Statistical spatial series modelling II: Some further results on unilateral lattice processes

1983 ◽  
Vol 15 (03) ◽  
pp. 562-584 ◽  
Author(s):  
Dag Tjøstheim
Keyword(s):  
Time Series ◽  
Half Space ◽  
Special Interest ◽  
Asymptotic Theory ◽  
Innovation Process ◽  
Asymptotic Results ◽  
Practical Applications ◽  
Spatial Series ◽  
Lattice Processes ◽  
Series Criteria

An asymptotic theory of estimation is developed for classes of spatial series F(x 1, · ··, xn ), where (x 1, · ··, xn ) varies over a regular cartesian lattice. Two classes of unilateral models are studied, namely half-space models and causal (quadrant-type) models. It is shown that a number of asymptotic results are common for these models. Of special interest for practical applications is the problem of determining how many parameters should be included to describe the degree of dependence in each direction. Here we are able to obtain weakly consistent generalizations of familiar time-series criteria under the assumption that the generating variables of the model are independently and identically distributed. For causal models we introduce the concepts of spatial innovation process and lattice martingale and use these to extend some of the asymptotic theory to the case where a certain type of dependence is permitted in the generating variables.

Statistical spatial series modelling II: Some further results on unilateral lattice processes

1983 ◽  
Vol 15 (3) ◽  
pp. 562-584 ◽  
Author(s):  
Dag Tjøstheim
Keyword(s):  
Time Series ◽  
Half Space ◽  
Special Interest ◽  
Asymptotic Theory ◽  
Innovation Process ◽  
Asymptotic Results ◽  
Practical Applications ◽  
Spatial Series ◽  
Lattice Processes ◽  
Series Criteria

An asymptotic theory of estimation is developed for classes of spatial series F(x1, · ··, xn), where (x1, · ··, xn) varies over a regular cartesian lattice. Two classes of unilateral models are studied, namely half-space models and causal (quadrant-type) models. It is shown that a number of asymptotic results are common for these models. Of special interest for practical applications is the problem of determining how many parameters should be included to describe the degree of dependence in each direction. Here we are able to obtain weakly consistent generalizations of familiar time-series criteria under the assumption that the generating variables of the model are independently and identically distributed. For causal models we introduce the concepts of spatial innovation process and lattice martingale and use these to extend some of the asymptotic theory to the case where a certain type of dependence is permitted in the generating variables.

Wald-Type Tests for Detecting Breaks in the Trend Function of a Dynamic Time Series

1997 ◽  
Vol 13 (6) ◽  
pp. 818-848 ◽  
Author(s):  
Timothy J. Vogelsang
Keyword(s):  
Time Series ◽  
Asymptotic Theory ◽  
Test Statistics ◽  
Asymptotic Results ◽  
Paper Test ◽  
Trend Function ◽  
Univariate Time Series ◽  
Macroeconomic Time Series ◽  
The Mean ◽  
Dynamic Time

In this paper, test statistics for detecting a break at an unknown date in the trend function of a dynamic univariate time series are proposed. The tests are based on the mean and exponential statistics of Andrews and Ploberger (1994, Econometrica 62, 1383–1414) and the supremum statistic of Andrews (1993, Econometrica 61, 821–856). Their results are extended to allow trending and unit root regressors. Asymptotic results are derived for both I(0) and I(1) errors. When the errors are highly persistent and it is not known which asymptotic theory (I(0) or I(1)) provides a better approximation, a conservative approach based on nearly integrated asymptotics is provided. Power of the mean statistic is shown to be nonmonotonic with respect to the break magnitude and is dominated by the exponential and supremum statistics. Versions of the tests applicable to first differences of the data are also proposed. The tests are applied to some macroeconomic time series, and the null hypothesis of a stable trend function is rejected in many cases.

Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

scholarly journals On nonparametric regression for bivariate circular long-memory time series

2021 ◽  
Author(s):  
Jan Beran ◽  
Britta Steffens ◽  
Sucharita Ghosh
Keyword(s):  
Time Series ◽  
Nonparametric Regression ◽  
Long Range ◽  
Long Memory ◽  
Regression Function ◽  
Confidence Bands ◽  
Long Range Dependence ◽  
Asymptotic Results ◽  
Memory Time ◽  
Circular Time Series

AbstractWe consider nonparametric regression for bivariate circular time series with long-range dependence. Asymptotic results for circular Nadaraya–Watson estimators are derived. Due to long-range dependence, a range of asymptotically optimal bandwidths can be found where the asymptotic rate of convergence does not depend on the bandwidth. The result can be used for obtaining simple confidence bands for the regression function. The method is illustrated by an application to wind direction data.

