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Behaviour of the Monotone Single Index Model Under Repeated Measurements

Author

Listed:
  • Fadoua Balabdaoui

    (Seminar for Statistics, ETH Zürich)

  • Cécile Durot

    (Modal’x, Université Paris Nanterre)

  • Hanna Jankowski

    (York University)

Abstract

The generalized linear model is an important method in the statistical toolkit. The isotonic single index model can be thought of as a further generalization whereby the link function is assumed to be monotone non-decreasing as opposed to known and fixed. Such a shape constraint is quite natural in many statistical problems, and is fulfilled by the usual generalized linear models. In this paper we consider inference in this model in the setting where repeated measurements of predictor values and associated responses are observed. This setting is encountered in medical studies and is very different from the one considered in the classical monotone single index model studied in the literature. Here, we use nonparametric maximum likelihood estimation to infer the unknown regression vector and link function. We present a detailed study of finite and asymptotic properties of this estimator and propose goodness-of-fit tests for the model. Through an extended simulation study, we show that the model has competitive predictive performance. We illustrate our estimation approach using a Leukemia data set.

Suggested Citation

  • Fadoua Balabdaoui & Cécile Durot & Hanna Jankowski, 2023. "Behaviour of the Monotone Single Index Model Under Repeated Measurements," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 324-350, February.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00250-7
    DOI: 10.1007/s13171-021-00250-7
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    References listed on IDEAS

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    1. Durot, Cécile, 2003. "A Kolmogorov-type test for monotonicity of regression," Statistics & Probability Letters, Elsevier, vol. 63(4), pages 425-433, July.
    2. Hayfield, Tristen & Racine, Jeffrey S., 2008. "Nonparametric Econometrics: The np Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i05).
    3. Maiying Kong & J. Jack Lee, 2006. "A Generalized Response Surface Model with Varying Relative Potency for Assessing Drug Interaction," Biometrics, The International Biometric Society, vol. 62(4), pages 986-995, December.
    4. Klein, Roger W & Spady, Richard H, 1993. "An Efficient Semiparametric Estimator for Binary Response Models," Econometrica, Econometric Society, vol. 61(2), pages 387-421, March.
    5. Yining Chen & Richard J. Samworth, 2016. "Generalized additive and index models with shape constraints," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 729-754, September.
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