Framework for constructing an optimal weighted score based on agreement

Abstract

In many medical studies, questionnaires or instruments with item ratings are often used to measure the health outcomes reflecting the disease status. Therefore, combining these item ratings to derive a score that reflects the disease status is an important issue. This paper proposes a new weighted method with the weights determined by maximizing the broad sense agreement Peng et al. [L. Peng, R. Li, Y. Guo, and A. Manatunga, A framework for assessing broad sense agreement between ordinal and continuous measurements, J. Am. Stat. Assoc. 106 (2011), pp. 1592–1601.] between the new score and the disease status measured by an ordinal scale. Theoretical and simulation results demonstrate the validity of our proposed optimal weights. We illustrate our method via an application to a post-traumatic stress disorder (PTSD) study.

1. Introduction

In many mental health studies, such as attention-deficit hyperactivity disorder, major depression, and posttraumatic stress disorder (PTSD), etc., instruments or questionnaires with multiple-item ratings are commonly utilized to screen and diagnose diseases tentatively [2,3,21]. These instruments or questionnaires are typically based on a structured questionnaire and an assigned rating system. Therefore, combining these item ratings to derive a global score that reflects the actual disease status is often necessary and an important issue. For example, posttraumatic stress disorder (PTSD) is a complex, heterogeneous disorder and is one of the most prevalent mental health disorders among combat veterans [6]. Establishing a diagnosis such as PTSD is challenging due to the variability of the disease phenotype and heterogeneity in patient population [4]. The best available diagnosis of PTSD (yes vs. no) is evaluated by the Clinician-Administered PTSD Scale (CAPS), which is considered to be the standard criterion measure in PTSD assessment [4]. However, the administration of the CAPS requires a structured interview by trained clinicians with working knowledge of PTSD and is very time-consuming. Therefore, wide applications of this instrument to large populations are costly and challenging. In contrast, the PTSD Symptom Scale (PSS) is a 17-item self-reported questionnaire to assess symptoms of PTSD and requires less assessment time than the CAPS [3]. However, the self-reported PSS consists of four subscales, including PSS intrusive, PSS avoidance, PSS negative, and PSS hyperarousal, and thus, how to combine these four subscales reasonably to construct a score that reflects the degree of disease severity is very important.

In the literature, a naive approach commonly used is to sum over all items to construct a total score. Foa et al. [3], Agresti [1], and McDowell [10]. This approach accounts for each subscale in the questionnaire with equal importance regarding the severity of the disease, and the corresponding total score may not best reflect the disease status. Furthermore, observing disease severity as an ordinal variable is common in practice. For example, in the melanoma and depression study [14], the degree of disease severity is often characterized by no depression, mild depression, and severe depression. Another example we consider in this manuscript is PTSD measured from CAPS (0= PTSD negative, 1= PTSD positive), which is considered the gold standard for PTSD diagnosis. Now, how to develop a general and objective analytic framework for addressing the problem of determining the contributions of each subscale to construct a score when the ordinal disease severity is available. To this end, a fundamental question is, what are the most desirable weights reflecting contributions from subscales to the disease severity? A common viewpoint in practice is that meaningful weights should produce high agreement or association between the newly weighted score and the established disease categories.

This paper aims to develop an analytical strategy for constructing a new score for an instrument by assigning and estimating weights for the ratings of items within a given instrument. The newly developed score is a weighted sum of ratings of items and provides the highest agreement with the actual disease status, as measured by gold or the best available standard known as the broad sense of agreement (BSA) [14]. In addition, we develop an inferential procedure for determining weights along with a statistical test determining whether a simple equal weighting of items is adequate.

The remainder of the article is organized as follows. In Section 2, we present our framework for constructing new weighted scores. Section 3 offers the estimator of the optimal weights and investigates its asymptotic properties. In Section 4, we conduct simulation studies to evaluate the proposed method. An application of our way to the Grady Trauma Study is presented in Section 5. Concluding comments are given in Section 6. Finally, proofs are relegated to the Appendix.

2. The proposed framework for constructing weight scores

We consider a continuous instrument with items grouped according to the item interpretations. Let $Z_1,\dots ,Z_M$ be the continuous scores from item groups $1,\dots ,M$. In addition, here we suppose the disease status of interest is captured by a categorical measurement Y, which takes values of $1,\dots ,L$ and is considered the gold standard. For instance, in our cross-sectional Grady trauma study of the PTSD dataset given in Section 5, $Z_1,\dots ,Z_4$ respectively denote the self-reported subtotal score of PSS intrusive, PSS avoidance, PSS negative, and PSS hyperarousal. Moreover, Y is the corresponding PTSD diagnosis measured from CAPS, $Y=0$ for PTSD negative, Y = 1 otherwise.

In the literature, a common approach for constructing a total score is by the arithmetic average over $Z_1,\dots ,Z_M$. This approach not only does not consider the different importance of $Z_1,\dots ,Z_M$ to the severity of the disease, but also can not best reflect the disease status. Here, we propose a weighted sum of $Z_1,\dots ,Z_M$ based upon the broad sense of agreement (BSA) proposed by Peng et al. [14]. We first give a brief introduction to the BSA.

2.1. Review of broad sense of agreement

Let X and Y be some continuous and ordinal variables that measure a common trait from the same subject, and their domains are ${\mathcal{D}}_X$ and ${\mathcal{D}}_Y=\{1,2,\dots ,L\}$. For example, in the cross-sectional Grady trauma study of PTSD, X may represent the average value of the subscores of PSS intrusive, PSS avoidance, PSS negative, and PSS hyperarousal, and Y is the corresponding PTSD diagnos. Let $X_{(\ast l)}$ denotes a randomly selected X given Y = l. By Peng et al. [14], X and Y are in perfect BSA if $X_{(\ast 1)}<\cdots <X_{(\ast L)}$ with probability 1, and in a perfect broad sense of disagreement if $X_{(\ast 1)}>\cdots >X_{(\ast L)}$ with probability 1. Equivalently, the perfect broad sense of agreement (BSA) entails that there exists an increasing (or decreasing) step function Ψ from ${\mathcal{D}}_X$ to ${\mathcal{D}}_Y$ such that $Y=\mathrm{\Psi }(X)$ with probability 1. Let $\parallel \mathit{v}\parallel$ denote the Euclidean norm of the vector $\mathit{v}$, and $\overrightarrow{\mathbf{R}}(x_1,\dots ,x_L)$ denote a mapping from $(x_1,\dots ,x_L{)}^T$ to its ranks $(r_1,\dots ,r_L{)}^T$ with $r_l=\sum _{j=1}^{L}I(x_l\ge x_j),l=1,2,\dots ,L$. Peng et al. [14] defines the broad sense agreement (BSA) measure between X and Y as follows:

$${\rho }_{\mathrm{bsa}}=1-\frac{E\{\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}(X_{\ast 1},\dots ,X_{\ast L}){\parallel }^2\}}{E\{\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}(X_{\ast 1},\dots ,X_{\ast L}){\parallel }^2|X\perp Y\}},$$ (1)

where $\overrightarrow{\mathbf{S}}=(1,2,\dots ,L{)}^T$, $E\{\cdot \}$ denotes the expectation and $E\{\cdot |X\perp Y\}$ denotes the expectation under the independence between X and Y. The BSA definition (1) quantifies the extent of departure from perfect BSA by using the distance between the observed ranks in X and the ranks expected under perfect BSA within a group of randomly selected observations, one from each category of Y. Thus, this new measure has a clear meaning and is easy to interpret. In [14], they further showed that $-1\le {\rho }_{\mathrm{bsa}}\le 1$, and ${\rho }_{\mathrm{bsa}}=1$ (or $-1$) corresponding to the perfect BSA agreement (or disagreement) scenario, and $E\{\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}(X_{\ast 1},\dots ,X_{\ast L}){\parallel }^2|X\perp Y\}=(L^3-L)/6$.

2.2. The proposed method to construct a score

Let $\mathit{Z}=(Z_1,\dots ,Z_M{)}^T$, and $\mathit{V}\equiv ({\mathit{Z}}^T,Y{)}^T$ denote an observation from the distribution P on the set $\mathcal{T}\subseteq {\mathbb{R}}^M\otimes \mathbb{R}$. We propose a new method by assigning weights to different continuous group scores, denoted as $X(\mathbf{w})\equiv (w_1,\dots ,w_{M-1},1-\sum _{k=1}^{M-1}{w}_k)\mathit{Z}$, where $\mathbf{w}\equiv (w_1,\dots ,w_{M-1})$, and $w_1,\dots ,w_{M-1}$ and $1-\sum _{k=1}^{M-1}{w}_k$ respectively denote the weights for item groups $1,\dots ,M$. Define $\mathbf{A}=\{(w_1,\dots ,w_{M-1}{)}^T:0\le w_k\le 1,k=1,\dots ,M-1,\sum _{k=1}^{M-1}{w}_k\le 1\}$. We can express the BSA between the categorical measurement Y and the continuous weighted score $X(\mathbf{w})$ as a function of weights $\mathbf{w}$, given by:

$${\rho }_{\mathrm{bsa}}(\mathbf{w})=1-\frac{E\{\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}(X_{(\ast 1)}(\mathbf{w}),\dots ,X_{(\ast L)}(\mathbf{w})){\parallel }^2\}}{C_L},\mathbf{w}\in \mathbf{A},$$ (2)

where $X_{(\ast l)}(\mathbf{w})=(w_1,\dots ,w_{M-1},1-\sum _{k=1}^{M-1}{w}_k){\mathit{Z}}_{(\ast l)}$, ${\mathit{Z}}_{(\ast l)}$ denotes a randomly selected $\mathit{Z}$ given Y = l, and $C_L=(L^3-L)/6$. If the categories do not have an intrinsic natural order, we may use the methods proposed by [8] to sort out the multiple categories. Following our view that a tangible score that measures the degree of severity of the disease should produce a high agreement with the ordinal scale Y, we propose to define the optimal weights as the vector $\mathbf{w}\in \mathbf{A}$ that maximizes ${\rho }_{\mathrm{bsa}}(\mathbf{w})$, i.e.

$${\mathbf{w}}_0=\arg \underset{\mathbf{w}\in \mathbf{A}}{\max }{\rho }_{\mathrm{bsa}}(\mathbf{w}).$$ (3)

