We establish the asymptotic theory of least absolute deviation estimators for AR(1) processes with autoregressive parameter satisfying
n
(
ρ
n
-
1
)
→
γ
for some fixed
γ
as
n
→
∞
, which is parallel ...to the results of ordinary least squares estimators developed by Andrews and Guggenberger (Journal of Time Series Analysis, 29, 203–212, 2008) in the case
γ
=
0
or Chan and Wei (Annals of Statistics, 15, 1050–1063, 1987) and Phillips (Biometrika, 74, 535–574, 1987) in the case
γ
≠
0
. Simulation experiments are conducted to confirm the theoretical results and to demonstrate the robustness of the least absolute deviation estimation.
On the integrated mean squared error of wavelet density estimation for linear processes Beknazaryan, Aleksandr; Sang, Hailin; Adamic, Peter
Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems,
07/2023, Letnik:
26, Številka:
2
Journal Article
Odprti dostop
Let
{
X
n
:
n
∈
N
}
be a linear process with density function
f
(
x
)
∈
L
2
(
R
)
. We study wavelet density estimation of
f
(
x
). Under some regular conditions on the characteristic function of ...innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions.
We consider regression estimation with modified ReLU neural networks in which network weight matrices are first modified by a function α before being multiplied by input vectors. We give an example ...of continuous, piecewise linear function α for which the empirical risk minimizers over the classes of modified ReLU networks with l1 and squared l2 penalties attain, up to a logarithmic factor, the minimax rate of prediction of unknown β-smooth function.
In this paper we investigate the local limit theorem for partial sums of linear sequences of the form Xj=∑i∈Zaiξj−i. Here (ai)i∈Z is a sequence of constants satisfying ∑i∈Zai2<∞ and (ξi)i∈Z are ...functions of a stationary Markov chain with mean zero and finite second moment. The Markov chain is assumed to satisfy one-sided lower psi-mixing condition.
By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for ...studying nonparametric regression with random field errors and strong limit theorems.
In this paper we study the convergence in distribution and the local limit theorem for the partial sums of linear random fields with i.i.d. innovations that have infinite second moment and belong to ...the domain of attraction of a stable law with index 0<α≤2 under the condition that the innovations are centered if 1<α≤2 and are symmetric if α=1. We establish these two types of limit theorems as long as the linear random fields are well-defined, the coefficients are either absolutely summable or not absolutely summable.
We study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel method to estimate the mutual ...information between the two random variables. The estimates enjoy a central limit theorem under some regular conditions on the distributions. The theoretical results are demonstrated by simulation study.
We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or ...long memory and to nonparametric regression with random field errors.
Memory properties of transformations of linear processes Sang, Hailin; Sang, Yongli
Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems,
04/2017, Letnik:
20, Številka:
1
Journal Article
Odprti dostop
In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (J Econ 110:113–133,
2002
) studied the polynomial transformations of Gaussian FARIMA(0,
d
..., 0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (Ann Stat 24:992–1024,
1996
; Ann Probab 25:1636–1669,
1997
) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(
p
,
d
,
q
) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.
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