Tests for the linear hypothesis in semi-functional partial linear regression models

Abstract

An empirical likelihood ratio testing method is proposed, in this paper, for semi-functional partial linear regression models. Two empirical likelihood ratio statistics are employed to test the linear hypothesis of parametric components, then we demonstrate that their asymptotic null distributions are standard Chi-square distributions with the degrees of freedom being independent of the nuisance parameters. We also verify the proposed statistics follow non-central Chi-square distributions under the alternative hypothesis, and their powers are derived. Furthermore, we apply the proposed method to test the significance of parametric components. In addition, a F-test statistic is introduced. Simulations are undertaken to demonstrate the proposed methodologies and the simulation results indicate that the proposed testing methods are workable. A real example is applied for illustration.

Appendix

To facilitate the proofs, the following technical conditions are imposed.

• C1. ${\mathscr {C}}$ is some given compact subset of ${\mathscr {H}}$.

• C2. The kernel $K(\cdot )$, supported on [0, 1], is strictly decreasing and Lipschitz continuous on $[0,\infty ]$. Its derivative ${K}'$ exists on [0, 1], and $\exists {{\theta }_{1}}>0$, such that $\forall x\in [0,1],-{K}'(x)>{{\theta }_{1}}$.

• C3. $\forall G(z)\in \left\{ {{m(z), {\mu } }_{1}}(z), \ldots ,{{\mu }_{p}}(z)\right\}$, $G(\cdot )$ satisfies the Lipschitz-type regularity condition: $\exists {a}_{1} >0$, and $C >0$, $\forall (u,v)\in {{{\mathscr {C}}}\times {{\mathscr {C}}}}$, $\left| G(u)-G(v) \right| \le Cd{{(u,v)}^{{a}_{1}}}$.

• C4. There exists a function $\phi$ and positive constants ${{\theta }_{2}}$, ${{\alpha }_{1}}$ and ${{\alpha }_{2}}$ such that $\int _{0}^{1}{\phi }(ht)dt/\phi (h)>{{\theta }_{2}}$, and for $\forall t\in {\mathscr {C}}$, $0<{{\alpha }_{1}}\phi (h)<F(h)<{{\alpha }_{2}}\phi (h)$, where $F(h)=P(Z\in B(t,h))$, $B(t,h)=\left\{ \left. s\in {\mathscr {C}} \right| d(s,t)\le h \right\}$.

• C5. $\exists {r}_{1}\ge 4$, $E{({| \varepsilon |}^{{r}_{1}})} +E{({|\breve{\mathbf{X }}_{1} |^{{r}_{1}}})} + , \ldots , + E{({|\breve{\mathbf{X }}_{p} |^{{r}_{1}}})} <\infty$.

• C6. $h {\rightarrow } 0$, $\phi (h){\rightarrow }0$, ${{n}}{{h}^{4{a}_{1}}}{\rightarrow }0$ as $n {\rightarrow } \infty$ and $\phi (h)>n^{(2/{r}_{1})+{a}_{2}-1}/(\log n)^{2}$, where the constant ${a}_{2}> 0$ and ${(2/{r}_{1})+{a}_{2}}> 1/2$.

• C7. $Var(\varepsilon )=\sigma _{\varepsilon }^{2}>0$, $\varSigma =E[\breve{\mathbf{X }}\breve{\mathbf{X }}^{T}]$ is a positive definite matrix. ${{\breve{\mathbf{X }}}_{i}}$ is independent of $\varepsilon _{i}$, $i=1, \ldots ,n$.

The above conditions are exactly the same as the conditions (7)–(13) used in Theorem 1 of Aneiros and Vieu (2006), except in (C5) we let ${r}_{1}\ge 4$, instead of ${r}_{1}\ge 3$. Condition (C5) is a usual assumption in the literature (Owen 2001).

The proofs of the main results rely on some preliminary lemmas. For this, we introduce some notations. Let $B(r)=\{{\varvec{\beta }}^{(1)}| \Vert {\varvec{\beta }}^{(1)}- {\varvec{\beta }}^{(1)}_{0}\Vert \le r$, where $r > 0$ and $r=O_{P}(n^{-1/2})$, where $\Vert \cdot \Vert$ represents the Euclidean norm.