3. Estimation procedure and inference

This section presents a nonparametric approach for estimating the optimal weights ${\mathbf{w}}_0$ in a clinical application. Let $Z_{\mathrm{ik}}$ be the observed continuous score of item group k for subject i. Let $Y_i$ denote the observed ordinal disease status for subject i. Define ${\mathit{Z}}_i=(Z_{i1},Z_{i2},\dots ,Z_{\mathrm{iM}}{)}^T$, and let $X_i(\mathbf{w})\equiv (w_1,\dots ,w_{M-1},1-\sum _{k=1}^{M-1}{w}_k){\mathit{Z}}_i$ be the severity score of the disease. Then the observed data can be summarized as ${\mathit{V}}_i=({\mathit{Z}}_i^T,Y_i{)}^T,i=1,\dots ,n$, which is a sample of independent observations of $({\mathit{Z}}^T,Y{)}^T$. Following [14], without loss of generality, we first arrange them as follows:

$$\begin{array}{rl}& ({\mathit{Z}}_1^T,Y_1=1),\dots ,({\mathit{Z}}_{n_1}^T,Y_{n_1}=1),\\ & ({\mathit{Z}}_{n_1+1}^T,Y_{n_1+1}=2),\dots ,({\mathit{Z}}_{n_1+n_2}^T,Y_{n_1+n_2}=2),\\ & ⋮\\ & ({\mathit{Z}}_{\sum _{l=1}^{L-1}{n}_l+1}^T,Y_{\sum _{l=1}^{L-1}{n}_l+1}=L),\dots ,({\mathit{Z}}_{\sum _{l=1}^L{n}_l}^T,Y_{\sum _{l=1}^L{n}_l}=L),\end{array}$$

where $n_l=\sum _{i=1}^{n}I(Y_i=l)$ and $\sum _{l=1}^L{n}_l=n$.

Define ${\tilde{\mathit{Z}}}_{s,t_s}={\mathit{Z}}_{\sum _{l=0}^{s-1}{n}_l+t_s},t_s=1,2,\dots ,n_s,s=1,2,\dots ,L$, and

$$W_n(\mathbf{w})={\left(\prod _{l=1}^L{n}_l\right)}^{-1}\sum _{j_1=1}^{n_1}\dots \sum _{j_L=1}^{n_L}\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}({\tilde{X}}_{1,j_1}(\mathbf{w}),\dots ,{\tilde{X}}_{L,j_L}(\mathbf{w})){\parallel }^2,$$

where ${\tilde{X}}_{l,j_l}(\mathbf{w})=(w_1,\dots ,w_{M-1},1-\sum _{k=1}^{M-1}{w}_k){\tilde{\mathit{Z}}}_{l,t_l}$. Note that ${\tilde{\mathit{Z}}}_{l,j_l}$ is the realization of ${\mathit{Z}}_{(\ast l)},l=1,2,\dots ,L$, it can be shown that $W_n(\mathbf{w})$ is an unbiased estimator of $E\{\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}(X_{(\ast 1)}(\mathbf{w}),\dots ,X_{(\ast L)}(\mathbf{w})){\parallel }^2\}$ in (2). Thus, it naturally leads to the estimator of the optimal weights ${\mathbf{w}}_0$ given by:

$$\hat{\mathbf{w}}=\arg \underset{\mathbf{w}\in \mathbf{A}}{\max }{\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w}),$$ (4)

where ${\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})=1-\frac{W_n(\mathbf{w})}{C_L}$. Note that the direct calculation of $W_n(\mathbf{w})$ involves the exhausted stratified sampling, which can cause prohibitive computational burden. To circumvent this issue, we write

$$\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}({\tilde{X}}_{1,j_1}(\mathbf{w}),\dots ,{\tilde{X}}_{L,j_L}(\mathbf{w})){\parallel }^2=2\sum _{l=1}^L{l}^2-2\sum _{l=1}^L\sum _{r=1}^{L}l\cdot I({\tilde{X}}_{r,j_r}(\mathbf{w})\le {\tilde{X}}_{l,j_l}(\mathbf{w})).$$

This leads to an equivalent expression of $W_n(\mathbf{w})$,

$$W_n(\mathbf{w})=\frac{L(L+1)(2L+1)}{3}-2\sum _{l=1}^L\sum _{r=1}^L\frac{l}{n_r{n}_l}\sum _{j_r=1}^{n_r}\sum _{j_l=1}^{n_l}I\{{\tilde{X}}_{r,j_r}(\mathbf{w})\le {\tilde{X}}_{l,j_l}(\mathbf{w})\}.$$ (5)

However, ${\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})$ is not a continuous function with respect to $\mathbf{w}$ since $W_n(\mathbf{w})$ involves indicator functions. To reduce possible numerical issues caused by the discontinuity, we further propose to use the sigmoid function $s(x)=[1+\text{exp}(-x){]}^{-1}$ as an approximation to the indicator functions in $W_n(\mathbf{w})$. One problem using $s(x)$ in place of indicator functions is that this approximation is poor when x is close to 0. As a remedy, we use the family of functions $s_n(x)=[1+\text{exp}(-x/{\sigma }_n){]}^{-1}$ as an approximation to the indicator function $I(x\ge 0)$ in $W_n(\mathbf{w})$, where ${\sigma }_n$ is a sequence of strictly positive and decreasing numbers satisfying $\underset{n\to \mathrm{\infty }}{lim}{\sigma }_n=0$ [9,15,16,20]. Thus we obtain that a smooth version of $W_n(\mathbf{w})$, which is equal to

$$W_{n,s}(\mathbf{w})=\frac{L(L+1)(2L+1)}{3}-2\sum _{l=1}^L\sum _{r=1}^L\frac{l}{n_r{n}_l}\sum _{j_r=1}^{n_r}\sum _{j_l=1}^{n_l}{s}_n({\tilde{X}}_{l,j_l}(\mathbf{w})-{\tilde{X}}_{r,j_r}(\mathbf{w})).$$

Replacing $W_n(\mathbf{w})$ by $W_{n,s}(\mathbf{w})$ in ${\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})$, we obtain a smooth estimator of the BSA between $X(\mathbf{w})$ and Y, ${\rho }_{\mathrm{bsa}}(\mathbf{w})$, which is given by

$${\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})=1-\frac{{\displaystyle W_{n,s}(\mathbf{w})}}{{\displaystyle C_L}}.$$ (6)

The corresponding estimator of optimal weights based on the smooth BSA is given by:

$${\hat{\mathbf{w}}}_s=\arg \underset{\mathbf{w}\in \mathbf{A}}{\max }{\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w}).$$ (7)

The maximization of (7) involves the tuning parameter ${\sigma }_n$. In the following simulation studies and real data analysis, we suggest a rule of thumb for choosing ${\sigma }_n$ [20]. Specifically, we first Initialize ${\sigma }_n^0={\sigma }_n$, where ${\sigma }_n$ is user specified and data independent, satisfying ${\sigma }_n\to 0$ as $n\to \mathrm{\infty }$. We use ${\sigma }_n=1/\sqrt{n}$ in our data analysis. Next, calculate $\hat{\mathbf{w}}$ by maximizing (6) with ${\sigma }_n^0$. Denote ${\sigma }_n^1$ as the largest constant such that 95% of the $|(X_i(\hat{\mathbf{w}})-X_j(\hat{\mathbf{w}}))/{\sigma }_n|$ is greater than 5. Lastly, set ${\sigma }_n=min({\sigma }_n^0,{\sigma }_n^1)$. In addition, note that the objective function ${\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})$ is differentiable concerning $\mathbf{w}$. Thus, we can efficiently solve the maximization problem in (7) by many existing algorithms, such as the Newton-Raphson algorithm, which can be finished using the R function $\mathrm{optim}()$ directly.

We investigate the asymptotic properties of ${\hat{\mathbf{w}}}_s$, and summarize its consistency and asymptotic normality in Theorems 3.1 and 3.2. We first introduce some notations and regularity conditions. Let ${\mathbf{\Omega }}_{n,L}=\{(j_1,\dots ,j_L):1\le j_l\le j_L,j_1,\dots ,j_L\text{are distinct}\}$, $p_l=\mathrm{Pr}(Y=l)$ for $l=1,\dots ,L$, and ${\gamma }_L=2/(C_L\cdot L!)$. For $(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}$, define

$$\begin{array}{rl}& \mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})=I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L)\\ & \times \sum _{k=1}^L\{Y_{m_k}^2-Y_{m_k}\sum _{r=1}^{L}I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))\},\\ & h({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})=1-\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}){\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1},\end{array}$$

where ${\mathbf{\Theta }}_L$ denotes all $L!$ permutations of $\{1,2,\dots ,L\}$. For each $\mathit{v}\in \mathcal{T}$ and for each $\mathbf{w}\in \mathit{A}$, define

$$\tau (\mathit{v},\mathbf{w})=\mathrm{Eh}(\mathit{v},{\mathit{V}}_2,\dots ,{\mathit{V}}_L,\mathbf{w})+\mathrm{Eh}({\mathit{V}}_1,\mathit{v},\dots ,{\mathit{V}}_L,\mathbf{w})+\cdots +\mathrm{Eh}({\mathit{V}}_1,{\mathit{V}}_2,\dots ,\mathit{v},\mathbf{w}).$$

Let ${\mathrm{\nabla }}_m$ denotes the mth partial derivative operator with respect to $\mathbf{w}$. The following regularity conditions are needed in deriving the asymptotic properties of ${\hat{\mathbf{w}}}_s$:

