Name: Paulo Roberto Prezotti Filho
Type: PhD thesis
Publication date: 26/02/2019

Namesort descending Role
Valdério Anselmo Reisen Advisor *

Examining board:

Namesort descending Role
Alexandre Renaux External Examiner *
Jane Meri Santos Internal Examiner *
Pascal Bondon Co advisor *
Paulo Jorge Canas Rodrigues External Examiner *
Pierre-Olivier External Examiner *
Valdério Anselmo Reisen Advisor *

Summary: manuscript deals with some extensions to time series taking integer values of the autoregressive periodic parametric model established for
series taking real values. The models we consider are
based on the use of the operator of Steutel and Van
Harn (1979) and generalize the stationary integer autoregressive process (INAR) introduced by Al-Osh &
Alzaid (1987) to periodically correlated counting series. These generalizations include the introduction of
a periodic operator, the taking into account of a more
complex autocorrelation structure whose order is higher than one, the appearance of innovations of periodic variances but also at zero inflation by relation
to a discrete law given in the family of exponential
distributions, as well as the use of explanatory covariates. These extensions greatly enrich the applicability domain of INAR type models. On the theoretical
level, we establish mathematical properties of our models such as the existence, the uniqueness, the periodic stationarity of solutions to the equations defining
the models. We propose three methods for estimating
model parameters, including a method of moments
based on Yule-Walker equations (YW), a conditional
least squares method, and a quasi-maximum likelihood method (QML) based on the maximization of
a Gaussian likelihood. We establish the consistency
and asymptotic normality of these estimation procedures. Monte Carlo simulations illustrate their behavior for different finite sample sizes. The models are
then adjusted to real data and used for prediction purposes. The first extension of the INAR model that we
propose consists of introducing two periodic operators
of Steutel and Van Harn, one modeling the partial autocorrelations of order one on each period and the
other capturing the periodic seasonality of the data.
Through a vector representation of the process, we
establish the conditions of existence and uniqueness
of a solution periodically correlated to the equations
defining the model. In the case WHERE the innovations
follow Poisson’s laws, we study the marginal law of
the process. As an example of real-world application,
we are adjusting this model to daily count data on
the number of people who received antibiotics for the
treatment of respiratory diseases in the Vitoria region ´
in Brazil. Because respiratory conditions are strongly
correlated with air pollution and weather, the correlation pattern of the daily numbers of people receiving antibiotics shows, among other characteristics,
weekly periodicity and seasonality. We then extend
this model to data with periodic partial autocorrelations of order higher than one. We study the statistical
properties of the model, such as mean, variance, marginal and joined distributions. We are adjusting this
model to the daily number of people receiving emergency service from the public hospital of the municipality of Vitoria for treatment of asthma. Finally, our ´
last extension deals with the introduction of innovations according to a Poisson law with zero inflation
whose parameters vary periodically, and on the addition of covariates explaining the logarithm of the intensity of the Poisson’s law. We establish some statistical properties of the model, and we use the QML
method to estimate its parameters. Finally, we apply
this modeling to daily data of the number of people
who have visited a hospital’s emergency department
for respiratory problems, and we use the concentration of a pollutant in the same geographical area as a

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