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Under certain conditions, AIC and LOOCV (leave-one-out cross validation) are asymptotically equivalent (Stone, 1977). Stone's paper is less than 4 pages long, but quite mathy, so I turn here for some assistance. I presume the equivalence holds when the loss function in LOOCV is exactly the same or somehow compatible with the loss function implied by the likelihood as used in the AIC.

Questions

  1. What happens if the loss function employed in LOOCV does not exactly correspond to the loss function implied by the likelihood in AIC?
    For example, say that the loss function in LOOCV is some form of tick function (quantile loss) while the likelihood is normal?
  2. Under what conditions, at least roughly, would the asymptotic equivalence hold?
    Simple, concrete examples as well as rigorous explications are welcome.

References

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