Exponential parametric distortion nonlinear measurement errors Models

Abstract This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with exponential parametric distortion measurement errors. To estimate parameters in the distortion functions, we propose nonlinear least squares and weighted nonlinear least squares estimation methods under two identifiability conditions. After obtaining calibrated variables, the nonlinear least squares based estimators are proposed to estimate the parameters in the regression model. We studied the asymptotic results of estimators, especially we discuss the difference between the parametric calibrations and nonparametric calibrations. The latter is conducted as if the parametric structures in distortion functions are unknown. Simulation studies demonstrate the performance of the proposed estimators.


Introduction
Measurement errors are common in many disciplines, such as medical research, health science and social science, due to improper instrument calibration or many other reasons.Obviously, measurement error is a serious problem as it usually causes biased analyses and leads to incorrect conclusions.Measurement error can have severe consequences, and it is therefore essential to assess the quality of observed variables, statistical procedures and models in order to reduce such errors.The statistical estimation and inference for measurement errors models are challenging, various statistical methods have been proposed to deal with measurement errors.Research on measurement errors models has been widely studied, in this paper, we propose the following exponential parametric distortion measurement errors nonlinear model: Y ¼ / P ðUÞY, / P ðUÞ ¼ expða 0 þ d 0 UÞ, e X r ¼ w P r ðUÞX r , w P r ðUÞ ¼ expðg r0 þ h r0 UÞ r ¼ 1, :::, p, 8 > < > : (1.1) where, Y is an unobservable response variable, X ¼ ðX 1 , X 2 , :::, X p Þ > is an unobservable continuous covariate vector (the superscript ">" denotes the transpose operator throughout this paper), e Y and e X are the observed response variable and covariate vector.The function gðÁ, ÁÞ is a known parametric function, and the parameter b 0 is an unknown q Â 1 parameter vector on a compact parameter space H b & R q : The model error e satisfies EðejXÞ ¼ 0 and Eðe 2 jXÞ < 1: The confounding variable U 2 R 1 is observable and independent of ðX > , YÞ > , and the parameters ða 0 , d 0 , g r0 , h r0 Þ > are unknown parameters to be estimated.In model Equation (1.1), the distortion functions / P ðUÞ, w P r ðUÞ amplify or shrink the unobserved Y, X r in exponential multiplicative fashions.
The existing multiplicative distortion measurement errors in the literature are focused on nonparametric distortion functions e Y ¼ /ðUÞY, e X r ¼ w r ðUÞX r , and /ðUÞ and w r ðUÞ are unknown smoothing function of the confounding variable U.There is no literature to consider the parametric distortion measurement error models Equation (1.1).The multiplicative distortion measurement data usually occur in biomedical research and health-related studies, and the research on the multiplicative distortion measurement errors under nonparametric distortion functions has been well studied in Cui et al. (2009); Delaigle, Hall, and Zhou (2016); Li, Zhang, and Feng (2016); Nguyen andS ¸ent€ urk (2007, 2008) ;Nguyen, Sent€ urk, and Carroll (2008); S ¸ent€ urk and M€ uller (2005); S ¸ent€ urk and Nguyen (2006Nguyen ( , 2009)); Zhao and Xie (2018) by assuming some identifiability conditions.Recently, Zhang, Yang, and Feng (2021) and Zhang (2022) proposed a parametric linear distortion measurement errors model and exponential parametric distortion measurement errors model for the estimation of correlation coefficient between two underlying variables, respectively.In literature, there are no papers to study the exponential distortion nonlinear regression models, and we tackle this important problem in this paper.
