A new goodness of fit test in the presence of uncertain parameters
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ORIGINAL ARTICLE
A new goodness of fit test in the presence of uncertain parameters Muhammad Aslam1 Received: 18 July 2020 / Accepted: 29 September 2020 © The Author(s) 2020
Abstract The Weibull distribution has been widely used in the areas of quality and reliability. The Anderson–Darling test has been popularly used either the data in hand follow the Weibull distribution or not. The existing Anderson–Darling test under classical statistics is applied when all the observations in quality and reliability work are determined, précised, and exact. In the areas of reliability and quality, the data may indeterminate, in-interval and fuzzy. In this case, the existing Anderson–Darling test cannot be applied for testing the assumption of the Weibull distribution. In this paper, we present the Anderson–Darling test under neutrosophic statistics. We present the methodology to fit the neutrosophic Weibull distribution on the data. We discuss the testing procedure with the help of reliability data. We present the comparisons of the proposed test with the existing Anderson–Darling the goodness of fit test under classical statistics. From the comparison, it is concluded that the proposed test is more informative than the existing Anderson–Darling test under an indeterminate environment. In addition, the proposed test gives information about the measure of indeterminacy. Keywords Neutrosophy · Neutrosophic numbers · Reliability · Weibull distribution · Classical statistics
Introduction The derivation of statistical methods is based on the assumption that a random variable or the data follow some specific distribution. According to Romeu [1] “when we assume that our data follow a specific distribution, we take a serious risk. If our assumption is wrong, then the results obtained may invalid”. For example, before testing a hypothesis, the suitable test statistic is chosen according to the nature of the data in hand. The tests based on normal distribution are chosen when the assumption of the normality is met; otherwise, the non-parametric tests are applied for testing the hypothesis. Two approaches have been widely used to checking the assumption of any distribution. An approach in which the assumption of the data is checked using the graphical properties is called the empirical procedure. Another approach which provides the more formal, a quantifiable, and reliable result is called the goodness of fit test. The goodness of fit tests is based on the cumulative distribution function (cdf) or the probability density function (pdf) of the underlying distribution. Arshad et al. [2] applied the Anderson–Darling
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Muhammad Aslam [email protected] Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia
test for testing the assumption of generalized Pareto distribution. Marsaglia and Marsaglia [3] and Razali and Wah [4] presented a study of the performance evaluation of this test. Jäntschi and Bolboac˘a [5] worked on the computational probabilities of Anderson–Darling test. Formen
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