A central limit theorem with application to inference in α-stable regression models

Marina Riabiz, Tohid Ardeshiri, Simon Godsill
Proceedings of the Time Series Workshop at NIPS 2016, PMLR 55:70-82, 2017.

Abstract

It is well known that the α-stable distribution, while having no closed form density function in the general case, admits a Poisson series representation (PSR) in which the terms of the series are a function of the arrival times of a unit rate Poisson process. In our previous work we have shown how to carry out inference for regression models using this series representation, which leads to a very convenient conditionally Gaussian framework, amenable to straightforward Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual terms have been approximated in our previous work by a Gaussian distribution with fully characterised moments. In this paper we present a new Central Limit Theorem (CLT) for the residual terms which serves to justify our previous approximation of the residual as Gaussian. Furthermore, we provide an analysis of the asymptotic convergence rate expressed in the CLT.

Cite this Paper

BibTeX
@InProceedings{pmlr-v55-riabiz16, title = {{A central limit theorem with application to inference in $\alpha$-stable regression models}}, author = {Riabiz, Marina and Ardeshiri, Tohid and Godsill, Simon}, booktitle = {Proceedings of the Time Series Workshop at NIPS 2016}, pages = {70--82}, year = {2017}, editor = {Anava, Oren and Khaleghi, Azadeh and Cuturi, Marco and Kuznetsov, Vitaly and Rakhlin, Alexander}, volume = {55}, series = {Proceedings of Machine Learning Research}, address = {Barcelona, Spain}, month = {09 Dec}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v55/riabiz16.pdf}, url = {https://proceedings.mlr.press/v55/riabiz16.html}, abstract = {It is well known that the α-stable distribution, while having no closed form density function in the general case, admits a Poisson series representation (PSR) in which the terms of the series are a function of the arrival times of a unit rate Poisson process. In our previous work we have shown how to carry out inference for regression models using this series representation, which leads to a very convenient conditionally Gaussian framework, amenable to straightforward Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual terms have been approximated in our previous work by a Gaussian distribution with fully characterised moments. In this paper we present a new Central Limit Theorem (CLT) for the residual terms which serves to justify our previous approximation of the residual as Gaussian. Furthermore, we provide an analysis of the asymptotic convergence rate expressed in the CLT.} }
Endnote
%0 Conference Paper %T A central limit theorem with application to inference in α-stable regression models %A Marina Riabiz %A Tohid Ardeshiri %A Simon Godsill %B Proceedings of the Time Series Workshop at NIPS 2016 %C Proceedings of Machine Learning Research %D 2017 %E Oren Anava %E Azadeh Khaleghi %E Marco Cuturi %E Vitaly Kuznetsov %E Alexander Rakhlin %F pmlr-v55-riabiz16 %I PMLR %P 70--82 %U https://proceedings.mlr.press/v55/riabiz16.html %V 55 %X It is well known that the α-stable distribution, while having no closed form density function in the general case, admits a Poisson series representation (PSR) in which the terms of the series are a function of the arrival times of a unit rate Poisson process. In our previous work we have shown how to carry out inference for regression models using this series representation, which leads to a very convenient conditionally Gaussian framework, amenable to straightforward Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual terms have been approximated in our previous work by a Gaussian distribution with fully characterised moments. In this paper we present a new Central Limit Theorem (CLT) for the residual terms which serves to justify our previous approximation of the residual as Gaussian. Furthermore, we provide an analysis of the asymptotic convergence rate expressed in the CLT.
RIS
TY - CPAPER TI - A central limit theorem with application to inference in α-stable regression models AU - Marina Riabiz AU - Tohid Ardeshiri AU - Simon Godsill BT - Proceedings of the Time Series Workshop at NIPS 2016 DA - 2017/02/16 ED - Oren Anava ED - Azadeh Khaleghi ED - Marco Cuturi ED - Vitaly Kuznetsov ED - Alexander Rakhlin ID - pmlr-v55-riabiz16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 55 SP - 70 EP - 82 L1 - http://proceedings.mlr.press/v55/riabiz16.pdf UR - https://proceedings.mlr.press/v55/riabiz16.html AB - It is well known that the α-stable distribution, while having no closed form density function in the general case, admits a Poisson series representation (PSR) in which the terms of the series are a function of the arrival times of a unit rate Poisson process. In our previous work we have shown how to carry out inference for regression models using this series representation, which leads to a very convenient conditionally Gaussian framework, amenable to straightforward Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual terms have been approximated in our previous work by a Gaussian distribution with fully characterised moments. In this paper we present a new Central Limit Theorem (CLT) for the residual terms which serves to justify our previous approximation of the residual as Gaussian. Furthermore, we provide an analysis of the asymptotic convergence rate expressed in the CLT. ER -
APA
Riabiz, M., Ardeshiri, T. & Godsill, S.. (2017). A central limit theorem with application to inference in α-stable regression models. Proceedings of the Time Series Workshop at NIPS 2016, in Proceedings of Machine Learning Research 55:70-82 Available from https://proceedings.mlr.press/v55/riabiz16.html.

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