Testing for the shape parameter of generalized extreme value distribution based on the $L_q$-likelihood ratio statistic

Abstract

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the $L_q$-likelihood ratio ($L_q$R) statistic. The $L_q$R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion $q$. For small and modest sample sizes, the $L_q$R statistic is still available when $q$ is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods.

Appendices

Appendix A: Definitions and Lemmas

First of all, some definitions are presented below.

• $l(\varvec{\theta };x)=\log [g(x;\varvec{\theta })], \dot{l}(\varvec{\theta };x)=\frac{\partial }{\partial \varvec{\theta }}l(\varvec{\theta };x), l^{(2)}(\varvec{\theta };x)=\frac{\partial ^2}{\partial \varvec{\theta }\partial \varvec{\theta }^T}l(\varvec{\theta };x)$;

• $l_{q_n}(\varvec{\theta };x)=l_{q_n}[g(x;\varvec{\theta })], \dot{l}_{q_n}(\varvec{\theta };x)=\frac{\partial }{\partial \varvec{\theta }}l_{q_n}(\varvec{\theta };x), l^{(2)}_{q_n}(\varvec{\theta };x)=\frac{\partial ^2}{\partial \varvec{\theta }\partial \varvec{\theta }^T}l_{q_n}(\varvec{\theta };x)$;

• $K_n(\varvec{\theta })=E_{\varvec{\theta }_0}[\dot{l}_{q_n}(\varvec{\theta };x)\dot{l}_{q_n}(\varvec{\theta };x)^T], J_n(\varvec{\theta })=E_{\varvec{\theta }_0}[l^{(2)}_{q_n}(\varvec{\theta };x)]$, where $E_{\varvec{\theta }_0}[\cdot ]$ denote the expectation with respect to the true p.d.f $g(x;\varvec{\theta }_0)$;

• ${\varvec{\theta }}=(\xi ,\varvec{\phi }^T)^T, \varvec{\phi }=(\mu ,\sigma )^T, \widehat{\varvec{\theta }}_n= (\widehat{\xi },\widehat{\varvec{\phi }}^T)^T$ is the $\text{ ML}_\mathrm{q}\text{ E}$ under the whole parameter space $\varvec{\Theta }$ and $\bar{\varvec{\theta }}= (\xi _0,\bar{\varvec{\phi }}^T)^T$ is the $\text{ ML}_\mathrm{q}\text{ E}$ under $\varvec{\Theta }_0$ of the hypothesis test (2) and (4);

• $\Delta \varvec{\theta }=\widehat{\varvec{\theta }}_n-\bar{\varvec{\theta }}=(\widehat{\xi }-\xi _0,\widehat{\varvec{\phi }}^T-\bar{\varvec{\phi }}^T)^T \doteq (\Delta \widehat{\xi },\Delta \bar{\varvec{\phi }}^T)^T$;

• $\dot{L}_{q_n}(\varvec{\theta })=(\dot{L}_{q_{n}}^1(\varvec{\theta }),\dot{L}_{q_{n}}^2(\varvec{\theta })^T)^T$, where $\dot{L}_{q_{n}}^1(\varvec{\theta })=\frac{\partial }{\partial \xi }L_{q_n}(\varvec{\theta }), \dot{L}_{q_{n}}^2(\varvec{\theta })=\frac{\partial }{\partial \varvec{\phi }}L_{q_n}(\varvec{\theta })$;

• $$\begin{aligned} L_{q_n}^{(2)}(\varvec{\theta })=\left(\begin{array}{l l}L_{q_n11}(\varvec{\theta })&L_{q_n12}(\varvec{\theta })\\ L_{q_n21}(\varvec{\theta })&L_{q_n22}(\varvec{\theta })\end{array} \right), \end{aligned}$$

  where $L_{q_n11}(\varvec{\theta })=\frac{\partial ^2}{\partial \xi ^2}L_{q_n}(\varvec{\theta }), L_{q_n12}(\varvec{\theta })=L_{q_n21}(\varvec{\theta })=\frac{\partial ^2}{\partial \xi \partial \varvec{\phi }}L_{q_n}(\varvec{\theta }), L_{q_n22}(\varvec{\theta })=\frac{\partial ^2}{\partial \varvec{\phi }\partial \varvec{\phi }^T}L_{q_n}(\varvec{\theta })$;

• $$\begin{aligned}{}[L_{q_n}^{(2)}(\varvec{\theta })]^{-1}=\left(\begin{array}{l l}L_{q_n}^{11}(\varvec{\theta })&L_{q_n}^{12}(\varvec{\theta })\\ L_{q_n}^{21}(\varvec{\theta })&L_{q_n}^{22}(\varvec{\theta })\end{array} \right), \end{aligned}$$

  where the partitioning form of $[L_{q_n}^{(2)}(\varvec{\theta })]^{-1}$ is the same as that of $L_{q_n}^{(2)}(\varvec{\theta })$;

• $h(\varvec{\theta })=\xi -\xi _0, \dot{h}(\varvec{\theta })=(1,0,0)^T, h^{(2)}(\varvec{\theta })=\varvec{0}_{3\times 3}$;

• $P=I^{-1}(\varvec{\theta }_0)-I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0)(\dot{h}^T(\varvec{\theta }_0) I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0))^{-1}\dot{h}^T(\varvec{\theta }_0)I^{-1}(\varvec{\theta }_0)$;

• Let $M_n(\varvec{\theta })=2[L_{q_n}(\bar{\varvec{\theta }})-L_{q_n}({\varvec{\theta }})]$ and $D_n(\varvec{\beta })=M_n(n^{-1/2}\varvec{\beta }+\varvec{\theta }_0)$, where $\varvec{\beta }=n^{1/2}(\varvec{\theta }-\varvec{\theta }_0)$;

• $D(\varvec{\beta })=(\varvec{Z}-\varvec{\beta })^TI(\varvec{\theta }_0)(\varvec{Z}-\varvec{\beta })-\varvec{Z}^T(I-PI(\varvec{\theta }_0))^TI(\varvec{\theta }_0)(I-PI(\varvec{\theta }_0))\varvec{Z}$, where $\varvec{Z}\sim N(\varvec{0},I^{-1}(\varvec{\theta }_0))$ and $I$ is a $3\times 3$ unit matrix;

• $S_n=\{\varvec{\beta }: h(n^{-1/2}\varvec{\beta }+\varvec{\theta }_0)\ge 0\}, S_0=\{\varvec{\beta }: \dot{h}^T(\varvec{\theta }_0)\varvec{\beta }\ge 0\}$.

