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On the population least‐squares criterion in the monotone single index model

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  • Fadoua Balabdaoui
  • Cécile Durot
  • Christopher Fragneau

Abstract

Monotone single index models have gained increasing popularity over the past decades due to their flexibility and versatile use in diverse areas. Semi‐parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.

Suggested Citation

  • Fadoua Balabdaoui & Cécile Durot & Christopher Fragneau, 2021. "On the population least‐squares criterion in the monotone single index model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(4), pages 408-436, November.
    • Handle: RePEc:bla:stanee:v:75:y:2021:i:4:p:408-436
    DOI: 10.1111/stan.12240
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    References listed on IDEAS

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    1. Fadoua Balabdaoui & Piet Groeneboom & Kim Hendrickx, 2019. "Score estimation in the monotone single‐index model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 46(2), pages 517-544, June.
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