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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.

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