This paper intends to start research on nonlinear models involved with the exponential parametric distortion measurement errors.Because the model Equation (1.1) contains parametric structures, the parameters in the distortion functions should be estimated at first.Under nonzero mean conditions and nonzero absolute mean conditions, we propose the nonlinear least squares and weighted nonlinear least squares estimation methods to estimate these parameters, and obtain the calibrated the variables : After obtaining these calibrated variables, namely, the nonlinear least squares estimators, the weighted nonlinear least squares estimators, the absolute nonlinear least squares estimators and the weighted absolute nonlinear least squares estimators of b 0 are proposed.We further propose a least squares estimator by using the absolute logarithmic transformation, coupling with the identifiability conditions.To make a comparison, we treat the exponential parametric distortions as unknown smoothing functions, and we used the proposed conditional mean calibration and conditional absolute mean calibration in literature to obtain calibrated variables and the nonlinear least squares estimators of b 0 : The estimators of parameters in the distortion functions and models are shown to be root-n consistent estimators The paper is organized as follows.In Section 2, we propose the parametric calibration estimation methods for the unobserved variables and nonlinear least squares estimators of the parameter b 0 , and derive related asymptotic results.In Section 3, we make a comparison with the nonparametric calibration estimation methods.In Section 4, we report the results of simulation studies.All technical proofs of the asymptotic results are given in the on-line "Supplementary Material".

Nonlinear least squares estimation
In this subsection, we first calibrate unobserved Y and X by using the observed i:i:d: sample f e Y i , e X ri , U i , r ¼ 1, :::, pg n i¼1 : To ensure identifiability, an assumption (assumption-M1) is needed: and "no-average-distortion" conditions hold: The assumption-M1 for the parametric distortion functions are analogous to the nonparametric distortion functions in Cui et al. (2009); S ¸ent€ urk and M€ uller (2005,2006); Zhang, Lin and Li (2019).Without assumption-M1, the parameters ða 0 , d 0 , g r0 , h r0 Þ > and other parameters (such as the parameter b 0 ) usually can not be estimated consistently.
The "no-average-distortion" conditions Equation (2.1) are also analogous to the classical additive measurement errors: The exponential parametric distortions disappear and the corresponding response variable or covariate are observed exactly without distortion measurement errors.The expectations taking values one in Equation (2.1) are assumed for the sake of simplicity and identifiability.If one assumed that E½expða 0 þ d 0 UÞ ¼ £ 0 > 0 such that £ 0 is known but not equal one, we can transform it as E½expða 0? þ d 0 UÞ ¼ 1 with a 0? ¼ a 0 À log ð£ 0 Þ: Since £ 0 is known, the parameter a 0 can be identified from a 0? : Under the identifiability conditions Equation (2.1) and the independence condition between U and ðY, X > Þ > , we have From Equations (2.2) and (2.3), we have Followed from Zhang (2022), using Equation (2.4), we can use the classical nonlinear least squares estimation to estimate unknown parameters ða 0 , d 0 , g r0 , h r0 Þ > , r ¼ 1, :::, p, Then, the unobserved covariates fY i , X ri , r ¼ 1, :::, pg n i¼1 can be calibrated as follows: Using Equation (2.7), the nonlinear least squares estimator bP is obtained by solving where XP, i ¼ ð XP 1 , i , :::, XP p , i Þ > and @gð X P, i , bÞ is the partial derivative of g with respect to b r (the r-th component of b).When Equation (2.8) has no closed form solution, one may solve it by the Newton-Raphson iterative method.
We now list the assumptions needed in the following theorems.
(C1) For some s ! 4, The matrices R a 0 d 0 , R g r0 h r0 , r ¼ 1, :::, p and R 0 in theorems are all positive-definite matrices.Moreover, H b , the partial derivatives @ l 1 þl 2 þl 3 þl 4 gðx, bÞ @ l 1 b t 1 @ l 2 b t 2 @ l 3 x s 1 @ l 4 x s 2 exist, and for some positive constant C.Moreover, when 1 l 1 þ l 2 2, and l 3 þ l 4 ¼ 0, it also requires The kernel function KðÁÞ is a symmetric bounded density function supported on ½ÀA, A satisfying a Lipschitz condition.KðÁÞ also has second-order continuous bounded derivatives, satisfying 0 < Ð s 2 KðsÞds < 1: (C5) As n ! 1, the bandwidth h satisfies log 2 n nh 2 !0 and nh 4 !0: (C6) The density function f U ðuÞ of the random variable U is bounded away from 0 and satisfies the Lipschitz condition of order 1 on ½U L , U R , where ½U L , U R denotes the compact support of U.
