This dissertation studies a class of exponential family models whose canonical parameters are specified as linear functionals of an unknown infinite-dimensional slope function. The optimal rates of convergence for estimating the slope function and its linear functionals are established. The problem of estimating linear functionals of the slope function is regarded as prediction problem.
The minimax lower bounds for rates of convergence for the estimation problem and the prediction problem are derived by a construction of a sequence of least favorable distributions and applications of Assouad's lemma and Le Cam's method.
The estimators that achieve the optimal rates are constructed based on the constrained maximum likelihood estimation (CMLE) whose dimension grows with sample size. Effectively, the CMLE was used by Hall and Horowitz (2007) and Cai and Hall (2006) to achieve the optimal minimax rates in the linear case. However, for general exponential families, new difficulties emerge in the analysis of the CMLE as a result of the presence of an asymptotic bias term. In order to overcome the extra hurdles, a change-of-measure argument is developed, inspired by the ideas from Le Cam's theory of asymptotic equivalence. More precisely, the change-of-measure argument approximates the original infinite-dimensional model with a sequence of finite-dimensional models whose dimensions increase with sample size. The sequence of approximations is chosen so that the CMLE of the original model is the MLE for the approximations, and hence the asymptotic bias term is eliminated automatically.
Practically, a functional logistic regression model is applied to the economic problem of predicting the occurrence of U.S. recessions from U.S. Treasury zero-coupon yield curves.
|Advisor:||Pollard, David, Zhou, Harrison H.|
|School Location:||United States -- Connecticut|
|Source:||DAI-B 72/02, Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Statistics, Economics|
|Keywords:||Compact operators, Constrained maximum likelihood estimation, Exponential families, Yield curves|
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