Randomized versus non-randomized hypergeometric hypothesis testing with crisp and fuzzy hypotheses
- PDF / 831,853 Bytes
- 37 Pages / 439.37 x 666.142 pts Page_size
- 79 Downloads / 190 Views
Randomized versus non-randomized hypergeometric hypothesis testing with crisp and fuzzy hypotheses Nataliya Chukhrova1
· Arne Johannssen1
Received: 9 April 2017 / Revised: 28 August 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper is concerned with fuzzy hypothesis testing in the framework of the randomized and non-randomized hypergeometric test for a proportion. Moreover, we differentiate between a test of significance and an alternative test to control the type I error or both error types simultaneously. In contrast to classical (non-)randomized hypothesis testing, fuzzy hypothesis testing provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. In particular, various types of hypotheses with user-specified membership functions can be formulated. Additionally, the proposed test methods are compared via a comprehensive case study, which demonstrates the high flexibility of fuzzy hypothesis testing in practical applications. Keywords Fuzzy statistics · Fuzzy hypotheses · Hypergeometric test · Hypothesis testing for a proportion · Randomized test · Test of significance · Alternative test · Creditworthiness
1 Introduction Applications of fuzzy set theory in theory of hypothesis testing is a popular field of research, which comprises modeling of fuzzy hypotheses, fuzzily formulated data, fuzzy α- and β-levels as also fuzzy p values. Fuzzy hypothesis testing for crisp data was proposed by Arnold (1996, 1998) and within the framework of the Neyman–Pearson lemma (Taheri and Behboodian 1999; Torabi et al. 2006; Torabi and Behboodian 2007). Fuzzily formulated data was considered by several researchers as mixed data (Saade and Schwarzlander 1990; Saade 1994), imprecise data (Grzegorzewski 2000, 2002) or fuzzy data (Son et al. 1992; Taheri and Behboodian 2002). In addition, hypothesis
B
Nataliya Chukhrova [email protected] Arne Johannssen [email protected]
1
Faculty of Business Administration, University of Hamburg, 20146 Hamburg, Germany
123
N. Chukhrova, A. Johannssen
testing with fuzzy specifications of α- and β-levels or the fuzzy p value was presented for example by Filzmoser and Viertl (2004) and Parchami et al. (2010, 2012). In this paper we use modeling of fuzzy hypotheses introduced by Arnold (1996) in the framework of a test procedure, which unites both test methods, a test of significance (controlling the type I error) and an alternative test (controlling both error types). Moreover, it provides various types of hypothesis formulation (crisp vs. fuzzy, complementary vs. non-complementary) with user-specified membership functions. Such an approach has numerous advantages: a higher flexibility in hypothesis testing and practical applications, a gradual consideration of the indifference zone between the hypotheses in compliance with expert opinion or user priorities and the control of the type I error for user-defined sample size or a simultaneous control of both error types
Data Loading...
