Statistical inference for the functional quadratic quantile regression model

Abstract

In this paper, we develop statistical inference procedures for functional quadratic quantile regression model in which the response is a scalar and the predictor is a random function defined on a compact set of R. The functional coefficients are estimated by functional principal components. The asymptotic properties of the resulting estimators are established under mild conditions. In order to test the significance of the nonlinear term in the model, we propose a rank score test procedure. The asymptotic properties of the proposed test statistic are established. The proposed method provides a highly efficient and robust alternative to the least squares method, and can be conveniently implemented using existing R software package. Finally, we examine the performance of the proposed method for finite sample sizes by Monte Carlo simulation studies and illustrate it with a real data example.

Appendix

In this section, we present the proofs of the results stated in Sect. 3. In what follows, for two sequences of positive numbers $a_n$ and $b_n$, $a_n\lesssim b_n$ satisfies that $a_n/b_n$ is uniformly bounded and $a_n \asymp b_n$ if $a_n \lesssim b_n$ and $b_n\lesssim a_n$. Let C denote a positive constant which might take a different value at the different place. First, we give the following three lemmas which are useful to prove the main results.

Lemma 1

Let $\eta _1$ and $\eta _n$ be the smallest and largest eigenvalues of $\frac{1}{n}{\varvec{X}}{\varvec{X}}^T.$ Under conditions C1–C6, we have $\eta _1=\hat{\lambda }_m, \eta _n=\hat{\lambda }_1.$

Lemma 2

Let $\tau _1$ and $\tau _n$ be the smallest and largest eigenvalues of $\frac{1}{n}{\varvec{Z}}{\varvec{Z}}^T.$ Under conditions C1–C6, we have $\tau _1=\lambda _m^2+o_p(1), \tau _n=\lambda _1^2+o_p(1),$

The proof of Lemmas 1 and 2 is similar to the proof of Lemma 2 in Yu et al. (2017).

Lemma 3

Let

$$\begin{aligned} R_{i}= & {} \sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j} +\sum _{k=1}^m\sum _{l=1}^m\varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle \\&-\int _{0}^{1}\beta (t)X_i(t)dt-\int _{0}^{1}\int _{0}^{1}\varphi (t,s)X_i(t)X_i(s)dtds. \end{aligned}$$

Under conditions C1–C4 and $C6'$, we have $\Vert R_{i}\Vert ^2=O_p(\delta _n^2),$ where

$$\begin{aligned} \delta _n^2=n^{-\frac{2b-1}{a+2b}}+\sum \limits _{k=1}^m\sum \limits _{l=m+1}^\infty \varphi ^2_{kl}+\sum _{k=1}^m\sum \limits _{l=1}^m \varphi ^2_{kl} \frac{k^2}{n}+\sum \limits _{k=m+1}^\infty \sum \limits _{l=m+1}^\infty \varphi ^2_{kl}. \end{aligned}$$

Proof of Lemma 3

Invoking the fact $\Vert \phi _{j}-{\hat{\phi }}_{j}\Vert ^2=O_p(n^{-1}j^2)$ and (4)–(6), one has

$$\begin{aligned} \Vert R_{i}\Vert ^{2}= & {} \left\| \sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j} +\sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle \right. \\&\left. -\int _{0}^{1}\beta (t)X_i(t)dt-\int _{0}^{1}\int _{0}^{1}\varphi (t,s)X_i(t)X_i(s)dtds\right\| ^{2}\\\le & {} \Vert R_{i1}\Vert ^{2}+\Vert R_{i2}\Vert ^{2}. \end{aligned}$$

where

$$\begin{aligned} R_{i1}=\sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j}-\int _{0}^{1}\beta (t)X_i(t)dt, \end{aligned}$$

and

$$\begin{aligned} R_{i2}=\sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\int _{0}^{1}\int _{0}^{1}\varphi (t,s)X_i(t)X_i(s)dtds. \end{aligned}$$