Lemma 1

Suppose that conditions C1–C7 hold, and in addition, let ${\varvec{\beta }}^{T}_{0}=({\varvec{\beta }}^{(1)T}_{0}, {\varvec{\beta }}^{(2)T}_{0})$ be the true value of ${\varvec{\beta }}^{T}=({\varvec{\beta }}^{(1)T}, {\varvec{\beta }}^{(2)T})$, and if $\underset{n\rightarrow \infty }{\mathop {\lim }}{ \sqrt{n} \mathbf A {\varvec{\beta }}_{0} }$ exists, we have

$$\begin{aligned} \begin{array}{lll} \frac{1}{\sqrt{n}}\sum \nolimits _{i=1}^{n} { [{\psi }_{i}({\varvec{\beta }}^{(1)}_{0})-\varvec{\mu }] } {\mathop {\longrightarrow }\limits ^{d}}N(0,\sigma _{\varepsilon }^{2}\varSigma ), \end{array} \end{aligned}$$

where $\varvec{\mu }= {\varSigma }({\varvec{\beta }}_{0}-\mathbf A ^{*}\varvec{\beta }^{(1)}_{0})$.

Proof

Observe that

$$\begin{aligned} \begin{array}{lll} {\psi }_{i}({\varvec{\beta }}^{(1)}_{0}) &{}=&{}\tilde{\mathbf{X }}_{i}({{Y}_{i}} -\sum \nolimits _{j=1}^{n}{{{w}_{h,n}}({{Z}_{i}},{{Z}_{j}}){{Y}_{j}}} -\tilde{\mathbf{X }}_{i}^{T}{} \mathbf A ^{*}\varvec{\beta }^{(1)}_{0})\\ &{}=&{} \tilde{\mathbf{X }}_{i}(\tilde{m}({{Z}_{i}}) + {{\varepsilon }_{i}} -\sum \nolimits _{j=1}^{n}{ {{w}_{h,n}}({{Z}_{i}},{{Z}_{j}}) {{\varepsilon }_{j}} } +\tilde{\mathbf{X }}_{i}^{T}({\varvec{\beta }}_{0}-\mathbf A ^{*}\varvec{\beta }^{(1)}_{0}) ) \end{array} \end{aligned}$$

where $\tilde{m}({{Z}_{i}})=m({{Z}_{i}}) -\sum \nolimits _{j=1}^{n}{ {{w}_{h,n}}({{Z}_{i}},{{Z}_{j}}) m({{Z}_{j}}) }$.

Then

$$\begin{aligned} \begin{array}{lll} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ [{\psi }_{i}({\varvec{\beta }}^{(1)}_{0})-\varvec{\mu }] } &{}=&{}\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{m}({{Z}_{i}})} - \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\sum \limits _{j=1}^{n}{ {{w}_{h,n}}({{Z}_{i}},{{Z}_{j}}) {{\varepsilon }_{j}} } }\\ &{} &{}+ \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}{{\varepsilon }_{i}} } + \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ (\tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T}({\varvec{\beta }}_{0}-\mathbf A ^{*} \varvec{\beta }^{(1)}_{0})-\varvec{\mu }) }\\ &{}\equiv &{}S_{n1}+S_{n2}+S_{n3}+S_{n4}. \end{array} \end{aligned}$$

By Eqs. (20)–(23) of Aneiros and Vieu (2006), we have $S_{n1}=o_{P}(1)$, $S_{n2}=o_{P}(1)$, and $S_{n3}=\sum \nolimits _{i=1}^{n}{ \breve{\mathbf{X }}_{i}{{\varepsilon }_{i}} }/\sqrt{n}+o_{P}(1)$.

Consider the last term $S_{n4}= [\sum \nolimits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} }/n-{\varSigma }]\sqrt{n}({\varvec{\beta }}_{0}-\mathbf A ^{*} \varvec{\beta }^{(1)}_{0})$. Note that $\sqrt{n}({\varvec{\beta }}_{0}-\mathbf A ^{*} \varvec{\beta }^{(1)}_{0})=(0, (\mathbf A _{2}^{-1} \sqrt{n}{} \mathbf A {\varvec{\beta }}_{0})^{T})^{T}$, where 0 is a $1\times k$ vector and $\mathbf A _{2}^{-1} \sqrt{n}{} \mathbf A {\varvec{\beta }}_{0}=O_{P}(1)$ since $\mathop {\lim }\nolimits _{{n \rightarrow \infty }} \sqrt{n} A{\varvec{\beta }}_{0}$ exists. In addition, by Lemma 7 of Aneiros and Vieu (2006), we get $\sum \nolimits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} }/n {\mathop {\longrightarrow }\limits ^{P}}{\varSigma }$. These yield that $S_{n4}=o_{P}(1)$. Thus,

$$\begin{aligned} \begin{array}{lll} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ [{\psi }_{i}({\varvec{\beta }}^{(1)}_{0})--\varvec{\mu }] } =\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ \breve{\mathbf{X }}_{i}{{\varepsilon }_{i}} } +o_{P}(1) \equiv \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}{ L_{i} }+o_{P}(1). \end{array} \end{aligned}$$

