MU-estimation and smoothing

JOURNAL OF MULTIVARIATE ANALYSIS

卷:76

Article

Liu, ZJ ; Rao, CR

关键词: data depth;    discrepancy measure;    estimating equation;    kernel;    multivariate median;    M-estimation;    MU-estimation;    U-statistic;   

DOI  :  10.1006/jmva.2000.1916

来源: Elsevier

【 摘 要 】

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist. 43, 1449-1458), the parameter under estimation is the value of 0 which minimizes the expectation of what is called a discrepancy measure (DM) delta (X, 0) which is a function of 0 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 0 which minimizes the expectation of a DM of the type delta (X-1,..., X-m, 0) where X-1,..., X-m 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 0, 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. (C) 2001 Academic Press.

【 授权许可】

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