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In this paper the linear regression model y = X -f U is considered, where the design matrix X is deterministic, and V is the covariance matrix of the random error term U. The regression coefficient β is to be estimated in combining the observation with some prior knowledge about β and V. The prior information is represented by a suitable fuzzy set. Applying the weighted mean squared error criterion we define optimal linear estimators generalizing the 'classical' linear minimax estimators. Under a very mild assumption it is shown that such optimal linear estimators exist; furthermore, sufficient conditions for their uniqueness are given. The approach is illustrated by an example, and in a specific case, the generalized least squares estimator turns out to be optimal.

We study a static portfolio selection problem, in which future returns of securities are given as fuzzy sets. In contrast to traditional analysis, we assume that investment decisions are not based on statistical expectation values, but rather on maximal and minimal potential returns resulting from the so-called α-cuts of these fuzzy sets. By aggregating over all α-cuts and assigning weights for both best and worst possible cases we get a new objective function to derive an optimal portfolio. Allowing for short sales and modelling α-cuts in ellipsoidal shape, we obtain the optimal portfolio as the unique solution of a simple optimization problem. Since our model does not include any stochastic assumptions, we present a procedure, which turns the data of observable returns as well as experts' expectations into fuzzy sets in order to quantify the potential future returns and the investment risk.