Distortion risk measures are used intensively in finance and insurance applications. We present a new method to construct new distortion functions. The method is based on the transformation of an existing non-negative random variable whose distribution function, named the generating distribution, may contain more than one parameter. The coherency of the resulting risk measure is ensured by restricting the parameter space on which the distortion function is concave. We study the cases when the generating distributions are exponentiated exponential and Gompertz distributions. Closed-form expressions for risk measures are derived for uniform, exponential, and Lomax losses. Numerical and graphical results are presented to demonstrate the effects of parameter values on the risk measures. We then propose a simple estimator of the risk measure and conduct simulation studies to compare and demonstrate the performance of the proposed estimator.