# Example and counterexample for Stone's (1977) assumption

Stone (1977) considers the problem of the choice of predicting density for $$y$$ given $$x$$ from a prescribed class of formal predicting densities $$\{f(y|x,\alpha,S), \alpha \in \mathscr{A}\}$$ whose members are indexed by the choice parameter $$\alpha$$. He shows that AIC and LOOCV (leave-one-out cross validation) are asymptotically equivalent provided that the following assumption holds:

The conditional distribution of $$y$$ given $$x$$ in the distribution $$P$$ is $$f(y|x,\theta^*)$$ for some unique $$\theta^* \in \Theta$$, that is, the conventional model $$\{f(y|x,\theta),\theta \in \Theta)\}$$ is true.

I am having a hard time understanding this formal requirement and using it in applications.

Could anyone illustrate when this assumption holds vs. when it fails by an example and a counterexample?

References

Simply said, if your data come from a $$\text{Pois}(\lambda^\ast)$$ distribution, and your hat contains all possible Poisson distributions, then you are good. But if your hat only contains Poisson distributions with parameters that differ by at least $$\epsilon>0$$ from $$\lambda^\ast$$, then of course you will not get there. (If your hat contains everything except the one true value, $$\mathbb{R}\setminus\{\lambda^\ast\}$$, then I presume that the asymptotic result still holds, because we can get arbitrarily close to $$\lambda^\ast$$. But that is not a counterexample to the statement.)
The other part of the statement is uniqueness: there must be only a single parameter $$\theta^\ast$$ in the hat that gives the true distribution $$f(y|x,\theta)$$, so we can asymptotically converge to it. The counterexample to uniqueness would be a hat containing a slice of the real plane $$\{(\theta,\tau)\in\mathbb{R}^2|0<\theta<\tau\}$$, where a parameter vector $$(\theta,\tau)$$ parameterizes a $$\text{Pois}(\tau-\theta)$$ distribution. Then of course a single Poisson will be parameterized by many different pairs $$(\theta,\tau)$$... but they will be indistinguishable, since they all yield the same PMF.
I honestly don't see (yet) how this is important. I would assume that non-uniqueness of $$\theta^\ast$$ would simply mean that we would approach the solution space arbitrarily closely but might oscillate wildly within it - but this would, again, not make any difference in terms of what we can actually observe, just make the mathematics more opaque.