Analysis For Uncertain Statistics And Its Applications

Some information and knowledge are usually represented by human language like "about 100km","Roughly 80kg"," low speed", "middle age" and "big size". A lot of survey showed that those imprecise quantities behave neither like randomness nor like fuzziness. Uncertainty theory is a branch of axiomatic mathematics based on normality, monotonicity, self-duality, countable subadditivity and product measure axioms. It is a mathematical tool in the research of human uncertain system. Uncertain statistics is a methodology for collecting and interpreting expert's experimental data by uncertainty theory and provides help for the decision makers by the corresponding statistical models.Uncertainty distribution is one of the important concepts of uncertainty theory and used to describe uncertain variables in easy-to-use way. It is a key step that builds the appropriate uncertainty distribution for applications in the real world. However, it is hard to obtain the precise uncertainty distribution in the experiment. So it is a feasible way to approximate uncertainty distribution by the empirical uncertainty distribution or estimation of uncertain measure.Based on uncertainty theory, this dissertation builds some inequalities on uncertain measure, presents Delphi method for obtaining uncertainty distribution and proposes the method of moments for estimating the unknown parameters. The main content of the dissertation and contributions are as follows:1. Some mathematical properties of convergence in mean square for the sequences of uncertain variables are discussed. After that, a necessary and sufficient condition of convergence in mean square for the sequences of uncertain variables is provided. On the other hand, uncertain second moment process is defined and its some related properties are introduced based on uncertainty theory.2. In order to apply uncertain measure in the real life and build essential limit theorems for the sequences of uncertain variables, a class of inequalities on uncertain measure are constructed by using the moments of uncertain variables. These inequalities may give the upper bounded or lower bounded estimates of some uncertain measure.3. In order to estimate the uncertainty distribution for an uncertain variable via multiple experts'experimental data, a new method by combination of individual empirical uncertainty distribution and Delphi is presented. At the same time, some real examples are given to verify this method.4. Based on the expert's experimental data, an uncertain method of moments is presented by the empirical moment defined in this dissertation for estimating the unknown parameters, and a numerical method is designed to find moment estimates of unknown parameters.5. According to the expert's experimental data, a step empirical uncertainty distribution is defined to describe some uncertain variables. Based on step empirical uncertainty distribution, empirical mean, empirical variance and empirical k-th moments are defined and another method of moments is presented to estimate the unknown parameters of uncertainty distribution.