Profile Least Squares Estimators in the Monotone Single Index Model

Fadoua Balabdaoui () and Piet Groeneboom ()
Additional contact information
Fadoua Balabdaoui: Seminar für Statistik ETH
Piet Groeneboom: Delft University of Technology, DIAM

A chapter in Advances in Contemporary Statistics and Econometrics, 2021, pp 3-22 from Springer

Abstract: Abstract We consider least squares estimators of the finite regression parameter $$\boldsymbol{\alpha }$$ α in the single index regression model $$Y=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})+\varepsilon $$ Y = ψ ( α T X ) + ε , where $$\boldsymbol{X}$$ X is a d-dimensional random vector, $${\mathbb E}(Y|\boldsymbol{X})=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})$$ E ( Y | X ) = ψ ( α T X ) , and $$\psi $$ ψ is a monotone. It has been suggested to estimate $$\boldsymbol{\alpha }$$ α by a profile least squares estimator, minimizing $$\sum _{i=1}^n(Y_i-\psi (\boldsymbol{\alpha }^T\boldsymbol{X}_i))^2$$ ∑ i = 1 n ( Y i - ψ ( α T X i ) ) 2 over monotone $$\psi $$ ψ and $$\boldsymbol{\alpha }$$ α on the boundary $$\mathcal {S}_{d-1}$$ S d - 1 of the unit ball. Although this suggestion has been around for a long time, it is still unknown whether the estimate is $$\sqrt{n}$$ n -convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed $$\boldsymbol{\alpha }$$ α , but using a different global sum of squares, is $$\sqrt{n}$$ n -convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.

Date: 2021
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-73249-3_1

Ordering information: This item can be ordered from
http://www.springer.com/9783030732493

DOI: 10.1007/978-3-030-73249-3_1

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().