• (C1) $p_l>0$ for $l=1,\dots ,L$.
• (C2) ${\mathbf{w}}_0$ is an interior point of the set $\mathit{A}$ and is the unique maximizer of ${\rho }_{\mathrm{bsa}}(\mathbf{w})$.
• (C3)The support of $\mathit{Z}$ is not contained in any proper linear subspace of ${\mathbb{R}}^M$.
• (C4)The tuning parameter ${\sigma }_n$ satisfies: $n{\sigma }_n^2\to 0$, as $n\to \mathrm{\infty }$.
• (C5)Let $\mathcal{N}$ denote a neighborhood of ${\mathbf{w}}_0$.
• (i)For each $\mathit{v}\in \mathcal{T}$, all mixed second partial derivatives with respect to $\mathbf{w}$ of $\tau (\mathit{v},\mathbf{w})$ exist on $\mathcal{N}$. The partial derivatives of the conditional density of $\mathit{Z}$ given $\mathit{X}({\mathbf{w}}_0)$ exist and are bounded.
• (ii)
  For all $\mathit{v}\in \mathcal{T}$ and $\mathbf{w}\in \mathit{A}$, there exists an integrable function $M(\mathit{v})$ such that

  $$\parallel {\mathrm{\nabla }}_2\tau (\mathit{v},\mathbf{w})-{\mathrm{\nabla }}_2\tau (\mathit{v},{\mathbf{w}}_0)\parallel \le M(\mathit{v})|\mathbf{w}-{\mathbf{w}}_0|.$$
• (iii) $E\Vert {\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0){\Vert }^2<\mathrm{\infty }$, where the symbol $\Vert \cdot \Vert$ denote the matrix norm: $\Vert (c_{\mathrm{ij}})\Vert =(\sum _{i,j}{c}_{\mathrm{ij}}^2{)}^{1/2}$.
• (iv) $E|{\mathrm{\nabla }}_2|\tau (\mathit{V},{\mathbf{w}}_0)<\mathrm{\infty }$, where $|{\mathrm{\nabla }}_2|\tau (\mathit{v},\mathbf{w})=\sum _{i_1,i_2}|\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{w}_{i_1}\mathrm{\partial }{w}_{i_2}}\tau (\mathit{v},\mathbf{w})|$.
• (v)The matrix $E[{\mathrm{\nabla }}_2\tau (\mathit{V},{\mathbf{w}}_0)]$ is negative definite.

The conditions above are commonly adopted in the literature [5,7,18–20]. These conditions ensure the consistency and asymptotic normality of ${\hat{\mathbf{w}}}_s$. Under these regularity conditions, we establish the consistency and asymptotic normality in the following two Theorems.

Theorem 3.1

Assume that the regularity conditions (C1)–(C4) hold. Then

$${\hat{\mathbf{w}}}_s\stackrel{p}{⟶}{\mathbf{w}}_0.$$

Theorem 3.2

Assume that the regularity conditions (C1)–(C5) hold. Then

$$\sqrt{n}({\hat{\mathbf{w}}}_s-{\mathbf{w}}_0)\stackrel{d}{⟶}N(\mathbf{0},[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}{\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0)[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}),$$

where ‘ $\stackrel{d}{⟶}$’ denote the convergence in distribution, ${\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0)=\frac{1}{L}E[{\mathrm{\nabla }}_2\tau (\mathit{V},{\mathbf{w}}_0)]$ and ${\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0)=E{\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0)[{\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0){]}^T$.

Suggested by a referee, next we also give the asymptotic distribution of the estimated agreement parameters, ${\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)$.

Theorem 3.3

Assume that the regularity conditions (C1)–(C5) hold. Then

$$\begin{array}{rl}& \sqrt{n}\{{\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)-{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)\}\stackrel{d}{⟶}\\ & N(0,[{\mathrm{\nabla }}_1{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0){]}^T[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}{\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0)[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}[{\mathrm{\nabla }}_1{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)]).\end{array}$$

The proofs of Theorems 3.1–3.3 are provided in Appendix. Next we propose using the jacknife method to estimate the asymptotic variance of ${\hat{\mathbf{w}}}_s$ due to the complication of the sandwich matrix $[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}{\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0)[{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0){]}^{-1}$. The jacknife variance estimator of ${\hat{\mathbf{w}}}_s$ is given as below:

$$\frac{{\displaystyle n-1}}{{\displaystyle n}}\sum _{i=1}^n({\hat{\mathbf{w}}}_s^{-i}-n^{-1}\sum _{k=1}^n{\hat{\mathbf{w}}}_s^{-k}){({\hat{\mathbf{w}}}_s^{-i}-n^{-1}\sum _{k=1}^n{\hat{\mathbf{w}}}_s^{-k})}^T,$$ (8)

where ${\hat{\mathbf{w}}}_s^{-i}$ denotes the estimator of ${\mathbf{w}}_0$ based on the original data excluding ${\mathit{V}}_i$.

4. Simulation studies

We conduct simulation studies to examine the performance of our proposed method. We consider the following three scenarios.

For the first scenario, Y is an ordinal random variable taking values $1,2,\dots ,L$ with equal probabilities. Let ${\mathbf{1}}_M$ denote a $M\times 1$ vector with all elements are equal to 1, $\mathbf{\Sigma }=({\sigma }_{\mathrm{ij}}{)}_{M\times M}$ denote a $M\times M$ matrix with the element ${\sigma }_{\mathrm{ij}}={\rho }^{|i-j|}$ if $i\ne j$, ${\sigma }_{\mathrm{ij}}=1$ otherwise. The conditional distribution of ${\mathit{Z}}_i=(Z_{i1},Z_{i2},\dots ,Z_{\mathrm{iM}}{)}^T$ given Y = l is specified as the following two cases: multivariate normal distribution ${\mathit{N}}_M(l{\mathbf{1}}_M,\mathbf{\Sigma })$, or multivariate t distribution $t_5(l{\mathbf{1}}_M,\mathbf{\Sigma })$. We set L = 3, M = 2 or 4, and $\rho =0$, 0.3, or 0.7. Note that $\rho =0$ denotes that the variables in ${\mathit{Z}}_i=(Z_{i1},Z_{i2},\dots ,Z_{\mathrm{iM}}{)}^T$ are independent, and when ρ becomes larger, the correlation between the variables in ${\mathit{Z}}_i=(Z_{i1},Z_{i2},\dots ,Z_{\mathrm{iM}}{)}^T$ becomes stronger. The simulation results based on 1000 replications are presented in Table 1, where ${\mathbf{w}}_0$ denotes the true weight, and BIAS, SE, ASE, and CP denote the empirical bias, the empirical standard deviations, the average estimated standard deviation and the empirical coverage probability of 95% confidence interval respectively. The results show that the proposed approach yields very small empirical biases in all cases. The SE's and the corresponding ASE's are close for all sample sizes. The CP's are close to the nominal value 95%. Moreover, the different values of ρ and L have no obvious impacts on the performances.

Table 1.

Simulation results for the first scenario.

				Normal					t
ρ	M	n	Est	$w_0$	BIAS	SE	ASE	CP	$w_0$	BIAS	SE	ASE	CP
0	2	200	${\hat{w}}_1$	0.500	−0.001	0.052	0.055	96.2%	0.500	0.000	0.067	0.075	95.1%
			${\hat{w}}_2$	0.500	0.001	0.052	0.055	96.1%	0.500	0.000	0.067	0.075	95.4%
		400	${\hat{w}}_1$	0.500	−0.000	0.035	0.037	96.7%	0.500	0.002	0.048	0.050	93.2%
			${\hat{w}}_2$	0.500	0.000	0.035	0.037	96.4%	0.500	−0.002	0.048	0.050	95.3%
	4	200	${\hat{w}}_1$	0.250	−0.002	0.040	0.043	97.4%	0.250	−0.002	0.052	0.058	96.0%
			${\hat{w}}_2$	0.250	0.000	0.041	0.043	97.6%	0.250	−0.002	0.052	0.058	97.2%
			${\hat{w}}_3$	0.250	−0.000	0.039	0.043	97.4%	0.250	0.004	0.051	0.058	96.7%
			${\hat{w}}_4$	0.250	0.002	0.041	0.043	97.1%	0.250	0.000	0.054	0.057	97.1%
		400	${\hat{w}}_1$	0.250	0.000	0.028	0.029	96.7%	0.250	−0.002	0.038	0.041	97.2%
			${\hat{w}}_2$	0.250	−0.000	0.028	0.030	97.8%	0.250	0.006	0.038	0.041	95.1%
			${\hat{w}}_3$	0.250	0.000	0.029	0.029	97.2%	0.250	−0.001	0.037	0.040	98.2%
			${\hat{w}}_4$	0.250	−0.000	0.029	0.029	97.5%	0.250	−0.003	0.036	0.040	96.1%
0.3	2	200	${\hat{w}}_1$	0.500	−0.003	0.077	0.079	95.9%	0.500	0.000	0.098	0.107	94.2%
			${\hat{w}}_2$	0.500	0.003	0.077	0.079	95.0%	0.500	0.000	0.098	0.107	94.5%
		400	${\hat{w}}_1$	0.500	0.001	0.051	0.054	96.1%	0.500	−0.007	0.064	0.075	96.0%
			${\hat{w}}_2$	0.500	−0.001	0.051	0.054	96.4%	0.500	0.007	0.064	0.075	95.0%
	4	200	${\hat{w}}_1$	0.294	0.001	0.058	0.061	96.2%	0.294	−0.001	0.073	0.088	96.3%
			${\hat{w}}_2$	0.206	−0.003	0.064	0.069	97.4%	0.206	0.001	0.08	0.097	96.8%
			${\hat{w}}_3$	0.206	−0.000	0.066	0.068	97.0%	0.206	0.004	0.082	0.096	96.2%
			${\hat{w}}_4$	0.294	0.002	0.059	0.061	95.9%	0.294	−0.004	0.073	0.088	96.4%
		400	${\hat{w}}_1$	0.294	−0.000	0.039	0.042	97.6%	0.294	0.000	0.050	0.058	97.8%
			${\hat{w}}_2$	0.206	0.001	0.045	0.046	96.8%	0.206	0.000	0.060	0.064	96.6%
			${\hat{w}}_3$	0.206	−0.001	0.044	0.046	97.1%	0.206	0.002	0.059	0.064	95.1%
			${\hat{w}}_4$	0.294	0.001	0.039	0.042	97.1%	0.294	−0.002	0.051	0.059	96.6%
0.7	2	200	${\hat{w}}_1$	0.500	−0.009	0.141	0.152	95.8%	0.500	−0.005	0.179	0.198	93.3%
			${\hat{w}}_2$	0.500	0.009	0.141	0.152	93.7%	0.500	0.005	0.179	0.198	93.6%
		400	${\hat{w}}_1$	0.500	0.002	0.098	0.104	95.6%	0.500	−0.005	0.120	0.132	94.7%
			${\hat{w}}_2$	0.500	−0.002	0.098	0.104	95.9%	0.500	0.005	0.120	0.132	93.4%
	4	200	${\hat{w}}_1$	0.385	0.002	0.119	0.134	97.2%	0.385	0.003	0.152	0.185	95.5%
			${\hat{w}}_2$	0.115	0.001	0.153	0.172	96.7%	0.115	−0.002	0.191	0.235	96.3%
			${\hat{w}}_3$	0.115	−0.003	0.157	0.175	95.8%	0.115	−0.008	0.193	0.235	97.2%
			${\hat{w}}_4$	0.385	−0.000	0.117	0.135	97.2%	0.385	0.007	0.146	0.189	95.8%
		400	${\hat{w}}_1$	0.385	−0.002	0.082	0.088	97.0%	0.385	0.005	0.105	0.132	96.2%
			${\hat{w}}_2$	0.115	0.006	0.108	0.114	95.4%	0.115	−0.005	0.133	0.172	95.7%
			${\hat{w}}_3$	0.115	−0.003	0.103	0.115	97.6%	0.115	0.001	0.137	0.176	94.9%
			${\hat{w}}_4$	0.385	−0.001	0.082	0.088	97.5%	0.385	−0.001	0.109	0.136	95.4%