These are mild conditions that are satisfied in most practical situations.Condition (C1) is recently used in the study of exponential parametric distortion measurement error models for estimating the unknown distortion functions, see for example, Zhang (2022).Conditions (C2)-(C3) are used for the nonlinear least squares estimator (Cui et al. 2009;Wu 1981;Zhang, Lin and Li 2019;Zhang, Yang and Li 2020).Conditions (C4)-(C6) are used in the study of asymptotic results for the nonparametric calibration procedures.if we consider the distortions expða 0 þ d 0 UÞ and expðg r0 þ h r0 UÞ are fully unknown and non parametrical.These are the regular conditions for the kernel smoothing methods to guarantee the asymptotic normality of the proposed estimators in the models (Li, Zhang, and  In the following, we define A 2 ¼ AA > for any matrix or vector A and Theorem 1. Suppose Condition (C1) and assumption-M1 hold, as n ! 1, we have where, Remark.Theorem 1 is analogous to Theorem 2.1 in Zhang (2022).The asymptotic expressions in Theorem 1 will be used to construct asymptotic results of estimators or asymptotic expressions under the exponential parametric identifiability conditions Equation (2.1).
Theorem 3. Suppose Conditions (C1)-(C3) and assumption-M1 hold, as n ! 1, we have Remark.The first term K À1 0 K 0e K À1 0 is the usual asymptotic variance for the nonlinear least squares estimator when the data are exactly observed, i.e., expða 0 þ d 0 UÞ 1, expðg r0 þ h r0 UÞ 1, r ¼ 1, :::, p: If the model error e is further independent of X, this term reduces to Eðe 2 ÞK À1 0 : The second term K À1 0 K / P w P K À1 0 is dedicated to the parametric multiplicative distortion measurement errors involved in the response variable and covariates.

Absolute nonlinear least estimation
From Equations (2.5) and (2.6), it is seen that the nonlinear least estimators may not be workable when EðYÞ Þ when EðX r Þ ¼ 0 for some r 2 1, :::, p f g: As a remedy, the ideas of conditional absolute mean calibration (Delaigle, Hall, and Zhou 2016;Zhang 2021a;Zhang, Xu and Wei, 2020;Zhao and Xie 2018) can be adopted here to estimate parameters in the distortion functions.To ensure identifiability, an assumption (assumption-M2) is needed: (2.17) Based on Equations (2.15)-(2.17), the absolute nonlinear least squares estimators of unknown parameters ða 0 , d 0 , g r0 , h r0 Þ > , r ¼ 1, :::, p, are obtained as Then, the unobserved covariates Y i , X ri , r ¼ 1, :::, p f g n i¼1 can be calibrated as follows: Using Equation (2.20), the nonlinear least squares estimator bjPj is obtained by solving where XjPj, i ¼ ð XjPj 1 , i , :::, XjPj p , i Þ > and @gð X jPj, i , bÞ is the partial derivative of g with respect to b r : Theorem 7. Suppose Condition (C1) and assumption-M2 hold, as n ! 1, we have and MðWÞ is a function of W, satisfying E ½MðWÞ 2 È É < 1. Suppose Condition (C1) and assumption-M2 hold, as n ! 1, we have Theorem 9. Suppose Conditions (C1)-(C3) and assumption-M2 hold, we have where EðjYjÞ , U Þ: Similarly, when the covariates X r 's satisfy jX r j EðjX r jÞ ¼ X r EðX r Þ with probability one, we have The asymptotic results of Theorems 7-9 are the same with Theorems 1-3.In other words, the estimators ðâ, d, ĝr , ĥr , bP Þ > are asymptotically equivalent to ðâ A , dA , ĝA r , ĥA r , bjPj Þ > as the sample size n goes to infinity.