First, we consider $R_{i1}.$

$$\begin{aligned} \Vert R_{i1}\Vert ^{2}= & {} \left\| \sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j}-\int _{0}^{1}\beta (t)X_i(t)dt\right\| ^2\\= & {} \left\| \sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j}-\sum _{j=1}^{\infty }\langle X_{i},\phi _{j}\rangle \beta _{j}\right\| ^{2}\\\le & {} \left\| \sum _{j=1}^{m}\langle X_{i},\hat{\phi }_{j}\rangle \beta _{j}-\sum _{j=1}^{m}\langle X_{i},\phi _{j}\rangle \beta _{j}\right\| ^{2} +\left\| \sum _{j=m+1}^{\infty }\langle X_{i},\phi _{j}\rangle \beta _{j}\right\| ^{2}\\\le & {} \sum _{j=1}^{m}\left\| \hat{\phi }_{j}-\phi _{j}\right\| ^{2}|\beta _{j}|^{2}+\sum _{j=m+1}^{\infty }|\beta _{j}|^{2}\\= & {} \sum _{j=1}^{m}O_{p}\left( n^{-1} j^{2-2b}\right) +\sum _{j=m+1}^{\infty }j^{-2b}\\= & {} O_{p_{}}\left( n^{-\frac{2b-1}{a+2b}}\right) +O\left( n^{-\frac{2b-1}{a+2b}}\right) \\= & {} O_{p}\left( n^{-\frac{2b-1}{a+2b}}\right) . \end{aligned}$$

Next, we consider $R_{i2}$.

$$\begin{aligned} \Vert R_{i2}\Vert ^{2}= & {} \left\| \sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\int _{0}^{1}\int _{0}^{1}\varphi (t,s)X_i(t)X_i(s)dtds\right\| ^{2}\\= & {} \left\| \sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\sum _{k=1}^\infty \sum _{l=1}^\infty \varphi _{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\= & {} \left\| \sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\sum _{k=1}^m\sum _{l=1}^m \varphi _{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\&+\left\| \sum _{k=1}^m\sum _{l=m+1}^\infty \varphi _{kl}\langle X_{i},{\phi }_{l}\rangle \langle X_{i},{\phi }_{k}\rangle \right\| ^{2} +\left\| \sum _{k=m+1}^\infty \sum _{l=1}^m \varphi _{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\&+\left\| \sum _{k=m+1}^\infty \sum _{l= m+1}^\infty \varphi _{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\= & {} R_{i21}+R_{i22}+R_{i23}+R_{i24}. \\ R_{i21}= & {} \left\| \sum _{k=1}^m\sum _{l=1}^m\varphi _{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\sum _{k=1}^m\sum _{l=1}^m{\varphi }_{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\= & {} \left\| \sum _{k=1}^m\sum _{l=1}^m{\varphi }_{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},\hat{\phi }_{k}\rangle -\sum _{k=1}^m\sum _{l=1}^m{\varphi }_{kl}\langle X_{i}, \hat{\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\&+\left\| \sum _{k=1}^m\sum _{l=1}^m{\varphi }_{kl}\langle X_{i},\hat{\phi }_{l}\rangle \langle X_{i},{\phi }_{k}\rangle -\sum _{k=1}^m\sum _{l=1}^m{\varphi }_{kl}\langle X_{i}, {\phi }_{l}\rangle \langle X_{i}, {\phi }_{k}\rangle \right\| ^{2}\\\le & {} \sum _{k=1}^m\sum _{l=1}^m{\varphi }^2_{kl}\Vert \langle X_{i}, \hat{\phi }_{l}\rangle \Vert ^2 \Vert \langle X_{i},\hat{\phi }_{k}\rangle -\langle X_{i}, {\phi }_{k}\rangle \Vert ^{2}\\&+\sum _{k=1}^m\sum _{l=1}^m{\varphi }^2_{kl}\Vert \langle X_{i}, {\phi }_{k}\rangle \Vert ^2 \Vert \langle X_{i},\hat{\phi }_{l}\rangle -\langle X_{i}, {\phi }_{l}\rangle \Vert ^{2}\\= & {} 2\sum _{k=1}^m\sum \limits _{l=1}^m{\varphi }^2_{kl} \frac{k^2}{n}. \end{aligned}$$

