## Bernoulli

• Bernoulli
• Volume 22, Number 1 (2016), 325-344.

### Sharp oracle inequalities and slope heuristic for specification probabilities estimation in discrete random fields

#### Abstract

We study the problem of estimating the one-point specification probabilities in non-necessary finite discrete random fields from partially observed independent samples. Our procedures are based on model selection by minimization of a penalized empirical criterion. The selected estimators satisfy sharp oracle inequalities in $L_{2}$-risk.

We also obtain theoretical results on the slope heuristic for this problem, justifying the slope algorithm to calibrate the leading constant in the penalty. The practical performances of our methods are investigated in two simulation studies. We illustrate the usefulness of our approach by applying the methods to a multi-unit neuronal data from a rat hippocampus.

#### Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 325-344.

Dates
Revised: June 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620852

Digital Object Identifier
doi:10.3150/14-BEJ660

Mathematical Reviews number (MathSciNet)
MR3449785

Zentralblatt MATH identifier
1342.60077

#### Citation

Lerasle, Matthieu; Takahashi, Daniel Y. Sharp oracle inequalities and slope heuristic for specification probabilities estimation in discrete random fields. Bernoulli 22 (2016), no. 1, 325--344. doi:10.3150/14-BEJ660. https://projecteuclid.org/euclid.bj/1443620852

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#### Supplemental materials

• Supplement to “Sharp oracle inequalities and slope heuristic for specification probabilities estimation in discrete random fields”. On this supplementary material available on-line, we prove the probabilistic tools needed in the proofs of the main results. The second part provides additional simulation results. The last one is devoted to the extension of all our results to the Küllback loss.