  Consider the spiked Wigner model \[ X = \sum_{i = 1}^k \lambda_i u_i u_i^\top
+ \sigma G, \] where $G$ is an $N \times N$ GOE random matrix, and the
eigenvalues $\lambda_i$ are all spiked, i.e. above the Baik-Ben Arous-P\'ech\'e
(BBP) threshold $\sigma$. We consider AIC-type model selection criteria of the
form \[ -2 \, (\text{maximised log-likelihood}) + \gamma \, (\text{number of
parameters}) \] for estimating the number $k$ of spikes. For $\gamma > 2$, the
above criterion is strongly consistent provided $\lambda_k > \lambda_{\gamma}$,
where $\lambda_{\gamma}$ is a threshold strictly above the BBP threshold,
whereas for $\gamma < 2$, it almost surely overestimates $k$. Although AIC
(which corresponds to $\gamma = 2$) is not strongly consistent, we show that
taking $\gamma = 2 + \delta_N$, where $\delta_N \to 0$ and $\delta_N \gg
N^{-2/3}$, results in a weakly consistent estimator of $k$. We also show that a
certain soft minimiser of AIC is strongly consistent.