Thus $R_{i21}=O_p\left( \sum _{k=1}^m\sum _{l=1}^m \varphi ^2_{kl} \frac{k^2}{n}\right)$. Similarly, $R_{i22}=R_{i23}=O_p\left( \sum _{k=1}^m\right. \left. \sum _{l=m+1}^\infty {\varphi }^2_{kl}\right) ,$ and $R_{i24}=O_p\left( \sum _{k=m+1}^\infty \sum _{l=m+1}^\infty {\varphi }^2_{kl}\right) .$ Therefore, we have

$$\begin{aligned} \Vert R_{i2}\Vert ^2=O_p\left( \sum \limits _{k=1}^m\sum \limits _{l=m+1}^\infty {\varphi }^2_{kl}\right) +O_p\left( \sum _{k=1}^m\sum \limits _{l=1}^m{\varphi }^2_{kl} \frac{k^2}{n}\right) +O_p\left( \sum \limits _{k=m+1}^\infty \sum \limits _{l=m+1}^\infty {\varphi }^2_{kl}\right) . \end{aligned}$$

According the convergence of rate of $R_{i1}$ and $R_{i2}$, one has $\Vert R_{i}\Vert ^2=O_{p}\left( \delta _n^2\right) .$$\square$

Lemma 4

Under $H_0$ and conditions C1–C5 and $C6'$, we have

$$\begin{aligned} \sup \limits _{\Vert {\hat{\beta }}-\beta _0\Vert< n^{-\frac{ b-a}{a+2b} } ,\Vert {\hat{\mu }}-\mu _0\Vert < n^{-\frac{ b-a}{a+2b} }} \Vert {\varvec{V}}_n^*-{\varvec{V}}_n\Vert =o_p(n), \end{aligned}$$

(15)

where ${\varvec{V}}_n^*=(1-\tau )\tau {\varvec{H}}^T {\varvec{H}}.$

The proof of Lemma 4 is similar to these of He et al. (2002).

Lemma 5

Under $H_0$ and conditions C1–C5 and $C6'$, we have

$$\begin{aligned} \sup \limits _{\Vert {\hat{\beta }}-\beta _0\Vert< n^{-\frac{ b-a}{a+2b} } ,\Vert {\hat{\mu }}-\mu _0\Vert < n^{-\frac{ b-a}{a+2b} }} \Vert {\varvec{S}}_n^*-{\varvec{S}}_n\Vert =o_p(n^{1/2}), \end{aligned}$$

where ${\varvec{S}}_n^*=(I-{\varvec{P}}){\varvec{Z}} {\varvec{\psi }}({\varvec{\varepsilon }}).$

The proof of Lemma 5 is similar to the Theorem 4.1 in Wei and He (2006).

Proof of Theorem 1

Denote ${\varvec{P}}_Z= {\varvec{Z}}({\varvec{Z}}^T{\varvec{Z}})^{-1}{\varvec{Z}}^T, {\varvec{X}}^*=({\varvec{I}}-{\varvec{P}}_Z){\varvec{X}}$, ${\varvec{X}}^* =({\varvec{X}}^*_1,\dots ,{\varvec{X}}^*_n)^T$, ${\varvec{S}}_n={{\varvec{X}}^*}^T{\varvec{X}}^*$, ${\varvec{\beta }}=(\beta _1,\dots ,\beta _m)^T$, ${\varvec{\gamma }}=\text {vech}(\{\varphi _{kl}(2-\delta _{kl}),1\le k \le l\le m \}^T).$

