Chi-square test for continuous distributions with location and scale parameters

The problem of testing the hypothesis that the distribution function of independent equally distributed random variables is $G[(x-\theta_1)/\theta_2]$ is considered; $\theta_1$ and $\theta_2$ being unknown parameters. A statistic which is a modification of Pearson's $\chi^2$ is proposed whose limit distribution is chi-square with $(k-1)$ degrees of freedom, $k$ being the number of cells (it means that the number of degrees of freedom does not depend on the number of unknown parameters). In the statistic the maximum likelihood estimations of $\theta_1$ and $\theta_2$ based on the original observations are used. A similar result is obtained for the quantile test.