Residual errors in shear measurements, after corrections for instrument systematics and atmospheric effects, can impact cosmological parameters derived from weak lensing observations. Here we combine convergence maps from our suite of ray-tracing simulations with random realizations of spurious shear. This allows us to quantify the errors and biases of the triplet $(Omega_m,w,sigma_8)$ derived from the power spectrum (PS), as well as from three different sets of non-Gaussian statistics of the lensing convergence field: Minkowski functionals (MF), low--order moments (LM), and peak counts (PK). Our main results are: (i) We find an order of magnitude smaller biases from the PS than in previous work. (ii) The PS and LM yield biases much smaller than the morphological statistics (MF, PK). (iii) For strictly Gaussian spurious shear with integrated amplitude as low as its current estimate of $sigma^2_{sys}approx 10^{-7}$, biases from the PS and LM would be unimportant even for a survey with the statistical power of LSST. However, we find that for surveys larger than $approx 100$ deg$^2$, non-Gaussianity in the noise (not included in our analysis) will likely be important and must be quantified to assess the biases. (iv) The morphological statistics (MF,PK) introduce important biases even for Gaussian noise, which must be corrected in large surveys. The biases are in different directions in $(Omega_m,w,sigma_8)$ parameter space, allowing self-calibration by combining multiple statistics. Our results warrant follow-up studies with more extensive lensing simulations and more accurate spurious shear estimates.
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