On and Off Semiparametric Models
The information lower bound theory of semiparametric models gives a
fairly systematic treatment concerning efficient estimation when a
semiparametric models holds. In some cases, especially problems involving
missing data, efficient estimators (i.e. estimators achieving the bounds)
are difficult to construct and evaluate, while various ad hoc estimators
are available which are much easier to compute and study, but which are
inefficient "on the model". What happens "off the model" when the
model fails to hold, perhaps just by a small amount? Is it possible for
the inefficient estimators to be more efficient on neighborhoods of the
semiparametric model than the "efficient estimators (on the model)"?
In this talk I will review some of the (old and new) literature on this
problem and present several recent results concerning estimation on
neighborhoods of semiparametric models. The main concern will be stability
of efficiency properties on local neighborhoods of semiparametric models.
Tracking Predictable Drifting Parameters of a Time Series
When analyzing data that arrive sequentially over time, it is important to detect changes in the underlying model which can then be adjusted accordingly. Such problems arise in many engineering, econometric and biomedical applications.
Consider a time series such that each observation has a (potentially different) conditional law, given the full past of the process. Suppose we are interested in certain predictable characteristics (e.g. a parameter, a functional) of these conditional distributions. We would therefore like to track a time-varying quantity, a predictable process with respect to the natural filtration.
The parametric formulation is the simplest particular case of this setting: the observations are independent and have a fixed distribution. The simplest nonparametric formulation deals with independent observations and a time-varying parameter. Markov chains with time varying transition laws are the next level of complexity.
Since data arrive in a successive manner, conventional methods based on samples of fixed size are not easy to use. Alternatively, stochastic recursive algorithms allow fast updating of parameter or state estimates at each instant as each new datum arrives. Our algorithm is based on the existence of a gain function - a (vector-valued) function of the previous estimate, a new observation and the prehistory of the time series -- which, roughly speaking, "pushes" the previous estimate towards the true parameter being tracked. A gain function, together with a step sequence and new observations from the model, can be used to iteratively adjust the current estimate, resulting in a recursive tracking algorithm.
To motivate our results we consider the problem of tracking conditional quantiles. We generalize the standard quantile regression model since we do not require observations to be independent. We also address the tracking of multidimensional analogs of the median, points of centrality or symmetry of multivariate distributions.
Based on a joint work with Eduard Belitser.
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