Lemma 5.1

If $\varvec{\theta }_n^*$ is the value such that $E_{\varvec{\theta }_0}[\dot{l}_{q_n}(\varvec{\theta }_n^*;x)]=0$, then we have

$$\begin{aligned} \varvec{\theta }^*_n=\varvec{\theta }_0+\mathcal O (1-q_n). \end{aligned}$$

 

Proof

First we denote $A(\varvec{\theta })=E_{\varvec{\theta }_0}[\dot{l}_{q_n}(\varvec{\theta };x)]$, then from the definition of $\varvec{\theta }_n^*$, it is easily seen that $A(\varvec{\theta }_n^*)=0$. By Taylor’s theorem, we can obtain that

$$\begin{aligned} 0=A\left(\varvec{\theta }_n^*\right)=A\left(\varvec{\theta }_0\right) +\triangledown _{\varvec{\theta }}A\left(\breve{\varvec{\theta }}\right) \left(\varvec{\theta }_n^*-\varvec{\theta }_0\right)\!, \end{aligned}$$

(15)

where $\breve{\varvec{\theta }}=\alpha \varvec{\theta }_n^*+(1-\alpha )\varvec{\theta }_0, 0\le \alpha \le 1$. Now will prove that $A(\varvec{\theta }_0)=\mathcal O (1-q_n)$ and the eigenvalue of $\triangledown _{\varvec{\theta }}A(\breve{\varvec{\theta }})$ is bounded away from zero.

We denote $B(y,\varvec{\theta }_0)=E_{\varvec{\theta }_0}[\dot{l}_{1-y}(\varvec{\theta }_0;x)]$, where $B(1-q_n,\varvec{\theta }_0)=A(\varvec{\theta }_0)$. then substitute the expression of $\dot{l}_{1-y}(\varvec{\theta };x)$ into $B(y,\varvec{\theta }_0)$, we have

$$\begin{aligned} \Vert B(y,\varvec{\theta }_0)\Vert&= \Vert E_{\varvec{\theta }_0}[\dot{l}(\varvec{\theta }_0;x)g^{y}(\varvec{\theta }_0;x)]\Vert \nonumber \\&\le \{E_{\varvec{\theta }_0}[\Vert \dot{l}(\varvec{\theta }_0;x)\Vert ^2]\}^{\frac{1}{2}}\{E_{\varvec{\theta }_0} [\Vert g^{y}(\varvec{\theta }_0;x)\Vert ^2]\}^{\frac{1}{2}} \end{aligned}$$

(16)

where $\Vert \cdot \Vert$ denote the $l_2$ norm. The last inequality in (16) holds according the Cauchy–Schwarz inequality. As the eigenvalues of the Fisher information matrix $E_{\varvec{\theta }_0}[\dot{l}(\varvec{\theta }_0;x)\dot{l}(\varvec{\theta }_0;x)^T]$ are bounded, we can derive that $\{E_{\varvec{\theta }_0}[\Vert \dot{l}(\varvec{\theta }_0;x)\Vert ^2]\}^{\frac{1}{2}}$ is also bounded. By substituting the expression of $g(\varvec{\theta }_0;x)$ into $E_{\varvec{\theta }_0} [\Vert g^{y}(\varvec{\theta }_0;x)\Vert ^2]$, it can be easily verified that,

$$\begin{aligned} E_{\varvec{\theta }_0}\left[\Vert g^{y}\left(\varvec{\theta }_0;x\right)\Vert ^2\right]^{\frac{1}{2}}<\infty ,\;\quad \forall \; 0\le y\le 1. \end{aligned}$$

Thus, for any given $0\le y\le 1, B(y,\varvec{\theta }_0)$ is bounded. Then by Taylor’s theorem, we derive the Taylor’s series expansion at $y=0$,

$$\begin{aligned} B\left(y,\varvec{\theta }_0\right)= B\left(0,\varvec{\theta }_0\right)+\triangledown _yB \left(0,\varvec{\theta }_0\right)y+o(y), \end{aligned}$$

(17)

where, according to the property of the MLE, we have $B(0,\varvec{\theta }_0)=0$. As $B(y,\varvec{\theta }_0)$ is bounded, then

$$\begin{aligned} \left\Vert\triangledown _yB\left(0,\varvec{\theta }_0\right)\right\Vert&= \Vert E_{\varvec{\theta }_0}\left\{ \dot{l}(\varvec{\theta }_0;x)\log \left[g\left(\varvec{\theta }_0;x\right)\right]\right\} \nonumber \\&\le \left\{ E_{\varvec{\theta }_0}\left\{ \left\Vert\dot{l}(\varvec{\theta }_0;x)\right\Vert^2\right\} \right\} ^{\frac{1}{2}} \left\{ E_{\varvec{\theta }_0}\left\{ \left\Vert\log \left[g(\varvec{\theta }_0;x)\right]\right\Vert^2\right\} \right\} ^{\frac{1}{2}} \end{aligned}$$

(18)

Similar to the argument below (16), by substituting $g(\varvec{\theta }_0;x)$ into $E_{\varvec{\theta }_0} [\Vert \log [g(\varvec{\theta }_0;$ $x)]\Vert ^2]$, we can obtain that $\triangledown _yB(0,\varvec{\theta }_0)$ is bounded. Therefore, it can be obviously found that $B(y,\varvec{\theta }_0)=\mathcal O (y)$, in other word,

$$\begin{aligned} A\left(\varvec{\theta }_0\right)=\mathcal O (1-q_n). \end{aligned}$$

(19)

Next we consider the term $\triangledown _{\varvec{\theta }}A(\breve{\varvec{\theta }})$ in the right hand of (15). Then it can be derived that

$$\begin{aligned} \triangledown _{\varvec{\theta }}A\left(\breve{\varvec{\theta }}\right)= \triangledown _{\varvec{\theta }}E_{\varvec{\theta }_0}\left[\dot{l}_{q_n}\left(\varvec{\theta };x\right)\right]|_{\varvec{\theta }=\breve{\varvec{\theta }}} =E_{\varvec{\theta }_0}\left[l^{(2)}_{q_n}\left(\breve{\varvec{\theta }};x\right)\right]=J_n\left(\breve{\varvec{\theta }}\right)\!. \end{aligned}$$

(20)

According to the definition of $\breve{\varvec{\theta }}$ and the fact $\lim \nolimits _{n\rightarrow \infty }\varvec{\theta }_n^*=\varvec{\theta }_0$, it can be found that $\breve{\varvec{\theta }}\rightarrow \varvec{\theta }_0$ as $n\rightarrow \infty$. As $J_n({\varvec{\theta }})$ is continuous at $\varvec{\theta }_0$, then for any given $\epsilon _1>0$, there exists an integer $N_1>0$, as $n\ge N_1$, we have $\Vert J_n(\breve{\varvec{\theta }})-J_n({\varvec{\theta }}_0)\Vert <\epsilon _1$. Now we consider the expression of $J_n({\varvec{\theta }}_0)$. By substituting the expression of $g(\varvec{\theta }_0;x)$ into $J_n({\varvec{\theta }}_0)$, it can be derived that

$$\begin{aligned} J_n\left({\varvec{\theta }}_0\right)&= E_{\varvec{\theta }_0}\left[l^{(2)}\left({\varvec{\theta }}_0;x\right)g^{1-q_n}\left(\varvec{\theta }_0;x\right)\right] \nonumber \\&+(1-q_n)E_{\varvec{\theta }_0}\left[\dot{l}\left({\varvec{\theta }}_0;x\right) \dot{l}^T\left({\varvec{\theta }}_0;x\right)g^{1-q_n} \left(\varvec{\theta }_0;x\right)\right]. \end{aligned}$$

