mathematics Article Robust Estimation and Tests for Parameters of Some Nonlinear Regression Models Pengfei Liu 1,2,† , Mengchen Zhang 3,*,† , Ru Zhang 1,4,*,† and Qin Zhou 1,2,† 1 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China; [email protected] (P.L.); [email protected] (Q.Z.) 2 Research Institute of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, China 3 Department of Public Administration and Policy, College of Public Affairs, National Taipei University, Taipei 237, Taiwan 4 Jiangsu Provincial Key Laboratory of Educational Big Data Science and Engineering, Jiangsu Normal University, Xuzhou 221116, China * Correspondence: [email protected] (M.Z.); [email protected] (R.Z.) † The author order is alphabetically sorted. These authors contributed equally to this work. And the submit author is Ru Zhang. Abstract: This paper uses the median-of-means (MOM) method to estimate the parameters of the nonlinear regression models and proves the consistency and asymptotic normality of the MOM estimator. Especially when there are outliers, the MOM estimator is more robust than nonlinear least squares (NLS) estimator and empirical likelihood (EL) estimator. On this basis, we propose hypothesis testing Statistics for the parameters of the nonlinear regression models using empirical likelihood method, and the simulation performance shows the superiority of MOM estimator. We apply the MOM method to analyze the top 50 data of GDP of China in 2019. The result shows that MOM method is more feasible than NLS estimator and EL estimator. Citation: Liu, P.; Zhang, M.; Zhang, Keywords: median-of-means (MOM); nonlinear regression (NR); empirical likelihood (EL); R.; Zhou, Q. Robust Estimation and hypothesis testing (HT) Tests for Parameters of Some Nonlinear Regression Models. Mathematics 2021, 9, 599. https:// doi.org/10.3390/math9060599 1. Introduction A nonlinear regression model refers to a regression model in which the relationship Academic Editor: Vasile Preda between variables is not linear. Nonlinear regression model has been widely used in various disciplines. For instance, Hong [1] applied a nonlinear regression model to the economic Received: 15 February 2021 system prediction; Wang et al. [2] studied the application of nonlinear regression model Accepted: 5 March 2021 Published: 11 March 2021 in the detection of protein layer thickness; Chen et al. [3] utilized a nonlinear regression model in the price estimation of surface-to-air missiles; Archontoulis and Miguez [4] used Publisher’s Note: MDPI stays neutral a nonlinear regression model in agricultural research. with regard to jurisdictional claims in The principle of median-of-means (MOM) was firstly introduced by Alon, Matias, published maps and institutional affil- and Szegedy [5] in order to approximate the frequency moment with space complexity. iations. Lecue´ and Lerasle [6] proposed new estimators for robust machine learning based on MOM estimators of the mean of real-valued random variables. These estimators achieved optimal rates of convergence under minimal assumptions on the dataset. Lecue´ et al. [7] proposed MOM minimizers estimator based on MOM method. The MOM minimizers estimator is very effective when the instantaneous hypothesis may have been corrupted Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. by some outliers. Zhang and Liu [8] applied MOM method to estimate the parameters in This article is an open access article multiple linear regression models and AR error models of repeated measurement data. distributed under the terms and For unknown parameters of a nonlinear regression model, Radchenko [9] proposed an conditions of the Creative Commons estimator named nonlinear least square to approximate the unknown parameters. Ding [10] Attribution (CC BY) license (https:// introduced the empirical likelihood (EL) estimator of the parameter of the nonlinear creativecommons.org/licenses/by/ regression model based on the empirical likelihood method. However, when there are 4.0/). outliers, the general methods are more sensitive and easily affected by the outliers based Mathematics 2021, 9, 599. https://doi.org/10.3390/math9060599 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 599 2 of 16 on Gao and Li [11]. On the basis of the study of Zhang and Liu [8], this paper applies the MOM method to estimate the parameters of the nonlinear regression models and receives more robust results. The paper is organized as follows: In Section2, we review the definition of the nonlinear regression model and introduce the MOM method specifically. We prove the consistency and asymptotic properties of the MOM estimator. In Section3, we introduce a new test method based on the empirical likelihood method for the median. Section4 illustrates the superiority of the MOM method with simulation studies. A real application to GDP data is given in Section5, and the conclusion is discussed in the last section. 