Mu-Estimating and Smoothing

In the traditional M-estimation theory developed by Huber 1964, the parameter under estimation is the value of theta which minimizes the expectation of what is called a discrepancy measure DISM 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 multivariate median defined by Oja 1983 and Liu 1990, as the value of theta which minimizes the expectation of a DISM of the type delta X1,... , Xm, theta where X1,... , Xm are independent copies of the underlying random variable X. Arcones et al 1994 studied the estimation of such parameters. We call the associated M-estimation MU-estimation or mu-estimation for convenience. When a DISM is not a differentiable function of theta, some complexities arise in studying the properties of estimations as well as in their computation. In such a case we introduce a new method of smoothing the DISM with a kernel function and using it in estimation. It is seen that smoothing allows up to develop an elegant approach to the study of asymptotic properties and computation of estimations.