In this paper we study the properties of nonparametric tests for testing the null hypothesis of no change against one sided and two sided alternatives in scale parameter at unknown point. We first propose two types of nonparametric tests based on linear rank statistics and rank-like statistics, respectively. For these statistics, we drive the asymptotic distributions under the null and contiguous alternatives. The main theoretical tools used for derivation are the stochastic process representation of the test statistic and the Brownian bridge approximation. We evaluate the Pitman efficiencies of the test for the contiguous alternatives, and also compute empirical power by Monte Carlo simulation.