Integral-type tests for goodness-of-fit

A family of integral-type goodness-of-fit tests is investigated. This family includes some existing tests, such as the Cram,r-von Mises test and Anderson-Darling test, etc. The asymptotic distributions of the tests in the family under the null and local alternative hypotheses are established. The almost sure convergence under a fixed underlying distribution is obtained. Furthermore, simulations are conducted to compare the powers of the tests in the family. Simulation results show that for different alternatives, the more powerful tests are different, and the parameter lambda has great influence on the tests in small sample cases.