Model selection criteria that represent a compromise between AIC and BIC I am very familiar with the ideas and formula of the two popular model selection criteria AIC/AICc and BIC. When I use them for practical problems in chemometrics, the use of AIC/AICc often gives models with too many undesirable variables selected, but BIC seems to choose two few variables. My question is whether there are any other criteria that are calculable as EASILY as AIC and BIC, but represent a middle ground between the two in terms of penalizing model complexity.  I know there are a wide range of other information criteria (e.g., DIC, GIC, and HIC). But what I look for is one that is almost as simple as AIC or BIC. For example, is there a AIC/BIC-like criterion that uses a different factor for the penalty term?

There is the Hannan–Quinn information criterion (HQIC):
$$
\text{HQ}=-2L_{\text{max}}+2k\ln(\ln(n))
$$
where $L_{\text{max}}$ is the maximized log-likelihood, $k$ is the number of parameters, and $n$ is the number of observations (the sample size). The penalty is smaller than BIC's $k\ln(n)$ but larger than AIC's $2k$. HQIC is consistent for model selection like BIC, and its penalty is the minimal possible penalty to achieve consistency. It is not efficient like AIC but not very far from that since $\ln(\ln(n))$ is not that much bigger than 1 (1.5 for $n=100$, 1.9 for $n=1000$, 2.2 for $n=10000$). According to Wikipedia

HQC is asymptotically very well-behaved. Van der Pas and Grünwald prove that model selection based on a modified Bayesian estimator, the so-called switch distribution, in many cases behaves asymptotically like HQC, while retaining the advantages of Bayesian methods such as the use of priors etc.

And obviously, HQIC is just as easy to calculate as AIC or BIC.
