Ann. Inst. Statist. Math. Vol. 40, No. 4, 627-639 (1988) ON THE LOSS OF INFORMATION DUE TO FUZZINESS IN EXPERIMENTAL OBSERVATIONS MARIA ANGELESGIL Departamento de Matem~ticas, Universidad de Oviedo, 33071 Oviedo, Spain (Received April ! I, 1987; revised January 28, 1988) Abstract. The absence of exactness in the observation of the outcomes of a random experiment always entails a loss of information about the experimental distribution. This intuitive assertion will be formally proved in this paper by using a mathematical model involving the notions of fuzzy information and fuzzy information system (as intended by Tanaka, Okuda and Asai) and Zadeh's probabilistic definition. On the basis of this model we are first going to consider a family of measures of information enclosing some well-known measures, such as those defined by Kagan, Kullback-Leibler and Matusita, and then to establish methods for removing the loss of information due to fuzziness by increasing suitably the number of experimental observations. Key words and phrases: Fuzzy information, fuzzy information system, non-parametric measures of directed divergence, probability of a fuzzy event, random experiment. 1. Introduction An experiment is the process by which an observation is made. When a random experiment is performed it results in one outcome that cannot be previously predicted. In this way, in a random experiment one must distinguish two fundamental elements: the sample space (set consisting of all possible experimental outcomes) and the ability to observe the outcomes. Given a random experiment and a sample from it, the aim of the Statistical Information Theory is to quantify the information contained in the sample and to use this information in making inferences about the experiment. When one tries to state such a quantification it is usually assumed that the ability to observe allows the statistician to identify each observable event with a subset of the sample space. In this paper, we will suppose that the person responsible for observa- 627