(21)

Denote $H_1(y)=E_{\varvec{\theta }_0}[l^{(2)}({\varvec{\theta }}_0;x)g^{y}(\varvec{\theta }_0;x)]$ and $H_2(y)=E_{\varvec{\theta }_0}[\dot{l}({\varvec{\theta }}_0;x)\dot{l}^T({\varvec{\theta }}_0;x)$ $g^{y}(\varvec{\theta }_0;x)]$. As $H_1(y)$ and $H_2(y)$ are both continuous at $y=0$, then for any given $\epsilon _2$, there exits an integer $N_2>0$, as $n\ge N_2$, we have $\Vert H_1(1-q_n)-H_1(0)\Vert <\epsilon _2$ and $\Vert H_2(1-q_n)-H_2(0)\Vert <\epsilon _2$, where $-H_1(0)$ and $H_2(0)$ are both the Fisher information matrix of $G(x;\varvec{\theta }_0), I(\varvec{\theta }_0)$.

Without loss of generality, we suppose that there exits an integer $N_3>0$, as $n\ge N_3, 1-q_n<\frac{1}{6}$. Now we set $\epsilon _1=\frac{1}{6}\Vert H_1(0)\Vert , \epsilon _2=\frac{1}{7}\Vert H_1(0)\Vert$, then there exits an integer $N_0>\max \{N_1, N_2, N_3\}$, as $n\ge N_0$, it can be derived that

$$\begin{aligned} \Vert J_n(\breve{\varvec{\theta }})-H_1(0)\Vert&\le \Vert J_n(\breve{\varvec{\theta }})-J_n({\varvec{\theta }}_0)\Vert +\Vert J_n({\varvec{\theta }}_0)-H_1(0)\Vert \nonumber \\&\le \epsilon _1+(2-q_n)\epsilon _2+(1-q_n)\Vert H_2(0)\Vert \le \frac{1}{2}\Vert H_1(0)\Vert \end{aligned}$$

(22)

As a result, it is easily proved that the eigenvalue of $\triangledown _{\varvec{\theta }}A(\breve{\varvec{\theta }})$ is bounded away from zero.

Rearrange (15) and we can get the result below

$$\begin{aligned} \varvec{\theta }_n^*=\varvec{\theta }_0-[\triangledown _{\varvec{\theta }}A(\breve{\varvec{\theta }})]^{-1}A(\varvec{\theta }_0). \end{aligned}$$

(23)

As the properties of $A(\varvec{\theta }_0)$ and $\triangledown _{\varvec{\theta }}A(\breve{\varvec{\theta }})$ we derived above, it can be obviously found that

$$\begin{aligned} \varvec{\theta }_n^*=\varvec{\theta }_0+\mathcal O (1-q_n), \end{aligned}$$

(24)

which establishes the lemma. $\square$

 

Lemma 5.2

As $1-q_n\sim cn^{-\alpha }$ for some positive constant $c$, if $\alpha \ge \frac{1}{2}$, we have

$$\begin{aligned} \Delta \widehat{\xi }&= -L_{q_n}^{11}\left(\bar{\varvec{\theta }}\right)\dot{L}_{q_n}^{1}(\bar{\varvec{\theta }})+\mathcal O _p\left(n^{-1}\right);\, \\ \Delta \bar{\varvec{\phi }}&= -\left[L_{q_n22}(\bar{\varvec{\theta }})\right]^{-1} L_{q_n21}\left(\bar{\varvec{\theta }}\right)\Delta \widehat{\xi }+\mathcal O _p\left(n^{-1}\right)\!. \end{aligned}$$

 

Proof

By Taylor’s theorem, we derive the Taylor’s series expansion of $L_{q_n}(\varvec{\theta })$ at $\varvec{\theta }=\bar{\varvec{\theta }}$,

$$\begin{aligned} \dot{L}_{q_n}\left(\widehat{\varvec{\theta }}_n\right)=\dot{L}_{q_n}\left(\bar{\varvec{\theta }}\right)+ L^{(2)}_{q_n}\left(\bar{\varvec{\theta }}\right)\Delta \varvec{\theta }+R_n, \end{aligned}$$

(25)

where $R_n=\frac{1}{2}\Delta \varvec{\theta }^TL_{q_n}^{(3)}(\breve{\varvec{\theta }})\Delta \varvec{\theta }, \breve{\varvec{\theta }}=\alpha \widehat{\varvec{\theta }}_n+(1-\alpha )\bar{\varvec{\theta }}, 0\le \alpha \le 1.$ By Lemma 5.1, it can be easily obtained that

$$\begin{aligned} R_n=\frac{1}{2}\sqrt{n}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_n^*+\mathcal O (1-q_n)\right)^T\frac{1}{n}L_{q_n}^{(3)}(\breve{\varvec{\theta }}) \sqrt{n}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_n^*+\mathcal O (1-q_n)\right) \end{aligned}$$

(26)

Ferrari and Yang (2010) presented that $\sqrt{n}(\widehat{\varvec{\theta }}_n-\varvec{\theta }_n^*)=\mathcal O _p(1)$ and $\frac{1}{n}L_{q_n}^{(3)}(\breve{\varvec{\theta }})=\mathcal O _p(1)$, as $1-q_n\sim cn^{-\alpha },\alpha \ge \frac{1}{2}$, we have $R_n=\mathcal O _p(1)$.

Since $\dot{L}_{q_n}(\widehat{\varvec{\theta }}_n)=0$, by substituting the definition of $\dot{L}_{q_n}(\varvec{\theta })$ and $\dot{L}^{(2)}_{q_n}(\varvec{\theta })$, we can derive that

$$\begin{aligned} \left(\begin{array}{l l}\dot{L}^1_{q_n}(\bar{\varvec{\theta }})\\ \dot{L}^2_{q_n}(\bar{\varvec{\theta }})\end{array}\right) =-\left(\begin{array}{l l}L_{q_n11}(\bar{\varvec{\theta }})&L_{q_n12}(\bar{\varvec{\theta }})\\ L_{q_n21}(\bar{\varvec{\theta }})&L_{q_n22}(\bar{\varvec{\theta }})\end{array} \right)\left(\begin{array}{l l}\Delta \widehat{\xi }\\ \Delta \bar{\varvec{\phi }}\end{array} \right)+\mathcal O _p(1). \end{aligned}$$

(27)

