We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of
the squared distance between the empirical characteristic function of the standardized data and the
characteristic function of the standard symmetric stable distribution with the characteristic exponent
R estimated from the data. We treat R as an unknown parameter, but for theoretical simplicity we
also consider the case that R is fixed. For estimation of parameters and the standardization of data
we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator
(EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the
characteristic function process with parameters estimated by MLE and EISE. For the case of MLE,
the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the
test statistic is obtained using complex integration. Simulation studies show that the asymptotic dis
tribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance
function of the characteristic function process with parameters estimated by an efficient estimator for
general distributions.
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