–562.
  • Radchenko, P. (2004). Asymptotics under nonstandard conditions. Ph.D. thesis, Yale Univ. Available at
  • Radchenko, P. (2005). Reweighting the lasso. In 2005 Proceedings of the American Statistical Association [CD-ROM]. American Statistical Association, Alexandria, VA. Available at
  • Radchenko, P. (2006). Mixed-rates asymptotics (extended version). Available at
  • Rotnitzky, A., Cox, D., Bottai, M. and Robins, J. (2000). Likelihood-based inference with singular information matrix. Bernoulli 6 243–284.
  • Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • Van de Geer, S. (1999). Empirical Processes in M-Estimation. Cambridge Univ. Press.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.