## 3 Results  
Given the findings contained in Tables 1-4, which show that the multiple \(d\) is stable across sample sizes and also across scenarios, as well as within each scenario, based on the underlying data that show the multiple moves within one-tenth to the left or right of its average, it can be selected to be used in the \(d\ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.063 & 0.056; 0.069 \\
40 & 0.059 & 0.053; 0.066 \\
50 & 0.058 & 0.051; 0.064 \\
60 & 0.056 & 0.050; 0.062 \\
70 & 0.055 & 0.049; 0.061 \\
80 & 0.050 & 0.049; 0.061 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Type I error probability alpha for the test (lower \(d=-1.96\), upper \(d=1.96\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 2) and percentiles for alpha. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.063 & 0.056; 0.069 \\
40 & 0.060 & 0.053; 0.066 \\
50 & 0.058 & 0.051; 0.064 \\
60 & 0.056 & 0.050; 0.063 \\
70 & 0.055 & 0.049; 0.062 \\
80 & 0.055 & 0.048; 0.061 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Type I error probability alpha for the test (lower \(d=-1.96\), upper \(d=1.96\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 3) and percentiles for alpha. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.063 & 0.056; 0.069 \\
40 & 0.060 & 0.053; 0.066 \\
50 & 0.058 & 0.051; 0.064 \\
60 & 0.056 & 0.050; 0.063 \\
70 & 0.055 & 0.049; 0.062 \\
80 & 0.055 & 0.049; 0.062 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Type I error probability alpha for the test (lower \(d=-1.92\) upper \(d=2.00\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 4), and percentiles for alpha. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Low \(d\)** & **Average Upper \(d\)** \\ \hline
30 & \(-1.95\) & 1.97 \\
40 & \(-1.95\) & 1.97 \\
50 & \(-1.95\) & 1.97 \\
60 & \(-1.95\) & 1.97 \\
70 & \(-1.95\) & 1.97 \\
80 & \(-1.95\) & 1.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average \(d\)s leading to the 2.5%, 97.5% percentiles (\(k_{1},k_{2},\theta_{1}\) from scenario 6). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.069 & 0.059; 0.090 \\
40 & 0.064 & 0.055; 0.081 \\
50 & 0.061 & 0.053; 0.076 \\
60 & 0.060 & 0.052; 0.073 \\
70 & 0.058 & 0.051; 0.070 \\
80 & 0.058 & 0.050; 0.068 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Type I error probability alpha for the test (lower \(d=-1.92\) upper \(d=2.00\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 4), and percentiles for alpha. It can be seen from Table 5 that for any considered sample size and almost (95%) any within-scenario 1 set-up of \(k_{1},k_{2},\theta_{1}\), the error probability is between the reasonable levels of 0.049 and 0.07. The average alpha is also not far from 0.05. Similar results follow. The tables clearly suggest that in a majority of the analyzed situations (in 95% of them), the testing procedure returns a probability of type I error between approximately 0.05 and 0.09. In light of these probabilities, the method is quite usable. The average alphas also circle almost always around 0.06 regardless of the scenario and sample size and regardless of the specific within-scenario parameter combination. Another set of results in Table 11 suggests which data sample sizes might be sought in order for the procedure to have a reasonable power of at least 0.7. Nevertheless, the table shows the average test power given the sample size, scenario, and constant \(c\) in \(H_{1}:k_{2}\theta_{2}=k_{1}\theta_{1}+c\), the constant being fixed within a scenario. The power is averaged across different within-scenario \(k_{1},k_{2},\theta_{1}\) set-ups.