Peter Stahlecker

Minimax estimation in linear regression with singular covariance structure and convex polyhedral constraints

Abstract Let the parameter of the linear regression model be restricted to a compact and convex polyhedron. On the basis of this additional information an estimator is shown to exist which minimizes the maximal weighted risk. This exact minimax solution turns out to be not feasible, but can be characterized as a limit point of approximate and operational minimax solutions. The results obtained are valid for general quadratic loss and possibly singular covariance matrix.

Another view of the Kuks–Olman estimator

Abstract In linear regression biased estimators like ridge estimators, Kuks–Olman estimators, Bayes, and minimax estimators are mainly used in order to circumvent difficulties caused by multicollinearity. Up to now, the application of the minimax principle to the weighted scalar mean squared error yields explicit solutions solely in specific cases, where, e.g., ridge estimators or Kuks–Olman estimators are obtained. In this paper we introduce a new objective function in such a way that we always get an explicit minimax solution which, in an important special case, can be interpreted as a Kuks–Olman estimator. Our functional may be viewed as a measure of relative rather than absolute squared error.

A portfolio selection model using fuzzy returns

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.

Linear regression analysis using the relative squared error

Abstract In order to determine estimators and predictors in a generalized linear regression model we apply a suitably defined relative squared error instead of the most frequently used absolute squared error. The general solution of a matrix problem is derived leading to minimax estimators and predictors. Furthermore, we consider an important special case, where an analogon to a well-known relation between estimators and predictors holds and where generalized least squares estimators as well as Kuks–Olman and ridge estimators play a prominent role.

Dropping variables versus use of proxy variables in linear regression

In this paper we investigate under which conditions it is preferable to use proxies or to omit variables from the linear regression model with respect to the matrix mean square error criterion. Furthermore, some attention is paid to the admissibility of the proxies-based least squares estimator.

Monopolistic price setting under fuzzy information

Abstract We consider a profit maximizing monopolistic firm that sets prices in the presence of improper information about the demand for its products. This information deficiency is viewed as being vagueness rather than stochastic uncertainty and modelled by a family of fuzzy sets. Applying a defuzzification strategy that explicitly takes the firm’s attitude towards deviations of actual demand from its most possible values into account, we derive explicit solutions to the maximization problem of a single-product and a general multi-product firm. Furthermore, some comparative static results are established.

Precautionary saving and fuzzy information

Abstract We consider a two-period life-cycle model where uncertainty about future labour income is modelled by a fuzzy set. Applying a defuzzification strategy that explicitly takes the individual’s behaviour towards risk into account, we show that pessimistic individuals engage in precautionary savings even if marginal utility is not convex, e.g. in case of a quadratic utility function.

Minimax adjustment technique and fuzzy information

Abstract In this paper the minimax adjustment technique is generalized to fuzzy information sets. Using a quadratic loss function and specific ellipsoidal constraints the case of fuzzy information can be reduced to the case of crisp information. Here, the minimax adjustment technique is equivalent to a projection method: furthermore, a characterization of the solution is given being not far away from an explicit representation. Applications to statistics and to economics are presented.

Some Further Results on the Use of Proxy Variables in Prediction

In econometric analysis, occasionally some of the regressors are not available. In this paper, the authors study the implications of two strategies: either cancel these regressors from the model or use proxy variables instead. It is analyzed which of both strategies leads to an improvement in conditional prediction of the systematic part in terms of the mean square error criterion. Furthermore, some characterizations for admissibility are given. Copyright 1993 by MIT Press.

Competitive supply behavior when price information is fuzzy

The theory of fuzzy sets is applied to the output decisions of a price-taking firm facing imprecise information about expected future prices. Accepting risk resulting from the randomness of prices, the manager is interested in expected profits only. Since the set of possible expected-price vectors is fuzzy, a suitable defuzzification strategy is defined in analogy to the pessimism-optimism index proposed by L. Hurwicz. It depends on the managers willingness to accept “surprises” resulting from a deviation of the true expected prices from the values that guided output decisions. Despite a linear cost function, well specified solutions to the optimization problem are possible without resorting to capacity constraints.