  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
$\alpha$ estimated from the data. We treat $\alpha$ as an unknown parameter,
but for theoretical simplicity we also consider the case that $\alpha$ 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 distribution 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.
