Recent Developments in the Study of Single-Index Type Models

With Mouli Banerjee, University of Michigan

Recent Developments in the Study of Single-Index Type Models

Single-index type models are popular in statistics, biostatistics and economics as they alleviate the curse of dimensionality to a considerable extent while allowing for broad classes of models through the dependence on an unknown link function. Various classes of single index models are known: in regular models under a fixed dimension setting [i.e. fixed number of regression parameters], the regression parameter is √n estimable; under current status type censoring of the response variable in a linear regression model — which leads to the binary choice model — one gets a generalized single-index model where the rate of estimation is at most n^{1/3}, a problem well-studied in the econometrics literature (by Manski and subsequent authors). Single index structures with a discontinuous link function arise in models involving change-planes in multidimensional space — hyperplanes that separate two (or more) response or survival regimes — and are relevant to applications in personalized medicine and dynamic treatment regimes. Here, rates of estimation can easily exceed √n in the finite dimensional case.

I will talk about some of my recent work in the above class of models in growing and high dimensional settings focusing on how growing dimensions introduce significantly new challenges at both theoretical and computational levels and present some recent results on convergence, minimax-optimal rates, and inference, as well as future challenges.

The talk is based on joint work with Ya’acov Ritov, Hamid Eftekhari, Zhiyuan Lu and Debarghya Mukherjee.

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