Robust Estimation and Robust Parameter

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.