Let $\left\{Y_i,X_i,U_i\right\}$, $i=1,\cdots ,n$ be independent and identically distributed (i.i.d.) copies of $\left\{Y,X,U\right\}$, the sample version of model (1.1) is given by

$Y_i=\text{exp}\left(X_i^{\text{T}}\beta +f\left(U_i\right)\right){ϵ}_i.$ (2.1)

To address the estimating issue of $\beta$ and $f(\cdot )$ simultaneously, we adopt the B-spline basis functions approximation to covert the estimation problem of model (2.1) to the problem of regression coefficients estimating in the linear combinations framework. We first provide a brief review about the construction of these basis functions. Without loss of generality, we assume that the compact support set of $U$ is ${\mathcal{S}}_U=\left[0,1\right]$. Let $B\left(u\right)={\left(B_m\left(u\right):1\le m\le J_n\right)}^{\text{T}}$ be the B-splines basis functions of order $q$, where $J_n=N_n+q$ and $N_n$ is the number of interior knots for a knot sequence given as

${\xi }_1=\cdots =0={\xi }_q<{\xi }_{q+1}<\cdots <{\xi }_{N_n+q}<1={\xi }_{N_n+q+1}=\cdots ={\xi }_{N_n+2q}.$

For $q\le \iota \le J_n$, this knot sequence satisfies

$\underset{q\le \iota \le J_n}{\text{max}}\left|{\xi }_{\iota +1}-{\xi }_{\iota }\right|/\underset{q\le \iota \le J_n}{\text{min}}\left|{\xi }_{\iota +1}-{\xi }_{\iota }\right|\le M,$

for some constants $0<M<\infty$. We refer the interested readers to Huang [17] for more details. Under some smoothness assumptions, we can approximate $f(\cdot )$ by

$f\left(U_i\right)\approx \underset{m=1}{\overset{J_n}{{\displaystyle \sum }}}{B}_m\left(U_i\right){\gamma }_m=B^{\text{T}}\left(U_i\right)\gamma ,$ (2.2)

where $\gamma ={\left({\gamma }_m:1\le m\le J_n\right)}^{\text{T}}$.

Follows Theorem 5.9 of Schumaker [18] , the spline $f(\cdot )$ is monotonically non-decreasing on ${\mathcal{S}}_U$ if non-decreasing constraints are imposed on the coefficients $\gamma ={\left({\gamma }_m:1\le m\le J_n\right)}^{\text{T}}$, i.e.,

${\gamma }_1\le {\gamma }_2\le \cdots \le {\gamma }_{J_n},$

this ensures the nondecreasing property of nonparametric function $f(\cdot )$.

Based on the above analysis, we can rewrite model (2.1) as

$Y_i\approx \text{exp}\left(X_i^{\text{T}}\beta +B^{\text{T}}\left(U_i\right)\gamma \right){ϵ}_i,$ (2.3)

which transforms the model (2.1) into an almost equivalent linear multiplicative model. Let ${\Pi }_i={\left(X_i^{\text{T}},B^{\text{T}}\left(U_i\right)\right)}^{\text{T}}$, then following Chen et al. [6] , we estimate the related coefficients $\vartheta ={\left({\beta }^{\text{T}},{\gamma }^{\text{T}}\right)}^{\text{T}}$ by minimizing

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left\{\left|\frac{Y_i-\text{exp}\left({\Pi }_i^{\text{T}}\vartheta \right)}{Y_i}\right|\times \left|\frac{Y_i-\text{exp}\left({\Pi }_i^{\text{T}}\vartheta \right)}{\text{exp}\left({\Pi }_i^{\text{T}}\vartheta \right)}\right|\right\},$ (2.4)

subject to the constraint ${\gamma }_1\le {\gamma }_2\le \cdots \le {\gamma }_{J_n}$, which is equivalent to solve the following

${ℒ}_n\left(\vartheta \right)\equiv \underset{i=1}{\overset{n}{{\displaystyle \sum }}}\left\{Y_i\exp \left(-{\Pi }_i^{\text{T}}\vartheta \right)+Y_i^{-1}\exp \left({\Pi }_i^{\text{T}}\vartheta \right)\right\},$ (2.5)

with constraint ${\gamma }_1\le {\gamma }_2\le \cdots \le {\gamma }_{J_n}$.

