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Local quadratic estimation of the curvature in a functional single index model

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  • Zi Ye
  • Giles Hooker

Abstract

The nonlinear responses of species to environmental variability can play an important role in the maintenance of ecological diversity. Nonetheless, many models use parametric nonlinear terms which pre‐determine the ecological conclusions. Motivated by this concern, we study the estimate of the second derivative (curvature) of the link function in a functional single index model. Since the coefficient function and the link function are both unknown, the estimate is expressed as a nested optimization. We first estimate the coefficient function by minimizing squared error where the link function is estimated with a Nadaraya‐Watson estimator for each candidate coefficient function. The first and second derivatives of the link function are then estimated via local‐quadratic regression using the estimated coefficient function. In this paper, we derive a convergence rate for the curvature of the nonlinear response. In addition, we prove that the argument of the linear predictor can be estimated root‐n consistently. However, practical implementation of the method requires solving a nonlinear optimization problem, and our results show that the estimates of the link function and the coefficient function are quite sensitive to the choices of starting values.

Suggested Citation

  • Zi Ye & Giles Hooker, 2020. "Local quadratic estimation of the curvature in a functional single index model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1307-1338, December.
    • Handle: RePEc:bla:scjsta:v:47:y:2020:i:4:p:1307-1338
    DOI: 10.1111/sjos.12481
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    References listed on IDEAS

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    1. Escabias, M. & Aguilera, A.M. & Valderrama, M.J., 2007. "Functional PLS logit regression model," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4891-4902, June.
    2. Stoker, Thomas M, 1986. "Consistent Estimation of Scaled Coefficients," Econometrica, Econometric Society, vol. 54(6), pages 1461-1481, November.
    3. Gareth M. James, 2002. "Generalized linear models with functional predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 411-432, August.
    4. Shujie Ma, 2016. "Estimation and inference in functional single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 181-208, February.
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    1. Zi Ye & Giles Hooker & Stephen P. Ellner, 2021. "Generalized Single Index Models and Jensen Effects on Reproduction and Survival," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(3), pages 492-512, September.

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