## Law of the iterated logarithm and invariance principle for $M$-estimators

HTML articles powered by AMS MathViewer

- by Xuming He and Gang Wang PDF
- Proc. Amer. Math. Soc.
**123**(1995), 563-573 Request permission

## Abstract:

We prove the law of the iterated logarithm for a general class of M-estimators which covers in particular robust M-estimators and S-estimators of multivariate location-scatter. We also obtain an almost sure invariance principle (Bahadur-type representation) for these estimators.## References

- N. H. Bingham,
*Variants on the law of the iterated logarithm*, Bull. London Math. Soc.**18**(1986), no. 5, 433–467. MR**847984**, DOI 10.1112/blms/18.5.433 - Dennis D. Boos and R. J. Serfling,
*A note on differentials and the CLT and LIL for statistical functions, with application to $M$-estimates*, Ann. Statist.**8**(1980), no. 3, 618–624. MR**568724** - Raymond J. Carroll,
*On almost sure expansions for $M$-estimates*, Ann. Statist.**6**(1978), no. 2, 314–318. MR**464470** - Yuan Shih Chow and Henry Teicher,
*Probability theory*, Springer-Verlag, New York-Heidelberg, 1978. Independence, interchangeability, martingales. MR**513230** - Dorota M. Dąbrowska,
*Kaplan-Meier estimate on the plane: weak convergence, LIL, and the bootstrap*, J. Multivariate Anal.**29**(1989), no. 2, 308–325. MR**1004341**, DOI 10.1016/0047-259X(89)90030-4 - P. L. Davies,
*Asymptotic behaviour of $S$-estimates of multivariate location parameters and dispersion matrices*, Ann. Statist.**15**(1987), no. 3, 1269–1292. MR**902258**, DOI 10.1214/aos/1176350505 - Herold Dehling,
*Complete convergence of triangular arrays and the law of the iterated logarithm for $U$-statistics*, Statist. Probab. Lett.**7**(1989), no. 4, 319–321. MR**980708**, DOI 10.1016/0167-7152(89)90115-6 - David A. Freedman,
*On tail probabilities for martingales*, Ann. Probability**3**(1975), 100–118. MR**380971**, DOI 10.1214/aop/1176996452
F. A. Greybill, - Peter Hall,
*Laws of the iterated logarithm for nonparametric density estimators*, Z. Wahrsch. Verw. Gebiete**56**(1981), no. 1, 47–61. MR**612159**, DOI 10.1007/BF00531973 - Frank R. Hampel, Elvezio M. Ronchetti, Peter J. Rousseeuw, and Werner A. Stahel,
*Robust statistics*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. The approach based on influence functions. MR**829458** - Xuming He and Douglas G. Simpson,
*Robust direction estimation*, Ann. Statist.**20**(1992), no. 1, 351–369. MR**1150348**, DOI 10.1214/aos/1176348526 - C. C. Heyde,
*An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes*, J. Appl. Probability**10**(1973), 146–157. MR**365962**, DOI 10.1017/s0021900200042157 - Peter J. Huber,
*The behavior of maximum likelihood estimates under nonstandard conditions*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 221–233. MR**0216620** - Peter J. Huber,
*Robust statistics*, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981. MR**606374** - H. K. Iverson and R. H. Randles,
*The effects on convergence of substituting parameter estimates into $U$-statistics and other families of statistics*, Probab. Theory Related Fields**81**(1989), no. 3, 453–471. MR**983092**, DOI 10.1007/BF00340061 - Jana Jurečková and Pranab Kumar Sen,
*Invariance principles for some stochastic processes relating to $M$-estimators and their role in sequential statistical inference*, Sankhyā Ser. A**43**(1981), no. 2, 190–210. MR**666380** - J. Kiefer,
*Old and new methods for studying order statistics and sample quantiles.*, Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) Cambridge Univ. Press, London, 1970, pp. 349–357. MR**0286228** - Hendrik P. Lopuhaä,
*On the relation between $S$-estimators and $M$-estimators of multivariate location and covariance*, Ann. Statist.**17**(1989), no. 4, 1662–1683. MR**1026304**, DOI 10.1214/aos/1176347386
J. Marcinkiewicz and A. Zygmund, - Ricardo Antonio Maronna,
*Robust $M$-estimators of multivariate location and scatter*, Ann. Statist.**4**(1976), no. 1, 51–67. MR**388656** - Herbert Robbins,
*Statistical methods related to the law of the iterated logarithm*, Ann. Math. Statist.**41**(1970), 1397–1409. MR**277063**, DOI 10.1214/aoms/1177696786 - P. Rousseeuw and V. Yohai,
*Robust regression by means of S-estimators*, Robust and nonlinear time series analysis (Heidelberg, 1983) Lect. Notes Stat., vol. 26, Springer, New York, 1984, pp. 256–272. MR**786313**, DOI 10.1007/978-1-4615-7821-5_{1}5 - Pranab Kumar Sen,
*Sequential nonparametrics*, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981. Invariance principles and statistical inference. MR**633884** - Robert J. Serfling,
*Approximation theorems of mathematical statistics*, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980. MR**595165** - William F. Stout,
*Almost sure convergence*, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR**0455094** - V. Strassen,
*An invariance principle for the law of the iterated logarithm*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**3**(1964), 211–226 (1964). MR**175194**, DOI 10.1007/BF00534910

*Matrices with application in statistics*, 2nd ed., Wadsworth, Belmont, CA, 1983.

*Sur les fonctions independantes*, Fund. Math.

**29**(1937), 60-90.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 563-573 - MSC: Primary 62F35; Secondary 60F15, 60F17
- DOI: https://doi.org/10.1090/S0002-9939-1995-1231036-7
- MathSciNet review: 1231036