For the second scenario, Y is an ordinal random variable taking values $1,2,\dots ,L$ with nonequal probabilities $1/\sum _{l=1}^{L}l,2/\sum _{l=1}^{L}l,\dots ,L/\sum _{l=1}^{L}l$ respectively. The other configurations are the same to the first scenario. The simulation results are reported in Table 2. Again, we find that BIAS's are very small, the SE's and the corresponding ASE's are close, and the CP's are close to the nominal value 95%.

Table 2.

Simulation results for the second scenario.

				Normal					t
ρ	M	n	Est	$w_0$	BIAS	SE	ASE	CP	$w_0$	BIAS	SE	ASE	CP
0	2	200	${\hat{w}}_1$	0.500	−0.001	0.061	0.061	96.1%	0.500	−0.003	0.073	0.086	96.4%
			${\hat{w}}_2$	0.500	0.001	0.061	0.061	95.7%	0.500	0.003	0.073	0.086	96.2%
		400	${\hat{w}}_1$	0.500	−0.000	0.042	0.042	96.0%	0.500	0.000	0.052	0.056	94.6%
			${\hat{w}}_2$	0.500	0.000	0.042	0.042	96.3%	0.500	0.000	0.052	0.056	95.6%
	4	200	${\hat{w}}_1$	0.250	0.000	0.044	0.048	97.5%	0.250	0.000	0.060	0.068	96.1%
			${\hat{w}}_2$	0.250	−0.000	0.044	0.047	97.2%	0.250	−0.001	0.059	0.067	96.9%
			${\hat{w}}_3$	0.250	−0.000	0.046	0.047	96.7%	0.250	0.001	0.058	0.068	97.6%
			${\hat{w}}_4$	0.250	0.000	0.045	0.048	97.5%	0.250	0.000	0.057	0.068	96.6%
		400	${\hat{w}}_1$	0.250	−0.000	0.031	0.032	98.0%	0.250	0.000	0.042	0.046	96.7%
			${\hat{w}}_2$	0.250	−0.000	0.031	0.032	96.9%	0.250	−0.003	0.042	0.046	98.2%
			${\hat{w}}_3$	0.250	0.001	0.031	0.032	97.3%	0.250	0.003	0.040	0.045	96.7%
			${\hat{w}}_4$	0.250	0.000	0.032	0.032	96.6%	0.250	0.000	0.042	0.045	96.9%
0.3	2	200	${\hat{w}}_1$	0.500	−0.003	0.087	0.090	96.1%	0.500	−0.007	0.108	0.122	94.5%
			${\hat{w}}_2$	0.500	0.003	0.087	0.090	95.2%	0.500	0.007	0.108	0.122	94.7%
		400	${\hat{w}}_1$	0.500	0.000	0.058	0.061	96.6%	0.500	−0.003	0.076	0.088	94.4%
			${\hat{w}}_2$	0.500	0.000	0.058	0.061	96.3%	0.500	0.003	0.076	0.088	94.6%
	4	200	${\hat{w}}_1$	0.294	−0.003	0.063	0.068	97.5%	0.294	0.000	0.084	0.097	96.8%
			${\hat{w}}_2$	0.206	0.002	0.073	0.077	96.3%	0.206	0.000	0.091	0.108	96.2%
			${\hat{w}}_3$	0.206	0.002	0.074	0.077	96.8%	0.206	0.001	0.092	0.107	96.5%
			${\hat{w}}_4$	0.294	−0.000	0.062	0.069	97.6%	0.294	−0.001	0.084	0.098	96.4%
		400	${\hat{w}}_1$	0.294	−0.003	0.044	0.047	97.3%	0.294	0.007	0.056	0.068	97.0%
			${\hat{w}}_2$	0.206	0.001	0.049	0.052	97.6%	0.206	−0.003	0.062	0.073	97.3%
			${\hat{w}}_3$	0.206	0.001	0.050	0.052	96.5%	0.206	−0.001	0.063	0.074	97.0%
			${\hat{w}}_4$	0.294	0.002	0.044	0.046	95.4%	0.294	−0.003	0.056	0.066	97.7%
0.7	2	200	${\hat{w}}_1$	0.500	−0.005	0.155	0.178	95.3%	0.500	0.010	0.193	0.229	94.9%
			${\hat{w}}_2$	0.500	0.005	0.155	0.178	95.0%	0.500	−0.010	0.193	0.229	95.4%
		400	${\hat{w}}_1$	0.500	0.002	0.110	0.115	95.1%	0.500	0.003	0.138	0.159	95.4%
			${\hat{w}}_2$	0.500	−0.002	0.110	0.115	95.7%	0.500	−0.003	0.138	0.159	94.7%
	4	200	${\hat{w}}_1$	0.385	0.008	0.128	0.160	97.4%	0.385	0.005	0.164	0.218	95.9%
			${\hat{w}}_2$	0.115	−0.014	0.180	0.206	97.9%	0.115	−0.008	0.210	0.270	97.7%
			${\hat{w}}_3$	0.115	0.003	0.183	0.206	95.7%	0.115	0.007	0.207	0.267	96.2%
			${\hat{w}}_4$	0.385	0.004	0.132	0.161	97.1%	0.385	−0.004	0.169	0.214	96.5%
		400	${\hat{w}}_1$	0.385	0.001	0.093	0.103	97.2%	0.385	0.016	0.112	0.150	95.5%
			${\hat{w}}_2$	0.115	−0.007	0.115	0.134	96.5%	0.115	−0.014	0.141	0.189	97.0%
			${\hat{w}}_3$	0.115	0.006	0.121	0.134	96.0%	0.115	0.006	0.150	0.193	97.3%
			${\hat{w}}_4$	0.385	0.001	0.095	0.102	96.8%	0.385	−0.009	0.113	0.152	97.9%

In the last scenario, we consider a heteroscedasticity case. The settings are same to the first and second scenario except that $\mathbf{\Sigma }=({\sigma }_{\mathrm{ij}}{)}_{M\times M}$ is a $M\times M$ matrix, where the diagonal elements ${\sigma }_{\mathrm{ii}}=i$, but the off-diagonal elements ${\sigma }_{\mathrm{ij}}={\rho }^{|i-j|}$ for $i\ne j$. That is, when $k\ne l$, the variables $Z_{\mathrm{ik}}$ and $Z_{\mathrm{il}}$ have different variances. The simulation results are presented in Tables 3 and 4. Table 3 is for the case when Y is an ordinal random variable taking values $1,2,\dots ,L$ with equal probabilities, and Table 4 is for the unequal probabilities case. Like in the first scenario and the second scenario, we consistently observe good performance of the proposed method in the heteroscedasticity case. In summary, our results indicate the good and broad utility of the proposed approach.

Table 3.

Simulation results for the third scenario with Y taking values $1,2,\dots ,L$ with equal probabilities.