Let

$$\begin{aligned} {\varvec{\theta }}\left( \begin{array}{c} \mu \\ {\varvec{\beta }} \\ {\varvec{\gamma }} \end{array} \right) =\left( \begin{array}{c} \theta _{1}\\ {\varvec{\theta }}_{2}\\ {\varvec{\theta }}_{3} \end{array} \right) =\left( \begin{array}{c} \sqrt{n}(\mu -\mu _0)\\ {\varvec{S}}_{n}^{\frac{1}{2}}({\varvec{\beta }}-{\varvec{\beta }}_{0}) \\ {\varvec{H}}_{m}^{\frac{1}{2}}({\varvec{\gamma }}-{\varvec{\gamma }}_{0})+{\varvec{H}}_{m}^{-\frac{1}{2}}{\varvec{Z}}{\varvec{X}}(\beta -\beta _{0}) \end{array} \right) , \end{aligned}$$

where ${\varvec{H}}_{m}={\varvec{Z}}^{T}{\varvec{Z}}$. Let $\hat{{\varvec{\theta }}}={\varvec{\theta }}({\hat{\mu }},\hat{{\varvec{\beta }}},\hat{{\varvec{\gamma }}})=({\hat{\theta }}_1,\hat{{\varvec{\theta }}}_{1}^{T},\hat{{\varvec{\theta }}}_{2}^{T})^{T}.$ Notice that the suffix 0 here means the true value of the parameter, and for simplicity we omit it where there is no misunderstanding. Now we show $\Vert \hat{{\varvec{\theta }}}\Vert =O_{p}( \frac{1}{\lambda _m})$. To do so, we standardize $\tilde{{\varvec{X}}}_{i}={\varvec{S}}_n^{-\frac{1}{2}} {\varvec{X}}_i^*$, $\tilde{{\varvec{Z}}}_{i}={\varvec{H}}_m^{-\frac{1}{2}} {\varvec{Z}}_i$, and let $R_i$ be defined in Lemma 3, then one has

$$\begin{aligned}&\sum _{i=1}^n\rho _\tau \left( Y_i-\mu -\sum _{k=1}^m {\hat{x}}_{ik}\beta _k-\sum _{k=1}^m\sum _{l=k}^m(2-\delta _{kl})\varphi _{kl}{\hat{x}}_{ik}{\hat{x}}_{il}\right) \\&\quad =\sum _{i=1}^n\rho _\tau \left( Y_i-\mu -{\varvec{X}}_i^T{\varvec{\beta }}-{\varvec{Z}}_i^T{\varvec{\gamma }}\right) \\&\quad =\sum _{i=1}^n\rho _\tau \left( \varepsilon _i-\theta _1/\sqrt{n}-\tilde{{\varvec{X}}}_{i}^{T}{\varvec{\theta }}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\theta }}_{3}-R_{i})\right) \end{aligned}$$

which is minimized at $\hat{{\varvec{\theta }}}$.

Invoking Lemmas 1 and 2, for any $\kappa >0$, similar arguments to these of Lemma 1 of Cardot et al. (2005), there exists $L_{\kappa }$ such that

$$\begin{aligned}&P\left\{ \inf \limits _{\Vert {\varvec{\theta }}\Vert>L_{\kappa }\lambda _m}\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _i-\theta _1/\sqrt{n}-\tilde{{\varvec{X}}}_{i}^{T}{\varvec{\theta }}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\theta }}_{3}-R_{i}\right)>\sum _{i=1}^{n}\rho _{\tau }(\varepsilon _{i} -R_{i})\right\} \\&\quad >1-\kappa . \end{aligned}$$

On the other hand, we have

$$\begin{aligned}&\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i}-{\hat{\theta }}_1/\sqrt{n} -\tilde{{\varvec{X}}}_{i}^{T}\hat{{\varvec{\theta }}}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\hat{\theta }}}_{3}-R_{i}\right) \nonumber \\&\quad =\inf \limits _{{\varvec{\theta }}\in R^{m(m+1)/2}} \sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _i-\theta _1/\sqrt{n}-\tilde{{\varvec{X}}}_{i}^{T}{\varvec{\theta }}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\theta }}_{3}-R_{i}\right) . \end{aligned}$$

(16)

Thus, we have

$$\begin{aligned} \sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i}-{\hat{\theta }}_1/\sqrt{n} -\tilde{{\varvec{X}}}_{i}^{T}\hat{{\varvec{\theta }}}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\hat{\theta }}}_{3}-R_{i}\right) < \sum _{i=1}^{n}\rho _{\tau }(\varepsilon _i-R_{i}). \end{aligned}$$

