Robust Estimation and Robust Parameter

Abstract

This chapter is addressed to the problem of defining the parameter in a semiparametric situation. Suppose, for example, that the observation X is assumed to be expressed as $X=\theta +\varepsilon$, where $\theta$ is the parameter to be estimated and $\varepsilon$ is the error whose distribution is not specified by a finite number of parameters. Although the distribution of $\varepsilon$ is not specified, it must satisfy some condition to guarantee that the observation be ‘unbiased’ in one sense or another. Usual assumption of ‘unbiasedness’ in the sense that the expectation of $\varepsilon$ being zero, is not necessarily appropriate, since it sometimes happens that $\varepsilon$ may not have the expectation. In this chapter the problem is discussed by considering the parameter as a functional of the distribution function of X.

The content of this chapter was presented at the IMS 1967 Annual Meeting as an invited paper titled ‘Robust estimation and robust parameter’ (mimeographed) in a session on robust estimation. That has been developed and published as Bickel and Lehmann (1975) Descriptive statistics for nonparametric models. I. Introduction. Ann. Statist. 3, 1038–1044.