				Normal					t
ρ	M	n	Est	$w_0$	BIAS	SE	ASE	CP	$w_0$	BIAS	SE	ASE	CP
0	2	200	${\hat{w}}_1$	0.667	−0.001	0.054	0.055	96.7%	0.667	0.000	0.066	0.076	94.8%
			${\hat{w}}_2$	0.333	0.001	0.054	0.055	94.6%	0.333	0.000	0.066	0.076	95.1%
		400	${\hat{w}}_1$	0.667	−0.000	0.035	0.037	95.9%	0.667	−0.001	0.046	0.051	94.5%
			${\hat{w}}_2$	0.333	0.000	0.035	0.037	96.2%	0.333	0.001	0.046	0.051	93.5%
	4	200	${\hat{w}}_1$	0.480	0.002	0.050	0.056	97.0%	0.480	−0.002	0.063	0.082	97.0%
			${\hat{w}}_2$	0.240	0.002	0.044	0.047	97.0%	0.240	0.002	0.056	0.071	96.0%
			${\hat{w}}_3$	0.160	−0.003	0.037	0.041	98.1%	0.160	0.000	0.048	0.061	97.1%
			${\hat{w}}_4$	0.120	−0.001	0.034	0.036	96.9%	0.120	0.000	0.043	0.055	95.8%
		400	${\hat{w}}_1$	0.480	−0.000	0.035	0.037	96.7%	0.480	−0.004	0.046	0.053	97.2%
			${\hat{w}}_2$	0.240	−0.000	0.030	0.032	97.0%	0.240	0.002	0.036	0.046	96.4%
			${\hat{w}}_3$	0.160	0.000	0.027	0.028	95.9%	0.160	0.001	0.034	0.040	94.7%
			${\hat{w}}_4$	0.120	−0.000	0.024	0.024	96.0%	0.120	0.001	0.03	0.035	95.5%
0.3	2	200	${\hat{w}}_1$	0.708	−0.000	0.067	0.072	95.2%	0.708	0.000	0.082	0.084	97.0%
			${\hat{w}}_2$	0.292	0.000	0.067	0.072	95.2%	0.292	0.000	0.082	0.084	96.8%
		400	${\hat{w}}_1$	0.708	0.001	0.047	0.048	96.4%	0.708	0.001	0.056	0.058	96.9%
			${\hat{w}}_2$	0.292	−0.001	0.047	0.048	95.7%	0.292	−0.001	0.056	0.058	97.9%
	4	200	${\hat{w}}_1$	0.525	0.004	0.064	0.072	96.1%	0.525	−0.002	0.081	0.086	97.2%
			${\hat{w}}_2$	0.193	−0.003	0.058	0.064	96.3%	0.193	0.001	0.069	0.076	97.2%
			${\hat{w}}_3$	0.152	−0.002	0.046	0.052	97.5%	0.152	0.001	0.061	0.063	96.0%
			${\hat{w}}_4$	0.131	0.001	0.040	0.044	96.3%	0.131	−0.002	0.049	0.052	96.8%
		400	${\hat{w}}_1$	0.525	0.000	0.044	0.048	97.1%	0.525	−0.003	0.055	0.059	97.0%
			${\hat{w}}_2$	0.193	−0.000	0.038	0.043	97.7%	0.193	0.003	0.049	0.053	96.1%
			${\hat{w}}_3$	0.152	−0.000	0.030	0.034	96.5%	0.152	−0.001	0.039	0.043	97.2%
			${\hat{w}}_4$	0.131	0.000	0.028	0.029	96.7%	0.131	0.000	0.034	0.036	96.7%
0.7	2	200	${\hat{w}}_1$	0.812	−0.004	0.090	0.102	95.7%	0.812	0.004	0.120	0.125	95.6%
			${\hat{w}}_2$	0.188	0.004	0.090	0.102	96.0%	0.188	−0.004	0.120	0.125	97.0%
		400	${\hat{w}}_1$	0.812	−0.001	0.065	0.066	95.9%	0.812	0.003	0.080	0.083	96.0%
			${\hat{w}}_2$	0.188	0.001	0.065	0.066	94.2%	0.188	−0.003	0.080	0.083	96.4%
	4	200	${\hat{w}}_1$	0.650	−0.006	0.095	0.116	96.9%	0.650	0.006	0.126	0.134	96.6%
			${\hat{w}}_2$	0.118	0.004	0.084	0.101	96.1%	0.118	-0.006	0.115	0.121	97.8%
			${\hat{w}}_3$	0.115	−0.002	0.065	0.076	95.6%	0.115	−0.004	0.085	0.089	97.2%
			${\hat{w}}_4$	0.117	0.003	0.053	0.062	95.5%	0.117	0.004	0.066	0.073	96.2%
		400	${\hat{w}}_1$	0.650	−0.002	0.069	0.077	96.1%	0.650	−0.004	0.082	0.084	96.7%
			${\hat{w}}_2$	0.118	0.001	0.059	0.069	95.9%	0.118	0.005	0.078	0.079	97.2%
			${\hat{w}}_3$	0.115	0.001	0.045	0.050	95.7%	0.115	0.003	0.057	0.057	97.7%
			${\hat{w}}_4$	0.117	−0.000	0.038	0.042	96.5%	0.117	−0.004	0.048	0.049	96.6%

Table 4.

Simulation results for the third scenario with Y taking values $1,2,\dots ,L$ with different probabilities.

				Normal					t
ρ	M	n	Est	$w_0$	BIAS	SE	ASE	CP	$w_0$	BIAS	SE	ASE	CP
0	2	200	${\hat{w}}_1$	0.667	−0.000	0.060	0.065	95.4%	0.667	0.001	0.074	0.078	97.2%
			${\hat{w}}_2$	0.333	0.000	0.060	0.065	94.8%	0.333	−0.001	0.074	0.078	96.8%
		400	${\hat{w}}_1$	0.667	0.000	0.041	0.043	96.0%	0.667	−0.002	0.052	0.052	95.2%
			${\hat{w}}_2$	0.333	−0.000	0.041	0.043	96.2%	0.333	0.002	0.052	0.052	96.8%
	4	200	${\hat{w}}_1$	0.480	−0.000	0.058	0.065	96.7%	0.480	−0.002	0.076	0.078	97.4%
			${\hat{w}}_2$	0.240	0.000	0.051	0.055	97.1%	0.240	−0.004	0.061	0.067	97.0%
			${\hat{w}}_3$	0.160	0.000	0.044	0.047	96.8%	0.160	0.002	0.055	0.057	97.0%
			${\hat{w}}_4$	0.120	−0.000	0.039	0.041	96.9%	0.120	0.004	0.052	0.050	96.0%
		400	${\hat{w}}_1$	0.480	−0.001	0.041	0.043	97.3%	0.480	0.004	0.052	0.053	97.0%
			${\hat{w}}_2$	0.240	0.002	0.034	0.037	96.7%	0.240	−0.003	0.042	0.046	97.6%
			${\hat{w}}_3$	0.160	−0.001	0.030	0.033	97.0%	0.160	−0.001	0.036	0.039	97.5%
			${\hat{w}}_4$	0.120	−0.000	0.027	0.028	97.1%	0.120	0.001	0.033	0.034	96.7%
0.3	2	200	${\hat{w}}_1$	0.708	0.001	0.073	0.087	96.6%	0.708	0.001	0.096	0.099	96.8%
			${\hat{w}}_2$	0.292	−0.000	0.070	0.082	96.5%	0.292	−0.001	0.096	0.099	96.4%
		400	${\hat{w}}_1$	0.708	−0.000	0.054	0.063	96.9%	0.708	−0.004	0.064	0.067	97.6%
			${\hat{w}}_2$	0.292	0.000	0.053	0.056	95.6%	0.292	0.004	0.064	0.067	96.2%
	4	200	${\hat{w}}_1$	0.525	0.001	0.071	0.083	96.7%	0.252	0.000	0.092	0.099	96.8%
			${\hat{w}}_2$	0.193	−0.001	0.067	0.075	96.9%	0.193	−0.001	0.080	0.089	97.6%
			${\hat{w}}_3$	0.152	0.001	0.052	0.059	96.4%	0.152	0.001	0.065	0.071	97.2%
			${\hat{w}}_4$	0.131	−0.002	0.047	0.050	96.5%	0.131	−0.001	0.055	0.059	97.8%
		400	${\hat{w}}_1$	0.525	−0.001	0.049	0.055	96.9%	0.252	0.001	0.068	0.065	96.8%
			${\hat{w}}_2$	0.193	0.001	0.044	0.049	97.5%	0.193	−0.001	0.059	0.058	96.0%
			${\hat{w}}_3$	0.152	−0.001	0.036	0.039	96.5%	0.152	0.001	0.044	0.048	97.2%
			${\hat{w}}_4$	0.131	0.000	0.030	0.033	97.4%	0.131	−0.002	0.038	0.040	97.4%
0.7	2	200	${\hat{w}}_1$	0.812	−0.002	0.108	0.120	95.0%	0.812	0.000	0.135	0.138	96.8%
			${\hat{w}}_2$	0.188	0.002	0.108	0.120	95.8%	0.188	0.000	0.135	0.138	96.6%
		400	${\hat{w}}_1$	0.812	−0.001	0.072	0.080	95.8%	0.812	−0.007	0.090	0.092	96.8%
			${\hat{w}}_2$	0.188	0.001	0.072	0.080	95.4%	0.188	0.007	0.090	0.092	95.2%
	4	200	${\hat{w}}_1$	0.650	0.000	0.116	0.139	96.9%	0.650	0.000	0.141	0.154	96.4%
			${\hat{w}}_2$	0.118	−0.006	0.101	0.122	96.9%	0.118	0.000	0.123	0.135	96.8%
			${\hat{w}}_3$	0.115	0.004	0.078	0.092	95.8%	0.115	0.001	0.090	0.099	96.8%
			${\hat{w}}_4$	0.117	0.002	0.062	0.075	96.4%	0.117	−0.001	0.074	0.080	97.0%
		400	${\hat{w}}_1$	0.650	0.003	0.082	0.092	96.7%	0.650	0.001	0.105	0.103	96.8%
			${\hat{w}}_2$	0.118	−0.004	0.073	0.078	96.3%	0.118	0.000	0.089	0.09	96.0%
			${\hat{w}}_3$	0.115	0.001	0.050	0.058	96.0%	0.115	0.001	0.062	0.069	97.2%
			${\hat{w}}_4$	0.117	−0.001	0.043	0.049	95.8%	0.117	−0.002	0.051	0.055	97.2%

5. Data example

We apply our method to a cross-sectional Grady trauma study of PTSD. This research was approved by the Trauma-Related Health Sequelae at Grady Memorial Hospital, Emory University (IRB number: 00002114, 00078593). The current PTSD diagnosis (Y ) is obtained from CAPS (0= PTSD negative, 1= PTSD positive), considered the gold standard for PTSD diagnosis. Data collected from a 17-item self-reported version of PSS are used in our analysis. The self-reported PSS consists of four subscales, including PSS intrusive $(Z_1)$, PSS avoidance $(Z_2)$, PSS negative $(Z_3)$, and PSS hyerarousal $(Z_4)$. Specifically, PSS intrusive and PSS negative each include four items with a subtotal score ranging from 0 to 12, PSS avoidance consists of 3 items with a subtotal score ranging from 0 to 9, and PSS avoidnumb includes six items with a subtotal score ranging from 0 to 18. A total of 942 participants with records on all variables (CAPS and the four PSS subscales) are included for further analysis. Among these participants, 433 (46.0 $\mathrm{\%}$) have a positive PTSD diagnosis.

Our goal for this analysis is to develop a new score based on four PSS subscales with the highest agreement with the current diagnosis of PTSD. To make the contributions of the four PSS scores comparable, we standardize them using the Z-score approach. The newly scaled PSS scores are calculated by subtracting the mean from the original score and then dividing it by the original score's standard deviation. As a preliminary result, we first present the descriptive statistics of four PSS subscales (intrusive, avoidance, negative, and hyerarousal) between the positive PTSD group vs. the negative PTSD group. As seen in Figure 1, the PSS subscales are different among those with PTSD vs. those without PTSD.

Figure 1.

The box plot of PSS intrusive, PSS avoidance, PSS negative, and PSS hyerarousal between positive PTSD group (‘1’) vs. the negative PTSD group (‘0’).

