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T. Holynski:
"Tests For Normality Based On Approximated Probability-Weighted Empirical Transforms";
Talk: SIMSTAT 2019 - 10th International Workshop on Simulation and Statistics, Universität Salzburg (invited); 09-02-2019 - 09-06-2019; in: "10th International Workshop on Simulation and Statistics - Workshop booklet", (2019), 38 - 39.

Goodness-of-fit tests based on transforms, such as characteristic functions and Laplace transforms, are regularly developed and improved over recent years. In particular, a new 38 research path is paved by the notion of the probability-weighted characteristic function (PWCF) introduced by S. Meintanis and his collaborators (2014, 2016). In a classical transform-based test, the test statistic has the form of L-2 type functional that measures the discrepancy between the model- and the empirical transform. To maximize the power
against specific alternatives, its integrand is equipped with a function downweighting the discrepancy in appropriate regions of the transform domain. The motivation behind the PWCF is to alleviate the problem of the optimal choice of that weight function by introducing a data-driven weight already ´within´ the transform. Unfortunately, the expressions
for parametric PWCFs of the standard densities are not available in closed forms. Hence, the wide-spread use of the method may be inhibited by necessity of double numerical integration: to obtain the values of the transform and those of the test statistic. In this
study, we focus on testing for normality in the proposed way. To enlarge our toolkit, we introduce also the probability weighted Laplace transform (PWLT) and discuss its properties. We stress that while the standard bilateral Laplace transform of the normal
density is unbounded, the corresponding PWLT is bounded; this is important for many procedures in which the always bounded characteristic function has been favored. Next, we show that in the normal case both the probability-weighted transforms can be approximated so that the test statistics are easier to compute. With this advantage, large-scale simulations are conducted for power assessments and comparisons that are missing in literature. As the distributions of the test statistics are hard to derive analytically, to estimate the critical points the parametric bootstrap is used.

http://datascience.sbg.ac.at/SimStatSalzburg2019/SimStat2019BoA.pdf