In the present article, we have studied the estimation of the reciprocal of scale parameter , that is, hazard rate of a two parameter exponential distribution based on a doubly censored sample. This estimation problem has been investigated under a general class of bowl-shaped scale invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of scale equivariant estimator is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for three special loss functions. A simulation study is conducted to compare the risk performance of the proposed estimators. Finally, we analyze a real data set.