Name: Higor Henrique Aranda Cotta

Type: PhD thesis

Publication date: 22/08/2019

Advisor:

Name | Role |
---|---|

Valdério Anselmo Reisen | Advisor * |

Examining board:

Name | Role |
---|---|

Alexandre Renaux | External Examiner * |

Glaura da Conceição Franco | External Examiner * |

Márton Ispány | External Examiner * |

Neyval Costa Reis Jr. | Internal Examiner * |

Pascal Bondon | Advisor * |

Taciana Toledo de Almeida Albuquerque | Internal Examiner * |

Valdério Anselmo Reisen | Advisor * |

Wilfredo Omar Palma Manriquez | External Examiner * |

Summary: This manuscript proposes new robust estimation methods for the autocovariance and autocorrelation matrices functions of stationary multivariates time series that may have random additives outliers. These

functions play an important role in the identification

and estimation of time series model parameters. Random additive outliers can impact the level of one or

more components of the multivariate vector. This increases the overall variability of the series, which has

an impact on the periodogram matrix and leads to

a decrease in the values of the autocorrelation matrix function. We first propose new estimators of the

autocovariance and of autocorrelation matrices functions constructed using a spectral approach considering the periodogram matrix periodogram which is the

natural estimator of the spectral density matrix. As in

the case of the classic autocovariance and autocorrelation matrices functions estimators, these estimators are affected by aberrant observations. Thus, any

identification or estimation procedure using them is directly affected, which leads to erroneous conclusions.

To mitigate this problem, we propose the use of robust

statistical techniques to create estimators resistant to

aberrant random observations.

As a first step, we propose new estimators of autocovariance and autocorrelation functions of univariate

time series. The time and frequency domains are linked by the relationship between the autocovariance

function and the spectral density. As the periodogram

is sensitive to aberrant data, we get a robust estimator by replacing it with the M-periodogram. The

M-periodogram is obtained by replacing the Fourier

coefficients related to periodogram calculated by the

standard least squares regression with the ones calculated by the M-robust regression. The asymptotic

properties of estimators are established. Their performances are studied by means of numerical simulations for different sample sizes and different scenarios of contamination. The empirical results indicate

that the proposed methods provide close values of

those obtained by the classical autocorrelation function when the data is not contaminated and it is resistant to different contamination scenarios. Thus, the

estimators proposed in this thesis are alternative methods that can be used for time series with or without

outliers.

The estimators obtained for univariate time series are

then extended to the case of multivariate series. This

extension is simplified by the fact that the calculation

of the cross-periodogram only involves the Fourier coefficients of each component from the univariate series. Again, the duality relationship between time and

frequency domains is considered via the link between

the autocovariance matrix function and the spectral

density matrix stationary multivariate time series. The

M-periodogram matrix is a robust periodogram matrix

alternative to build robust estimators of the autocovariance and autocorrelation matrices functions. The

asymptotic properties are studied and numerical experiments are performed. As an example of an application with real data, we use the proposed functions

to adjust an autoregressive model by the Yule-Walker

method to Pollution data collected in the Vitoria re- ´

gion Brazil (particles smaller than 10 micrometers in

diameter, PM10).

Finally, the robust estimation of the number of factors in large factorial models is considered in order

to reduce the dimensionality. It is well known that the

values random additive outliers affect the covariance

and correlation matrices and the techniques that depend on the calculation of their eigenvalues and eigenvectors, such as the analysis principal components and the factor analysis, are affected. Thus, in

the presence of outliers, the information criteria proposed by Bai & Ng (2002) tend to overestimate the

number of factors. To alleviate this problem, we proposeto replace the standard covariance matrix with

the robust covariance matrix proposed in this manuscript. Our Monte Carlo simulations show that, in the

absence of contamination, the standard and robust

methods are equivalent. In the presence of outliers,

the number of estimated factors increases with the

non-robust methods while it remains the same using

robust methods. As an application with real data, we

study pollutant concentrations PM10 measured in the

Ile-de-France region of France.