THE POWER Of THE CONTINUOUS DISTRIBUTIONS

We study the power of the Cauchy and Logistics distributions and derive the formula of the power for both distributions (at least) in two steps: (1) create sufficient statistics and rejection region in getting the bound of the area, and (2) use of the rejection area or the definition of the power. In this concept, we must define the hypothesis testing of the parameter (shape or scale) distribution. Furthermore, R code is used to compute the values of the power and to plot the curves. The results show that Cauchy distribution depends on the bound of rejection area and parameters (location and scale). The size increases as the scale parameter increases, but not for rejection area. The size changes as the parameter scale changes, and the size is constant and less than 0.05 (close to level of significance 0.05). We thus accept this size as the minimum size (close to 0.05). In the context of Logistics distribution, the result shows that the power of Logistics distribution increases as the rejection area increases, and the highest curve occurs when this area is 10. Generally, the size is constant and it does not significantly change the shape on several rejection areas. We note here that the highest size is around 1.0 when the rejection area is 10. This size is not reasonable being far away from 0.05. Therefore, we must choose the small size (less than 0.05) with the target to choose the maximum power and minimum size.