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Vietnam Journal of Mathematics 37:2&3 (2009) 315-337 

A Survey of Some Current Statistical Research Problems

Nguyen Trung Hung

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.

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...

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.

2000 Mathematics Subject Classification: 62-01, 62G05

Keywords: Censored, hidden Markov, measurement-error, missing, random set, unobserved data.

 

 

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