Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

Chen, Yining

doi:10.1111/sjos.12092

Cited by 5 publications

(12 citation statements)

References 56 publications

(122 reference statements)

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“…Thus reduces to the conditional log‐likelihood of the sequence

{\{X_{i}\}}_{i = 1}^{n}

. The maximizer

((),, \overset{true^}{β}, \overset{true^}{f})

of is exactly the estimator proposed in Chen and Samworth (), where consistency results were established. We now turn to the general case of non‐causal/non‐invertible models.…”

Section: Asymptotic Resultsmentioning

confidence: 89%

“…We state the relevant consistency result shown in Chen and Samworth () for comparison. Proposition 4.3 For causal‐invertible ARMA models, assume that P 0 ∈ 𝒫 and the parameter space Θ is compact.…”

Section: Asymptotic Resultsmentioning

confidence: 95%

“…The main result is: Theorem 4.1 In , suppose Z t satisfies the following condition,

L ((), \sum_{k = - \infty}^{\infty} d_{k} Z_{t - k}) ⩽ L ((), Z_{t}),

for any geometrically decaying sequence d k with

\sum_{k = - \infty}^{\infty} d_{k}^{2} ⩾ 1

and the equality holding if and only if only one d k is non‐zero. Then,

\overset{β}{true} \vec{true} β_{0} normaland \int true∣ \overset{f}{true} - Π ((), P_{0}) true∣ normald x \vec{true} 0 normalas n true\to 0 .

Remark 4.2 In the causal‐invertible case, Chen and Samworth () did not require condition because they excluded non‐causal/non‐invertible models. So if one expands the family of models to be non‐causal/non‐invertible, then a condition like is required even if the true model is causal‐invertible.…”

Section: Asymptotic Resultsmentioning

confidence: 99%

“…Kreiss () proposed one‐step adaptive estimators for ARMA models, in which the error density is estimated by a kernel density estimator. Chen and Samworth () studied semi‐parametric time series models including causal‐invertible ARMA processes, in which the distribution of the noise satisfies minor conditions and it has been shown that the semi‐parametric estimation procedure produces consistent estimators of the ARMA parameters. Moreover, the estimate of the noise distribution consistently estimates the log‐concave projection of the true density.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, if the noise density is log‐concave, then the density estimator is consistent. Inspired by Chen and Samworth (), we apply the log‐concave projection method to the non‐causal/non‐invertible ARMA models. We show the consistency of the estimators for both the coefficients and the density under mild conditions.…”

Section: Introductionmentioning

confidence: 99%

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Semi‐Parametric Estimation for Non‐Gaussian Non‐Minimum Phase ARMA Models

Davis

Zhang

2017

Journal Time Series Analysis

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We consider inference for the parameters of general autoregressive moving average (ARMA) models which are possibly non‐causal/non‐invertible (also referred to as non‐minimum phase) and driven by a non‐Gaussian distribution. For non‐minimum phase models, the observations can depend on both the past and future shocks in the system. The non‐Gaussianity constraint is necessary to distinguish between causal‐invertible and non‐causal/non‐invertible models. Many of the existing estimation procedures adopt quasi‐likelihood methods by assuming a non‐Gaussian density function for the noise distribution that is fully known up to a scalar parameter. To relax such distributional restrictions, we borrow ideas from non‐parametric density estimation and propose a semi‐parametric maximum likelihood estimation procedure, in which the noise distribution is projected onto the space of log‐concave measures. We show the maximum likelihood estimators (MLEs) in this semi‐parametric setting are consistent. In fact, the MLE is robust to the misspecification of log‐concavity in cases where the true distribution of the noise is close to its log‐concave projection (Cule and Samworth, 2010; Dümbgen et al., 2011). We derive a lower bound for the best asymptotic variance of regular estimators at rate n−12 for autoregressive (AR) models and construct a semi‐parametric efficient one‐step estimator. The estimation procedure is illustrated via a simulation study and an empirical example illustrating the methodology is also provided.

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“…Thus reduces to the conditional log‐likelihood of the sequence

{\{X_{i}\}}_{i = 1}^{n}

. The maximizer

((),, \overset{true^}{β}, \overset{true^}{f})

of is exactly the estimator proposed in Chen and Samworth (), where consistency results were established. We now turn to the general case of non‐causal/non‐invertible models.…”

Section: Asymptotic Resultsmentioning

confidence: 89%

Section: Asymptotic Resultsmentioning

confidence: 95%

“…The main result is: Theorem 4.1 In , suppose Z t satisfies the following condition,

L ((), \sum_{k = - \infty}^{\infty} d_{k} Z_{t - k}) ⩽ L ((), Z_{t}),

for any geometrically decaying sequence d k with

\sum_{k = - \infty}^{\infty} d_{k}^{2} ⩾ 1

and the equality holding if and only if only one d k is non‐zero. Then,

\overset{β}{true} \vec{true} β_{0} normaland \int true∣ \overset{f}{true} - Π ((), P_{0}) true∣ normald x \vec{true} 0 normalas n true\to 0 .

Section: Asymptotic Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semi‐Parametric Estimation for Non‐Gaussian Non‐Minimum Phase ARMA Models

Davis

Zhang

2017

Journal Time Series Analysis

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Semiparametric integer‐valued autoregressive models on ℤ

Liu

Zhu

2021

Can J Statistics

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In the analysis of real integer‐valued time series data, we often encounter negative values and negative correlations. For integer‐valued autoregressive time series, there are many parametric models to choose from, but some of them are relatively complex. With little information about the background of real data, we hope that a simple and effective semiparametric model can be used to obtain more information that usually cannot be provided by parametric models, such as the confidence interval of the innovation distribution. But the only existing semiparametric model based on thinning operators can only deal with non‐negative data with positive correlation coefficients. In addition, it has two drawbacks: first, an initial distribution of the innovation is required, but different initial values may lead to different results; second, the confidence interval of the innovation distribution is not available, which is essential in low‐valued data. To overcome these drawbacks, we propose a rounded semiparametric autoregressive model with a log‐concave innovation, which can deal with ℤ‐valued time series with autoregressive coefficients of arbitrary sign. The consistencies of the estimators for the parametric and nonparametric parts of the model are also discussed. We illustrate the superior performance of the proposed model based on three real datasets.

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R-estimation in semiparametric dynamic location-scale models

Hallin

Vecchia

2017

Journal of Econometrics

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Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

Cited by 5 publications

References 56 publications

Semi‐Parametric Estimation for Non‐Gaussian Non‐Minimum Phase ARMA Models

Semi‐Parametric Estimation for Non‐Gaussian Non‐Minimum Phase ARMA Models

Semiparametric integer‐valued autoregressive models on ℤ

R-estimation in semiparametric dynamic location-scale models

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