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Tests for location-scale families based on the empirical characteristic function. (English) Zbl 1080.62010
Summary: A class of procedures is presented for using random samples to test the fit of location-scale families distributions \(F(\cdot;\theta_1,\theta_2)\) such that \(Z = (X - \theta_1)/\theta_2\) has distribution \(F(\cdot;0,1)\) for all \(\theta_1 \in \operatorname{Re}\), \(\theta_2 > 0\). Working with empirically standardized data, \(\{\widehat Z_j = (X_j-{\widehat\theta}_1)/ {\widehat\theta}_2\}_{j=1}^n\), the test statistic is a measure of \(\mathcal L_2\) distance between the empirical characteristic function, \({\widehat\phi}_n(t) = n^{-1} \sum_{j=1}^n \exp(it{\widehat Z}_j)\), \(t \in \operatorname{Re}\), and the c.f. of \(Z\) under the null hypothesis, \(\phi_{0}(t)\). The closed-form test statistic is derived by integrating over \(\operatorname{Re}\) the product of a weight function times \(| {\widehat\phi}_n - \phi_0|^2 \). Using as weight function for each location-scale family the squared modulus of \(\phi_{0}\) itself presents a unified test procedure. Included as special cases are well-known tests for normal and Cauchy laws. Small-sample powers are compared with those of Anderson-Darling tests for each of seven univariate location-scale families.

62F03 Parametric hypothesis testing
62G30 Order statistics; empirical distribution functions
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