MU-estimation and smoothing

In the M-estimation theory developed by Huber (1964, Ann. Math. Statist.43, 1449-1458), the parameter under estimation is the value of θ which minimizes the expectation of what is called a discrepancy measure (DM) δ(X, θ) which is a function of θ 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 θ which minimizes the expectation of a DM of the type δ(X1, ..., Xm, θ) 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 μ-estimation for convenience). When a DM is not a differentiable function of θ, 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