Raffaello Seri

The Analytic Hierarchy Process and the Theory of Measurement

The analytic hierarchy process (AHP) is a decision-making procedure widely used in management for establishing priorities in multicriteria decision problems. Underlying the AHP is the theory of ratio-scale measures developed in psychophysics since the middle of the last century. It is, however, well known that classical ratio-scaling approaches have several problems. We reconsider the AHP in the light of the modern theory of measurement based on the so-called separable representations recently axiomatized in mathematical psychology. We provide various theoretical and empirical results on the extent to which the AHP can be considered a reliable decision-making procedure in terms of the modern theory of subjective measurement.

Empirical Properties of Group Preference Aggregation Methods Employed in AHP: Theory and Evidence

We study various methods of aggregating individual judgments and individual priorities in group decision making with the AHP. The focus is on the empirical properties of the various methods, mainly on the extent to which the various aggregation methods represent an accurate approximation of the priority vector of interest. We identify five main classes of aggregation procedures which provide identical or very similar empirical expressions for the vectors of interest. We also propose a method to decompose in the AHP response matrix distortions due to random errors and perturbations caused by cognitive bias predicted by the mathematical psychology literature. We test the decomposition with experimental data and find that perturbations in group decision making caused by cognitive distortions are more important than those caused by random errors. We propose methods to correct systematic distortions.

Quadrature rules and distribution of points on manifolds

We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

Estimation in Discrete Parameter Models

In some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points. In this case, we speak of discrete parameter models. Even though the problem is quite old and has interesting connections with testing and model selection, asymptotic theory for these models has hardly ever been studied. Therefore, we discuss consistency, asymptotic distribu- tion theory, information inequalities and their relations with efficiency and superefficiency for a general class of m-estimators.

Measurement by subjective estimation: Testing for separable representations

Studying how individuals compare two given quantitative stimuli, sayd 1 andd 2 , is a fundamental problem. One very common way to address it is throughratio estimation, that is to ask individuals not to give values tod 1 andd 2 , but rather to give their estimates of the ratiop=d 1 /d 2 . Several psychophysical theories (the best known being Stevens’ power-law) claim that this ratio cannot be known directly and that there are cognitive distortions on the apprehension of the different quantities. These theories result in the so-calledseparable representations[Luce, R. D. (2002). A psychophysical theory of intensity proportions, joint presentations, and matches.Psychological Review, 109, 520‐532; Narens, L. (1996). A theory of ratio magnitude estimation.Journal of Mathematical Psychology, 40, 109‐788], which include Stevens’ model as a special case. In this paper we propose a general statistical framework that allows for testing in a rigorous way whether the separable representation theory is grounded or not. We conclude in favor of it, but reject Stevens’ model. As a byproduct, we provide estimates of the psychophysical functions of interest. c

The Diffusion of Complementary Technologies: An Empirical Test

We analyze the simultaneous diffusion of multiple process technologies that are related. A new econometric model is used to examine the presence of complementarities, testing for strong one-step-ahead non-causality and strong simultaneous independence. Results indicate significant complementarities between CAD and CNC technologies. Prior adoption of either of the two technologies has a large effect on the posterior adoption of the other one; in addition, simultaneous adoption is found to be more likely than adoption of the two technologies in isolation. Consistent with the presence of complementarities, we also find evidence of substantial price cross-effects: a decrease in the price of CAD (or CNC) increases the adoption probability of CNC (or CAD). Lastly, the increase in the likelihood of adopting the complementary technology turns out to depend on several plant-specific moderating factors.

Bootstrap confidence sets for the Aumann mean of a random closed set

The objective is to develop a reliable method to build confidence sets for the Aumann mean of a random closed set as estimated through the Minkowski empirical mean. First, a general definition of the confidence set for the mean of a random set is provided. Then, a method using a characterization of the confidence set through the support function is proposed and a bootstrap algorithm is described, whose performance is investigated in Monte Carlo simulations.

Approximation of Stochastic Programming Problems

In Stochastic Programming, the aim is often the optimization of a criterion function that can be written as an integral or mean functional with respect to a probability measure \(\mathbb{P}\). When this functional cannot be computed in closed form, it is customary to approximate it through an empirical mean functional based on a random Monte Carlo sample. Several improved methods have been proposed, using quasi-Monte Carlo samples, quadrature rules, etc. In this paper, we propose a result on the epigraphical approximation of an integral functional through an approximate one. This result allows us to deal with Monte Carlo, quasi-Monte Carlo and quadrature methods. We propose an application to the epi-convergence of stochastic programs approximated through the empirical measure based on an asymptotically mean stationary (ams) sequence. Because of the large scope of applications of ams measures in Applied Probability, this result turns out to be relevant for approximation of stochastic programs through real data.

Numerical properties of generalized discrepancies on spheres of arbitrary dimension

Quantifying uniformity of a configuration of points on the sphere is an interesting topic that is receiving growing attention in numerical analysis. An elegant solution has been provided by Cui and Freeden [J. Cui, W. Freeden, Equidistribution on the sphere, SIAM J. Sci. Comput. 18 (2) (1997) 595-609], where a class of discrepancies, called generalized discrepancies and originally associated with pseudodifferential operators on the unit sphere in R^3, has been introduced. The objective of this paper is to extend to the sphere of arbitrary dimension this class of discrepancies and to study their numerical properties. First we show that generalized discrepancies are diaphonies on the hypersphere. This allows us to completely characterize the sequences of points for which convergence to zero of these discrepancies takes place. Then we discuss the worst-case error of quadrature rules and we derive a result on tractability of multivariate integration on the hypersphere. At last we provide several versions of Koksma-Hlawka type inequalities for integration of functions defined on the sphere.

Computational aspects of Cui-Freeden statistics for equidistribution on the sphere

Abstract. In this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (SIAM J. Sci. Comput., 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations.