Pedagogy of chi‐square goodness of fit test for continuous distributions

Abstract

Chi‐square goodness of fit testing to examine whether or not it is reasonable to assume that a random sample of the data comes from a specific probability density was one of the topics covered in an undergraduate engineering probability course. In the absence of details on this topic in engineering probability books, a Matlab® demo was created to facilitate the link between theory and practice. The step‐by‐step procedure to determine the closest fit among a number of continuous densities has been demonstrated involving binning (fixed width and fixed population), parameter estimation, and computation of the test statistic, degrees of freedom and the P values. The cautionary aspects of the test regarding the variability in test results have been illustrated by choosing a smaller size data through permutation. The pedagogical aspects of procedure demonstrated suggest that it may be used to fill the gaps in textbooks devoted to probability and statistics.

Pedagogy of diversity and data analytics: Theory to practice

“…Prior work by author had demonstrated the pedagogy of receiver operating characteristic curves and the use of the area under the ROC curve in machine vision applications . Author had also shown the use of parametric hypothesis testing, specifically χ 2 testing of goodness fit, in the context of machine vision applications . The Matlab based module on diversity relied on these two demos as the building blocks.…”

“…As stated earlier, students had completed assignments on ROC analysis , χ 2 tests followed by the simple assignment on diversity described earlier. These reported studies offered opportunities to demonstrate the use of signal processing algorithms to improve the metrics of the receiver, namely increasing the area under the ROC curve, positive predictive values, and the performance index and decreasing the error rates . The data used in the demo consisted of 70 samples of target absent and 60 samples of target present .…”

“…The first step was to undertake goodness of fit testing to determine the closest fit to each of these two sets. For the examples done in class, the densities tested were gamma, Rayleigh, Nakagami, Weibull, and Rician and in each case, target absent and target present, the null hypothesis associated with the density having the lowest value of the χ 2 test statistic (being smaller than the critical value), is not rejected and used as the closest fit to the data .…”

“…among others alongside the traditional topics listed earlier. Some of the preliminary work in this regard was reported earlier . A natural progression of these efforts is an examination of methods to improve the performance of machine vision or medical diagnostics system using diversity techniques.…”

“…By week 5, students had completed individual assignments covering confusion matrix, transition matrix, ROC curves, area under the ROC curve (AUC) and performance indices such as error rates and positive predictive values, both based on the Neyman‐Person optimal threshold . For the next homework, every student was provided with a data set of 100 samples and the assignment consisted of goodness of fit test ( χ 2 ) to determine the closest density fit among a number of densities namely Rayleigh, Rician, Nakagami, gamma and Weibull . Using the concept of multiple hypotheses testing covered in class, students were required to obtain the closest fit based on the lowest value of the χ 2 test statistic (less than the critical value).…”