By Condition (C7), $E(L_{i})=0$ and $var(L_{i}) =\sigma _{\varepsilon }^{2}\varSigma$. In addition, ${{L}_{i}}'s$ are i.i.d.. Then the result follows from the Central Limit Theorem. $\square$

Lemma 2

Under the conditions of Lemma 1, ${{\max }_{1\le i\le n}}\, \Vert {{\psi }_{i}({\varvec{\beta }}^{(1)})} \Vert ={o}_{p}({n}^{1/2})$ uniformly for ${\varvec{\beta }}^{(1)}\in B(r)$.

Proof

Similarly to Lemma 1, we have

$$\begin{aligned} \begin{array}{lll} \underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \underset{1\le i\le n}{\mathop {\max }}\, \Vert {{\psi }_{i}({\varvec{\beta }}^{(1)})}\Vert &{}\le &{}\underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}{{\varepsilon }_{i}}\Vert + \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}{\hat{m}}({{Z}_{i}})\Vert \\ &{}&{}+ \underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} (\mathbf A ^{*}{\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0})\Vert . \end{array} \end{aligned}$$

where ${\hat{m}}({{Z}_{i}})= \sum \nolimits _{j=1}^{n}{ {{w}_{h,n}}({{Z}_{i}},{{Z}_{j}}) ({m}({{Z}_{j}})+{{\varepsilon }_{j}})}$. We will prove that the first and second terms of the right-hand side of this equation are ${o}_{p}({n}^{1/2})$ and the third term is ${o}_{p}(1)$.

Let $c_{n}={{O}_{p}}([\log n/(n\phi (h))]^{1/2}+{h}^{a_{1}})$. Note that $c_{n}={{o}_{p}}({{n}^{1/2}})$ [by condition (C6)].

We need the following equation in the processes of the proofs. By Lemma 3 of Owen (1990), we obtain that

$$\begin{aligned} { \begin{array}{lll} \underset{1\le i\le n}{\mathop {\max }}\,\Vert \zeta _{i} \Vert ={{o}_{p}}({{n}^{1/2}}) \end{array} } \end{aligned}$$

(17)

where $\zeta _{i}\ge 0, i=1, \ldots , n$ are independent and identically distributed random variables with $E(\zeta _{i}^{2})< \infty$.

By (17) and Corollary 4.1 of Ferraty et al. (2004), we have

$$\begin{aligned} { \begin{array}{lll} \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i} \Vert \le \underset{1\le i\le n}{\mathop {\max }}\, \Vert \breve{\mathbf{X }}_{i} \Vert + \underset{1\le i\le n}{\mathop {\max }}\, \Vert \varvec{\mu }_{X}({Z}_{i})-\hat{\varvec{\mu }}_{X}({Z}_{i}) \Vert ={{o}_{p}}({{n}^{1/2}}), \end{array} } \end{aligned}$$

(18)

Note that ${\hat{m}}(\cdot )$ is the kernel estimator of ${m}(\cdot )$ in the nonparametric model ${Y}_{i}^{**} = {m}({Z}_{i})+{\varepsilon }_{i}$, where ${Y}_{i}^{**}={Y}_{i}-\mathbf X _{i}^{T}\varvec{\beta }$. Thus, by Corollary 4.1 of Ferraty et al. (2004) and (18), we have $\underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}{\hat{m}}({{Z}_{i}}) \Vert ={o}_{p}({n}^{1/2}){O}_{p}(c_{n}) ={o}_{p}({n}^{1/2})$.

