Inverse Problems with Missing Data

• Activity Number: 4
• Type: Invited
• Date/Time: Sunday, August 4, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Sam Efromovich*+
• Companies: UTDallas
• Keywords: Adaptation ; Sharp Minimax ; Regression ; Density

Abstract

Minimax nonparametric estimation of regression and probability density in so-called inverse problems, whose ill-posedness implies dramatically slower rates of convergence, is well known for the case of complete data, and the corresponding literature is vast. On the other hand, for a practically important case of data with missing observations the literature is next to nonexistence. What can be done, in terms of sharp minimax estimation, if some observations are missed? Asymptotic theory of adaptive and efficient estimation, numerical simulations and applications are presented and discussed. Particular examples include: (i) Nonparametric regression with measurement errors in predictors where either responses or measurements of the predictor may be missed; (ii) Density deconvolution with missed observations. The latter is the case of the missing not at random (MNAR). In general the problem has no solution, and we discuss what extra information is needed for its solution.