As $\dot{L}^2_{q_n}(\bar{\varvec{\theta }})=0$, we have $0=-L_{q_n21}(\bar{\varvec{\theta }})\Delta \widehat{\xi }-L_{q_n22}(\bar{\varvec{\theta }})\Delta \bar{\varvec{\phi }} +\mathcal O _p(1)$. As $J_n(\varvec{\theta })$ is continuous at $\varvec{\theta }=\varvec{\theta }_0$, similar to the argument in Lemma 5.1, it can be derived that $J_n(\bar{\varvec{\theta }})=\mathcal O (1)$. Then the fact $L_{q_n}^{(2)}(\bar{\varvec{\theta }})=\mathcal O _p(n)$ follows. By taking some simple arrangements, we have

$$\begin{aligned} \Delta \bar{\varvec{\phi }}=-\left[L_{q_n22}\left(\bar{\varvec{\theta }}\right)\right]^{-1}L_{q_n21}(\bar{\varvec{\theta }}) \Delta \widehat{\xi }+\mathcal O _p\left(n^{-1}\right). \end{aligned}$$

(28)

The second result of the lemma follows. Meanwhile, we also have

$$\begin{aligned} \left(\begin{array}{l }\Delta \widehat{\xi }\\ \Delta \bar{\varvec{\phi }}\end{array}\right) =-\left(\begin{array}{l l}L_{q_n}^{11}(\bar{\varvec{\theta }})&L_{q_n}^{12}(\bar{\varvec{\theta }})\\ L_{q_n}^{21}(\bar{\varvec{\theta }})&L_{q_n}^{22}(\bar{\varvec{\theta }})\end{array} \right)\left(\begin{array}{c }\dot{L}^1_{q_n}(\bar{\varvec{\theta }})\\ \varvec{0}\end{array} \right)+\mathcal O _p\left(n^{-1}\right)\!. \end{aligned}$$

(29)

Then we have

$$\begin{aligned} \Delta \widehat{\xi }=-L_{q_n}^{11}\left(\bar{\varvec{\theta }}\right) \dot{L}_{q_n}^{1}\left(\bar{\varvec{\theta }}\right)+\mathcal O _p\left(n^{-1}\right). \end{aligned}$$

(30)

This establishes the lemma. $\square$

 

Lemma 5.3

As $1-q_n\sim cn^{-\alpha }$, where $c>0, \alpha >\frac{1}{2}$, for the hypothesis test (4), suppose that $\varvec{\theta }_0\in \varvec{\Theta }_0$, we have

$$\begin{aligned} n^{1/2}\left(\bar{\varvec{\theta }}-\varvec{\theta }_0\right)=n^{-1/2}P\dot{L}_{q_n}\left(\varvec{\theta }_0\right)+o_p(1). \end{aligned}$$

 

Proof

Denote that $\varvec{\psi }=(\varvec{\theta }^T,\lambda )^T, \bar{\varvec{\psi }}=(\bar{\varvec{\theta }}^T,\bar{\lambda })^T, \varvec{\psi }_0=(\varvec{\theta }_0^T,\lambda _0)^T$ and $T_{q_n}(\varvec{\psi })=L_{q_n}(\varvec{\theta })+\lambda h(\varvec{\theta })$. By Taylor’s theorem, we derive the Taylor’s series expansion of $T_{q_n}(\varvec{\psi })$ at $\varvec{\psi }=\varvec{\psi }_0$,

$$\begin{aligned} \dot{T}_{q_n}\left(\bar{\varvec{\psi }}\right)=\dot{T}_{q_n}\left(\varvec{\psi }_0\right)+T^{(2)}_{q_n}(\varvec{\psi }_0)(\bar{\varvec{\psi }}-\varvec{\psi }_0)+R_n, \end{aligned}$$

(31)

where $R_n=(\bar{\varvec{\psi }}-\varvec{\psi }_0)^TT^{(3)}_{q_n}(\breve{\varvec{\psi }})(\bar{\varvec{\psi }}-\varvec{\psi }_0), \breve{\varvec{\psi }}=\alpha \bar{\varvec{\psi }}+(1-\alpha )\varvec{\psi }_0, 0\le \alpha \le 1.$ According to the definition of $h(\varvec{\theta })$ and the fact $\bar{\varvec{\psi }}-\varvec{\psi }_0=\mathcal O _p(n^{-1/2}), n^{-1}T^{(3)}_{q_n}(\breve{\varvec{\psi }})=\mathcal O _p(1)$, we have $R_n=\mathcal O _p(1)$.

Since $\lambda _0=0, h({\varvec{\theta }}_0)=0$ and $\dot{T}_{q_n}(\bar{\varvec{\psi }})=0$, then by taking some arrangements of (31), it can be written as

$$\begin{aligned} \left(\begin{array}{c}\dot{L}_{q_n}({\varvec{\theta }}_0)\\ 0\end{array}\right)&= -\left(\begin{array}{c c}L_{q_n}^{(2)}({\varvec{\theta }}_0)&\dot{h}({\varvec{\theta }}_0)\\ \dot{h}^T({\varvec{\theta }}_0)&0\end{array} \right)\left(\begin{array}{l l}\bar{\varvec{\theta }}-\varvec{\theta }_0\\ \bar{\lambda }-\lambda _0\end{array} \right)+\mathcal O _p(1).\end{aligned}$$

(32)

$$\begin{aligned} \sqrt{n}\left(\begin{array}{l l}\bar{\varvec{\theta }}-\varvec{\theta }_0\\ \bar{\lambda }-\lambda _0\end{array} \right)&= n^{-\frac{1}{2}}\left(\begin{array}{l l}P&Q\\ Q&R\end{array} \right)\left(\begin{array}{c}\dot{L}_{q_n}({\varvec{\theta }}_0)\\ 0\end{array}\right) +o_p(1), \end{aligned}$$

(33)

where $P=I^{-1}(\varvec{\theta }_0)-I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0)(\dot{h}^T(\varvec{\theta }_0) I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0))^{-1}\dot{h}^T(\varvec{\theta }_0)I^{-1}(\varvec{\theta }_0)$. The last equality holds by applying the matrix theory and the fact $-n^{-1}L_{q_n}^{(2)}({\varvec{\theta }}_0)=I(\varvec{\theta }_0)+o_p(1)$. As a result, we can derive that

$$\begin{aligned} \sqrt{n}\left(\bar{\varvec{\theta }}-\varvec{\theta }_0\right)= n^{-\frac{1}{2}}P\dot{L}_{q_n}\left({\varvec{\theta }}_0\right)+o_p(1), \end{aligned}$$

(34)

Thus the desired result follows. $\square$

 

Lemma 5.4

As $1-q_n\sim cn^{-\alpha }$, where $c>0, \alpha >\frac{1}{2}$, for any fixed $\varvec{\beta }, D_n(\varvec{\beta })$ converges to $D(\varvec{\beta })$ in distribution, that is,

$$\begin{aligned} D_n(\varvec{\beta })\mathop {\longrightarrow }\limits ^{d}D(\varvec{\beta })\quad as\quad n\rightarrow \infty . \end{aligned}$$

(35)

 

Proof

First we consider the function series $M_n(\varvec{\theta })$ instead of $D_n(\varvec{\beta })$.

