컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0251 |
|---|---|
| 분류(Section) | Special Session |
| 분과(Session) | Structured nonparametric and high-dimensional statistics (SS-15) |
| 영문제목 (Title(Eng.)) |
Optimal estimation of sparse high-dimensional additive models |
| 저자(Author(s)) |
Karl Gregory1, Enno Mammen2, Martin Wahl3 University of South Carolina1, Heidelberg University2, Humboldt-Universit\"at zu Berlin3 |
| 초록본문(Abstract) | In this talk we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + \varepsilon$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + \varepsilon$, for which the additive components $f_2,\dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ in the additive model and state finite-sample bounds for the difference between our estimator and some smoothing estimators $\hat f_1^{\text{(oracle)}}$ in the oracle model which satisfy mild conditions. In an asymptotic setting these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimators; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of $f_2,\dots,f_q$ has no effect on estimation accuracy. Our first step is to estimate all of the components in the additive model with undersmoothing using a group-Lasso estimator. We then construct pseudo responses $\hat Y$ by evaluating a desparsified modification of our undersmoothed estimator of $f_1$ at the design points. In the second step the smoothing method of the oracle estimator $\hat f_1^{\text{(oracle)}}$ is applied to a nonparametric regression problem with ``responses'' $\hat Y$ and covariates $X_1$. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We also present simulation results demonstrating close-to-oracle performance of our estimator in practical applications. The main results of the paper are also important for understanding the behavior of the presmoothing estimator when the resmoothing step is omitted. |
| 분류기호 (MSC number(s)) |
62G08, 62G20 |
| 키워드(Keyword(s)) | nonparametric curve estimation, additive models, penalization, Lasso, variable selection, dimension reduction |
| 강연 형태 (Language of Session (Talk)) |
English |