Combining Eq. (16), one has

$$\begin{aligned}&P\left\{ \inf \limits _{\Vert {\varvec{\theta }}\Vert>\frac{L_{\kappa }}{\lambda _{m}}}\sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _i-\theta _1/\sqrt{n}-\tilde{{\varvec{X}}}_{i}^{T}{\varvec{\theta }}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\theta }}_{3}-R_{i}\right) \right. \\&\quad \left.> \sum _{i=1}^{n}\rho _{\tau }\left( \varepsilon _{i}-{\hat{\theta }}_1/\sqrt{n} -\tilde{{\varvec{X}}}_{i}^{T}\hat{{\varvec{\theta }}}_{2}-\tilde{{\varvec{Z}}}_{i}^{T}{\varvec{\hat{\theta }}}_{3}-R_{i}\right) \right\} >1-\kappa , \end{aligned}$$

Thus, $\Vert \hat{{\varvec{\theta }}}\Vert =O_{p}\left( \frac{1}{\lambda _m}\right) .$ By the definition of ${\varvec{\theta }}$, we have $\Vert \hat{\mu }-\mu _0\Vert =O_{p}\left( {\frac{\lambda _m}{\sqrt{n}}} \right) .$

Combining Lemma 1 and the definition of $\hat{{\varvec{\theta }}}$, one has

$$\begin{aligned} \Vert \hat{{\varvec{\beta }}}-{\varvec{\beta }}_{0}\Vert =\left\| {\varvec{S}}_{n}^{-\frac{1}{2}}\hat{{\varvec{\theta }}}_{2}\right\| =O_{p}\left( (\lambda _m n)^{-\frac{1}{2}}\Vert \hat{{\varvec{\theta }}}_{2}\Vert \right) =O_{p}\left( {\frac{1}{ \sqrt{\lambda _m^3 n}}} \right) . \end{aligned}$$

By Lemmas 1 and 2, we have

$$\begin{aligned} \Vert \hat{{\varvec{\gamma }}}-{\varvec{\gamma }}_{0}\Vert \le \frac{1}{\lambda _m\sqrt{n}}\left\| {\hat{ {\varvec{\theta }}}}_{3}\right\| +\frac{1}{\lambda _m\sqrt{n}}\left\| {\varvec{H}}_{m}^{-\frac{1}{2}}{\varvec{Z}}{\varvec{X}}(\hat{{\varvec{\beta }}}-{\varvec{\beta }}_{0})\right\| =O_{p}\left( {\frac{1}{ \lambda _m^2\sqrt{n}}} \right) . \end{aligned}$$

Now, we consider the convergence rate of the proposed estimators ${\hat{\beta }}(t)$ and ${\hat{\varphi }}(t,s).$

$$\begin{aligned} \Vert \hat{\beta } -\beta _{0} \Vert ^{2}= & {} \left\| \sum _{j=1}^{m}\hat{\beta }_{j}\hat{\phi }_{j}-\sum _{j=1}^{\infty }\beta _{j0}\phi _{j}\right\| ^{2}\\\le & {} 2\left\| \sum _{j=1}^{m}\hat{\beta }_{j}\hat{\phi }_{j}-\sum _{j=1}^{m}\beta _{j0}\phi _{j}\right\| ^{2} +2\left\| \sum _{j=m+1}^{\infty }\beta _{j0}\phi _{j}\right\| ^{2}\\\le & {} 4\left\| \sum _{j=1}^{m}(\hat{\beta }_{j}-\beta _{j0})\hat{\phi }_{j}\right\| ^{2} +4\left\| \sum _{j=1}^{m}\beta _{j0}(\hat{\phi }_{j}-\phi _{j})\right\| ^{2} +2\sum _{j=m+1}^{\infty }\beta _{j0}^{2}\\= & {} R_{n1}+R_{n2}+R_{n3}. \end{aligned}$$

