## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 2107-2136.

### Root-$n$ consistent estimation of the marginal density in semiparametric autoregressive time series models

Lionel Truquet

#### Abstract

In this paper, we consider the problem of estimating the marginal density in some autoregressive time series models for which the conditional mean and variance have a parametric specification. Under some regularity conditions, we show that a kernel type estimate based on the residuals can be root-$n$ consistent even if the noise density is unknown. Our results substantially extend those existing in the literature. Our assumptions are carefully checked for some standard time series models such as ARMA or GARCH processes. Asymptotic expansion of our estimator is obtained by combining some martingale type arguments and a coupling method for time series which is of independent interest. We also study the uniform convergence of our estimator on compact intervals.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 2107-2136.

Dates
Revised: April 2018
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.bj/1560326439

Digital Object Identifier
doi:10.3150/18-BEJ1047

Mathematical Reviews number (MathSciNet)
MR3961242

Zentralblatt MATH identifier
07066251

#### Citation

Truquet, Lionel. Root-$n$ consistent estimation of the marginal density in semiparametric autoregressive time series models. Bernoulli 25 (2019), no. 3, 2107--2136. doi:10.3150/18-BEJ1047. https://projecteuclid.org/euclid.bj/1560326439

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#### Supplemental materials

• Supplement to “Root-$n$ consistent estimation of the marginal density in semiparametric autoregressive time series models”. We provide additional proofs and a simulation study for adequation tests.