By using the same argument as (18), we obtain ${\mathop {\max }}_{1\le i\le n}\Vert \tilde{\mathbf{X }}_{i}{{\varepsilon }_{j}} \Vert \le {\mathop {\max }}_{1\le i\le n} \Vert \breve{\mathbf{X }}_{i}{{\varepsilon }_{j}} \Vert + {\mathop {\max }}_{1\le i\le n}\Vert \varvec{\mu }_{X}({Z}_{i})-\hat{\varvec{\mu }}_{X}({Z}_{i}) \Vert {\mathop {\max }}_{1\le i\le n} \Vert {{\varepsilon }_{j}} \Vert ={o}_{p}({n}^{1/2})$, and

$$\begin{aligned} { \begin{array}{lll} \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} \Vert ={{o}_{p}}({{n}^{1/2}}). \end{array} } \end{aligned}$$

(19)

Consider the third term

$$\begin{aligned} \begin{array}{lll} &{}&{}\underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} (\mathbf A ^{*}{\varvec{\beta }}^{(1)}-{\varvec{\beta }}_{0}) \Vert \\ &{}&{}\qquad \le \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} \Vert (\underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \Vert \mathbf A ^{*}({\varvec{\beta }}^{(1)}-{\varvec{\beta }}_{0}^{(1)}) \Vert +\Vert \mathbf A ^{*}{\varvec{\beta }}_{0}^{(1)}-{\varvec{\beta }}_{0} \Vert ) \\ &{}&{}\qquad ={o}_{p}({n}^{1/2})({O}_{p}({n}^{-1/2})+{O}_{p}({n}^{-1/2})) ={o}_{p}(1). \end{array} \end{aligned}$$

The last line due to (18) and that $\underset{n\rightarrow \infty }{\mathop {\lim }}{ \sqrt{n}(\mathbf A ^{*}{\varvec{\beta }}_{0}^{(1)}-{\varvec{\beta }}_{0}) }$ exists (see the proof of Lemma 1). The proof of this lemma is completed. $\square$

Lemma 3

Under the conditions of Lemma 1, $1/n\sum \nolimits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}^{(1)})} {{\psi }_{i}^{T}({\varvec{\beta }}^{(1)})} }$ converges to $\sigma _{\varepsilon }^{2}\varSigma$ in probability as $n{\rightarrow }\infty$ uniformly for ${\varvec{\beta }}^{(1)}\in B(r)$.

Proof

It follows from Lemma 1 that

$$\begin{aligned} \begin{array}{lll} 1/n\sum \nolimits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}^{(1)}_{0})} {{\psi }_{i}^{T}({\varvec{\beta }}^{(1)}_{0})} } = 1/n\sum \nolimits _{i=1}^{n}{ {{L}_{i}}{{L}_{i}}^{T} } +o_{P}(1). \end{array} \end{aligned}$$

This together with the law of large numbers yields $1/n\sum \nolimits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}^{(1)}_{0})} {{\psi }_{i}^{T}({\varvec{\beta }}^{(1)}_{0})} }\xrightarrow {P}\sigma _{\varepsilon }^{2}\varSigma$. In addition,

$$\begin{aligned} \begin{array}{lll} &{}&{}\underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \Vert \frac{1}{n}\sum \limits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}^{(1)})} {{\psi }_{i}^{T}({\varvec{\beta }}^{(1)})} } - \frac{1}{n}\sum \limits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}_{0}^{(1)})} {{\psi }_{i}^{T}({\varvec{\beta }}^{(1)}_{0})} } \Vert \\ &{}&{}\qquad \quad \le 2\underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \Vert \frac{1}{n}\sum \limits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} \mathbf A ^{*}({\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0}) {{\psi }_{i}^{T}({\varvec{\beta }}_{0}^{(1)})} } \Vert \\ &{}&{}\qquad \qquad + \underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \Vert \frac{1}{n}\sum \limits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} \mathbf A ^{*}({\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0}) ({\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0})^{T} \mathbf A ^{*T}\tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} } \Vert \\ &{}&{}\qquad \quad \le 2\underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \underset{1\le i\le n}{\mathop {\max }}\, \Vert \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T} \mathbf A ^{*}({\varvec{\beta }}^{(1)}-{\varvec{\beta }}_{0}) \Vert \Vert \frac{1}{n}\sum \limits _{i=1}^{n}{ {{\psi }_{i}^{T}({\varvec{\beta }}_{0}^{(1)})} } \Vert \\ &{}&{}\qquad \qquad + \frac{1}{n}\sum \limits _{i=1}^{n}{ \Vert \tilde{\mathbf{X }}_{i}\Vert ^{4} \underset{{\varvec{\beta }}^{(1)}\in B(r)}{\mathop {\sup }} \Vert \mathbf A ^{*}({\varvec{\beta }}^{(1)}-{\varvec{\beta }}_{0}) \Vert ^{2} }\\ &{}&{}\qquad \quad ={o}_{p}({n}^{1/2}){O}_{p}({n}^{-1/2}){O}_{p}(1) + {O}_{p}(1)({O}_{p}({n}^{-1/2}))^{2} \\ &{}&{}\qquad \quad ={o}_{p}(1) \end{array} \end{aligned}$$

The last second line due to Lemma 1, Condition (C5) and Eq. (19). The proof of this lemma is completed. $\square$

The following lemma states the asymptotic normality of $\tilde{\varvec{\beta }}^{(1)}$.

