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Series estimation for single‐index models under constraints
Author(s) -
Dong Chaohua,
Gao Jiti,
Peng Bin
Publication year - 2019
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/anzs.12274
Subject(s) - estimator , mathematics , hermite polynomials , monte carlo method , parametric statistics , series (stratigraphy) , function (biology) , constraint (computer aided design) , index (typography) , parameter space , function space , parametric model , mathematical optimization , statistics , mathematical analysis , computer science , paleontology , geometry , evolutionary biology , world wide web , biology
Summary In this paper, a semi‐parametric single‐index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pending issue in the literature. Meanwhile, the link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimisation with the constraint of identification condition for the index parameter, which addresses an important problem in the literature of single‐index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimates are demonstrated through an extensive Monte Carlo study and an empirical example.