The fuzzy rating method has been introduced in psychometric studies as a tool, which allows the capture of and accurate reflection of the diversity, subjectivity, and imprecision inherent in human responses to many questionnaires. The lack of statistical techniques for in-depth analysis of these responses has been, for years, the appearance of an important barrier. At present, this barrier is being overcome thanks to new statistical techniques. In this way, the information from fuzzy rating method-based responses can be suitably explored and exploited. This paper aims to formally endorse some of the main statistical benefits of using free-response format fuzzy rating scale-based questionnaires instead of using the closed-response format involving fuzzy linguistic representations.Index Terms-Fuzzy linguistic representation, fuzzy numbers, fuzzy rating method, questionnaires, random fuzzy sets, statistical analysis of fuzzy data.
This work explores the mathematical properties of a distribution introduced by Basu & Jones (2004), and applies it to model the stellar initial mass function (IMF). The distribution arises simply from an initial lognormal distribution, requiring that each object in it subsequently undergoes exponential growth but with an exponential distribution of growth lifetimes. This leads to a modified lognormal with a power-law tail (MLP) distribution, which can in fact be applied to a wide range of fields where distributions are observed to have a lognormal-like body and a power-law tail. We derive important properties of the MLP distribution, like the cumulative distribution, the mean, variance, arbitrary raw moments, and a random number generator. These analytic properties of the distribution can be used to facilitate application to modeling the IMF. We demonstrate how the MLP function provides an excellent fit to the IMF compiled by Chabrier (2005) and how this fit can be used to quickly identify quantities like the mean, median, and mode, as well as number and mass fractions in different mass intervals.
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