$$\begin{aligned} M_n(\varvec{\theta })&= 2\left[L_{q_n}(\bar{\varvec{\theta }})-L_{q_n}({\varvec{\theta }})\right]\nonumber \\&= 2\left[L_{q_n}\left(\bar{\varvec{\theta }}\right)-L_{q_n} \left(\widehat{\varvec{\theta }}_n\right)\right]-2\left[L_{q_n}({\varvec{\theta }})-L_{q_n}(\widehat{\varvec{\theta }}_n)\right]\nonumber \\&= A_1-A_2. \end{aligned}$$

(36)

By taking Taylor’s series expansion of $L_{q_n}({\varvec{\theta }})$ at $\varvec{\theta }=\widehat{\varvec{\theta }}_n$, it is known that,

$$\begin{aligned} A_2=-\left[n^{1/2}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_0\right)-\varvec{\beta }\right]^T I(\varvec{\theta }_0)\left[n^{1/2}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_0\right)-\varvec{\beta }\right]+o_p(1). \end{aligned}$$

(37)

By taking Taylor’s series expansion of $L_{q_n}(\bar{\varvec{\theta }})$ at $\varvec{\theta }=\widehat{\varvec{\theta }}_n$ and applying Lemma 5.3, we have

$$\begin{aligned} A_1&= -\sqrt{n}\left(\bar{\varvec{\theta }}-\widehat{\varvec{\theta }}_n\right)^TI(\varvec{\theta }_0)\sqrt{n}\left(\bar{\varvec{\theta }}- \widehat{\varvec{\theta }}_n\right)+o_p(1)\nonumber \\&= -\left[(PI(\varvec{\theta }_0)-I)\sqrt{n}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_0\right)+o_p(1)\right]^TI(\varvec{\theta }_0)\nonumber \\&\left[(PI(\varvec{\theta }_0)-I)\sqrt{n}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_0\right)+o_p(1)\right]+o_p(1), \end{aligned}$$

(38)

where $I$ is a $3\times 3$ unit matrix. According to the fact $J_n(\widehat{\varvec{\theta }}_n)\rightarrow -I(\varvec{\theta }_0), K_n(\widehat{\varvec{\theta }}_n)\rightarrow I(\varvec{\theta }_0)$ as $n\rightarrow \infty$ and $\sqrt{n}(J_n^{-1}(\widehat{\varvec{\theta }}_n)K_n(\widehat{\varvec{\theta }}_n)J_n^{-1}(\widehat{\varvec{\theta }}_n))^{-1/2} (\widehat{\varvec{\theta }}_n-\varvec{\theta }_0)\mathop {\longrightarrow }\limits ^{d}N(\varvec{0},I)$ (see Ferrari and Yang 2010), through appealing to Theorem 5.1 of Billingsley (1968) and combining (36)–(38), Lemma 5.4 has been proved. $\square$

 

Lemma 5.5

As $1-q_n\sim cn^{-\alpha }$, where $c>0, \alpha >\frac{1}{2}$, the stochastic processes $\{D_n(\varvec{\beta }), \varvec{\beta }\in \mathcal D \}$ converge in distribution to $\{D(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$, that is,

$$\begin{aligned} \{D_n(\varvec{\beta }), \varvec{\beta }\in \mathcal D \}\mathop {\longrightarrow }\limits ^{d}\{D(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}\quad as\,n\rightarrow \infty . \end{aligned}$$

(39)

Furthermore, the test statistic for (4) $L_{q_n}R=-D_n(\widetilde{\varvec{\beta }}_n)$ converges in distribution to $-D(\widetilde{\varvec{\beta }})$, where $\widetilde{\varvec{\beta }}_n$ and $\widetilde{\varvec{\beta }}$ are the optimal solution under $S_n$ and $S_0$ respectively.

 

Proof

According to the theory on the convergence of stochastic processes (see Prakasa Rao 1975), $\{D_n(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$ converges in distribution to $\{D(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$ if and only if the following two conditions are satisfied:

• (1) Any finite dimensional distribution of process $\{D_n(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$ converges weakly to the corresponding finite dimensional distribution of $\{D(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$;

• (2) For any $\epsilon >0$ it holds that

  $$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{h\rightarrow 0}P\left\{ \sup _{\Vert \varvec{\beta }^{(1)}-\varvec{\beta }^{(2)}\Vert \le h}|D_n(\varvec{\beta }^{(1)})- D_n(\varvec{\beta }^{(2)})|>\epsilon , \varvec{\beta }^{(1)},\varvec{\beta }^{(2)}\in \mathcal D \right\} =0. \nonumber \\ \end{aligned}$$

  (40)

Now we check the two conditions, first, for condition (1), by Cramér-Wold Theorem, it suffices to show that for any $\alpha _1,\ldots ,alpha_r\in \mathbb{R }$ and any $\varvec{\beta }^{(1)},\ldots ,\varvec{\beta }^{(r)}\in \mathcal D$, we have

$$\begin{aligned} \sum _{j=1}^r\alpha _jD_n(\varvec{\beta }^{(j)})\mathop {\longrightarrow }\limits ^{d}\sum _{j=1}^r \alpha _jD\left(\varvec{\beta }^{(j)}\right). \end{aligned}$$

(41)

This convergence result follows as the same way as Lemma 5.4. The details is omitted here.

Next we discuss the condition (2), for $\varvec{\beta }^{(1)},\varvec{\beta }^{(2)}$ in $\mathcal D$, we have

$$\begin{aligned}&D_n(\varvec{\beta }^{(1)})-D_n\left(\varvec{\beta }^{(2)}\right)\nonumber \\&=2\left[L_{q_n}\left(\bar{\varvec{\theta }}\right)-L_{q_n}\left(n^{-1/2}\varvec{\beta }^{(1)}+\varvec{\theta }_0\right)\right] -2\left[L_{q_n}\left(\bar{\varvec{\theta }}\right)-L_{q_n}\left(n^{-1/2}\varvec{\beta }^{(2)}+\varvec{\theta }_0\right)\right]\nonumber \\&=2\left[L_{q_n}\left(n^{-1/2}\varvec{\beta }^{(2)}+\varvec{\theta }_0\right)-L_{q_n}\left({\varvec{\theta }}_0\right)\right] -2\left[L_{q_n}\left(n^{-1/2}\varvec{\beta }^{(1)}+\varvec{\theta }_0\right)-L_{q_n}\left({\varvec{\theta }}_0\right)\right]. \nonumber \\ \end{aligned}$$

(42)

By Taylor’s Theorem, for $i=1,2$, it can be obtained that

$$\begin{aligned} L_{q_n}\left(n^{-1/2}\varvec{\beta }^{(i)}+\varvec{\theta }_0\right)-L_{q_n}({\varvec{\theta }}_0)&= n^{-1/2}\dot{L}^T_{q_n}({\varvec{\theta }}_0)\varvec{\beta }^{(i)} -\frac{1}{2}\varvec{\beta }^{(i)^{\prime }}I\left(\varvec{\theta }_0\right)\varvec{\beta }^{(i)}\nonumber \\&\quad +o_p(1)\Vert \varvec{\beta }^{(i)}\Vert ^2 +n^{-1/2}\Vert \varvec{\beta }^{(i)}\Vert ^3\mathcal O _p(1).\qquad \quad \end{aligned}$$