Absolute logarithmic transformation with least squares estimation
We re-write the model Equation (1.1) by using the absolute logarithmic transformation: where a L0 ¼ a 0 þ E½log ðjYjÞ, g L r 0 ¼ g r0 þ E½log ðjX r jÞ, L ¼ log ðjYjÞ À E½log ðjYjÞ and L r ¼ log ðjX r jÞ À E½log ðjX r jÞ: From model Equation (2.27) and the independence condition between U and ðY, X > Þ > , the classical least squares (LS) estimation methods can be directly used to estimate d 0 and h r0 , r ¼ 1, :::, p: Although LS methods can obtain estimators âL0 and ĝL r 0 , due to the identifiability, the parameters a 0 , g r0 's can not be extracted from them.The identifiability condition Equation (2.1) will help to estimate the parameters a 0 , g r0 's.The advantage of the model Equation (2.27) is that we do not need to impose the nonzero-mean condition EðYÞ Q p r¼1 EðX r Þ 6 ¼ 0: The estimators of d 0 and h r0 , r ¼ 1, :::, p are obtained as According to Equation (2.1), we have Together with Equations (2.28) and (2.29), we have In the following, Theorem 13 shows that the estimators ðâ L , dL , ĝL r , ĥL r Þ > are root-n consistent.As claimed in Zhang (2022), the estimators ðâ L , dL , ĝL r , ĥL r Þ > can be used as the initial values for the nonlinear squares estimation methods Equations (2.5)-(2.6),(2.18), and (2.19) and also for the weighted nonlinear squares estimation methods Equations (2.12)-(2.13)and (2.24)-(2.25).The root-n initial values makes the nonlinear least squares iteration fast when iteration algorithms are needed in practice.Zhang (2022) suggested estimators ðâ L , dL , ĝL r , ĥL r Þ > for initial values rather than studying themselves to obtain logarithmic transformation based calibration method.Here, we study this method and present the related asymptotic results.Using Equations (2.28) and (2.30)-(2.31), the unobserved covariates Y i , X ri , r ¼ 1, f :::, pg n i¼1 can be calibrated as follows: Using Equation (2.32), the nonlinear least squares estimator bL is obtained by solving where XL, i ¼ ð XL 1 , i , :::, XL p , i Þ > and @gð X L, i , bÞ @b r is the partial derivative of g with respect to b r : for some s ! 4, r ¼ 1, :::, p, as n ! 1, we have and MðWÞ is a function of W, satisfying E ½MðWÞ 2 È É < 1. Suppose conditions in Theorem 13 hold, as n ! 1, we have Theorem 15.Suppose conditions in Theorem 13 and Conditions (C2)-(C3) hold, we have ffiffiffi where It is noted that there exists biases /M ðuÞ À / P ðuÞ, ŵM r ðuÞ À w P r ðuÞ with convergence order Oðh 2 Þ in the nonparametric estimates /M ðuÞ and ŵM r ðuÞ: Under the Condition (C4), we have ffiffiffi n p h 2 !0, so these biases terms have convergence order Oðh 2 Þ ¼ oðn À1=2 Þ and they are even faster the parametric convergence rate Oðn À1=2 Þ: For example, the parametric biases terms â À a 0 , d À d 0 in Equation (2.7) with convergence rate Oðn À1=2 Þ are involved in Theorems 2 and 3.While the nonparametric biases terms /M ðuÞ À / P ðuÞ, ŵM r ðuÞ À w P r ðuÞ can be analytically controlled in an asymptotical way under nh 4 !0, and they will not have impact on the asymptotic result of bM in the following theorem.
