A local limit theorem for linear random fields Fortune, Timothy; Peligrad, Magda; Sang, Hailin
Journal of time series analysis,
September-November 2021, 2021-09-00, 20210901, Letnik:
42, Številka:
5-6
Journal Article
Recenzirano
Odprti dostop
In this article, we establish a local limit theorem for linear fields of random variables constructed from i.i.d. innovations each with finite second moment. When the coefficients are absolutely ...summable we do not restrict the region of summation. However, when the coefficients are only square‐summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, that is, for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.
Kernel Entropy Estimation for Linear Processes Sang, Hailin; Sang, Yongli; Xu, Fangjun
Journal of time series analysis,
July 2018, 2018-07-00, 20180701, Letnik:
39, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Let {X
n
:n∈N}be a linear process with bounded probability density function f(x). We study the estimation of the quadratic functional ∫ R
f
2(x)dx. With a Fourier transform on the kernel function and ...the projection method, it is shown that, under certain mild conditions, the estimator
2
/
(
n
(
n
−
1
)
h
n
)
∑
1
≤
i
<
j
≤
n
K
(
(
X
i
−
X
j
)
/
h
n
)
has similar asymptotical properties as the i.i.d. case studied in Giné and Nickl if the linear process {X
n
:n∈N}has the defined short range dependence. We also provide an application to
L
2
2
divergence and the extension to multi‐variate linear processes. The simulation study for linear processes with Gaussian and α‐stable innovations confirms our theoretical results. As an illustration, we estimate the
L
2
2
divergences among the density functions of average annual river flows for four rivers and obtain promising results.
In this paper we propose a variable bandwidth kernel regression estimator for
i
.
i
.
d
. observations in ℝ
2
to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of
...O
(
h
n
4
) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.
We propose a new covariance matrix called Gini covariance matrix (GCM), which is a natural generalization of univariate Gini mean difference (GMD) to the multivariate case. The extension is based on ...the covariance representation of GMD by applying the multivariate spatial rank function. We study properties of GCM, especially in the elliptical distribution family. In order to gain the affine equivariance property for GCM, we utilize the transformation–retransformation (TR) technique and obtain an affine equivariant version GCM that turns out to be a symmetrized M-functional. The influence function of those two GCM’s are obtained and their estimation has been presented. Asymptotic results of estimators have been established. A closely related scatter Kotz functional and its estimator are also explored. Finally, asymptotical efficiency and finite sample efficiency of the TR version GCM are compared with those of sample covariance matrix, Tyler-M estimator and other scatter estimators under different distributions.
We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under ...mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O
P
(N
−1/2
) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391-422. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.
Standard Gini covariance and Gini correlation play important roles in measuring the dependence between random variables with heavy tailed distributions. However the asymmetry of Gini covariance and ...correlation brings a substantial difficulty in interpretation. In this article we propose a symmetric Gini-type covariance and a symmetric Gini correlation (ρg) based on the joint rank function. The proposed correlation ρg is more robust than the Pearson correlation but less robust than the Kendall's τ correlation. We establish the relationship between ρg and the linear correlation ρ for a class of random vectors in the family of elliptical distributions, which allows us to estimate ρ based on estimation of ρg. The asymptotic normality of the resulting estimators of ρ is studied through two approaches: one based on influence function and the other based on U-statistics and the delta method. We compare asymptotic efficiencies of the symmetric Gini, regular Gini, Pearson and Kendall's τ linear correlation estimators under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure pg shows superior finite sample performance, which makes it attractive in applications. La covariance de Gini et la corrélation de Gini sont importantes pour mesurer la dépendance entre des variables aléatoires dont la distribution possède des queues lourdes. L'asymétrie des statistiques de Gini cause toutefois des difficultés substantielles d'interprétation. Les auteurs présentent une version symétrique de la covariance et de la corrélation de Gini définies à partir de la fonction conjointe des rangs spatiaux. La corrélation ρg proposée s'avère plus robuste que la corrélation de Pearson, mais moins que τ le de Kendall. Les auteurs établissent le lien entre ρg et la corrélation linéaire ρ pour une classe de vecteurs aléatoires dont la distribution est elliptique, permettant d'estimer ρ à partir de l'estimé de ρg. Ils montrent la normalité asymptotique de l'estimé ainsi obtenu en suivant deux approches : l'une basée sur les fonctions d'influence, la seconde sur les U-statistiques et la méthode delta. Les auteurs comparent l'efficacité asymptotique des corrélations de Gini (classique et symétrique), Pearson et Kendall sous différentes distributions. En plus d'établir un compromis raisonnable entre la robustesse et l'efficacité, le coefficient ρg qu'ils proposent offre de meilleures performances sur des échantillons finis, lui conférant des avantages pratiques.
In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the ...self-normalized version of this theorem. The study is motivated by models arising in economic applications where often the linear processes have long memory, and the innovations have heavy tails.
This paper addresses the following classical question: Given a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process ...satisfy the central limit theorem? We study the question for several classes of dependent random variables. For independent and identically distributed random variables we show that the central limit theorem for the linear process is equivalent to the fact that the variables are in the domain of attraction of a normal law, answering in this way an open problem in the literature. The study is also motivated by models arising in economic applications where often the innovations have infinite variance, coefficients are not absolutely summable, and the innovations are dependent.
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