Lemma 4

Under the conditions of Lemma 1, we have

$$\begin{aligned} \begin{array}{lll} \sqrt{n}(\tilde{\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0}) {\mathop {\longrightarrow }\limits ^{d}}N(0,V), \end{array} \end{aligned}$$

where $V=\mathbf A ^{*T}\varSigma \mathbf A ^{*}/\sigma _{\varepsilon }^{2}$.

Proof

Applying the Taylor expansion to $Q_{1n}(\tilde{\varvec{\beta }}^{(1)}, {\tilde{\lambda }})$ and $Q_{2n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})$ at $({\varvec{\beta }}^{(1)}_{0}, 0)$ and expressing them in matrix form, we get

$$\begin{aligned} { \left( \begin{array}{ccc} Q_{1n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})\\ Q_{2n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})\\ \end{array} \right) = \left( \begin{array}{ccc} Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0) \\ Q_{2n}({\varvec{\beta }}^{(1)}_{0}, 0) \\ \end{array} \right) + \mathbf B _{n} \left( \begin{array}{ccc} {\tilde{\lambda }}\\ \tilde{\varvec{\beta }}^{(1)}-{\varvec{\beta }}_{0}\\ \end{array} \right) +o_{P}({\varDelta }_{n}), } \end{aligned}$$

(20)

where ${\varDelta }_{n}=\Vert \tilde{\varvec{\beta }}^{(1)} -{\varvec{\beta }}^{(1)}_{0}\Vert +\Vert {\tilde{\lambda }}-0 \Vert$, $\frac{\partial {Q_{1n}({\varvec{\beta }}^{(1)},0) } }{\partial {{\varvec{\beta }}^{(1)T}} }=\frac{1}{n}\sum \nolimits _{i=1}^{n}{ \frac{\partial {{\psi }_{i}({\varvec{\beta }}^{(1)})}}{\partial {{\varvec{\beta }}^{(1)}T} }}=\frac{1}{n}\sum \nolimits _{i=1}^{n}{ \tilde{\mathbf{X }}_{i}\tilde{\mathbf{X }}_{i}^{T}{} \mathbf A ^{*} }$, $\frac{\partial {Q_{1n}({\varvec{\beta }}^{(1)},0) } }{\partial {{ \lambda }^{T}} }=-\frac{1}{n}\sum \nolimits _{i=1}^{n}{ {{\psi }_{i}({\varvec{\beta }}^{(1)})}{\psi }_{i}^{T}({\varvec{\beta }}^{(1)}) }$, $\frac{\partial {Q_{2n}({\varvec{\beta }}^{(1)},0) } }{\partial {{ \lambda }^{T}} }=-\frac{1}{n}\sum \nolimits _{i=1}^{n}{ {\psi }_{i}^{T}({\varvec{\beta }}^{(1)}) }$, $\frac{\partial {Q_{2n}({\varvec{\beta }}^{(1)},0) } }{\partial {{\varvec{\beta }}^{(1)T}}}=0$ and

$$\begin{aligned} \mathbf B _{n}= \begin{pmatrix} \frac{\partial {Q_{1n} } ({\varvec{\beta }}^{(1)}_{0}, 0)}{\partial { \lambda }^{T}} &{}\quad \frac{\partial {Q_{1n}}({\varvec{\beta }}^{(1)}_{0}, 0) }{\partial {{\varvec{\beta }}^{(1)T}}}\\ \frac{\partial {Q_{2n} } ({\varvec{\beta }}^{(1)}_{0}, 0)}{\partial { \lambda }^{T}} &{}\quad \frac{\partial {Q_{2n}}({\varvec{\beta }}^{(1)}_{0}, 0) }{\partial { \varvec{\beta }}^{(1)T} }\\ \end{pmatrix} {{\mathop {\longrightarrow }\limits ^{P}}} \begin{pmatrix} -B_{11} &{} B_{12}\\ B_{21} &{} 0 \\ \end{pmatrix} \equiv \mathbf B \end{aligned}$$

where $B_{11}=\sigma _{\varepsilon }^{2}\varSigma$, $B_{12}=\varSigma \mathbf A ^{*}$, $B_{21}=B_{12}^{T}$ (by Lemmas 1 and 3).