(43)

As a result, when $n\rightarrow \infty$ and $\Vert \varvec{\beta }^{(1)}-\varvec{\beta }^{(2)}\Vert \rightarrow 0$, for any given $\epsilon >0$, we can derive that

$$\begin{aligned} \left|\varvec{\beta }^{(1)^{\prime }}I(\varvec{\theta }_0)\varvec{\beta }^{(1)}-\varvec{\beta }^{(2)^{\prime }}I(\varvec{\theta }_0)\varvec{\beta }^{(2)}\right|<\frac{\epsilon }{4}, \end{aligned}$$

(44)

and then the following result holds with probability approaching one

$$\begin{aligned} \left|D_n\left(\varvec{\beta }^{(1)}\right)-D_n\left(\varvec{\beta }^{(2)}\right)\right|\le 2\left|n^{-1/2}\dot{L}^T_{q_n}\left({\varvec{\theta }}_0\right)\left(\varvec{\beta }^{(1)}-\varvec{\beta }^{(2)}\right)\right|+\frac{\epsilon }{2}. \end{aligned}$$

(45)

As $\mathcal D$ is compact and $n^{-1/2}\dot{L}^T_{q_n}({\varvec{\theta }}_0)=\mathcal O _p(1)$, we have

$$\begin{aligned}&\!\! \!\lim _{n\rightarrow \infty }\sup _{h\rightarrow 0}P\bigg \{\sup _{\Vert \varvec{\beta }^{(1)}-\varvec{\beta }^{(2)}\Vert \le h}|D_n\left(\varvec{\beta }^{(1)}\right)-D_n\left(\varvec{\beta }^{(2)}\right)|>\epsilon , \varvec{\beta }^{(1)},\varvec{\beta }^{(2)}\in \mathcal D \bigg \}\nonumber \\&\!\!\!\le \lim _{n\rightarrow \infty }\sup _{h\rightarrow 0}P\bigg \{\sup _{\Vert \varvec{\beta }^{(1)}\!-\!\varvec{\beta }^{(2)}\Vert \le h}2\left|n^{-1/2}\dot{L}^T_{q_n}\left({\varvec{\theta }}_0\right)\left(\varvec{\beta }^{(1)}\!-\!\varvec{\beta }^{(2)}\right)\right|>\frac{\epsilon }{2}, \varvec{\beta }^{(1)},\varvec{\beta }^{(2)}\in \mathcal D \bigg \}\quad \nonumber \\&\!\!\!=0. \end{aligned}$$

(46)

Since the two conditions are satisfied, $\{D_n(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$ converges in distribution to $\{D(\varvec{\beta }),\varvec{\beta }\in \mathcal D \}$. Then by the similar arguments as has been applied in Theorem 4(1) (see Liu 2007), where the continuous mapping theorem (see Billingsley 1968) is used, the test statistic for (4) $L_{q_n}R=-D_n(\widetilde{\varvec{\beta }}_n)$ converges in distribution to $-D(\widetilde{\varvec{\beta }})$, where $\widetilde{\varvec{\beta }}_n$ and $\widetilde{\varvec{\beta }}$ are the optimal solution under $S_n$ and $S_0$ respectively. Therefore Lemma 5.5 has been proved. $\square$

 

Appendix B: Proofs of Theorems

1.1 Proof of Theorem 2.1

By Taylor’s theorem, we derive the Taylor’s series expansion of $L_{q_n}(\varvec{\theta })$ at $\varvec{\theta }=\bar{\varvec{\theta }}$, then the test statistic $L_{q_n}$R can be written as

$$\begin{aligned} L_{q_n}R&= 2\left\{ L_{q_n}(\widehat{\varvec{\theta }}_n)-L_{q_n}(\bar{\varvec{\theta }})\right\} \nonumber \\&= 2\dot{L}_{q_n}(\bar{\varvec{\theta }})^T\Delta \varvec{\theta }+\Delta \varvec{\theta }^TL_{q_n}^{(2)}(\bar{\varvec{\theta }}) \Delta \varvec{\theta }+R_n, \end{aligned}$$

(47)

where $R_n=\frac{1}{3}n^{-\frac{1}{2}}\sum _i^3\sum _j^3\sum _k^3n^{-1}L_{q_ni,j,k}(\breve{\varvec{\theta }}) \sqrt{n}\Delta \varvec{\theta }_i\sqrt{n}\Delta \varvec{\theta }_j\sqrt{n}\Delta \varvec{\theta }_k.$ As $n^{-1}$ $L_{q_ni,j,k}(\breve{\varvec{\theta }})$ is bounded and $\sqrt{n}\Delta \varvec{\theta }=\mathcal O _p(1)$. Then we have $R_n=\mathcal O _p(n^{-\frac{1}{2}})$. By Lemma 5.2 and the partitioned matrix result $L_{q_n}^{11}(\bar{\varvec{\theta }})=[L_{q_n11}(\bar{\varvec{\theta }})-L_{q_n12}(\bar{\varvec{\theta }})L^{-1}_{q_n22}(\bar{\varvec{\theta }})$ $L_{q_n21}(\bar{\varvec{\theta }})]^{-1}$, (47) can be written as

$$\begin{aligned} L_{q_n}R=-\Delta \widehat{\xi }\left[L_{q_n}^{11}\left(\bar{\varvec{\theta }}\right)\right]^{-1}\Delta \widehat{\xi }+\mathcal O _p\left(n^{-\frac{1}{2}}\right) \end{aligned}$$

(48)

According to the asymptotical normality of the $\text{ ML}_\mathrm{q}\text{ E}$ showed in Ferrari and Yang (2010),

$$\begin{aligned} V^{-\frac{1}{2}}_n\left(\varvec{\theta }_n^*\right)\sqrt{n}\left(\widehat{\varvec{\theta }}_n-\varvec{\theta }_n^*\right) \mathop {\longrightarrow }\limits ^{d}N\left(\varvec{0},\varvec{I}\right),\quad n\rightarrow \infty , \end{aligned}$$

(49)

where $V_n(\varvec{\theta })=J_n^{-1}(\varvec{\theta })K_n(\varvec{\theta })J_n^{-1}(\varvec{\theta })$. As the null hypothesis is true, by Lemma 5.1 and the continuity of $K_n(\varvec{\theta }),J_n(\varvec{\theta })$ at $\varvec{\theta }$, it can be found that $V_n(\varvec{\theta }_n^*)\rightarrow I^{-1}({\varvec{\theta }}_0)$ as $n\rightarrow \infty$. Furthermore, we have

$$\begin{aligned} \left[I^{11}\left({\varvec{\theta }}_0\right)\right]^{\frac{1}{2}} \sqrt{n}\left(\widehat{\xi }-\xi _0+\xi _0-\xi ^*\right) \mathop {\longrightarrow }\limits ^{d}N(0,1),\quad n\rightarrow \infty , \end{aligned}$$

(50)

where $\xi ^*$ is the first component of $\varvec{\theta }_n^*$. As the null hypothesis of (2) is true, then $\bar{\varvec{\theta }}$ converges to the true parameter $\varvec{\theta }_0$ in probability. Thus (48) can be written as

$$\begin{aligned} L_{q_n}R=\sqrt{n}\Delta \widehat{\xi }\left[I^{11}\left(\varvec{\theta }_0\right)+o_p(1)\right]^{-1}\sqrt{n}\Delta \widehat{\xi }+ \mathcal O _p\left(n^{-\frac{1}{2}}\right) \end{aligned}$$

(51)

If $\alpha >\frac{1}{2}$, we have $[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}(\widehat{\xi }-\xi _0+\xi _0-\xi ^*) =[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}\Delta \widehat{\xi }$, combine (50) and (51), it can be derived that $L_{q_n}$R is asymptotically $\chi ^2$-distributed with one degree of freedom.