2. Median-of-Means Method Applies to Nonlinear Regression Model We consider the following nonlinear regression model introduced by Wu [12] yi = g(q, xi) + #i i = 1, ..., T. (1) T where q = (q1, ..., qk) is a fixed k × 1 unknown parameter column vector. xi is the i-th “fixed” input vector with observation yi. g(q, xi) is a known functional form (usually 2 nonlinear). #i are i.i.d errors with 0 mean and s unknown variance. According to Zhang and Liu [8], MOM estimator of q is produced by the following steps: Step I: We seperate (yi, xi), i = 1, ..., T into g groups. The number of observations in each group is n = T/g (Usually for the convenience of calculation, we assume that T is always divisible by g). We discuss the choice of grouping number g. According to the 1 suggestion by Emilien et al. [13], g = d8 × log( z )e for any z 2 (0, 1), where dhe is the ceiling function. In fact, the structure of observations is always unknown, and the diagnosis of outliers is complicated. Therefore, we usually set z = pC for some constant C regardless T of outliers. Step II: We estimate the parameter q in each group j, 1 ≤ j ≤ g by the nonlinear least (j) (j) (j) T square estimator qb = (qb1 , ..., qbk ) . MOM MOM MOM T Step III: The MOM estimator of qb = (qb1 , ..., qbk ) is defined, where MOM (1) (g) qbq = median(qbq , ..., qbq ), q = 1, ..., k. The asymptotic properties of qbMOM are summarized in the following theorems. Their proofs are postponed to AppendixA. Theorem 1. For some constant C and any positive integer g, we suppose the following: (I) For certain 0 < a < b < ¥ and any q1, q2 2 Q, Q is is an open interval (finite or infinite) of the real axis E1. jn(q1, q2) = ¥ for q1 6= q2 if at least one of the points q1, q2 is −¥ or ¥ ). i = 1, ..., n. n 2 1 2 2 a(q1 − q2) ≤ jn(q1, q2) = ∑[g(q1, xi) − g(q2, xi)] ≤ b(q1 − q2) n i=1 s Suppose Ej#1j < ¥ for some s ≥ 2. For n ≥ N0 and sufficiently large positive r, c does not depend on n and r. 0 00 (II) g (q0, xi), g (q0, xi) exist for all q0 near qq, q = 1, ..., k, the true value qq is in the interior of q0, and n 1 0 2 lim sup ∑[(g (qq, xi)) ] = S 0 < S < ¥ n!¥ n i=1 (III) There exits qu 2 Q as n ! ¥ and jqu − qqj ! 0. n 0 2 ∑i=1f(g (qn, xi)) g lim 0 = 1 n!¥ n 2 ∑i=1f(g (qq, xi)) g Mathematics 2021, 9, 599 3 of 16 (IV) There exits a d > 0 such that 1 n ¶2g(q , x ) lim sup f 0 i g2 < ¥ n!¥ n ∑ ¶q2 i=1 jq0−qqj≤d 0 for all i = 1,...,n, where s = fqq 2 Q, jq0 − qqj ≤ dg. According to conditions I ∼ IV, for any fixed x > 0, we can get MOM C P(jqb − qqj ≥ x) ≤ . (2) q (T/g)g/5 Theorem 2. (1) Suppose g is fixed and s 6= 0. Let Q1, Q2, ..., Qg be i.i.d standard normal random variables. When T ! ¥, p n d −! medianfQ , ..., Qgg. (3) − 1 1 bsnS 2 (2) Suppose T/g2 ! ¥ as g ! ¥ and s 6= 0. Afterwards the following asymptotic normal holds p T d p −! 2/pN(0, 1). (4) − 1 bsnS 2 3. Empirical Likelihood Test Based on MOM Method In Section2, this paper uses the MOM method to estimate the parameters of the nonlinear regression model. In this section, we consider the hypothesis test that q equals a given value parameter based on the empirical likelihood method. (1) (2) (j) Because different groups are disjoint, qbq , qbq , ..., qbq , j = 1, ..., g, q = 1, ..., k are i.i.d. We treat them as a sample and apply empirical likelihood. For each j, we say (j) −1/2 Tn,j = I(qbq ≤ qq). Obviously, ETn,j ≈ 0.5. In fact, ETn,j − 0.5 = O(n ) ( by the process of proof of Theorem in the AppendixA. Given restrictive conditions, the empirical likelihood ratio of q is g g g R(q) = maxf∏ gwjj ∑ wjTn,j = 0.5, wj ≥ 0, ∑ wj = 1g. (5) j=1 j=1 j=1 Using the Lagrange multiplier to find the maximum point we obtained the following equation. 1 1 wj = g 1 + l(Tn,j − 0.5) where l = l(q) satisfies the equation g 1 Tn,j − 0.5 0 = . (6) ∑ + ( − ) g j=1 1 l Tn,j 0.5 Theorem 3. According to Theorem 2 and Owen [14], as g, n ! ¥, we have d 2 −2logR(q) −! c1. (7) Using the Theorem 3, the rejection region for the hypothesis with significance level a (0 < a < 1) H0 : q = q0 vs. H1 : q 6= q0 can be constructed as 2 R := f−2logR(q) > c1(a)g Mathematics 2021, 9, 599 4 of 16 2 2 where c1(a) is the upper a-th quantile of c1.