## 2 Materials and Methods
At this stage, the averaging, which by its very nature always hides a large amount of information, is not an imprecision in the whole procedure, as this stage of analysis aims primarily to suggest which multiple \(d\) could be used in the testing. Its suitability is only scrutinized later. Once the average \(ds\) are identified, the focus can be shifted to the formula  
\[d\cdot\widehat{\varpi}\widehat{r}^{1/2}, \tag{13}\]  
where  
\[H_{0}:k_{1}\theta_{1}=k_{2}\theta_{2}. \tag{5}\]0  
and  
\[H_{0}:k_{1}\theta_{1}=k_{2}\theta_{2}. \tag{5}\]1  
The term \((n+1)^{-1}\) is used instead of the usual \((n-1)^{-1}\) since it gives the sample variance that optimizes (minimizes) the general MSE criterion [16] of estimate quality. As outlined, the property of interest will then be the frequency of cases when the testing procedure, based on the percentiles calculated from (13)-(15), incorrectly rejects \(H_{\rm o}\). This type I error frequency depends again on the specific set-up of \(k_{1},k_{2},\theta_{1}\), as well as the selected sample size. Hence again, within the given scenario and sample size, the average type I error probability is identified but this time also together with the 2.5% and 97.5% percentiles of the observed \(k_{1},k_{2},\theta_{1}\)-dependent type I error probabilities so that the spread of the probabilities is seen, and the scenario-based aggregation does not mask the suitability of the test when it comes to its type I error. Each type I error is identified through thousands of employed tests. At the final stage, the power of the test using the estimated percentiles is also the subject of interest. The six scenarios and the sample sizes are used again, but now, the null hypothesis does not hold, which is conveyed by introducing a constant to differentiate \(E(X)\) from \(E(Y)\): \(k_{2}\theta_{2}=k_{1}\theta_{1}+\) const. Since the test power differs within a scenario due to different parameter combinations, given the sample size and constant, the average percentile and sample size power is considered. For each scenario-sample size combination, a specific and fixed constant is used. More interestingly, however, the method is also compared to another methodology regarding its test power. This is performed for specific parameter set-ups and sample sizes, and no averaging is involved. If the method proves to be viable, the resulting algorithm can be described as follows:  
1. Calculate the sample averages and variances for the two obtained random samples where the sample variances are evaluated according to (15). 2. Calculate \(\hat{k}_{1},\hat{k}_{2},\hat{\theta}_{1}\) estimates: For instance, by estimating \(E(X)=k\theta\) with \(\overline{x}\) and \(var(X)=k\theta^{2}\) with the sample variance \(s^{2}\), the common estimates \(\hat{\theta}=s^{2}/\overline{x}\) and \(\hat{k}=\overline{x}^{2}/s^{2}\) are obtained for both samples. This helps detect which scenario is worked with. 3. Evaluate the expression \(\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\widehat{ \varpi}=n^{-1}(\widehat{\varpi}\widehat{\varpi}(X)+\widehat{\varpi}\widehat{r }(Y))\) using the two MSE-minimizing sample variances from step 1. \(n\) is the size of any of the two equally large samples; the \(d\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\) is calculated twice with two generally different multiples \(d\) and the same \(\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\widehat{\varpi}\) so that two percentile estimates \(\hat{p}_{2.5}\) and \(\hat{p}_{97.5}\) are obtained. The multiples \(d\) are selected from the tables below according to the \(k_{1},k_{2},\theta_{1}\) scenario the practitioner is in, as suggested by the \(\hat{k}_{1},\hat{k}_{2},\hat{\theta}_{1}\) estimates. 4. Perform the test: If \(\hat{p}_{2.5}<\overline{x}-\overline{y}<\hat{p}_{97.5}\), the hypothesis of equal means is accepted; otherwise, it is rejected. The next section shows the results. Where necessary, a comment is attached, although the contents are self-explanatory through the table headlines.