Next, we apply our method by considering four PSS subscales and estimate the optimal weights by (7). As shown in Table 5, the optimal weights, computed by (7) respectively, for PSS intrusive, PSS avoidance, PSS negative, and PSS hyerarousal are ${\hat{\mathbf{w}}}_s=(0.2281,0.1199,0.0991,0.5529{)}^T$, and the corresponding broad sense agreement ${\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)=0.4895$. The weight estimates suggest a more dominating role of PSS hyerarousal in contributing to the PTSD diagnosis.

Table 5.

Optimal weights estimation of four PSS subscales for a cross-sectional Grady trauma study of PTSD.

PSS subscale	${\hat{w}}_s$	SE	CI
PSS intrusive	0.2281	0.1162	(0.0004, 0.4557)
PSS avoidance	0.1199	0.0936	(0.0000, 0.3034)
PSS negative	0.0991	0.0946	(0.0000, 0.2845)
PSS hyerarousal	0.5529	0.1040	(0.3668, 0.7567)

Note: ${\hat{w}}_s$: the estimated weights, SE: the standard error of ${\hat{w}}_s$, CI: 95% confidence interval.

We also calculate the BSA between the average value of the four subscales of the PSS and CAPS-based PTSD diagnosis and the resulting broad sense agreement 0.4772, which is smaller than ${\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)=0.4895$. We also test the hypothesis $H_0:\mathbf{w}=(1/4,1/4,1/4,1/4{)}^T$ v.s. $H_1:\mathbf{w}\ne (1/4,1/4,1/4,1/4{)}^T.$ The test statistic yields a P-value of 0.0013, which suggests equal weighting across the four subscales may not be appropriate.

To gain insight into the problem, we conduct a logistic regression for predicting the current PTSD diagnosis based on PSS intrusive, PSS avoidance, PSS negative, and PSS hyerarousal. Results show that PSS intrusive (estimate=0.2117, se=0.1046, p-value=0.043) and PSS hyerarousal (estimate=0.6710, se=0.1062, p-value<0.001) are significantly associated with PTSD diagnosis, while the association of PSS avoidance (estimate=0.1150, se=0.0967, p-value=0.234) and PSS negative (estimate=0.1545, se=0.1018, p-value=0.129) with PTSD diagnosis are not significant. This is consistent with the weight estimates, which indicate relatively minor contributions of PSS avoidance and PSS negative to PTSD diagnosis.

6. Concluding remarks

In this article, we propose a new method to derive a weighted score that can provide the highest agreement between the new score and the disease status. The proposed method offers an objective scientific way of constructing a new score that best aligns with the disease status. The weights are determined by the proposed methods and indicate the strength of each item subscale's contribution to the overall score. For a new patient, rather than totaling the scores of each of the four items, one can use the weighted score to represent the severity of the disease status better.

A valuable by-product of our method is a test for testing the hypothesis that the contribution of an item to the score is zero or that the contributions of different items are equal. In practice, computational difficulties become intensive when too many items exist. In this case, we recommend grouping items based on clinical guidance to reduce the number of item groups M. Another important question is the required number of subjects to estimate the weighted score when the ordinal disease severity is available. We note that The estimated weights follow a multivariate normal distribution asymptotically with the mean of true weights and variance-covariance matrix. The problem of calculating sample size for weights is complicated because estimated weights are correlated, and the estimated variance cannot be explicitly written. Simulation studies offer better insights for determining the required width of confidence interval weights for specified sample sizes, which may interest future studies.

Appendix.

Proof of Theorem 3.1

We divide the proof of this Theorem into two steps.

Step 1. First, we prove that $\hat{\mathbf{w}}\stackrel{p}{⟶}{\mathbf{w}}_0$. According to the Theorem 2.1 in [11] and the condition (C3), what remains for consistency is to show

$$\begin{array}{r}\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})-{\rho }_{\mathrm{bsa}}(\mathbf{w})|\stackrel{p}{⟶}0.\end{array}$$ (A1)

By the similar proof of [14,22], we have

$$\begin{array}{rl}{W}_n(\mathbf{w})& ={\left(\prod _{l=1}^L{n}_l\right)}^{-1}\sum _{j_1=1}^{n_1}\dots \sum _{j_L=1}^{n_L}\parallel \overrightarrow{\mathbf{S}}-\overrightarrow{\mathbf{R}}({\tilde{X}}_{1,j_1}(\mathbf{w}),\dots ,{\tilde{X}}_{L,j_L}(\mathbf{w})){\parallel }^2\\ & ={\left(\prod _{l=1}^L{n}_l\right)}^{-1}\frac{1}{L!}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathrm{\Theta }}_L)\\ & \times \sum _{k=1}^L{\{Y_{m_k}-\sum _{r=1}^{L}I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))\}}^2\\ & =2{\left(\prod _{l=1}^L{n}_l\right)}^{-1}\frac{1}{L!}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathrm{\Theta }}_L)\\ & \times \sum _{k=1}^L\{Y_{m_k}^2-Y_{m_k}\sum _{r=1}^{L}I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))\}\\ & =2{\left(\prod _{l=1}^L{n}_l\right)}^{-1}\frac{1}{L!}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}).\end{array}$$

Therefore,

$${\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})=\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}\{1-\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})M_{n,L}^{\ast }\},$$

where $M_{n,L}^{\ast }=2(\genfrac{}{}{0ex}{}{n}{L})/\{C_L\cdot \prod _{l=1}^L{n}_l\}$.

Define

$$\begin{array}{rl}{\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})& =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}h({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}),\\ {\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})& =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})\{M_{n,L}^{\ast }-{\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1}\}.\end{array}$$

Then ${\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})={\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})$. Thus it follows that

$$\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})-{\rho }_{\mathrm{bsa}}(\mathbf{w})|\le \underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\rho }_{\mathrm{bsa}}(\mathbf{w})|+\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})|.$$ (A2)

For the second item on the right-hand side of (A2), note that the uniform boundedness of $\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})$ and the convergence of $M_{n,L}^{\ast }\stackrel{p}{\to }{\gamma }_L(\prod _{l=1}^L{p}_l{)}^{-1}$ shown in [14], we have $\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})|\stackrel{p}{\to }0$. For the first item on the right-hand side of (A2). Let

$$\mathcal{F}=\{h({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}):\mathbf{w}\in \mathbf{A}\}.$$

According to the analogous subgraph set arguments as [18] and Lemma 2.14 in [13], we can show that it is Euclidean for a constant envelope. Thus, by applying Corollary 7 in [19], we get

$$\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\rho }_{\mathrm{bsa}}(\mathbf{w})|\stackrel{p}{\to }0.$$

Combining these arguments, we conclude the proof of (A1) and also the proof of $\hat{\mathbf{w}}\stackrel{p}{⟶}{\mathbf{w}}_0$.

Step 2. Next, we prove that ${\hat{\mathbf{w}}}_s\stackrel{p}{⟶}{\mathbf{w}}_0$. By Theorem 2.1 in [11], it is sufficient to show that

$$\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})|\stackrel{p}{⟶}0.$$ (A3)

Similarly, since ${\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})={\hat{\rho }}_{\mathrm{bsa},s}^{[1]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[2]}(\mathbf{w})$, where

$$\begin{array}{rl}{\hat{\rho }}_{\mathrm{bsa},s}^{[1]}(\mathbf{w})& =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathrm{\Omega }}_{n,L}}{h}_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}),\\ {\hat{\rho }}_{\mathrm{bsa},s}^{[2]}(\mathbf{w})& =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathrm{\Omega }}_{n,L}}{\mathrm{\Psi }}_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})\{M_{n,L}^{\ast }-{\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1}\},\end{array}$$

and

$$\begin{array}{rl}& h_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})=1-{\mathrm{\Psi }}_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w}){\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1},\\ & {\mathrm{\Psi }}_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})=I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L)\\ & \times \sum _{k=1}^L\{Y_{m_k}^2-Y_{m_k}\sum _{r=1}^L{s}_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))\}.\end{array}$$

Hence

$$\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})|p\le \underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[1]}(\mathbf{w})|+\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[2]}(\mathbf{w})|.$$ (A4)

Note that the boundedness of ${\mathrm{\Psi }}_s({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})$ and $\mathrm{\Psi }({\mathit{V}}_{m_1},\dots ,{\mathit{V}}_{m_L},\mathbf{w})$ again and by the similar derivations as above, we have $\underset{\mathbf{w}\in \mathbf{A}}{sup}|{\hat{\rho }}_{\mathrm{bsa}}^{[2]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[2]}(\mathbf{w})|\stackrel{p}{\to }0$. But for the first item on the right hand side of (A4), since

$$\begin{array}{rl}& {\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[1]}(\mathbf{w})\\ & =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L){\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1}\\ & \times \sum _{k=1}^L[\{Y_{m_k}^2-Y_{m_k}\sum _{r=1}^{L}I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))\}\\ & -\{Y_{m_k}^2-Y_{m_k}\sum _{r=1}^L{s}_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))\}]\\ & =\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L){\gamma }_L{\left(\prod _{l=1}^L{p}_l\right)}^{-1}\\ & \times \sum _{k=1}^L\sum _{r=1}^L[I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))-s_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))].\end{array}$$

Thus there exists a large constant M such that for any $\eta >0$,

$$\begin{array}{rl}& |{\hat{\rho }}_{\mathrm{bsa}}^{[1]}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}^{[1]}(\mathbf{w})|\\ & \le M\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L)\\ & \times \sum _{k=1}^L\sum _{r=1}^L|I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))-s_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))|\\ & =T_{n1}(\mathbf{w})+T_{n2}(\mathbf{w}),\end{array}$$

where

$$\begin{array}{rl}& T_{n1}(\mathbf{w})=M\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L)\\ & \times \sum _{k=1}^L\sum _{r=1}^L|I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))-s_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))|\\ & \times I(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|\ge \eta ),\\ & T_{n2}(\mathbf{w})=M\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathbf{\Omega }}_{n,L}}I((Y_{m_1},\dots ,Y_{m_L})\in {\mathbf{\Theta }}_L),\\ & \times \sum _{k=1}^L\sum _{r=1}^L|I(X_{m_k}(\mathbf{w})\ge X_{m_r}(\mathbf{w}))-s_n(X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w}))|\\ & \times I(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|<\eta ).\end{array}$$

On the set $\{|x|\ge \eta \}$, we have $|s_n(x)-I(x\ge 0)|\le \text{exp}(-|x|/{\sigma }_n)\le \text{exp}(-\eta /{\sigma }_n)$. Thus when ${\sigma }_n\to 0$, $s_n(x)\to I(x\ge 0)$ uniformly on the set $\{|x|\ge \eta \}$. Therefore, $T_{n1}(\mathbf{w})$ converges to 0 uniformly over $\mathbf{A}$. The second term

$$T_{n2}(\mathbf{w})\le M\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathrm{\Omega }}_{n,L}}\sum _{k=1}^L\sum _{r=1}^{L}I(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|<\eta ).$$

Since the class of indicator functions is manageable, we have

$$\frac{1}{P(n,L)}\sum _{(m_1,\dots ,m_L)\in {\mathrm{\Omega }}_{n,L}}\sum _{k=1}^L\sum _{r=1}^{L}I(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|<\eta )$$

converges uniformly almost surely to $\sum _{k=1}^L\sum _{r=1}^{L}P(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|<\eta )$ by uniform convergence of U-processes [12]. Letting $\eta \to 0$, $\sum _{k=1}^L\sum _{r=1}^{L}P(|X_{m_k}(\mathbf{w})-X_{m_r}(\mathbf{w})|<\eta )\to 0$ uniformly over $\mathbf{A}$ as $n\to \mathrm{\infty }$. This completes the proof of Theorem 3.1.