Using Equation (2.35), the nonlinear least squares estimator bM is obtained by solving where XM, i ¼ ð XM 1 , i , :::, XM p , i Þ > and @gð X M, i , bÞ is the partial derivative of g with respect to b r : Analogous to Theorems 2 and 4, the Lemma B.2 in Zhang, Zhu, and Liang (2012) shows that The asymptotic expressions F / ðs, uÞ and G w r ðt, uÞ are different from ÀA / ðs, uÞ, À B w r ðt, uÞ in Theorem 1.This is because the CMC method treats / P ðUÞ and w P r ðUÞ as fully nonparametric distortion functions.The parameters in / P ðUÞ and w P r ðUÞ can not be estimated from nonparametric kernel smoothing methods Equation (2.33)-(2.34),and we only obtain the asymptotic expressions of nonparametric estimators of / P ðUÞ and w P r ðUÞ rather than the expressions of estimators of parameters involved in / P ðuÞ and w P r ðuÞ: Theorem 16.Suppose Conditions (C1)-(C6) and assumption-M1 hold, as n ! 1, we have ffiffiffi where,

Nonparametric conditional absolute mean calibration
Under the assumption-M2, we follow the nonparametric conditional absolute mean calibrations (Delaigle, Hall, and Zhou 2016;Zhang 2021a;Zhang, Xu and Wei 2020;Zhao and Xie 2018) to estimate the parametric distortion functions in a nonparametric way.The Nadaraya-Watson estimators are used to estimate / P ðuÞ and w P r ðuÞ: Define (2.37) Using Equations (2.36)-(2.37),we obtain the conditional absolute mean calibrated (CAMC) variables Ŷ jMj, i , XjMj r , i , r ¼ 1, :::, p Using Equation (2.38), the nonlinear least squares estimator bjMj is obtained by solving where XjMj, i ¼ ð XjMj 1 , i , :::, XjMj p , i Þ > and @gð X jMj, i , bÞ is the partial derivative of g with respect to b r : Following the Lemma B.2 in Zhang, Zhu, and Liang (2012), we have Theorem 17. Suppose Conditions (C1)-( C6) and assumption-M2 hold, we have where, Remark.When jYj EðjYjÞ ¼ Y EðYÞ and jX r j EðjX r jÞ ¼ X r EðX r Þ hold for r ¼ 1, :::, p with probability one, we have EðjX r jÞ , U Þ, r ¼ 1, :::, p: The asymptotic result of Theorem 17 is the same with the asymptotic result in Theorem 16.In other words, the estimators bM and bjMj are asymptotically equivalent as the sample size n goes to infinity.

Implementation
Simulation studies are conducted in this section to show the performance of our proposed methods.For the nonparametric CMC and CAMC calibration methods, the Epanechnikov kernel KðtÞ ¼ 0:75ð1 À t 2 Þ þ is used here.We use the rule of thumb: h ¼ rU n À1=3 , and rU is the sample deviation of U. The bandwidth h is chosen here to meet the Condition (C4) This method is fairly effective and easy to implement in practice.
Our numerical experience suggests that the numerical results were stable when we shifted several values around this data-driven bandwidth.
Example 1.We consider the model (4.1) 2000 realizations are generated and sample sizes are n ¼ 300, n ¼ 500 and n ¼ 1000, respectively.In this example, b The confounding variable U follows an uniform distribution U½0, 1, and the distortion functions are chosen as and The model error e is independent of ðX > , UÞ > and it follows from a normal distribution N(0, 1).
