State-of-the-art neural language models represented by Transformers are becoming increasingly complex and expensive for practical applications. Low-bit deep neural network quantization techniques provides a powerful solution to dramatically reduce their model size. Current low-bit quantization methods are based on uniform precision and fail to account for the varying performance sensitivity at different parts of the system to quantization errors. To this end, novel mixed precision DNN quantization methods are proposed in this paper. The optimal local precision settings are automatically learned using two techniques. The first is based on a quantization sensitivity metric in the form of Hessian trace weighted quantization perturbation. The second is based on mixed precision Transformer architecture search. Alternating direction methods of multipliers (ADMM) are used to efficiently train mixed precision quantized DNN systems. Experiments conducted on Penn Treebank (PTB) and a Switchboard corpus trained LF-MMI TDNN system suggest the proposed mixed precision Transformer quantization techniques achieved model size compression ratios of up to 16 times over the full precision baseline with no recognition performance degradation. When being used to compress a larger full precision Transformer LM with more layers, overall word error rate (WER) reductions up to 1.7% absolute (18% relative) were obtained.