Working Hyphoteses
This project is based on the increasing literature dealing with
regression models under nonstandard assumptions, the possibility
of computational implementation related to the use of MCMC
techniques and, on the other hand, on the implementation of
nonstandard computational techniques. Moreover, there is also a
need of solving problems related to local and regional data
especially in finance, environmental contamination and
epidemiology using up to date methodology.
Additionally, as mentioned previously, the project is a natural
follow up to Fondecyt projects executed in the last eight years and
also resulting from international cooperation, developed
independently and jointly by the authors in the last decade.
Objetivos (Generales y Específicos)
Objetivos Generales:
O1: Develop Bayesian parametric and semiparametric solutions
for decision and inference problems related to regression models,
discrete regression models (binary and Poisson), measurement
error models and comparative calibration models.
O2: Apply the results to problems related to finance (Chilean
stock market), environmental contamination (model described in
PM2.5 in Santiago, Chile) and health problems (such as diabetes,
and cancer related problems in the Chilean population).
O3: Construct random probability measures which deal with
different types of asymmetry on the data.
Transversal objectives:
OT1: Study computational strategies related to the
implementation of bayesian solutions in situations where no
explicit solutions are available
OT2: Accomplish simulation studies to evaluate the performance
of the models when to obtain results from theoretical
development is not possible.
Specific objectives
Specific objectives detailed according to the directions established
in the proposal are:
D0: Calibration, change points, diagnostic analysis in elliptical
and skew elliptical regression models:
a) Describe the results developed for the calibration problem in
regression models using a Bayesian nonparametric
approach.
b) Implement parametric and nonparametric bayesian solutions
for regression models under assumptions typically
considered in CAPM theory in the Chilean stock market.
c) Deal with the calibration problem considering families of
asymmetric distributions including comparative model
selection under different families of models and approaches
(parametric versus semiparametric).
d) Deal with cluster analysis problems and change points
problems considering asymmetric distributions with location
scale parameters. As a subproduct, we deal with the topic of
density Bayesian estimation using product partition models
and Dirichlet processes.
e) Compute divergence measures to evaluate robustness for
different of asymmetric families of distributions and apply
the results to simulation studies with finance applications .
D1: Calibration, change point problems and diagnostic analysis
considering parametric and semiparametric models with
applications in citogenetic dosimetry problems. The main
objectives are:
a) Develop binary calibration models considering parametric
and semiparametric models for the link function (special
symmetric distributions or the normal mixture class or the
symmetric uniform class), comparing the distinct modeling
alternatives with a view to dealing with the calibration
problem.
b) Develop Bayesian procedures for comparing parametric and
nonparametric alternatives to modeling.
c) Study bayesian solutions to calibration problems
considering mixtures of asymmetric distributions in the
scale parameter directed to Dirichlet processes and
comparing results with the consideration of link functions
following asymmetric Dirichlet processes.
d) Explore the possibility of extending the results to correlated
binary problems as, for example, the autologistic binary
regression model.
e) In 1)-4), conduct simulation studies and applications related
to citogenetic dosimetry problems..
D2: Quantile regression in measurement error models (MEM):
Inference and diagnostic problems related to environmental
contamination..
a) Obtain Bayesian solutions for parameter estimation in MEM
when the error terms follow mixtures of normals or spherical
distributions for the scale parameter of mixtures of Dirichlet
processes.
a) Compute measures related to the evaluation of influent
observations with parametric and semiparametric
alternatives.
b) Obtain bayesian solutions for parameter estimation in MEM
when the error term has a distribution in a class of
asymmetric distributions with zero p-quantile. Moreover,
the measurement error follows a symmetric distribution.
c) In b) compute measures to evaluate asymmetry of the error
term in the regression model.
a. To obtain bayesian solution for parameter estimation
in MEM when the error term of the regresión model
follows a mixture of distributions directed by a DP.
b. In e) compute measures related to the evaluation of
asymmetry with parametric and semiparametric
alternatives..
In a)-f) conduct studies elliptical symmetric and asymmetric
models with applications to environmental contamination
problems.
D3: Elliptical and skew-elliptical comparative calibration models
with applications in the Chilean stock market (Proposta de
Manuel con Maria Paz).
a)
b) Obtain bayesian solutions for estimation, hypothesis
testing and diagnostic analysis when the error term
follows a parametric family of distributions (as, for
example, the Student-t, power exponential, uniform,
etc.)
c) Obtain bayesian solutions for estimation, hypothesis
testing and diagnostic analysis when the error term
follows a mixture of distributions directed by a DP.
d) Evaluate influence analysis and error term symmetry
based on divergence measures based on Bayes factors
and predictive distributions for the error terms.
e) Explore bayesian solutions when the error terms
follows asymmetric distributions with zero median or,
in general, with zero o-quantile.
f) Conduct simulation studies and applications for the
Chilean stock market and measurement error models.
