Asymptotic law of a modified score statistic for the asymmetric power distribution with unknown location and scale parameters
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For an i.i.d. sample of observations, we study a modified score statistic that tests the goodness-of-fit of a given exponential power distribution against a family of alternatives, called the asymmetric power distribution. The family of alternatives was introduced in Komunjer (2007) and is a reparametrization of the skewed exponential power distribution from Fernández et al. (1995) and Kotz et al. (2001). The score is modified in the sense that the location and scale parameters (assumed to be unknown) are replaced by their maximum likelihood estimators. We find the asymptotic law of the modified score statistic under the null hypothesis (H_0) and under local alternatives, using the notion of contiguity. Our work generalizes and extends the findings of Desgagné & Lafaye de Micheaux (2018), where the data points were normally distributed under H_0. The special case where each data point has a Laplace distribution under H_0 is the hardest to treat and requires a recent result from Lafaye de Micheaux & Ouimet (2018) on a uniform law of large numbers for summands that blow up.
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