The statistical structure of neural population responses is a topic of intense interest in systems neuroscience. The signal eigenspectrum (eigenvalues ordered from high to low) is a fundamental summary statistic that characterizes the dimensionality and efficiency or redundancy of population tuning curves. However, the problem of characterizing signal eigenvalue spectra from limited, noisy neural response data is complicated by two factors. First, noise in the neural responses (aka ``noise correlations'') can interfere with the estimation of the eigenvalues of the signal distribution. Second, data limitations lead to large discrepancies between the true eigenvalues and those of the sample covariance, a limitation that is especially pronounced in large neural populations where dimensionality is high relative to the number of samples. Here we introduce a new method for estimating eigenspectra that overcomes these limitations. Our estimator extends recent results from random matrix theory on the unbiased estimation of the moments of the eigenvalue distribution. We show how these moments can be used to infer parametric models, including power laws, of the eigenvalue distribution. We compare our method to cvPCA, a recently proposed estimator that was used to argue that population encoding in mouse visual cortex follows a power law with slope near 1 [Stringer et al, 2019] optimally balancing the efficiency of independent tuning and the robustness of redundant tuning. However, we show that cvPCA can exhibit sizeable biases, which our estimator substantially reduces. We then apply our estimator to the publicly available mouse dataset from Stringer et al, and show that it returns a higher estimate of power law slope, suggesting a more rapid fall-off in tuning with dimension. Our work provides a principled and accurate method for inferring a fundamental characteristic of neural population tuning that we expect to be of wide theoretical and experimental interest.