Proof of Theorem 3.2

Without loss of generality, we just prove the case when L = 3. Define the functions ${\mathrm{\Gamma }}_{\mathrm{sn}}(\mathbf{w})={\hat{\rho }}_{\mathrm{bsa},s}(\mathbf{w})-{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)$, and ${\mathrm{\Gamma }}_0(\mathbf{w})={\rho }_{\mathrm{bsa}}(\mathbf{w})-{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)$. According to Theorem 2 of [18], to prove Theorem 2, we only need to show

$$\begin{array}{rl}{\mathrm{\Gamma }}_{\mathrm{sn}}(\mathbf{w})& =\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0)(\mathbf{w}-{\mathbf{w}}_0)+\frac{1}{\sqrt{n}}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathit{W}}_n({\mathbf{w}}_0)\\ & +o_p(|\mathbf{w}-{\mathbf{w}}_0{|}^2)+o_p(1/n),\end{array}$$ (A5)

uniformly in $o_p(1)$ neighborhood of ${\mathbf{w}}_0$, where ${\mathit{W}}_n({\mathbf{w}}_0)\stackrel{d}{\to }N(\mathbf{0},{\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0))$.

First, let

$$\begin{array}{r}{\tau }_s(\mathit{v},\mathbf{w})=E{h}_s^{\ast }(\mathit{v},{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})+E{h}_s^{\ast }({\mathit{V}}_1,\mathit{v},{\mathit{V}}_3,\mathbf{w})+E{h}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,\mathit{v},\mathbf{w}),\end{array}$$

where $h_s^{\ast }({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})=1-{\mathrm{\Psi }}_s({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})M_{n,3}^{\ast }.$ Under the condition (C4) and by similar derivations as (A.1) in [20], rewriting ${\tau }_s(\mathit{v},\mathbf{w})$ as an integral, changing variables, the Taylor expansion technique, and the convergence of $M_{n,3}^{\ast }\stackrel{p}{\to }{\gamma }_3(\prod _{l=1}^3{p}_l{)}^{-1}$ shown in [14], we can get

$${\mathrm{\nabla }}_1{\tau }_s(\mathit{v},\mathbf{w})={\mathrm{\nabla }}_1\tau (\mathit{v},\mathbf{w})+O({\sigma }_n),\text{and}{\mathrm{\nabla }}_2{\tau }_s(\mathit{v},\mathbf{w})={\mathrm{\nabla }}_2\tau (\mathit{v},\mathbf{w})+O({\sigma }_n).$$ (A6)

Define $f_s^{\ast }({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})=h_s^{\ast }({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})-h_s^{\ast }({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},{\mathbf{w}}_0)$. Using same notations as in [19], denote $P_n$ as the empirical measure that places mass $n^{-1}$ at each ${\mathit{V}}_i$, and denote $U_n^k$ as a random probability measure putting mass $1/n(n-1)\cdots (n-k+1)$ at each ordered k-tuple, k = 2, 3. Then by the similar technique as the Hoeffding decomposition [17], it follows that

$${\mathrm{\Gamma }}_{\mathrm{sn}}(\mathbf{w})={\mathrm{\Gamma }}_{s0}(\mathbf{w})+P_n{f}_{s1}^{\ast }(\cdot ,\mathbf{w})+U_n^2{f}_{s2}^{\ast }(\cdot ,\cdot ,\mathbf{w})+U_n^3{f}_{s3}^{\ast }(\cdot ,\cdot ,\cdot ,\mathbf{w}),$$ (A7)

where

$$\begin{array}{rl}{\mathrm{\Gamma }}_{s0}(\mathbf{w})& =E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w}),\\ f_{s1}^{\ast }(\mathit{v},\mathbf{w})& =E{f}_s^{\ast }(\mathit{v},{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})+E{f}_s^{\ast }({\mathit{V}}_1,\mathit{v},{\mathit{V}}_3,\mathbf{w})\\ & +E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,\mathit{v},\mathbf{w})-3{\mathrm{\Gamma }}_{s0}(\mathbf{w}),\\ f_{s2}^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w})& =E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})+E{f}_s^{\ast }({\mathit{v}}_2,{\mathit{V}}_3,{\mathit{v}}_1,\mathbf{w})+E{f}_s^{\ast }({\mathit{V}}_3,{\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w})\\ & -E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})-E{f}_s^{\ast }({\mathit{v}}_2,{\mathit{V}}_3,{\mathit{V}}_1,\mathbf{w})-E{f}_s^{\ast }({\mathit{V}}_3,{\mathit{v}}_1,{\mathit{V}}_2,\mathbf{w})\\ & -E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})-E{f}_s^{\ast }({\mathit{V}}_2,{\mathit{V}}_3,{\mathit{v}}_1,\mathbf{w})\\ & -E{f}_s^{\ast }({\mathit{V}}_3,{\mathit{V}}_1,{\mathit{v}}_2,\mathbf{w})+3{\mathrm{\Gamma }}_{s0}(\mathbf{w}),\\ f_{s3}^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})& =E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})-E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})-E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{v}}_3,\mathbf{w})\\ & -E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})+E{f}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})+E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})\\ & +E{f}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,{\mathit{v}}_3,\mathbf{w})-{\mathrm{\Gamma }}_{s0}(\mathbf{w}).\end{array}$$

In the following, we show

$$\begin{array}{rl}{\mathrm{\Gamma }}_{s0}(\mathbf{w})& =\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathbf{\Sigma }}^{\ast }({\mathbf{w}}_0)(\mathbf{w}-{\mathbf{w}}_0)+O({\sigma }_n|\mathbf{w}-{\mathbf{w}}_0|)+o(|\mathbf{w}-{\mathbf{w}}_0{|}^2),\\ & \text{uniformly in an}{o}_p(1)\text{neighborhood of}{\mathbf{w}}_0,\end{array}$$ (A8)

$$\begin{array}{rl}{P}_n{f}_{s1}^{\ast }(\cdot ,\mathbf{w})& =\frac{1}{\sqrt{n}}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathit{W}}_n({\mathbf{w}}_0)+O({\sigma }_n|\mathbf{w}-{\mathbf{w}}_0|)+o(|\mathbf{w}-{\mathbf{w}}_0{|}^2),\\ & \text{uniformly in an}{o}_p(1)\text{neighborhood of}{\mathbf{w}}_0,\end{array}$$ (A9)

and

$$U_n^2{f}_{s2}^{\ast }(\cdot ,\cdot ,\mathbf{w})+U_n^3{f}_{s3}^{\ast }(\cdot ,\cdot ,\cdot ,\mathbf{w})=o_p(1/n),$$ (A10)

uniformly in an $o_p(1)$ neighborhood of ${\mathbf{w}}_0.$

For the Equation (A8), recall the definition of ${\tau }_s(\mathit{v},\mathbf{w})$. By expanding ${\tau }_s(\mathit{v},\mathbf{w})$ about ${\mathbf{w}}_0$ and the Equation (A6), it follows that

$$\begin{array}{rl}{\tau }_s(\mathit{v},\mathbf{w})& ={\tau }_s(\mathit{v},{\mathbf{w}}_0)+(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathrm{\nabla }}_1{\tau }_s(\mathit{v},{\mathbf{w}}_0)+\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathrm{\nabla }}_2{\tau }_s(\mathit{v},{\mathbf{w}}^{\ast })(\mathbf{w}-{\mathbf{w}}_0)\\ & ={\tau }_s(\mathit{v},{\mathbf{w}}_0)+(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathrm{\nabla }}_1\tau (\mathit{v},{\mathbf{w}}_0)+\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathrm{\nabla }}_2\tau (\mathit{v},{\mathbf{w}}_0)(\mathbf{w}-{\mathbf{w}}_0)\\ & +\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T[{\mathrm{\nabla }}_2\tau (\mathit{v},{\mathbf{w}}^{\ast })-{\mathrm{\nabla }}_2\tau (\mathit{v},{\mathbf{w}}_0)](\mathbf{w}-{\mathbf{w}}_0)\\ & +O({\sigma }_n|\mathbf{w}-{\mathbf{w}}_0|)+o(|\mathbf{w}-{\mathbf{w}}_0{|}^2),\end{array}$$ (A11)

where ${\mathbf{w}}^{\ast }$ is between $\mathbf{w}$ and ${\mathbf{w}}_0$. Replacing $\mathit{v}$ with $\mathit{V}$, and taking expectation in (A11) on both sizes, and apply the Lipschitz condition (C4), we have

$$\begin{array}{rl}3\cdot {\mathrm{\Gamma }}_{s0}(\mathbf{w})& =(\mathbf{w}-{\mathbf{w}}_0{)}^{T}E{\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0)+\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^{T}E{\mathrm{\nabla }}_2\tau (\mathit{V},{\mathbf{w}}_0)(\mathbf{w}-{\mathbf{w}}_0)\\ & +O({\sigma }_n|\mathbf{w}-{\mathbf{w}}_0|)+o(|\mathbf{w}-{\mathbf{w}}_0{|}^2).\end{array}$$

Note that ${\mathbf{w}}_0$ is the maximizer of ${\mathrm{\Gamma }}_0(\mathbf{w})$. Thus $E{\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0)=\mathbf{0}$. This finishes the proof of the Equation (A8).

