Penalized power approach to compare the power of the tests when Type I error probabilities are different.

Many researchers have worked to improve the tests. They needed to demonstrate that their improved tests performed better than alternative tests. Furthermore, to obtain the suitable test is another problem among several alternatives under any specific condition. In such cases, Monte-Carlo simulation studies are used to compare the performance of tests in terms of power and Type I error probability. However, any comparison of the powers is invalid when Type I error probabilities are different. Although Zhang and Boos (1994) and Lloyd (2005a) proposed solutions to overcome this problem, their proposals have some deficiencies. Zhang and Boos's adjusted power could not be used for tests when they have same test statistics and Lloyd's intrinsic power is not useful when Type I error probability is lower than the nominal level. In this article, a new criterion is introduced, called penalized power, to compare the power of tests when Type I error probabilities are different. A Monte-Carlo simulation study is conducted to show the efficiency of penalized power over the alternatives. [ABSTRACT FROM AUTHOR]