## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 1 (2017), 355-385.

### Approximate group context tree

Alexandre Belloni and Roberto I. Oliveira

#### Abstract

We study a variable length Markov chain model associated with a group of stationary processes that share the same context tree but each process has potentially different conditional probabilities. We propose a new model selection and estimation method which is computationally efficient. We develop oracle and adaptivity inequalities, as well as model selection properties, that hold under continuity of the transition probabilities and polynomial $\beta$-mixing. In particular, model misspecification is allowed.

These results are applied to interesting families of processes. For Markov processes, we obtain uniform rate of convergence for the estimation error of transition probabilities as well as perfect model selection results. For chains of infinite order with complete connections, we obtain explicit uniform rates of convergence on the estimation of conditional probabilities, which have an explicit dependence on the processes’ continuity rates. Similar guarantees are also derived for renewal processes.

Our results are shown to be applicable to discrete stochastic dynamic programming problems and to dynamic discrete choice models. We also apply our estimator to a linguistic study, based on recent work by Galves et al. [*Ann. Appl. Stat.* **6** (2012) 186–209], of the rhythmic differences between Brazilian and European Portuguese.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 1 (2017), 355-385.

**Dates**

Received: February 2015

Revised: December 2015

First available in Project Euclid: 21 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1487667626

**Digital Object Identifier**

doi:10.1214/16-AOS1455

**Mathematical Reviews number (MathSciNet)**

MR3611495

**Zentralblatt MATH identifier**

06710514

**Subjects**

Primary: 62M05: Markov processes: estimation 62M09: Non-Markovian processes: estimation 62G05: Estimation

Secondary: 62P20: Applications to economics [See also 91Bxx] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Categorical time series group context tree dynamic discrete choice models dynamic programming model selection VLMC

#### Citation

Belloni, Alexandre; Oliveira, Roberto I. Approximate group context tree. Ann. Statist. 45 (2017), no. 1, 355--385. doi:10.1214/16-AOS1455. https://projecteuclid.org/euclid.aos/1487667626

#### Supplemental materials

- Supplement to “Approximate group context tree”. We provide additional discussion on the oracle context tree, omitted proofs from Section 5, a compendium of Martingale results, minimax rates for chain with infinite connections, and simulation results.Digital Object Identifier: doi:10.1214/16-AOS1455SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.