The Minimum Power Distortionless Response (MPDR) beamformer processes the input signal such that the output power is minimized and the signal arriving from a look direction is undistorted. The sample spatial correlation matrix, estimated from snapshots of the received signal, plays a central role in the calculation of MPDR array weights and the estimate of the received signal power from the desired look direction. When the number of available snapshots and the number of array sensors are of similar order, the true and estimated correlation matrices might differ significantly. This paper presents analytical characterization of the MPDR-based power estimators in the snapshot scarce regime and when diagonal loading is used. In this effort, some results from the theory of large dimensional random matrices are exploited. The derived characterizations are validated via simulations.