The asymptotic distribution of the test statistic is the chi-square distribution.
In this paper, optimization of statistical distribution of the test is proposed.
Furthermore, the geometrical parameters also influenced the product distribution of the test reactions.
The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms.
We studied the distribution of the test statistics, critical values, sample size and power calculations using the exact distribution of the test statistics as well as using normal approximations.
An asymptotic distribution of the test statistic is derived and the test performance in finite samples is studied using simulation.
Then, the asymptotic distribution of the test statistic is derived under the null hypothesis and under contiguous alternatives.
In particular, a bootstrap approximation method is proposed to determine the critical values for the null distribution of the test statistic containing unknown tail index.
The asymptotic null distribution of the test statistic was also derived based on the maxima of Brownian bridge under two different conditions.
Here, we propose a computationally fast score-test-based method that estimates the distribution of the test statistic by using the saddlepoint approximation.
It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.