D4: Hierarchical Poisson models for studying spatial-temporal
rates for the incidence of type I diabetes, lung and stomach cancer
in Chilean subpopulations between 1997 and 2005. It is our
intention to aggregate model comparison, robustness and
diagnostic analysis to such models.
a) Literature revision on problems related to the spatiotemporal rate (risk)of incidence or rare disease
b)
found in the literature, but very few papers deal with the approach
for the Chilean population. Some exceptions are the master thesis
by Nora Díaz and Massiel Orellana related the research problems
developed by Gloria Icaza). In this direction, we consider models
of the type:
c) Use Bayesian methodology to study spatio-temporal
distribution of insulin-dependent diabetes mellitus (IDDM)
among children less than 15 years old in the urban area of
Santiago-Chile.
a) Use bayesian methodology available in the literature, to study
spatio–temporal distribution of lung and stomach cancer
among Chilean population. Conduct a comparative study
between different modeling alternatives.
b) Explore aspects such as robustness and diagnostic analysis of
the proposed solutions considering model failure as, for
example, nonnormality, influent observations and so on.
c) Propose bayesian solutions for asymmetry in the distribution of
random effects.
a) It is our intention to develop Bayesian methodology to
implement a spatio-temporal bivariate Poisson model, with
random effects. Apply results to cancer data for specific
time intervals.
b) Develop Bayesian methodology to implement a spatiotemporal bivariate Poisson model, with application to
cancer data.
c) Conduct comparison analysis for the different alternative
modelations.
D5: Asymmetric Dirichlet processes: Properties, extensions and
semiparametric bayesian modeling applications. Consider
distribution families more flexible than the normal distribution
has been subject to great development in recent years. We follow
ideas presented in Iglesias et al. (2006).
a) Study alternative representation such as the ones in
Sethuraman (1994) for Polya Urn models.
b) Study properties of the proposed models such as moments,
density, conjugation, density estimation and so on.
c) Explore extensions for asymmetry in Rk, considering
mixtures of DP.
Metodology
The methodology used in the project is the one typically used in
such models and applications. That is, the developments related to
the theory, although being mathematical to a certain degree, there
is no specific one to follow. Except the one related to an extensive
bibliography review related to the topics in the project. Related to
the applications, we can say that there should be a close relation
and interaction with the main investigators in specific areas of
interest encompassing hypothesis formulation, depuration,
exploratory data analysis, computational implementation and
model fitting.
En cuanto a la implementación de los resultados, especialmente
cuando no sea posible obtener solucione analíticamente tratables,
dependerá de MCMC métodos
Working time table
During each year of the project duration it is planned to work
jointly in conducting undergraduate, master, and doctoral thesis,
in writing papers, presenting paper in congresses and meetings.
Specifically it is planned to work with the time table presented in
the following according to project objectives:
Dirección
D0
D1
D2
D3
D4
D5
Year Year
1
a)-b)
a)
a)-b)
a)-b)
a)-b)c)
a)-b)
Year 2
Year 3
Year 4
c)
b)
c)-d)
c)
d)
d)
c)
e)-f)
d)-e)
e)-f)
e)-f)
d)-e)
g)
g)
g)-h)
c)-d)
e)
e)-f)
Moreover, it is also planned to conduct periodical meetings
between the main investigators and also workshops involving
students that are working with the main investigators. It is also
expected to visit (and being visited) by the following foreigner
investigators Heleno Bolfarine from U. de Sao Paulo,
Brasil, Rosangela Loschi, Frederico Cruz, and RenatoAssunçao,
from Universidad Federal de Minas Gerais-Brasil and Fabrizio
Ruggeri de CNR-IMATI, Italia.
Titulo: Análisis bayesiano completo de modelos de regresión
bajo supuestos no estándar: errores en las variables, calibración
comparativa, modelos de regresión discretos y aplicaciones.
Title: Full bayesian analysis of regression models under
nonstandard assumptions: measurement error models,
comparative calibration, discrete regression models and
applications.
Summary
The main object of this project is to develop parametric and
semiparametric Bayesian solutions to problems related to
continuous and discrete regression models under nonstandard
conditions. This is understood as assumptions beyond normality
as, for example, the consideration of spherical normal
distributions (Student-t family, power exponential, discrete
uniform), distributions generated by mixtures of DP and families
of asymmetric distributions. With this in mind, it is our intention
to approach the following topics: I) calibration, change points,
diagnostic analysis in elliptical regression models with
application is CAPM. ii) Calibration, change points and
diagnostic in binary and skew elliptical regression models with
applications in citogenetic dosimetry problems. Iii) Quantile
regression in measurement error models: inference, and
diagnostic analysis in environmental contamination.
iv) Elliptical and skew elliptical comparative calibration models
with applications in the Chilean stock market.
v) Poisson spatio-temporal modeling for the study of rate of
incidence of type I diabetes, lung and stomach cancer in Chilean
subpopulations: model comparison analysis, robustness and
diagnostic analysis.
vi) Asymmetric DPs: properties, extensions and applications in
Bayesian nonparametric modeling.
The study includes in each specific topic sensitivity analysis,
model comparison, computational implementation aspects,
applications and theoretical developments related to the properties
Of the asymmetric distributions and symmetric and asymmetric
random probability measures.
Key words: bayesian analysis, regression models with and
without measurement errors, binary regression and Poisson,
calibration, comparative calibration models, symmetric and
asymmetric distributions, Dirichlet processes.