Bounds on the Maximum Contact Stress of an Indented Elastic Layer

1971 ◽  
Vol 38 (3) ◽  
pp. 608-614 ◽  
Author(s):  
Y. C. Pao ◽  
Ting-Shu Wu ◽  
Y. P. Chiu
Keyword(s):  
Half Space ◽  
Contact Stress ◽  
Elastic Layer ◽  
Theory Of Elasticity ◽  
Contact Conditions ◽  
Practical Applications ◽  
Method Of Solution ◽  
Elastic Layers ◽  
Maximum Contact ◽  
Contact Stress Distribution

This paper is concerned with the plane-strain problem of an elastic layer supported on a half-space foundation and indented by a cylinder. A study is presented of the effect of the contact condition at the layer-foundation interface on the contact stresses of the indented layer. For the general problem of elastic indenter or elastic foundation, the integral equations governing the contact stress distribution of the indented layer derived on the basis of two-dimensional theory of elasticity are given and a numerical method of solution is formulated. The limiting contact conditions at the layer-foundation interface are then investigated by considering two extreme cases, one with the indented layer in frictionless contact with the half space and the other with the indented layer rigidly adhered to the half space. Graphs of the bounds on the maximum normal stress occurring in indented elastic layers for the cases of rigid cylindrical indenter and rigid half-space foundation are obtained for possible practical applications. Some results of the elastic indenter problem are also presented and discussed.

The Vanishing Gradient Problem During Learning Recurrent Neural Nets and Problem Solutions

1998 ◽  
Vol 06 (02) ◽  
pp. 107-116 ◽  
Author(s):  
Sepp Hochreiter
Keyword(s):  
Time Series ◽  
Time Series Prediction ◽  
Time Lag ◽  
Neural Nets ◽  
Alternative Methods ◽  
Reasonable Time ◽  
Practical Applications ◽  
Learning Time ◽  
Long Time ◽  
Gradient Based

Recurrent nets are in principle capable to store past inputs to produce the currently desired output. Because of this property recurrent nets are used in time series prediction and process control. Practical applications involve temporal dependencies spanning many time steps, e.g. between relevant inputs and desired outputs. In this case, however, gradient based learning methods take too much time. The extremely increased learning time arises because the error vanishes as it gets propagated back. In this article the de-caying error flow is theoretically analyzed. Then methods trying to overcome vanishing gradients are briefly discussed. Finally, experiments comparing conventional algorithms and alternative methods are presented. With advanced methods long time lag problems can be solved in reasonable time.

scholarly journals SHORT MATURITY ASIAN OPTIONS FOR THE CEV MODEL

2018 ◽  
Vol 33 (2) ◽  
pp. 258-290 ◽  
Author(s):  
Dan Pirjol ◽  
Lingjiong Zhu
Keyword(s):  
Analytical Approximation ◽  
Time Averaging ◽  
Asian Options ◽  
Option Prices ◽  
Asymptotic Results ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Practical Applications ◽  
Rigorous Study

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the constant elasticity of variance (CEV) model. The leading order short maturity limit of the Asian option prices under the CEV model is obtained in closed form. We propose an analytical approximation for the Asian options prices which reproduces the exact short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with Monte Carlo simulations and benchmark test cases for option parameters relevant for practical applications.

scholarly journals Rayleigh-type waves on a coated elastic half-space with a clamped surface

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190111 ◽  
Author(s):  
J. Kaplunov ◽  
D. Prikazchikov ◽  
L. Sultanova
Keyword(s):  
Half Space ◽  
Numerical Solutions ◽  
Substrate Surface ◽  
Numerical Data ◽  
Elastic Half Space ◽  
Asymptotic Results ◽  
Homogeneous Half Space ◽  
Upper Face ◽  
Localized Waves ◽  
Perturbed Equation

Elastodynamics of a half-space coated by a thin soft layer with a clamped upper face is considered. The focus is on the analysis of localized waves that do not exist on a clamped homogeneous half-space. Non-traditional effective boundary conditions along the substrate surface incorporating the effect of the coating are derived using a long-wave high-frequency procedure. The derived conditions are implemented within the framework of the earlier developed specialized formulation for surface waves, resulting in a perturbation of the shortened equation of surface motion in the form of an integral or pseudo-differential operator. Non-uniform asymptotic formula for the speeds of the sought for Rayleigh-type waves, failing near zero frequency and the thickness resonances of a layer with both clamped faces, follow from the aforementioned perturbed equation. Asymptotic results are compared with the numerical solutions of the full dispersion relation for a clamped coated half-space. A similarity with Love-type waves proves to be useful for interpreting numerical data. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

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