The assumption of strict stationarity is pivotal when a double autoregressive (DAR) model is used. Yet, tractable tools are unavailable for testing strict stationarity in the DAR framework. In this article, we attempt to provide a procedure for this test. We formulate such testing problem as testing if a top Lyapunov exponent is negative or not and introduce the t-type tests. To achieve this goal, a consistent estimator of the associate top Lyapunov exponent without strict stationarity assumption is presented and a random weighting approach is suggested for variance estimation. It is shown that such ttype tests are consistent and powerful, and the resampling method is very useful in capturing the sampling variability of the variance. We also propose a global robust quasi-maximum likelihood estimation(QMLE) for parameters of interest, which weakens key assumptions on the commonly used QMLE. All estimators are shown to be consistent and asymptotically normal in both stationary and explosive situations. Their asymptotic variances can be consistently estimated via random weighting approach in a unified framework. .