An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

Jiří Anděl

Applications of Mathematics (1998)

  • Volume: 43, Issue: 5, page 389-398
  • ISSN: 0862-7940

Abstract

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Let 𝕖 t = ( e t 1 , ⋯ , e t p ) ' be a p -dimensional nonnegative strict white noise with finite second moments. Let h i j ( x ) be nondecreasing functions from [ 0 , ∞ ) onto [ 0 , ∞ ) such that h i j ( x ) ≤ x for i , j = 1 , ⋯ , p . Let 𝕌 = ( u i j ) be a p × p matrix with nonnegative elements having all its roots inside the unit circle. Define a process 𝕏 t = ( X t 1 , ⋯ , X t p ) ' by X t j = u j 1 h 1 j ( X t - 1 , 1 ) + ⋯ + u j p h p j ( X t - 1 , p ) + e t j for j = 1 , ⋯ , p . A method for estimating 𝕌 from a realization 𝕏 1 , ⋯ , 𝕏 n is proposed. It is proved that the estimators are strongly consistent.

How to cite

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Anděl, Jiří. "An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process." Applications of Mathematics 43.5 (1998): 389-398. <http://eudml.org/doc/33017>.

@article{Anděl1998,
abstract = {Let $\mathbb \{e\}_t=(e_\{t1\},\dots ,e_\{tp\})^\{\prime \}$ be a $p$-dimensional nonnegative strict white noise with finite second moments. Let $h_\{ij\}(x)$ be nondecreasing functions from $[0,\infty )$ onto $[0,\infty )$ such that $h_\{ij\}(x)\le x$ for $i,j=1,\dots ,p$. Let $\mathbb \{U\}=(u_\{ij\})$ be a $p\times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process $\mathbb \{X\}_t=(X_\{t1\},\dots ,X_\{tp\})^\{\prime \}$ by \[ X\_\{tj\}=u\_\{j1\}h\_\{1j\}(X\_\{t-1,1\})+\dots +u\_\{jp\}h\_\{pj\}(X\_\{t-1,p\})+ e\_\{tj\} \] for $j=1,\dots ,p$. A method for estimating $\mathbb \{U\}$ from a realization $\mathbb \{X\}_1,\dots ,\mathbb \{X\}_n$ is proposed. It is proved that the estimators are strongly consistent.},
author = {Anděl, Jiří},
journal = {Applications of Mathematics},
keywords = {autoregressive process; estimating parameters; multidimensional process; nonlinear process; nonnegative process; autoregressive processes; multidimensional processes; nonlinear processes; nonnegative processes},
language = {eng},
number = {5},
pages = {389-398},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process},
url = {http://eudml.org/doc/33017},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Anděl, Jiří
TI - An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process
JO - Applications of Mathematics
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 43
IS - 5
SP - 389
EP - 398
AB - Let $\mathbb {e}_t=(e_{t1},\dots ,e_{tp})^{\prime }$ be a $p$-dimensional nonnegative strict white noise with finite second moments. Let $h_{ij}(x)$ be nondecreasing functions from $[0,\infty )$ onto $[0,\infty )$ such that $h_{ij}(x)\le x$ for $i,j=1,\dots ,p$. Let $\mathbb {U}=(u_{ij})$ be a $p\times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process $\mathbb {X}_t=(X_{t1},\dots ,X_{tp})^{\prime }$ by \[ X_{tj}=u_{j1}h_{1j}(X_{t-1,1})+\dots +u_{jp}h_{pj}(X_{t-1,p})+ e_{tj} \] for $j=1,\dots ,p$. A method for estimating $\mathbb {U}$ from a realization $\mathbb {X}_1,\dots ,\mathbb {X}_n$ is proposed. It is proved that the estimators are strongly consistent.
LA - eng
KW - autoregressive process; estimating parameters; multidimensional process; nonlinear process; nonnegative process; autoregressive processes; multidimensional processes; nonlinear processes; nonnegative processes
UR - http://eudml.org/doc/33017
ER -

References

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