If $\alpha =\frac{1}{2}$, we have $[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}(\widehat{\xi }-\xi _0+\xi _0-\xi ^*) =[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}\Delta \widehat{\xi }+[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}(\xi _0-\xi ^*)$, where $[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}\Delta \widehat{\xi }\mathop {\longrightarrow }\limits ^{d}N(0,1)$ as $n\rightarrow \infty$ and $[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}(\xi _0-\xi ^*)=\mathcal O (1)$. As a result, if $\lim _{n\rightarrow \infty }n(\xi _0-\xi ^*)^2I^{11}({\varvec{\theta }}_0)\ne 0$, the test statistic $L_{q_n}$R is asymptotically noncentral $\chi ^2$-distributed with one degree of freedom and noncentral parameter $\delta =\lim _{n\rightarrow \infty }n(\xi _0-\xi ^*)^2I^{11}({\varvec{\theta }}_0)$.

If $\alpha <\frac{1}{2}$, it can be derived that $[I^{11}({\varvec{\theta }}_0)]^{\frac{1}{2}}\sqrt{n}(\widehat{\xi }-\xi _0+\xi _0-\xi ^*)=\mathcal O _p(n^{\frac{1}{2}-\alpha })$, therefore the statistic $L_{q_n}$R diverges as $n\rightarrow \infty$. Thus the desired result is obtained.

1.2 Proof of Theorem 2.2

Similar to the proof of Theorem 2.1, by Taylor’s theorem, we derive the Taylor’s series expansion of $L_{q_n}R$ at $\varvec{\theta }=\bar{\varvec{\theta }}$, then the test statistic $L_{q_n}R$ can be written as

$$\begin{aligned} L_{q_n}R=2\dot{L}_{q_n}\left(\bar{\varvec{\theta }}\right)^T\Delta \varvec{\theta }+\Delta \varvec{\theta }^TL_{q_n}^{(2)}\left(\bar{\varvec{\theta }}\right) \Delta \varvec{\theta }+R_n, \end{aligned}$$

(52)

where $R_n=\frac{1}{3}n^{-\frac{1}{2}}\sum _i^3\sum _j^3\sum _k^3n^{-1}L_{q_ni,j,k}(\breve{\varvec{\theta }}) \sqrt{n}\Delta \varvec{\theta }_i\sqrt{n}\Delta \varvec{\theta }_j\sqrt{n}\Delta \varvec{\theta }_k, \breve{\varvec{\theta }}=\alpha \widehat{\varvec{\theta }}_n+(1-\alpha )\bar{\varvec{\theta }}, 0\le \alpha \le 1.$ As the sequence of alternatives $H_n: \xi =\xi _0+h/\sqrt{n}$ are true, then we have $\sqrt{n}(\bar{\varvec{\theta }}-{\varvec{\theta }}_0)=\mathcal O _p(1)$. Consequently, it can be derived that $n^{-1}L_{q_ni,j,k}(\breve{\varvec{\theta }})=\mathcal O _p(1), \sqrt{n}(\widehat{\varvec{\theta }}-\bar{\varvec{\theta }})=\mathcal O _p(1)$ and then $R_n=\mathcal O _p(n^{-\frac{1}{2}})$. By Lemma 5.2 and the partitioned matrix result, the same result for (52) can also be obtained

$$\begin{aligned} L_{q_n}R=-\Delta \widehat{\xi }\left[L_{q_n}^{11}\left(\bar{\varvec{\theta }}\right)\right]^{-1}\Delta \widehat{\xi }+\mathcal O _p\left(n^{-\frac{1}{2}}\right). \end{aligned}$$

(53)

According to the asymptotical normality of the modified MSPE, when the alternatives are true, we have

$$\begin{aligned} \left[I^{11}\left({\varvec{\theta }}_0\right)\right]^{-\frac{1}{2}}\sqrt{n}\left(\widehat{\xi }-\xi _0-h/\sqrt{n}\right) \mathop {\longrightarrow }\limits ^{d}N(0,1),\quad n\rightarrow \infty . \end{aligned}$$

(54)

As the alternatives are true, then $\bar{\varvec{\theta }}$ converges to the true parameter $\varvec{\theta }_0$ in probability. (53) can be written as

$$\begin{aligned} L_{q_n}R&= \sqrt{n}\left(\widehat{\xi }-\xi _0-h/\sqrt{n}+h/\sqrt{n}\right)\left[-nL_{q_n}^{11}(\bar{\varvec{\theta }})\right]^{-1} \nonumber \\&\times \sqrt{n}\left(\widehat{\xi }-\xi _0-h/\sqrt{n}+h/\sqrt{n}\right) +\mathcal O _p\left(n^{-\frac{1}{2}}\right)\nonumber \\&= (\eta _n+h)\left[I^{11}(\varvec{\theta }_0)+o_p(1)\right]^{-1}(\eta _n+h)+ \mathcal O _p\left(n^{-\frac{1}{2}}\right)\nonumber \\&= \left\Vert\left[I^{11}(\varvec{\theta }_0)\right]^{-\frac{1}{2}}\eta _n+\left[I^{11}(\varvec{\theta }_0)\right]^{-\frac{1}{2}}h\right\Vert^2+ \mathcal O _p\left(n^{-\frac{1}{2}}\right)\!, \end{aligned}$$

(55)

where $\eta _n\mathop {\longrightarrow }\limits ^{d}N(0,I^{11}(\varvec{\theta }_0))$ as $n\rightarrow \infty$. According to the definition of the noncentral $\chi ^2$ distribution, when the alternatives are true, the proposed test statistic $L_{q_n}R$ is asymptotically noncentral $\chi ^2$-distributed with one degree of freedom and noncentral parameter $\delta =h^2[I^{11}({\varvec{\theta }}_0)]^{-1}$. Thus the desired result is obtained.

