Several new pivotal quantities are proposed to do the statistical inferences for the shape parameter for a new two-parameters lifetime distribution with bathtub shape or increasing failure rate function and the lognormal distribution under type ii progressive censoring and general type ii progressive censoring in this paper. The simulation results show that the new pivotal quantities are superior to those pivotal quantities proposed in the master thesis of Cheng-Kang Hsieh (2005). For these new pivotal quantities, the optimal one is chosen based on the minimum confidence length for confidence interval or the highest power for hypothesis testing. At last, two numerical examples are given to demonstrate all proposed methods in this paper.