  Quantum-enhanced parameter estimation has widespread applications in many
fields. An important issue is to protect the estimation precision against the
noise-induced decoherence. Here we develop a general theoretical framework for
improving the precision for estimating an arbitrary parameter by monitoring the
noise-induced quantum trajectorie (MQT) and establish its connections to the
purification-based approach to quantum parameter estimation. MQT can be
achieved in two ways: (i) Any quantum trajectories can be monitored by directly
monitoring the environment, which is experimentally challenging for realistic
noises; (ii) Certain quantum trajectories can also be monitored by frequently
measuring the quantum probe alone via ancilla-assisted encoding and error
detection. This establishes an interesting connection between MQT and the full
quantum error correction protocol. Application of MQT to estimate the level
splitting and decoherence rate of a spin-1/2 under typical decoherence channels
demonstrate that it can avoid the long-time exponential loss of the estimation
precision and, in special cases, recover the Heisenberg scaling.
