Introduction
In this report, we will critically analyze and summarize an interesting research work (Chachi, Taheri, and Viertl) related to testing statistical hypothesis when relevant data and confidence intervals are not precise. In classical Statistical inferences, we state a Null hypothesis about a precise parameter. Then, we test the validity of the Null hypothesis using classical Statistical tests using the concept of confidence intervals at some predefined level x (say 5%) based on uncertainty we can tolerate. If the value of the testing parameter lies in confidence interval 1-x, then the Null hypothesis cannot be rejected. If testing results shows that parameter lies outside confidence interval, then Null hypothesis can be rejected.
The proposed research work suggests method to test Null hypothesis of imprecise or fuzzy parameter based on Fuzzy intervals. Consequently, we will not obtain binary decision of accepting or rejecting Null hypothesis. Instead, we will have degree of acceptance for Null hypothesis. The novelty of this research work is providing testing framework for integration of statistical hypothesis and confidence intervals in fuzzy domain.
Summary and Analysis
In statistical environment, it is not always possible to gather precise data in some experimental setups. The testing parameters may be imprecise or fuzzy. Moreover, the nature of confidence intervals may also be vague. In such cases, there must be some technique to deal with testing of fuzzy parameters in fuzzy confidence intervals. The fuzzy confidence intervals and statistical parameters for testing have been considered separately in the past. The general overview of statistical methods applied in fuzzy environment is discussed in (Taheri). The research work under analysis propels the work of (Chachi, and Taheri) to provide fuzzy testing hypothesis using the direct correspondence between testing hypothesis and confidence intervals. The authors in (Chachi, Taheri, and Viertl) suggest testing method for determination of degree of membership of fuzzy parameter in fuzzy confidence interval. This is the direct measure of the extent to which Null hypothesis could be accepted.
The proposed fuzzy testing strategy (Chachi, Taheri, and Viertl) for Null hypothesis is implemented in a series of steps mentioned below. The authors assume that the collected data is normally distributed with unknown mean and known variance.
Firstly, the original fuzzy hypothesis is transformed into a series of precise problems (hypotheses) using the range of fuzzy parameter. Then, we derive set of fuzzy confidence intervals based on accurate samples of fuzzy parameter. In the next step, we determine whether the samples of fuzzy parameter lie in the combination of confidence intervals by applying testing hypotheses. The fourth and fifth steps are very important since they involve formulation of fuzzy confidence intervals for deriving the required fuzzy test to find out the degree of membership of fuzzy variables inside these intervals. The derivation of fuzzy intervals is based on the method proposed in (Chachi, and Taheri). The final step uses the results derived in previous steps to come up with fuzzy testing hypothesis based on fuzzy random samples.
The interpretation of results is very important. They tell us the degree of conviction to accept or reject the Null hypothesis. The greater the value of outcome of testing hypothesis, the easier the decision it will be about acceptability of Null hypothesis. It is straightforward to imply that the classical testing of Null hypothesis is a special case of the proposed fuzzy testing algorithm. If we assume crisp values of testing parameter, then we will end up with binary decision whether to accept or reject the Null hypothesis.
The research results (Chachi, Taheri, and Viertl) are backed by practical examples. In first example, the fuzzy testing algorithm is applied to estimate the life of a newly manufactured tire by a company that deals in tires and rubber. The test tells whether the life of tire exceeds or falls short of that manufactured by a competitor. Another example discusses the average lifetime of front disk brake pads for a set of randomly selected 40 cars. These are the cases of imprecise samples that are required to be tested in fuzzy environment since freshly manufactured items may survive more than their estimated life or may deteriorate rapidly. Thus, the proposed fuzzy testing strategy may be used to solve many industrial problems.
The research article (Chachi, Taheri, and Viertl) is organized very well into different sections with beautifully crafted presentation skills. The authors build up their methodology from the very basics of classical statistics. The use of statistical terms and symbols is concrete and vivid. The use of practical examples clearly validates the proposed results and methodology.
Conclusion
The aim of this report was to analyze and summarize a revolutionary research idea in realm of testing hypothesis in fuzzy statistical environment (Chachi, Taheri, and Viertl). The idea is applied to the cases in which we deal with imprecise data involving fuzzy decision variables. The proposed strategy is related to building a fuzzy test function to test fuzzy parameter's membership inside fuzzy confidence interval. The test outcome will tell us the degree to which the Null hypothesis could be accepted. The greater the output, the more we will be sure about our decision about validity of Null hypothesis. This algorithm is applied to two practical problems to show the effectiveness of the ideas for solving industrial scale issues.
References
Chachi, Jalal and S. Mahmoud Taheri. "Fuzzy Confidence Intervals for Mean of Gaussian Fuzzy Random Variables." Expert Systems with Applications (2011): 5240--5244.
Chachi, Jalal, Seyed Mahmoud Taheri, and Reinhard Viertl. "Testing Statistical Hypotheses Based on Fuzzy Confidence Intervals." Austrian Journal of Statistics (2016): 267--286.
Taheri, S Mahmoud. "Trends in Fuzzy Statistics." Austrian journal of statistics (2003): 239--257.
