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There is an increasing number of goodness-of-fit tests whose test statistics measure deviations between the empirical characteristic function and an estimated characteristic function of the distribution in the null hypothesis. With the aim of overcoming certain computational difficulties with the calculation of some of these test statistics, a transformation of the data is considered. To apply such a transformation, the data are assumed to be continuous with arbitrary dimension, but we also provide a modification for discrete random vectors. Practical considerations leading to analytic formulas for the test statistics are studied, as well as theoretical properties such as the asymptotic null distribution, validity of the corresponding bootstrap approximation, and consistency of the test against fixed alternatives. Five applications are provided in order to illustrate the theory. These applications also include numerical comparison with other existing techniques for testing goodness-of-fit.

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The error matrix has been adopted as both the “de facto” and the “de jure” standard way to report on the thematic accuracy assessment of any remotely sensed data product. This perspective assumes that the error matrix can be considered as a set of values following a unique multinomial distribution. However, the assumption of the underlying statistical model falls down when true reference data are available for quality control. To overcome this problem, a new method for thematic accuracy quality control is proposed, which uses a multinomial approach for each category and is called QCCS (quality control column set). The main advantage is that it allows us to state a set of quality specifications for each class and to test if they are fulfilled. These requirements can be related to the percentage of correctness in the classification for a particular class but also to the percentage of possible misclassifications or confusions between classes. In order to test whether such specifications are achieved or not, an exact multinomial test is proposed for each category. Furthermore, if a global hypothesis test is desired, the Bonferroni correction is proposed. All these new approaches allow a more flexible way of understanding and testing thematic accuracy quality control compared with the classical methods based on the confusion matrix. For a better understanding, a practical example of an application is included for classification with four categories.

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