Testing satistical hipotheses in fuzzy environment

In traditional statistics all parameters of the mathematical model and possible observations should be well defined. Sometimes such assumption appears too rigid for the real-life problems, especially while dealing with linguistic data or imprecise requirements. To relax this rigidity fuzzy methods are incorporated into statistics. We review hitherto existing achievements in testing statistical hypotheses in fuzzy environment, point out their advantages or disadvantages and practical problems. We propose also a formalization of that decision problem and indicate the directions of further investigations in order to construct a more general theory

Statystyczne sterowanie procesami o danych stochastycznie zależnych - Pułapki rozwiązań standardowych

Shewhart control charts are the most frequently used tools of statistical process control. In their standard form they are designed under the assumption that consecutive observations are statistically independent and described by the normal distribution. When these assumptions are not fulfilled statistical properties of the Shewhart control charts are different from those assumed for the design purposes. When consecutive observations are not independent the properties of some Shewhart control charts have been investigated only in the case of classic autoregression processes. Hryniewicz (2012) considered the influence of the type of dependence, described in terms of copulas, on the properties of the Shewhart charts for monitoring the mean value of the process. In this paper some results from Hryniewicz (2012) have been recalled. Some new results, obtained for the R -chart used for the control of the variability of a process, have been presented.Karty kontrolne Shewharta są najczęściej stosowanym narzędziem statystycznego sterowania procesami. W swojej podstawowej postaci są one projektowane przy założeniu, że kolejne obserwacje procesu są statystycznie niezależne, i że są opisane rozkładem normalnym. Jeśli powyższe założenia nie są spełnione, to własności statystyczne kart kontrolnych Shewharta różnią się od tych, które zakłada się w procesie projektowania. Gdy kolejne obserwacje nie są niezależne, własności niektórych kart kontrolnych Shewharta zostały zbadane dla przypadku klasycznych procesów autoregresji. Hryniewicz (2012) rozpatrywał wpływ typu zależności pomiędzy obserwacjami, opisanego za pomocą pojęcia kopuli, na własności karty Shewharta służącej do monitorowania wartości średniej procesu. Niektóre z własności karty Shewharta omawiane w tamtej pracy zostały przypomniane w niniejszym opracowaniu, które zawiera ponadto nowe wyniki dotyczące analogicznego zagadnienia w odniesieniu do karty kontrolnej R , służącej do sterowania zmiennością monitorowanych procesów

Possibilistic approach to Bayes decisions, Journal of Telecommunications and Information Technology, 2003, nr 3

The decision problems are considered when the prior probabilistic information about the state of nature and decision maker’s utility function are imprecisely defined. In such a case the risks (or the expected utility) of considered decisions are also imprecisely defined. We propose two-step procedure for finding the optimal decision. First, we order possible decisions using the l -average ranking method by Campos and Gonzalez. Then we use possibilistic possibility of dominance and necessity of strict dominance indices proposed byDubois and Prade for the comparison of consequences of the most promising solutions

Hadamard and Jensen inequalities for s-convex fuzzy processes

We give some inequalities of Hadamard and Jensen type for s-convex fuzzy processes. We also give some applications

Theoretical aspects o f the description of uncertainty

The paper deals with the basic formal problems related to the-description of uncertainty. Four basic approaches to the modeling of uncertainty have been described and compared: probability theory, possibility theory, Dempster-Schafer theory, and Walley’s theory of upper probabilities. Differences between these theories, both formal and practical, have been briefly discussed.Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 zostało dofinansowane ze środków MNiSW w ramach działalności upowszechniającej naukę

Merging fuzzy statistical data with imprecise prior information - application in solving complex decision problems

Solving complex decision problems requires the usage of information from different sources. Usually this information is uncertain and statistical or probabilistic methods are needed for its processing. However, in many cases a decision maker faces not only uncertainty of a random nature but also imprecision in the description of input data that is rather of linguistic nature. Therefore, there is a need to merge uncertainties of both types into one mathematical model. In the paper we present methodology of merging information from imprecisely reported statistical data and imprecisely formulated fuzzy prior information. Moreover, we also consider the case of imprecisely defined loss functions. The proposed methodology may be considered as the application of fuzzy statistical methods for the decision making in the systems analysis.The original publication is available at JAIST Press http://www.jaist.ac.jp/library/jaist-press/index.htmlIFSR 2005 : Proceedings of the First World Congress of the International Federation for Systems Research : The New Roles of Systems Sciences For a Knowledge-based Society : Nov. 14-17, 2040, Kobe, JapanSymposium 4, Session 2 : Meta-synthesis and Complex Systems Complex Problem Solving (I

Theoretical aspects o f the description of uncertainty

The paper deals with the basic formal problems related to the-description of uncertainty. Four basic approaches to the modeling of uncertainty have been described and compared: probability theory, possibility theory, Dempster-Schafer theory, and Walley’s theory of upper probabilities. Differences between these theories, both formal and practical, have been briefly discussed

On Some Laws of Large Numbers for Uncertain Random Variables

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables