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On the convergence of Newton's method when estimating higher dimensional parameters

Author

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  • Clarke, Brenton R.
  • Futschik, Andreas

Abstract

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.

Suggested Citation

  • Clarke, Brenton R. & Futschik, Andreas, 2007. "On the convergence of Newton's method when estimating higher dimensional parameters," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 916-931, May.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:5:p:916-931
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    References listed on IDEAS

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    1. B. Clarke & C. Heathcote, 1994. "Robust estimation ofk-component univariate normal mixtures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(1), pages 83-93, March.
    2. Karunamuni, R. J. & Mehra, K. L., 1990. "Improvements on strong uniform consistency of some known kernel estimates of a density and its derivatives," Statistics & Probability Letters, Elsevier, vol. 9(2), pages 133-140, February.
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