Thus

$$\begin{aligned} \mathbf B _{n}^{-1} {{\mathop {\longrightarrow }\limits ^{P}}} \mathbf B ^{-1} = \begin{pmatrix} -B_{11}^{-1}+B_{11}^{-1}B_{12}B_{22.1}^{-1}B_{21}B_{11}^{-1} &{}\quad -B_{11}^{-1}B_{12}B_{22.1}^{-1}\\ B_{22.1}^{-1}B_{21}B_{11}^{-1} &{}\quad B_{22.1}^{-1}\\ \end{pmatrix} , \end{aligned}$$

where $B_{22.1}=B_{21}B_{11}^{-1}B_{12} =\mathbf A ^{*T}\varSigma \mathbf A ^{*}/\sigma _{\varepsilon }^{2}$. We note that $\varSigma$ is positive definite, and $\mathbf A ^{*}$ is of rank k so that $B_{22.1}=\mathbf A ^{*T}\varSigma \mathbf A ^{*}$ is non-singular. This result will be required below.

Note that $Q_{1n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})=0$, $Q_{2n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})=0$ and $Q_{2n}({\varvec{\beta }}^{(1)}_{0}, 0)=0$, then we have

$$\begin{aligned} \begin{pmatrix} {\tilde{\lambda }}\\ \tilde{\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0}\\ \end{pmatrix} = \mathbf B ^{-1} \begin{pmatrix} Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0) \\ 0 \\ \end{pmatrix} +o_{P}({\varDelta }_{n}). \end{aligned}$$

By Lemma 1, $\sqrt{n}Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0)={n}^{-1/2}\sum \nolimits _{i=1}^{n}{ {\psi }_{i}({\varvec{\beta }}_{0}^{(1)}) }{\mathop {\rightarrow }\limits ^{d}}N(0,\sigma _{\varepsilon }^{2}\varSigma )$, then $Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0)=O_{P}({n}^{-1/2})$. This together with the above equation yields ${\varDelta }_{n}=O_{P}({n}^{-1/2})$. Hence, $\sqrt{n}(\tilde{\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0}) =B_{22.1}^{-1}B_{21}B_{11}^{-1}\sqrt{n}Q_{1n}({\varvec{\beta }}_{0},0) +o_{P}(1) {\mathop {\longrightarrow }\limits ^{d}}N(0,V)$, where

$$\begin{aligned} V=B_{22.1}^{-1}B_{21}B_{11}^{-1} (\sigma _{\varepsilon }^{2}\varSigma ) B_{11}^{-1}B_{12}B_{22.1}^{-1} =B_{22.1}^{-1}=\mathbf A ^{*T}\varSigma \mathbf A ^{*}/\sigma _{\varepsilon }^{2}. \end{aligned}$$

$\square$

Theorem 1 is a specific case of Theorem 2. Thus, we just give the proof of Theorem 2.

Proof of Theorem 2

From Lemma 4, we can obtain that $\tilde{\varvec{\beta }}^{(1)}- {\varvec{\beta }}^{(1)}_{0}=O_{P}({n}^{-1/2})$. Thus, $\tilde{\varvec{\beta }}^{(1)}$ satisfies Lemmas 2 and 3. By Lemma 2, we have ${\mathop {\max }}_{1\le i\le n} \Vert {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})}\Vert ={{o}_{p}}({{n}^{1/2}})$. That is ${{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})}$ satisfies Equation (2.5) of Owen (1990). Combining this with Lemma 3 and applying the same arguments as in Eqs. (2.11)–(2.14) of Owen (1990), we can get

$$\begin{aligned} { \begin{array}{lll} \left\| {\tilde{\lambda }} \right\| ={{O}_{p}}({{n}^{-1/2}}) \end{array} } \end{aligned}$$

(21)

Thus, ${\mathop {\max }}_{1\le i\le n}\Vert {\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})}\Vert ={{o}_{p}}({{n}^{1/2}}) {{O}_{p}}({{n}^{-1/2}})={{o}_{p}}(1)$.

