Density Estimation and Combination under Model Ambiguity

The empirical application to daily stock returns indicates that, during the phases of expansion, the best
model is the double Gamma distribution, while during the phases of recession is the Gaussian distribution.
Moreover, the combination of the Normal and the double Gamma, according to the weights obtained with
the described methodology, outperforms in- and out-of-sample all candidate models including the best one.
Most likely, this result is due to the fact that none of the candidate models is the true structure, as such
the models combination being a higher dimensional parametric alternative is able to approximate the data
more closely.

This suggests that in decision contexts characterized by high uncertainty, such that it can be hard: to
form specific priors, to conceive an exhaustive set of all possible models and/or to use the true complex
structure, the proposed approach can provide a better hedge against the lack of knowledge of the correct
model. Additionally, this methodology can also be used to form priors in training sample, before applying
more sophisticated Bayesian averaging techniques.

This approach can be further extended to conditional distributions to address more challenging and
complex prediction problems. I leave this problem to future research.

9 Appendix

9.1 Proof Theorem 1

The first step consists in showing that KIjn (θ) converges at least in probability to the contrast function

KIj(θ).

KIjn(θ) - KIj(θ) = (ln fcn (xi) -ln

i=1

f (θ, xi)fcn(xi) - (ln g(x) - ln f (θ, x)g(x)dx = (27)

))g(x)dx - lnf(θ,xi