Now we consider $R_{n1}$, by the fact that the sequences $\{{\hat{\phi }}_j\}$ forms an orthonormal basis in $L^2([0, 1])$, one has

$$\begin{aligned} R_{n1}= & {} \left\| \sum _{j=1}^{m}(\hat{\beta }_{j}-\beta _{j0})\hat{\phi }_{j}\right\| ^{2}\\\le & {} \sum _{j=1}^{m}(\hat{\beta }_{j}-\beta _{j0})^{2}\\\le & {} \Vert \hat{{\varvec{\beta }}} -{{\varvec{\beta }}_0}\Vert ^{2}. \end{aligned}$$

Next we consider $R_{n2}$ and $R_{n3}$,

$$\begin{aligned} R_{n2}\le & {} m\sum _{j=1}^{m}\Vert \hat{\phi }_{j}-\phi _{j}\Vert ^{2}\beta _{j0}^{2}\le n^{-1}m\sum _{j=1}^{m}j^{2}\beta _{j0}^{2}\\= & {} O_{p}\left( n^{-1}m\sum _{j=1}^{m}j^{2-2b}\right) =O_{p}\left( n^{-\frac{a+3b-1}{a+2b}}\right) , \end{aligned}$$

and

$$\begin{aligned} R_{n3}= 2\sum _{j=m+1}^{\infty }\beta _{j0}^{2}\le C\sum _{j=m+1}^{\infty }j^{-2b}=O\left( n^{-\frac{2b-1}{a+2b}}\right) . \end{aligned}$$

Therefore, one has

$$\begin{aligned} \Vert \hat{\beta }-\beta _0\Vert ^{2}=O_{p}\left( n^{-\frac{2(b-a)}{a+2b}}\right) . \end{aligned}$$

Note that

$$\begin{aligned} \Vert \hat{\varphi } -\varphi _{0} \Vert ^{2}= & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}\hat{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(t)- \sum _{k=1}^\infty \sum _{l=1}^{\infty }{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(t)\right) ^{2}dtds\\= & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}\hat{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(t)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(t)\right) ^{2}dtds\\&+\int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=m+1}^{\infty }{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(t)\right) ^{2}dtds\\&+\int _0^1\int _0^1\left( \sum _{k=m+1}^\infty \sum _{l=1}^{m}{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(t)\right) ^{2}dtds\\&+\int _0^1\int _0^1\left( \sum _{k=m+1}^\infty \sum _{l=m+1}^{\infty }{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(t)\right) ^{2}dtds\\= & {} T_{n1}+T_{n2}+T_{n3}+T_{n4}. \end{aligned}$$

Firstly, we consider $T_{n1}.$

$$\begin{aligned} T_{n1}= & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}\hat{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(s)\right) ^{2}dtds\\\le & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}\hat{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)\right) ^{2}dtds \\&+\int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t){\phi }_{l}(s)\right) ^{2}dtds \\&+\int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t){\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}{\phi }_{k}(t){\phi }_{l}(s)\right) ^{2}dtds \\= & {} T_{n11}+T_{n12}+T_{n13} \end{aligned}$$

For $T_{n11},$ by the orthogonality of $\{\hat{\phi }_{k}\},$ we have

$$\begin{aligned} T_{n11}= & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}\hat{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)\right) ^{2}dtds\\= & {} \sum _{k=1}^m\sum _{l=1}^{m}(\hat{\varphi }_{kl}-{\varphi }_{kl})^2\\= & {} \Vert \hat{{\varvec{\gamma }}}-\gamma \Vert ^2\\= & {} O_p\left( n^{-\frac{(2b-3a)}{a+2b}}\right) \end{aligned}$$

For $T_{n12},$ according to the Cauchy–Schwarz inequality, we have

$$\begin{aligned} T_{n12}= & {} \int _0^1\int _0^1\left( \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t)\hat{\phi }_{l}(s)- \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}\hat{\phi }_{k}(t){\phi }_{l}(s)\right) ^{2}dtds \\= & {} \sum _{k=1}^m\int _0^1\left( \sum _{l=1}^{m}{\varphi }_{kl}(\hat{\phi }_{l}(s)- {\phi }_{l}(s))\right) ^{2}ds \\\le & {} \sum _{k=1}^m\sum _{l=1}^{m}{\varphi }_{kl}^2\sum _{r=1}^{m}\int _0^1 \left( \hat{\phi }_{r}(s)- {\phi }_{r}(s)\right) ^{2}ds \\= & {} O_p\left( \frac{m^3}{n} \right) \end{aligned}$$