Simulation results are reported in Tables 1 and 2. In this example, the mean values of ðY, X > Þ > satisfy EðYÞEðX 1 ÞEðX 2 Þ 6 ¼ 0, so the proposed estimators in this paper can be used to estimate parameters in model Equation (4.1).In Table 1, we report the means, the standard errors and mean squares errors (MSE) for the nonlinear least squares estimators ðâ, d, ĝs , ĥs Þ > , the weighted nonlinear least squares estimators ðâ w , dw , ĝws , ĥws Þ > , the absolute nonlinear least squares estimators ðâ A , dA , ĝA s , ĥA s Þ > , the weighted absolute nonlinear least squares estimators ðâ jwj , djwj , ĝjwjs , ĥjwjs Þ > and the absolute logarithmic least squares estimators ðâ L , dL , ĝL s , ĥL s Þ > , s ¼ 1, 2. For the parameters ða 0 , d 0 Þ > , the estimators ðâ jwj , djwj Þ > perform the best, ðâ L , dL Þ > perform the second best, and the classical nonlinear least squares estimators ðâ, dÞ > perform the worst as they have large values of MSE even when the sample size n is large.For the parameters ðg s0 , h s0 Þ > , s ¼ 1, 2, the weighted absolute nonlinear least squares estimators ðĝ jwjs , ĥjwjs Þ > still perform the best.The estimators ðĝ A s , ĥA s Þ > , ðĝ ws , ĥws Þ > and ðĝ s , ĥs Þ > performs similarly, and the former are the best among the three estimators.The absolute logarithmic least squares estimators ðĝ L s , ĥL s Þ > , s ¼ 1, 2 perform the worst since their values of MSE are the largest.
In Table 2, we report the simulation results of the true nonlinear least squares estimators ð b1 , b2 Þ > , the nonlinear least squares estimators ð bP, 1 , bP, 2 Þ > , the weighted nonlinear least squares estimators ð bW, 1 , bW, 2 Þ > , the absolute nonlinear least squares estimators ð bjPj, 1 , bjPj, 2 Þ > , the weighted absolute nonlinear least squares estimators ð bjWj, 1 , bjWj, 2 Þ > , the nonlinear least squares estimators ð bL, 1 , bL, 2 Þ > by using absolute logarithmic transformation, CMC estimators ð bM, 1 , bM, 2 Þ > and CAMC estimators ð bjMj, 1 , bjMj, 2 Þ > : Among these estimators, we observed that the estimators ð bjWj, 1 , bjWj, 2 Þ > perform the best since their values of MSE are the smallest.All the values of MSE for the estimators of b 0 based on parametric calibrations or nonparametric calibrations are larger than the true estimators ð b1 , b2 Þ > : As indicated in Theorems 3,6,9,12,15,16,and 17, the parametric distortion functions / P ðUÞ, w P r ðUÞ, r ¼ 1, 2 increase the asymptotic variances of the proposed estimators.For the parameter b 0, 1 , the estimator bL, 1 performs the worst.For the parameter b 0, 2 , the classical nonlinear estimator bP, 2 perform the worst, and the CMC and CAMC estimators bM, 2 and bjMj, 2 performs better than the parametric calibrated estimator bP, 2 : This phenomenon shows that the nonparametric calibration procedures with nonparametric convergent rates can also be used to estimate parameters in the models, even the distortion functions have parametric structures (such as the exponential structures in this simulation).
The estimators ðĝ A s , ĥA s Þ > , perform slightly not better than ðâ jwjs , djwjs Þ > , and the absolute logarithmic least squares estimators ðĝ L s , ĥL s Þ > , s ¼ 1, 2 perform the worst since their values of MSE are the largest even when the sample size n ¼ 1000.

Discussions and further research
In this paper, we proposed several estimators for the parameters in nonlinear regression models under the exponential parametric distortion measurement errors.Different calibration procedures with assumptions of expectations of response variable and covariates are discussed, coupling with parametric calibrations and nonparametric calibrations.In future work, the hypothesis tests for parameters in the nonlinear regression models under the exponential parametric distortion measurement errors, and also the model checking problems can be considered.Other semi-parametric models such as partial linear models, single-index models and partial linear varying coefficient models with the parametric distortions can also be considered.Because of the limitation of space, we only study the first order estimation for the nonlinear regression models in this paper.The second order nonlinear least squares estimation procedures (Kim and Ma 2012;Salamh and Wang 2021;Wang and Leblanc 2008;Zhang 2021b) with parametric exponential distortion measurement errors will be covered in future work to increase the estimation efficiency of parameters in the mean function or variance function, although both the theoretical analysis and the implementation of the estimators with parametric, nonparametric or mixed calibrations will become increasingly complex.The research on this topic is ongoing.