Now we delve into the algorithm for solving (2.5), since minimization problem (2.5) requires constrained nonlinear programming, we use the constrOptim package in R software to ensure restrictive condition ${\gamma }_1\le {\gamma }_2\le \cdots \le {\gamma }_{J_n}$ and achieve this optimization. To use the constrOptim algorithm, we need to specify the gradient of the objective function ${ℒ}_n\left(\vartheta \right)$ in (2.5) given as,

$\frac{\partial {ℒ}_n\left(\vartheta \right)}{\partial \vartheta }=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\Pi }_i^{\text{T}}\left\{Y_i^{-1}\exp \left({\Pi }_i^{\text{T}}\vartheta \right)-Y_i\exp \left(-{\Pi }_i^{\text{T}}\vartheta \right)\right\}.$ (2.6)

Denote the final estimator in (2.5) as $\hat{\vartheta }$, we call it monotone constrained LPRE estimator, then the estimator for $f(\cdot )$ is given by $\hat{f}(\cdot )=B^{\text{T}}\left(U_i\right)\hat{\gamma }$.

The asymptotic properties of the proposed estimators are studied in this subsection. Let ${\beta }_0$ and $f_0(\cdot )$ be the true values of $\beta$ and $f(\cdot )$ in model (1.1). We use $\Vert \cdot \Vert$ to denote the $L^2$ norm for functions. For some positive series $a_n$ and $b_n$, $a_n\asymp b_n$ means ${\lim }_{n\to \infty }{a}_n/b_n=c$ for some nonzero constant $c>0$. The following regularity conditions are required.

(C1). The covariate $U_i$ has a continuous density $g_{U_i}\left(u\right)$ which is bounded away from 0 and infinity on ${\mathcal{S}}_U$ for every $i=1,\cdots ,n$.

(C2). $f(\cdot )\in C^{\left(r\right)}\left({\mathcal{S}}_U\right)$ for some integer $r\ge 2$, and the spline order satisfies $q\ge r$. $C^{\left(r\right)}\left({\mathcal{S}}_U\right)=\left\{\phi |{\phi }^{\left(r\right)}\in C\left({\mathcal{S}}_U\right)\right\}$ denotes the space of $r$-th order smooth function.

(C3). The matrix $E\left(X{X}^{\text{T}}\right)$ is nonsingular and its eigenvalues are uniformly bounded away from 0 and infinity.

(C4). The error $ϵ$ satisfies $E\left(ϵ+{ϵ}^{-1}|X,U\right)<\infty$.

(C5). The error $ϵ$ satisfies $E\left(ϵ|X,U\right)=E\left({ϵ}^{-1}|X,U\right)$.

Conditions (C1)-(C3) are standard for the nonparametric estimation in spline smoothing literature, which are essentially same as those in Shen et al. [19] and Guo et al. [20] Condition (C4) ensures the asymptotic normality of the estimator for parametric term, see Liu and Xia [9] and Hu [8] . Condition (C5) is needed for model identification and asymptotic variance of estimates, see Ming et al. [10] and Hu [8] .

Theorem 1 Under conditions (C1)-(C5), if $N_n\asymp n^{1/\left(2r+1\right)}$, then we have

${\Vert \hat{f}\left(u\right)-f_0\left(u\right)\Vert }^2=O_p\left(n^{-2r/\left(2r+1\right)}\right).$

Theorem 1 indicates that the nonparametric estimates obtained by our proposed method achieve the optimal convergence rates. The following Theorem 2 states that the estimator $\hat{\beta }$ is asymptotically normal.

Theorem 2 Under the same assumptions of Theorem 1, $\hat{\beta }$ converges in probability to the true value ${\beta }_0$, i.e.

$\sqrt{n}\left(\hat{\beta }-{\beta }_0\right)\stackrel{d}{\to }N\left(0,{\text{Σ}}^{-1}{\text{ΛΣ}}^{-1}\right),$

where $\stackrel{d}{\to }$ denotes convergence in distribution, $\text{Σ}=E\left\{X{X}^{\text{T}}\left(ϵ+{ϵ}^{-1}\right)\right\}$ and $\text{Λ}=E\left\{X{X}^{\text{T}}{\left(ϵ-{ϵ}^{-1}\right)}^2\right\}$.

Since the proofs of Theorems 1-2 follow along the same ideas as the proofs of Theorems in Ming et al. [10] although part of details differs, we omit the proofs in this paper.