Note the fact that $Q_{1n}(\tilde{\varvec{\beta }}^{(1)},{\tilde{\lambda }})=0$ and expanding it yields that

$$\begin{aligned} \begin{array}{lll} 0= & {} \sum \limits _{i=1}^{n}{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} } -\sum \limits _{i=1}^{n}{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} {{\psi }^{T}_{i}(\tilde{\varvec{\beta }}^{(1)})}{\tilde{\lambda }} } +\sum \limits _{i=1}^{n}{ \frac{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} {{\left[ {\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} \right] }^{2}} }{ 1+{\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} } }, \end{array} \end{aligned}$$

then we can obtain that

$$\begin{aligned} { \begin{array}{lll} {\tilde{\lambda }}={{\left[ \frac{1}{n}\sum \limits _{i=1}^{n}{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} {{\psi }^{T}_{i}(\tilde{\varvec{\beta }}^{(1)})} } \right] }^{-1}}\frac{1}{n}\sum \limits _{i=1}^{n}{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} }+{{o}_{p}}({{n}^{-1/2}}), \end{array} } \end{aligned}$$

(22)

where the term ${{o}_{p}}({{n}^{-1/2}})$ can be obtained similarly to Eq. (2.17) of Owen (1990). Since ${\mathop {\max }}_{1\le i\le n}\Vert {\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})}\Vert ={{o}_{p}}(1)$, we can apply the Taylor expansion to (11). Then

$$\begin{aligned} { \begin{array}{lll} {\hat{L}}(\tilde{\varvec{\beta }}^{(1)}) &{}=&{}\sum \limits _{i=1}^{n}{ {\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} } -\sum \limits _{i=1}^{n}{ {\tilde{\lambda }}^{T}{{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} {{\psi }_{i}^{T}(\tilde{\varvec{\beta }}^{(1)}){\tilde{\lambda }}} } +{{o}_{p}}(1)\\ &{}=&{}n[ \frac{1}{n}\sum \limits _{i=1}^{n}{ {\psi }_{i}(\tilde{\varvec{\beta }}^{(1)}) }]^{T}{B}_{11}^{-1}[\frac{1}{n}\sum \limits _{i=1}^{n}{ {\psi }_{i}(\tilde{\varvec{\beta }}^{(1)}) }] +{{o}_{p}}(1)\\ \end{array} } \end{aligned}$$

(23)

where we substitute Eqs. (22) and ${n}^{-1}\sum \nolimits _{i=1}^{n}{ {{\psi }_{i}(\tilde{\varvec{\beta }}^{(1)})} {{\psi }^{T}_{i}(\tilde{\varvec{\beta }}^{(1)})} }={B}_{11} {{o}_{p}}(1)$ (by Lemma 3) into the last line.

Moreover, note that ${n}^{-1}\sum \nolimits _{i=1}^{n}{{\psi }_{i} (\tilde{\varvec{\beta }}^{(1)})}=Q_{1n}(\tilde{\varvec{\beta }}^{(1)}, 0)$ and expanding $Q_{1n}(\tilde{\varvec{\beta }}^{(1)}, 0)$ yields

$$\begin{aligned} { \begin{array}{lll} \frac{1}{n}\sum \limits _{i=1}^{n}{{\psi }_{i} (\tilde{\varvec{\beta }}^{(1)})} &{}=&{}Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0)+\frac{\partial {Q_{1n}({\varvec{\beta }}^{(1)}_{0},0) } }{\partial {{\varvec{\beta }}^{(1)}} } (\tilde{\varvec{\beta }}^{(1)}-{\varvec{\beta }}^{(1)}_{0})\\ &{}=&{}(I_{p}-B_{12}B_{22.1}^{-1}B_{21}B_{11}^{-1})Q_{1n} ({\varvec{\beta }}^{(1)}_{0}, 0)+{{o}_{p}}(1). \\ \end{array} } \end{aligned}$$

(24)

