Estimating largest eigenvalue of $N_{d\gg 1}(0,\Sigma)$ from small data I am trying to estimate the largest eigenvalue of some $d$-dimensional normal distribution $N_d(0,\Sigma)$ from the sample data
$$X_1, \ldots, X_N \sim_{iid} N(0,\Sigma)$$
where $N$ is much smaller than $d$, say $N = 20$ and $d=2000$. (Recall that PCA works nicely for $N$ much larger than $d$, but may not work in my case [1].) There's no assumption on the real-valued covariance matrix $\Sigma$. However, by performing PCA anyways, our observation shows that the eigenvalues seem to decay log-linearly.
Question How do we estimate the largest eigenvalue?
Pointers to any analysis are appreciated too, for example, an analysis that quantifies how bad the estimation is by running PCA on this setting anyways.
Footnote

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*[1] However, a quick experiment in python actually suggests that this approach is stable enough! I still need a formal analysis though.


There is a history of answers to this kind of question following Horn's 1965 article on parallel analysis in principal component/s analysis (which has a confusing history of being called factor analysis, which Horn did). At any rate, in the 1960s Horn was interested in accounting for "sampling error and least-squares bias" in PCAs of the correlation matrix $\mathbf{R}$. Horn's solution was to make a quantitative, not analytic, estimation through PCA on $k$ matrices of random data of size $N\times d$, and to average the resulting $1^{\text{st}}$, $2^{\text{nd}}$, … $d^{\text{th}}$ eigenvalues (which he then subtracted from the eigenvalues of the PCA of the observed $\mathbf{R}$). This should also work for $\mathbf{\Sigma}$.
Because generating (pseudo) random numbers and because eigen-decomposition were both computationally expensive until about the mid 90s (and perhaps later, depending on the size of $N \times d$), there's a literature until about the turn of the millennium on estimating these average eigenvalues from $1$ to $d$ given $N$ and $d$ using computationally cheap formulas, while still using his parallel analysis approach to component retention decisions (Humphries and Montanelli, 1976; Allen and Hubbard, 1986; Lautenschlager, 1989; Longman, et al., 1989; Keeling, 2000). These methods were generally considered to be pretty lousy for the purposes of actual parallel analysis (e.g., see Zwick and Velicer, 1986; Hayton, et al., 2004), however, if, for your purposes, $\mathbf{\Sigma}$ is reasonably close to $\mathbf{R}$, you could probably use one of them.
There's also been a bit of a literature (Hayton, et al., 2004; Peres-eto, et al., 2005) about resampling/rerandomizing to generate the random data for parallel analysis, but this is unnecessary, since eigen-decomposition is not of the data per se, but of $\mathbf{R}$, and is insensitive to the distributional form of the data (Dinno, 2009).



References
Allen, S. J., & Hubbard, R. (1986). Regression equations for the latent roots of random data correlation matrices with unities on the diagonal. Multivariate Behavioral Research, 21(3), 393–396.
Dinno, A. (2009). Exploring the Sensitivity of Horn’s Parallel Analysis to the Distributional Form of Simulated Data. Multivariate Behavioral Research, 44(3), 362–388.
Hayton, J. C., Allen, D. G., & Scarpello, V. (2004). Factor Retention Decisions in Exploratory Factor Analysis: A Tutorial on Parallel Analysis. Organizational Research Methods, 7(2), 191–205.
Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30(2), 179–185.
Humphreys, L. G., & Montanelli, R. G. (1976). Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A Monte Carlo study. Psychometrika, 41(3), 341–348.
Keeling, K. B. (2000). A Regression Equation for Determining the Dimensionality of Data. Multivariate Behavioral Research, 35(4), 457–468.
Lautenschlager, G. J. (1989). A comparison of alternatives to conducting Monte Carlo analyses for determining parallel analysis criteria. Multivariate Behavioral Research, 24(3), 365–395.
Longman, R. S., Cota, A. A., Holden, R. R., & Fekken, G. C. (1989). A regression equation for the parallel analysis criterion in principal components analysis: Mean and 95th percentile eigenvalues. Multivariate Behavioral Research, 24(1), 59–69.
Montanelli, R. G., & Humphreys, L. G. (1976). Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A Monte Carlo study. Psychometrika, 41(3), 341–348.
Peres-Neto, P. R., Jackson, D. A., & Somers, K. M. (2005). How many principal components? Stopping rules for determining the number of non-trivial axes revisited. Computational Statistics & Data Analysis, 49(4), 974–997.
Zwick, W. R., & Velicer, W. F. (1986). A comparison of five rules for determining the number of factors to retain. Psychological Bulletin, 99(3), 432–442.
