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Vietnam
Journal of Mathematics 37:2&3 (2009) 315-337
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A Survey of Some Current Statistical Research
Problems
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Nguyen Trung Hung
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Abstract. The goal of this survey paper is to
point out where mathematical research problems in statistics come from! As
statistics is a science of uncertainty analysis, it is a methodology and a
tool for all real world problems in which uncertainty is a fact of life.
These include both physical and social domains.
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In a general sense, statistics is used for prediction purposes, either
through time series models or regression models. While a standard theory of
statistics is available, such as Bayesian statistics, U-statistics
and logistic regression, research arises when we embark on a specific
domain of applications. In each domain of applications, e.g. biology,
financial economics, mathematical finance, control engineering,... at least
two distinguished features show up, namely the structured equations due to
the theory itself, and more importantly, the type of data available. The
important field of econometrics is a typical example in which structured
equations come from economic principles, and data (time series, cross
section or panel data) are coarse in various possible ways, such as
missing, censored, hidden, partially observed...
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We will discuss research problems in theoretical statistics which arise
mainly from various types of coarse data, especially in economics.
These include statistical decision theory (based upon Von Neumann's utility
theory) in the context of financial investments and portfolio selection via
the theory of stochastic dominance and risk management, qualitative choice
problems with latent dependent variables (generalized linear models,
probit, logit and tobit regression models), James Heckman's sample
selection models, hidden Markov models in biology, indirect observed data
in auction theory (via Nash's equilibrium concepts in games with incomplete
information). The exposition is tutorial in nature.
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2000 Mathematics Subject Classification: 62-01, 62G05
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Keywords: Censored, hidden Markov,
measurement-error, missing, random set, unobserved data.
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Vietnam Academy of Science and Technology & Vietnamese
Mathematical Society
©
Copyright 2009 Vietnam
Journal of Mathematics. All rights reserved.
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