It has been advocated that medical imaging systems and reconstruction algorithms should be assessed and optimized by use of objective measures of image quality that quantify the performance of an observer at specific diagnostic tasks. One important source of variability that can significantly limit observer performance is variation in the objects to-be-imaged. This source of variability can be described by stochastic object models (SOMs). A SOM is a generative model that can be employed to establish an ensemble of to-be-imaged objects with prescribed statistical properties. In order to accurately model variations in anatomical structures and object textures, it is desirable to establish SOMs from experimental imaging measurements acquired by use of a well-characterized imaging system. Deep generative neural networks, such as generative adversarial networks (GANs) hold great potential for this task. However, conventional GANs are typically trained by use of reconstructed images that are influenced by the effects of measurement noise and the reconstruction process. To circumvent this, an AmbientGAN has been proposed that augments a GAN with a measurement operator. However, the original AmbientGAN could not immediately benefit from modern training procedures, such as progressive growing, which limited its ability to be applied to realistically sized medical image data. To circumvent this, in this work, a new Progressive Growing AmbientGAN (ProAmGAN) strategy is developed for establishing SOMs from medical imaging measurements. Stylized numerical studies corresponding to common medical imaging modalities are conducted to demonstrate and validate the proposed method for establishing SOMs.