A threshold AR(1) model

1984 ◽  
Vol 21 (2) ◽  
pp. 270-286 ◽  
Author(s):  
Joseph D. Petruccelli ◽  
Samuel W. Woolford
Keyword(s):  
Hypothesis Test ◽  
Threshold Model ◽  
Sufficient Conditions ◽  
Simulated Data ◽  
Small Sample ◽  
Regularity Conditions ◽  
Test Statistic ◽  
Asymptotically Normal ◽  
Least Squares Estimators ◽  
Necessary And Sufficient

We consider the model where φ1, φ2 are real coefficients, not necessarily equal, and the at,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ1 = φ2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.

A threshold AR(1) model

1984 ◽  
Vol 21 (02) ◽  
pp. 270-286 ◽  
Author(s):  
Joseph D. Petruccelli ◽  
Samuel W. Woolford
Keyword(s):  
Hypothesis Test ◽  
Threshold Model ◽  
Sufficient Conditions ◽  
Simulated Data ◽  
Small Sample ◽  
Regularity Conditions ◽  
Test Statistic ◽  
Asymptotically Normal ◽  
Least Squares Estimators ◽  
Necessary And Sufficient

We consider the model where φ 1, φ 2 are real coefficients, not necessarily equal, and the at ,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ 1 = φ 2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.

A multiple-threshold AR(1) model

1985 ◽  
Vol 22 (02) ◽  
pp. 267-279 ◽  
Author(s):  
K. S. Chan ◽  
Joseph D. Petruccelli ◽  
H. Tong ◽  
Samuel W. Woolford
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Regularity Conditions ◽  
Model Parameters ◽  
Asymptotically Normal ◽  
Least Squares Estimators ◽  
Multiple Threshold ◽  
Necessary And Sufficient ◽  
Strongly Consistent

We consider the model Zt = φ (0, k)+ φ(1, k)Zt –1 + at (k) whenever r k−1<Z t−1≦r k , 1≦k≦l, with r 0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at (k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at (k), t≧1} independent of {at (j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.

A multiple-threshold AR(1) model

1985 ◽  
Vol 22 (2) ◽  
pp. 267-279 ◽  
Author(s):  
K. S. Chan ◽  
Joseph D. Petruccelli ◽  
H. Tong ◽  
Samuel W. Woolford
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Regularity Conditions ◽  
Model Parameters ◽  
Asymptotically Normal ◽  
Least Squares Estimators ◽  
Multiple Threshold ◽  
Necessary And Sufficient ◽  
Strongly Consistent

We consider the model Zt = φ (0, k)+ φ(1, k)Zt–1 + at (k) whenever rk−1<Zt−1≦rk, 1≦k≦l, with r0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k), t≧1} independent of {at(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal.

A note on exceedances and rare events of non-stationary sequences

1993 ◽  
Vol 30 (04) ◽  
pp. 877-888 ◽  
Author(s):  
J. Hüsler
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Rare Events ◽  
Sufficient Conditions ◽  
Stationary Sequence ◽  
Poisson Processes ◽  
Regularity Conditions ◽  
Mixing Conditions ◽  
Necessary And Sufficient ◽  
Point Process Of Exceedances

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

scholarly journals Necessary and Sufficient Conditions for Inner Functions to Be inQK(p,p-2)-Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Atte Reijonen
Keyword(s):  
Blaschke Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inner Function ◽  
Regularity Conditions ◽  
Inner Functions ◽  
Necessary And Sufficient

Under some regularity conditions onK, inner functions inQK(p,p-2)-spaces are characterized in the following way: an inner function belongs toQK(p,p-2)if and only if it is a Blaschke product associated with{zn}satisfyingsupa∈D∑nK(1-|φa(zn)|)<∞, whereφa(z)=(a-z)/(1-a¯z). The result generalizes earlier theorems in (Essén et al., 2006) and (Pérez-González and Rättyä, 2009).

Geometric product form queueing networks with concurrent batch movements

1998 ◽  
Vol 30 (04) ◽  
pp. 1111-1129 ◽  
Author(s):  
Hideaki Yamashita ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Product Form ◽  
Regularity Conditions ◽  
Geometric Product ◽  
Batch Sizes ◽  
Network States ◽  
The One ◽  
Necessary And Sufficient

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

Geometric product form queueing networks with concurrent batch movements

1998 ◽  
Vol 30 (4) ◽  
pp. 1111-1129 ◽  
Author(s):  
Hideaki Yamashita ◽  
Masakiyo Miyazawa
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Product Form ◽  
Regularity Conditions ◽  
Geometric Product ◽  
Batch Sizes ◽  
Network States ◽  
The One ◽  
Necessary And Sufficient

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

A note on exceedances and rare events of non-stationary sequences

1993 ◽  
Vol 30 (4) ◽  
pp. 877-888 ◽  
Author(s):  
J. Hüsler
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Rare Events ◽  
Sufficient Conditions ◽  
Stationary Sequence ◽  
Poisson Processes ◽  
Regularity Conditions ◽  
Mixing Conditions ◽  
Necessary And Sufficient ◽  
Point Process Of Exceedances

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

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