Linear decompressors are the dominant methodology used in commercial test data compression tools. However, they are generally not able to exploit correlations in the test data, and thus the amount of compression that can be achieved with a linear decompressor is directly limited by the number of specified bits in the test data. The paper describes a scheme in which a non-linear decoder is placed between the linear decompressor and the scan chains. The nonlinear decoder uses statistical transformations that exploit correlations in the test data to reduce the number of specified bits that need to be produced by the linear decompressor. Given a test set, a procedure is presented for selecting a statistical code that effectively "compresses" the number of specified bits (note that this is a novel and different application of statistical codes from what has been studied before and requires new algorithms). Results indicate that the overall compression can be increased significantly using a small non-linear decoder produced with the procedure described in this paper.