The model under consideration is a first-order linear AR model \(x_t=\rho x_{t-1}+\varepsilon_t\), \(t=1,\dots,T\), where \(\{\varepsilon_t\}\) is a sequence of independent, exponentially distributed random variables with common positive scale parameter \(\lambda>0\). The distribution theory for inference on the autoregressive parameter \(\rho\) is given and the exact distribution of the maximum likelihood estimator \(\widehat{\rho}=\min_{1\leq t \leq T}(x_t/x_{t-1})\) of the autoregressive parameter is derived. Asymptotic properties of \(\widehat{\rho}\) are investigated. Particularly, it is proved that if \(\rho<1\), then the estimator \(\widehat{\rho}\) is asymptotically exponential and \(T\)-consistent; when \(\rho=1\), the asymptotic distribution is again exponential but \(T^2\)-consistent, while in the explosive case the estimator is \(\rho^T\)-consistent. In all cases the likelihood ratio test statistics for the simple hypothesis \(\rho=\rho_0\) are asymptotically uniformly distributed.