For the Equation (A9), note that according to the definition of ${\tau }_s(\mathit{v},\mathbf{w})$, we have

$$f_{s1}^{\ast }(\mathit{v},\mathbf{w})={\tau }_s(\mathit{v},\mathbf{w})-{\tau }_s(\mathit{v},{\mathbf{w}}_0)-3{\mathrm{\Gamma }}_{s0}(\mathbf{w}).$$ (A12)

Thus according to (A8) and (A11), we have

$$\begin{array}{rl}{P}_n{f}_{s1}^{\ast }(\cdot ,\mathbf{w})& =\frac{1}{\sqrt{n}}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathit{W}}_n({\mathbf{w}}_0)+\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{\mathit{D}}_n({\mathbf{w}}_0)(\mathbf{w}-{\mathbf{w}}_0)\\ & +R_n(\mathbf{w})+o(|\mathbf{w}-{\mathbf{w}}_0{|}^2)+O({\sigma }_n|\mathbf{w}-{\mathbf{w}}_0|),\end{array}$$ (A13)

where

$$\begin{array}{rl}& {\mathit{W}}_n({\mathbf{w}}_0)=\sqrt{n}{P}_n{\mathrm{\nabla }}_1\tau (\cdot ,{\mathbf{w}}_0),\\ & {\mathit{D}}_n({\mathbf{w}}_0)=P_n{\mathrm{\nabla }}_2\tau (\cdot ,{\mathbf{w}}_0)-E{\mathrm{\nabla }}_2\tau (\mathit{V},{\mathbf{w}}_0),\\ & R_n(\mathbf{w})=\frac{1}{2}(\mathbf{w}-{\mathbf{w}}_0{)}^T{P}_n({\mathrm{\nabla }}_2\tau (\cdot ,{\mathbf{w}}^{\ast })-{\mathrm{\nabla }}_2\tau (\cdot ,{\mathbf{w}}_0))(\mathbf{w}-{\mathbf{w}}_0).\end{array}$$

Deduce from (iii) in the regularity condition (C5) and note the fact that $E{\mathrm{\nabla }}_1\tau (\mathit{V},{\mathbf{w}}_0)=0$, it follows that ${\mathit{W}}_n({\mathbf{w}}_0)$ converges to $N(0,{\mathbf{\Sigma }}_{\ast }({\mathbf{w}}_0))$. According to (iv) in the regularity condition (C5) and a weak law of large numbers, ${\mathit{D}}_n({\mathbf{w}}_0)$ converges to zero in probability as n tends to infinity. Finally, deduce from (ii) in the regularity condition (C5), we get $|R_n(\mathbf{w})|\le |\mathbf{w}-{\mathbf{w}}_0{|}^3{P}_{n}M(\cdot )$. Thus, by the integrability of $M(\mathit{v})$ and a weak law of large numbers, we have $|R_n(\mathbf{w})|=o_p(|\mathbf{w}-{\mathbf{w}}_0{|}^2)$ uniformly over $o_p(1)$ neighborhoods of ${\mathbf{w}}_0$. This finishes the proof of (A9).

Lastly we consider the Equation (A10). Define ${\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})={\mathrm{\Psi }}_s({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},\mathbf{w})-{\mathrm{\Psi }}_s({\mathit{v}}_{m1},{\mathit{v}}_{m2},{\mathit{v}}_{m3},{\mathbf{w}}_0)$, and

$$\begin{array}{rl}{\mathrm{\Psi }}_{s2}({\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w})& =E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})+E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_2,{\mathit{V}}_3,{\mathit{v}}_1,\mathbf{w})+E{\mathrm{\Psi }}_s({\mathit{V}}_3,{\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w})\\ & -E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_2,{\mathit{V}}_3,{\mathit{V}}_1,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_3,{\mathit{v}}_1,{\mathit{V}}_2,\mathbf{w})\\ & -E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_2,{\mathit{V}}_3,{\mathit{v}}_1,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_3,{\mathit{V}}_1,{\mathit{v}}_2,\mathbf{w})\\ & +3E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w}),\\ {\mathrm{\Psi }}_{s3}({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})& =E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{v}}_3,\mathbf{w})\\ & -E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})+E{\mathrm{\Psi }}_s^{\ast }({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w})+E{\mathrm{\Psi }}_s({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})\\ & +E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,{\mathit{v}}_3,\mathbf{w})-E{\mathrm{\Psi }}_s^{\ast }({\mathit{V}}_1,{\mathit{V}}_2,{\mathit{V}}_3,\mathbf{w}).\end{array}$$

Let the class of functions ${\mathcal{F}}_1=\{{\mathrm{\Psi }}_{s2}({\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w}):\mathbf{w}\in \mathit{A}\}$, and ${\mathcal{F}}_2=\{{\mathrm{\Psi }}_{s3}({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w}):\mathbf{w}\in \mathit{A}\}$. Clearly, for $\mathbf{w}\in \mathit{A}$, $E{\mathrm{\Psi }}_{s2}({\mathit{V}}_1,{\mathit{v}}_2,\mathbf{w})=E{\mathrm{\Psi }}_{s2}({\mathit{v}}_1,{\mathit{V}}_2,\mathbf{w})=0$ and $E{\mathrm{\Psi }}_{s3}({\mathit{V}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})=E{\mathrm{\Psi }}_{s3}({\mathit{v}}_1,{\mathit{V}}_2,{\mathit{v}}_3,\mathbf{w})=E{\mathrm{\Psi }}_{s3}({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{V}}_3,\mathbf{w})=0$. That is, $U_n^2{\mathrm{\Psi }}_{s2}(\cdot ,\cdot ,\mathbf{w})$ and $U_n^3{\mathrm{\Psi }}_{s3}(\cdot ,\cdot ,\cdot ,\mathbf{w})$ are, respectively, the degenerate U-statistic of order 2 and 3 on $\mathcal{S}\otimes \mathcal{S}$ and $\mathcal{S}\otimes \mathcal{S}\otimes \mathcal{S}$.

Note that ${\mathrm{\Psi }}_{s2}({\mathit{v}}_1,{\mathit{v}}_2,{\mathbf{w}}_0)=0$ and ${\mathrm{\Psi }}_{s3}({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,{\mathbf{w}}_0)=0$. By the continuities of ${\mathrm{\Psi }}_{s2}$ and ${\mathrm{\Psi }}_{s3}$, it follows from the dominated convergence theorem that $E{\mathrm{\Psi }}_{s2}^2({\mathit{v}}_1,{\mathit{v}}_2,\mathbf{w})\to 0$ and $E{\mathrm{\Psi }}_{s3}^2({\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\mathbf{w})\to 0$ as $\mathbf{w}\to {\mathbf{w}}_0$. Moreover, by the Lemma 2.14 in [13], we can show that ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ are Euclidean for constant envelope. Therefore, by the Theorem 3 in [18], it follows that $U_n^2{\mathrm{\Psi }}_{s2}(\cdot ,\cdot ,\mathbf{w})=o_p(1/n)$ and $U_n^3{\mathrm{\Psi }}_{s3}(\cdot ,\cdot ,\cdot ,\mathbf{w})=o_p(1/n)$, uniformly in an $o_p(1)$ neighborhood of ${\mathbf{w}}_0$. Lastly, by the convergence of $M_{n,3}^{\ast }\stackrel{p}{\to }{\gamma }_3(\prod _{l=1}^3{p}_l{)}^{-1}$ again, and note that $U_n^2{f}_{s2}^{\ast }(\cdot ,\cdot ,\mathbf{w})=M_{n,3}^{\ast }\cdot U_n^2{\mathrm{\Psi }}_{s2}(\cdot ,\cdot ,\mathbf{w})$ and $U_n^3{f}_{s3}^{\ast }(\cdot ,\cdot ,\cdot ,\mathbf{w})=M_{n,3}^{\ast }\cdot U_n^3{\mathrm{\Psi }}_{s3}(\cdot ,\cdot ,\cdot ,\mathbf{w})$, we conclude the proof of (A10).

Proof of Theorem 3.3

According to the Taylor expansion, we have

$${\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)={\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)+[{\mathrm{\nabla }}_1{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0){]}^T({\hat{\mathbf{w}}}_s-{\mathbf{w}}_0)+o_p({\hat{\mathbf{w}}}_s-{\mathbf{w}}_0).$$

Thus, it follows that

$$\begin{array}{rl}& \sqrt{n}\{{\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)-{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)\}\\ & =\sqrt{n}\{{\hat{\rho }}_{\mathrm{bsa},s}({\hat{\mathbf{w}}}_s)-{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)\}+\sqrt{n}\{{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)-{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)\}\\ & =[{\mathrm{\nabla }}_1{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0){]}^T\sqrt{n}\{{\hat{\mathbf{w}}}_s-{\mathbf{w}}_0\}\\ & +\sqrt{n}\{{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)-{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)\}+o_p(\sqrt{n}\{{\hat{\mathbf{w}}}_s-{\mathbf{w}}_0\}).\end{array}$$

By some calculations, we get

$${\mathrm{\nabla }}_1{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)={\mathrm{\nabla }}_1{\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)+O_p({\sigma }_n),\text{and}{\hat{\rho }}_{\mathrm{bsa},s}({\mathbf{w}}_0)={\rho }_{\mathrm{bsa}}({\mathbf{w}}_0)+O_p({\sigma }_n).$$

Therefore, by Slutsky's theorem, we finish the proof of Theorem 3.3.

Funding Statement

This research project was supported by grants from National Institute of Health (R01MH079448, R01HL113548, and R01MH105561). Qiu's work was supported by the National Natural Science Foundation of China (NSFC) (12071164) and the Center for Applied Mathematics of Fujian Province (FJNU).

Disclosure statement

No potential conflict of interest was reported by the author(s).