Membership set estimators: size, optimal inputs, complexity and relations with least squares

In this paper, we study some fundamental properties of the membership set estimators. First, the size of the membership set S^N is derived if the noise is bounded by ε but otherwise unknown. Second, in the case when the noise is an independent and identically distributed random variable in the interval [-ε,ε], the probability distribution of the size of S^N is also obtained. We then derive optimality conditions on the input in order to minimize the size of this set. Finally, we study the relations between least squares and membership set estimators and we obtain necessary and sufficient conditions under which the least squares estimate lies in S^N