Substituting (24) into (23) yields that

$$\begin{aligned} \begin{array}{lll} {\hat{L}}(\tilde{\varvec{\beta }}^{(1)}) &{}=&{} \sqrt{n}Q_{1n}^{T}({\varvec{\beta }}^{(1)}_{0}, 0) {\varSigma }^{*} \sqrt{n}Q_{1n}({\varvec{\beta }}^{(1)}_{0}, 0) +o_{P}(1)\\ &{}=&{}\sqrt{n}Q_{1n}^{T}({\varvec{\beta }}^{(1)}_{0}, 0) B_{11}^{-1/2} {\varSigma }^{**} B_{11}^{-1/2}\sqrt{n}Q_{1n} ({\varvec{\beta }}^{(1)}_{0}, 0)+o_{P}(1), \end{array} \end{aligned}$$

where ${\varSigma }^{*}=(I_{p}-B_{11}^{-1}B_{12}B_{22.1}^{-1}B_{21}) {B}_{11}^{-1} (I_{p}-B_{12}B_{22.1}^{-1}B_{21}B_{11}^{-1}) ={B}_{11}^{-1/2}{\varSigma }^{**} {B}_{11}^{-1/2}$ and ${\varSigma }^{**}=I_{p}-{B}_{11}^{-1/2}B_{12}B_{22.1}^{-1} B_{21}{B}_{11}^{-1/2}$.

From the above compution we known that ${\varSigma }^{**}$ is an idempotent matrix. In additon, by Lemma 1, $B_{11}^{-1/2}\sqrt{n}[Q_{1n} ({\varvec{\beta }}^{(1)}_{0}, 0)- {\varSigma }({\varvec{\beta }}_{0}- \mathbf A ^{*}{\varvec{\beta }}^{(1)}_{0})]$ follows the standard normal distribution. These together with the result in Rao (1973, pp. 186) yield that

$$\begin{aligned} \begin{array}{lll} {\hat{L}}(\tilde{\varvec{\beta }}^{(1)}) =\sqrt{n}Q_{1n}^{T}({\varvec{\beta }}^{(1)}_{0}, 0) B_{11}^{-1/2} {\varSigma }^{**} B_{11}^{-1/2}\sqrt{n}Q_{1n} ({\varvec{\beta }}^{(1)}_{0}, 0)+o_{P}(1) {\mathop {\longrightarrow }\limits ^{d}}&\chi _{q}^{2}(c_{0}), \end{array} \end{aligned}$$

where the non-centrality parameter

$$\begin{aligned} \begin{array}{lll} c_{0} &{}=&{}\underset{n\rightarrow \infty }{\mathop {\lim }}{ [{\varvec{\beta }}_{0} -\mathbf A ^{*}{\varvec{\beta }}^{(1)}_{0}]^{T}{\varSigma } B_{11}^{-1/2} {\varSigma }^{**} {\varSigma }^{-1/2} B_{11}^{-1/2}{\varSigma }[{\varvec{\beta }}_{0} -\mathbf A ^{*}{\varvec{\beta }}^{(1)}_{0}] }\\ &{}=&{}\underset{n\rightarrow \infty }{\mathop {\lim }}{ [{\varvec{\beta }}_{0} -\mathbf A ^{*}{\varvec{\beta }}^{(1)}_{0}]^{T} ({\varSigma }- {\varSigma }{} \mathbf A ^{*}(\mathbf A ^{*T}\varSigma \mathbf A ^{*})^{-1}{} \mathbf A ^{*T}{\varSigma }) [{\varvec{\beta }}_{0} -\mathbf A ^{*}{\varvec{\beta }}^{(1)}_{0}] }/\sigma _{\varepsilon }^{2}. \end{array} \end{aligned}$$

and the degrees of freedom of ${\chi }^{2}$ is the rank of ${\varSigma }^{**}$. Then we will show that the rank of ${\varSigma }^{**}$ is q below. Moreover, it is well known that the rank of an idempotent matrix equals its trace. The trace of ${B}_{11}^{-1/2}B_{12}B_{22.1}^{-1} B_{21}{B}_{11}^{-1/2}$ :

$tr(B_{11}^{-1/2}B_{12}B_{22.1}^{-1} B_{21}B_{11}^{-1/2}) =tr(B_{22.1}^{-1}B_{21}B_{11}^{-1/2}B_{11}^{-1/2}B_{12}) =tr(B_{22.1}^{-1} B_{22.1})=tr(I_{k})=k$, where the last second equation due to the fact that $B_{22.1}=\mathbf A ^{*T}\varSigma \mathbf A ^{*}$ is an invertible matrix of order k. Then the rank of ${\varSigma }^{**}$ equals $tr(I_{p}- B_{11}^{-1/2}B_{12}B_{22.1}^{-1} B_{21}B_{11}^{-1/2})= p-tr(B_{11}^{-1/2}B_{12}B_{22.1}^{-1} B_{21}B_{11}^{-1/2})=p-k=q$. $\square$