1.3 Proof of Theorem 2.3

In Lemma 5.5, we have pointed out that, for the hypothesis test (4),

$$\begin{aligned} L_{q_n}R=-D_n\left(\widetilde{\varvec{\beta }}_n\right) \mathop {\longrightarrow }\limits ^{d}-D\left(\widetilde{\varvec{\beta }}\right)\!, \end{aligned}$$

(56)

where $\widetilde{\varvec{\beta }}_n$ and $\widetilde{\varvec{\beta }}$ are the optimal solution under $S_n$ and $S_0$ respectively. as a result, here we discuss the distribution of $-D(\widetilde{\varvec{\beta }})$ instead. According to the definition of $D(\varvec{\beta })$, we have

$$\begin{aligned} -D\left(\widetilde{\varvec{\beta }}\right)&= -\inf _{\varvec{\beta }\in S_0}\left(\varvec{Z}-\varvec{\beta }\right)^TI \left(\varvec{\theta }_0\right) \left(\varvec{Z}-\varvec{\beta }\right)+ \varvec{Z}^T\left(I-PI\left(\varvec{\theta }_0\right)\right)^T \nonumber \\&\times I \left(\varvec{\theta }_0\right)\left(I-PI\left(\varvec{\theta }_0\right)\right) \varvec{Z}\nonumber \\&= \left[\varvec{Z}^TI\left(\varvec{\theta }_0\right) \varvec{Z}-\inf _{\varvec{\beta }\in S_0}\left(\varvec{Z}-\varvec{\beta }\right)^TI\left(\varvec{\theta }_0\right)(\varvec{Z}-\varvec{\beta })\right]\nonumber \\&\quad -\left[\varvec{Z}^TI\left(\varvec{\theta }_0\right) \varvec{Z}-\varvec{Z}^T\left(I-PI(\varvec{\theta }_0)\right)^TI \left(\varvec{\theta }_0\right)\left(I-PI(\varvec{\theta }_0)\right)\varvec{Z}\right]\nonumber \\&\doteq A_1-A_2. \end{aligned}$$

(57)

From the definition of $\bar{\chi }^2$ distribution given by Shapiro (1988), we have

$$\begin{aligned} A_1\sim \bar{\chi }^2\left(I^{-1}\left(\varvec{\theta }_0\right),S_0\right)\!, \end{aligned}$$

(58)

and the polar cone of $S_0$ is given by

$$\begin{aligned} (S_0)^0=\left\{ \varvec{\beta }: \varvec{\beta }=-I^{-1}\left(\varvec{\theta }_0\right)\dot{h} \left(\varvec{\theta }_0\right)\gamma ,\gamma \in \mathbb{R }^+\right\} . \end{aligned}$$

(59)

According to the definition of the polar cone and the equation (3.5) in Shapiro (1988), $A_1$ can be written as

$$\begin{aligned} A_1=\inf _{\varvec{\beta }\in {S_0}^0}\left(\varvec{Z}-\varvec{\beta }\right)^TI \left(\varvec{\theta }_0\right)\left(\varvec{Z}-\varvec{\beta }\right)= \inf _{\gamma \in \mathbb{R }^+}(X-\gamma )^TQ(\varvec{\theta }_0)(X-\gamma )+A_3,\nonumber \\ \end{aligned}$$

(60)

where $X=(\dot{h}^T(\varvec{\theta }_0)I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0))^{-1}\dot{h}^T(\varvec{\theta }_0)\varvec{Z}, Q(\varvec{\theta }_0)=\dot{h}^T(\varvec{\theta }_0)I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0)$. According to the behavior of $\varvec{Z}$, we have the result $X\sim N(0,Q^{-1}(\varvec{\theta }_0))$. $A_3$ can be written as

$$\begin{aligned} A_3=\varvec{Z}^T\left[I\left(\varvec{\theta }_0\right)-\dot{h}\left(\varvec{\theta }_0\right) \left(\dot{h}^T\left(\varvec{\theta }_0\right)I^{-1}\left(\varvec{\theta }_0\right) \dot{h}\left(\varvec{\theta }_0\right)\right)^{-1} \dot{h}^T\left(\varvec{\theta }_0\right)\right]\varvec{Z}. \end{aligned}$$

(61)

Now we show that $A_2=A_3$, as $P=I^{-1}(\varvec{\theta }_0)-I^{-1}(\varvec{\theta }_0)\dot{h}(\varvec{\theta }_0)(\dot{h}^T(\varvec{\theta }_0) I^{-1}(\varvec{\theta }_0)$ $\dot{h}(\varvec{\theta }_0))^{-1}\dot{h}^T(\varvec{\theta }_0)I^{-1}(\varvec{\theta }_0)$, then

$$\begin{aligned} A_2&= \varvec{Z}^TI\left(\varvec{\theta }_0\right)\varvec{Z}-\varvec{Z}^T(I-PI(\varvec{\theta }_0))^TI\left(\varvec{\theta }_0\right)\left(I-PI\left(\varvec{\theta }_0\right)\right) \varvec{Z}\nonumber \\&= \varvec{Z}^TI\left(\varvec{\theta }_0\right) \varvec{Z}-\varvec{Z}^T\dot{h}\left(\varvec{\theta }_0\right) \left(\dot{h}^T\left(\varvec{\theta }_0\right)I^{-1} \left(\varvec{\theta }_0\right)\dot{h} \left(\varvec{\theta }_0\right)\right)^{-1} \dot{h}^T\left(\varvec{\theta }_0\right)\varvec{Z}=A_3.\nonumber \\ \end{aligned}$$

(62)

Thus, we can derive that, together with (57) and (60),

$$\begin{aligned} -D(\widetilde{\varvec{\beta }})=\inf _{\gamma \in \mathbb{R }^+}(X-\gamma )^TQ(\varvec{\theta }_0)(X-\gamma ). \end{aligned}$$

(63)

Consequently, $-D(\widetilde{\varvec{\beta }})\sim \bar{\chi }^2(Q^{-1}(\varvec{\theta }_0),(\mathbb{R }^+)^0)$, where $(\mathbb{R }^+)^0$ is the polar cone of $\mathbb{R }^+$. In other word,

$$\begin{aligned} Pr\{-D(\widetilde{\varvec{\beta }})\ge c\}=w_0Pr\left\{ \chi ^2_0\ge c\right\} +w_1Pr\left\{ \chi ^2_1\ge c\right\} , \end{aligned}$$

(64)

where the $\chi ^2_0$ is a degenerate distribution with all its probability mass at zero and $\chi ^2_1$ is a $\chi ^2$ distribution with one degree of freedom. According to Shapiro (1988) and Barmi and Dykstra (1999), the exact form of the weights here are as follows:

$$\begin{aligned} w_0=w_1=Pr\{X_\ge 0\}, \end{aligned}$$

(65)

where $X\sim N(0,Q^{-1}(\varvec{\theta }_0))$. Then it can be obviously found that $w_0=w_1=\frac{1}{2}$. As a result, the asymptotical behavior of the test statistic $L_{q_n}$R under the null hypothesis of (4) is asymptotically $\frac{1}{2}\chi ^2_0+\frac{1}{2}\chi ^2_1$.