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Robust Inference for Non-Linear Regression Models with Applications in Enzyme Kinetics

Suryasis Jana, Abhik Ghosh

TL;DR

This paper addresses robust inference for nonlinear regression models, with a focus on enzyme kinetics exemplified by Michaelis–Menten. It introduces minimum density power divergence estimators (MDPDE) with tuning α to balance efficiency and robustness, and derives their consistency, asymptotic normality, and influence functions. It further develops robust Wald-type tests based on the MDPDE, analyzes their robustness through second-order influence functions, and demonstrates favorable finite-sample performance under contamination. Applying the framework to the MM model shows that MDPDEs provide more reliable parameter estimates and model-validation tests than classical methods in the presence of outliers, as evidenced by real-data examples (ONPG and drug-concentration datasets). The work offers practical, data-driven guidance for selecting α and highlights implications for robust inference in nonlinear regression relevant to biochemical kinetics and broader applications.

Abstract

Despite linear regression being the most popular statistical modelling technique, in real-life we often need to deal with situations where the true relationship between the response and the covariates is nonlinear in parameters. In such cases one needs to adopt appropriate non-linear regression (NLR) analysis, having wider applications in biochemical and medical studies among many others. In this paper we propose a new improved robust estimation and testing methodologies for general NLR models based on the minimum density power divergence approach and apply our proposal to analyze the widely popular Michaelis-Menten (MM) model in enzyme kinetics. We establish the asymptotic properties of our proposed estimator and tests, along with their theoretical robustness characteristics through influence function analysis. For the particular MM model, we have further empirically justified the robustness and the efficiency of our proposed estimator and the testing procedure through extensive simulation studies and several interesting real data examples of enzyme-catalyzed (biochemical) reactions.

Robust Inference for Non-Linear Regression Models with Applications in Enzyme Kinetics

TL;DR

This paper addresses robust inference for nonlinear regression models, with a focus on enzyme kinetics exemplified by Michaelis–Menten. It introduces minimum density power divergence estimators (MDPDE) with tuning α to balance efficiency and robustness, and derives their consistency, asymptotic normality, and influence functions. It further develops robust Wald-type tests based on the MDPDE, analyzes their robustness through second-order influence functions, and demonstrates favorable finite-sample performance under contamination. Applying the framework to the MM model shows that MDPDEs provide more reliable parameter estimates and model-validation tests than classical methods in the presence of outliers, as evidenced by real-data examples (ONPG and drug-concentration datasets). The work offers practical, data-driven guidance for selecting α and highlights implications for robust inference in nonlinear regression relevant to biochemical kinetics and broader applications.

Abstract

Despite linear regression being the most popular statistical modelling technique, in real-life we often need to deal with situations where the true relationship between the response and the covariates is nonlinear in parameters. In such cases one needs to adopt appropriate non-linear regression (NLR) analysis, having wider applications in biochemical and medical studies among many others. In this paper we propose a new improved robust estimation and testing methodologies for general NLR models based on the minimum density power divergence approach and apply our proposal to analyze the widely popular Michaelis-Menten (MM) model in enzyme kinetics. We establish the asymptotic properties of our proposed estimator and tests, along with their theoretical robustness characteristics through influence function analysis. For the particular MM model, we have further empirically justified the robustness and the efficiency of our proposed estimator and the testing procedure through extensive simulation studies and several interesting real data examples of enzyme-catalyzed (biochemical) reactions.
Paper Structure (35 sections, 4 theorems, 92 equations, 10 figures, 12 tables)

This paper contains 35 sections, 4 theorems, 92 equations, 10 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Robust parameter estimation under general NLR models
  3. The MDPDE
  4. Asymptotic properties
  5. Influence function and sensitivity analysis of the proposed MDPDE
  6. Robust Wald-type tests based on the MDPDE
  7. General Wald-type tests of NLR model parameters
  8. Influence functions of the Wald-type tests
  9. Tests for scalar parameter against one-sided alternative
  10. Application to the Michaelis-Menten (MM) Model
  11. Asymptotic properties
  12. Influence functions
  13. Finite sample performances of the MDPDE
  14. Wald-type tests for validity of the Michaelis-Menten model
  15. On the choice of optimum tuning parameter $\alpha$
  16. ...and 20 more sections

Key Result

Theorem 2.1

Under the setup of the NLR model nlr, suppose that the true data-generating distributions belong to the corresponding parametric model families with the true parameter value being $\bm{\theta}_0 = \left(\bm{\beta}_0^T, \sigma_0^2\right)^T$ and Conditions (R1)-(R2) hold. Then, for any $\alpha\geq 0$,

Figures (10)

  • Figure 1: IFs of the MDPDEs of the parameters $\beta_1$, $\beta_2$, and $\sigma^2$ in the MM model, with contamination in all observations, at the same contamination point, i.e., for $\bm{t}^* = (t,\ldots,t)^T$.
  • Figure 2: Empirical levels (ELevel) of the robust Wald-type tests
  • Figure 3: Empirical powers (EPower) of the robust Wald-type tests
  • Figure 4: ONPG data and the plot of the fitted MM models using different methods
  • Figure 5: Drug concentration data and the plot of the fitted MM models using different methods
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Remark 2.1
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Lemma B.1
  • proof