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MU-Estimation and Smoothing

Author

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  • Liu, Z. J.
  • Rao, C. R.

Abstract

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449-1458), the parameter under estimation is the value of [theta] which minimizes the expectation of what is called a discrepancy measure (DM) [delta](X, [theta]) which is a function of [theta] and the underlying random variable X. Such a setting does not cover the estimation of parameters such as the multivariate median defined by Oja (1983) and Liu (1990), as the value of [theta] which minimizes the expectation of a DM of the type [delta](X1, ..., Xm, [theta]) where X1, ..., Xm are independent copies of the underlying random variable X. Arcones et al. (1994, Ann. Statist.22, 1460-1477) studied the estimation of such parameters. We call such an M-type MU-estimation (or [mu]-estimation for convenience). When a DM is not a differentiable function of [theta], some complexities arise in studying the properties of estimators as well as in their computation. In such a case, we introduce a new method of smoothing the DM with a kernel function and using it in estimation. It is seen that smoothing allows us to develop an elegant approach to the study of asymptotic properties and possibly apply the Newton-Raphson procedure in the computation of estimators.

Suggested Citation

  • Liu, Z. J. & Rao, C. R., 2001. "MU-Estimation and Smoothing," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 277-293, February.
    • Handle: RePEc:eee:jmvana:v:76:y:2001:i:2:p:277-293
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    References listed on IDEAS

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    1. Joel L. Horowitz, 1998. "Bootstrap Methods for Median Regression Models," Econometrica, Econometric Society, vol. 66(6), pages 1327-1352, November.
    2. Horowitz, Joel L, 1992. "A Smoothed Maximum Score Estimator for the Binary Response Model," Econometrica, Econometric Society, vol. 60(3), pages 505-531, May.
    3. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
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