On Adaptive Estimation in Stationary ARMA Processes

Kreiß, Jens-Peter

doi:10.1214/aos/1176350256

Cited by 253 publications

(177 citation statements)

References 9 publications

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“…We will therefore tacitly assume thatθ θ θ ∈ S k−1 (and thereforê ϑ ϑ ϑ = 1 m ⊗θ θ θ) is locally and asymptotically discrete throughout this section. Following Lemma 4.4 in Kreiss (1987), the local discreteness allows to replace in Part (iv) of Proposition 5.1 non-random perturbations of the form ϑ ϑ ϑ + n −1/2 ν ν ν (n) t (n) with t (n) such that ϑ ϑ ϑ + n −1/2 ν ν ν (n) t (n) still belongs to H 0 by a n 1/2 (ν ν ν (n) ) −1 -consistent estimatorθ ϑ ϑ := 1 m ⊗θ θ θ. Based on the asymptotic result of Proposition 5.1 and letting…”

Section: Proposition 51 Let Assumptions a B C And E Hold And Considermentioning

confidence: 99%

Efficient ANOVA for directional data

2015

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In this paper we tackle the ANOVA problem for directional data (with particular emphasis on geological data) by having recourse to the Le Cam methodology usually reserved for linear multivariate analysis. We construct locally and asymptotically most stringent parametric tests for ANOVA for directional data within the class of rotationally symmetric distributions. We turn these parametric tests into semi-parametric ones by (i) using a studentization argument (which leads to what we call pseudo-FvML tests) and by (ii) resorting to the invariance principle (which leads to efficient rank-based tests). Within each construction the semi-parametric tests inherit optimality under a given distribution (the FvML distribution in the first case, any rotationally symmetric distribution in the second) from their parametric antecedents and also improve on the latter by being valid under the whole class of rotationally symmetric distributions. Asymptotic relative efficiencies are calculated and the finite-sample behavior of the proposed tests is investigated by means of a Monte Carlo simulation. We conclude by applying our findings on a real-data example involving geological data.

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Section: Proposition 51 Let Assumptions a B C And E Hold And Considermentioning

confidence: 99%

Efficient ANOVA for directional data

2015

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“…This condition is however essential as it precisely allows to replace τ (4) and (5) by Lemma 4.4 in Kreiss (1987), where such replacements have been theoretically worked out in detail. Using this fact in combination with Slutsky's Lemma, we have the following crucial result.…”

Section: Construction Of Efficient Testing Proceduresmentioning

confidence: 99%

Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models

Koudou

Ley

2014

TEST

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The standard efficient testing procedures in the Generalized Inverse Gaussian (GIG) family (also known as Halphen Type A family) are likelihood ratio tests, hence rely on Maximum Likelihood (ML) estimation of the three parameters of the GIG. The particular form of GIG densities, involving modified Bessel functions, prevents in general from a closed-form expression for ML estimators, which are obtained at the expense of complex numerical approximation methods. On the contrary, Method of Moments (MM) estimators allow for concise expressions, but tests based on these estimators suffer from a lack of efficiency compared to likelihood ratio tests. This is why, in recent years, trade-offs between ML and MM estimators have been proposed, resulting in simpler yet not completely efficient estimators and tests. In the present paper, we do not propose such a trade-off but rather an optimal combination of both methods, our tests inheriting efficiency from an ML-like construction and simplicity from the MM estimators of the nuisance parameters. This goal shall be reached by attacking the problem from a new angle, namely via the Le Cam methodology.Besides providing simple efficient testing methods, the theoretical background of this methodology further allows us to write out explicitly power expressions for our tests. A Monte Carlo simulation study shows that, also at small sample sizes, our simpler procedures do at least as good as the complex likelihood ratio tests. We conclude the paper by applying our findings on two real-data sets.

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“…expansion (1.4) holds with φ(y) = r ρ (y). Efficient estimators for parameters of time series are constructed in Kreiss [9,10], Jeganathan [7], Drost et al [3] and Koul and Schick [8]. If we use an efficient estimatorρ forq w , then, by Theorem 2,q w (z) has influence function r z X 0 (ε 1 ) + S ρ 2 x (ψ * (ε 1 , z) − E[ψ * (ε, z)]) with r z = r ρ S ρ 2 xq (z).…”

Section: Efficiencymentioning

confidence: 99%

Plug-in estimators for higher-order transition densities in autoregression

Schίck

Wefelmeyer

2009

ESAIM: PS

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Abstract. In this paper we obtain root-n consistency and functional central limit theorems in weighted L1-spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estimator for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit the fact that the innovations have mean zero. If an efficient estimator for the autoregression parameter is used, then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes.Mathematics Subject Classification. 62M05, 62M10.

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On Adaptive Estimation in Stationary ARMA Processes

Cited by 253 publications

References 9 publications

Efficient ANOVA for directional data

Efficient ANOVA for directional data

Efficiency combined with simplicity: new testing procedures for Generalized Inverse Gaussian models

Plug-in estimators for higher-order transition densities in autoregression

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