Similar to $T_{n12},$ we have $T_{n12}=O_p\left( \frac{m^3}{n} \right) .$ Therefore, we conclude that

$$\begin{aligned} T_{n1}=O_p\left( \frac{m^3}{n} \right) +O_p\left( n^{-\frac{(2b-3a)}{a+2b}}\right) =O_p\left( n^{-\frac{(2b-3a)}{a+2b}}\right) . \end{aligned}$$

Via some elementary but tedious algebra, we can conclude that

$$\begin{aligned} T_{n2}= & {} O_p\left( \sum _{k=1}^m\sum _{l=m+1}^{\infty }{\varphi }_{kl}^2 \right) . \\ T_{n3}= & {} O_p\left( \sum _{k=m+1}^\infty \sum _{l=1}^{m}{\varphi }_{kl}^2 \right) . \\ T_{n4}= & {} O_p\left( \sum _{k=m+1}^\infty \sum _{l=m+1}^{\infty }{\varphi }_{kl}^2 \right) . \end{aligned}$$

Thus, we have

$$\begin{aligned} \Vert \hat{\varphi } -\varphi _{0} \Vert ^{2}= & {} O_p\left( n^{-\frac{(2b-3a)}{a+2b}}\right) +O_p\left( \sum _{k=1}^m\sum _{l=m+1}^{\infty }{\varphi }_{kl}^2 \right) +O_p\left( \sum _{k=m+1}^\infty \sum _{l=1}^{m}{\varphi }_{kl}^2 \right) \\&+O_p\left( \sum _{k=m+1}^\infty \sum _{l=m+1}^{\infty }{\varphi }_{kl}^2 \right) . \end{aligned}$$

Next we will show the asymptotic normality of $\hat{ \theta }_1$. Let $\theta _{1}^{*}=\frac{1}{f(0)\sqrt{n}} \sum _{i=0}^{n} \psi _{\tau }(\varepsilon _{i})$, according to the Lindeberg–Feller theorem, $\theta _{1}^{*}$ is asymptotically normal with variance-covariance $\frac{\tau (1-\tau )}{f^{2}(0)}$.

On the other hand, following arguments similar to those used in He and Shi (1996), we can establish that $\Vert \theta _{1}^{*}-\hat{ \theta }_{1}\Vert =o_{p}(1).$ Combining this, we conclude that

$$\begin{aligned} \sqrt{n}(\hat{\mu } -\mu _{0})\rightarrow N\left( 0,\frac{\tau (1-\tau )}{f^{2}(0)}\right) . \end{aligned}$$

This completes the proof of Theorem 1. $\square$

Proof of Theorem 2

We first define

$$\begin{aligned} T_n^*={{\varvec{S}}_n^*}^T{{\varvec{V}}_n^*}^{-1}{\varvec{S}}_n^*. \end{aligned}$$

By the central limit theorem, we have

$$\begin{aligned} A_n{{\varvec{V}}_n^*}^{-1/2}{\varvec{S}}_n^* {\mathop {\longrightarrow }\limits ^{L}}N(0,G) \end{aligned}$$

where $A_n$ is a $q\times m$ matrix such that $A_nA^T_n\rightarrow G$, and G is a $q\times q$ nonegative symmetric matrix. By the definition of the chi-squared distribution, one has

$$\begin{aligned} {{\varvec{S}}_n^*}^T{{\varvec{V}}_n^*}^{-1/2}{A_n}^{1/2}G^{-1}A_n{{\varvec{V}}_n^*}^{-1/2}{\varvec{S}}_n^* {\mathop {\longrightarrow }\limits ^{L}}\chi ^2(q), \end{aligned}$$

(17)

Combining (17) with the continuous mapping theorem, one has

$$\begin{aligned} \frac{T_n^*-\frac{m(m+1)}{2}}{\sqrt{m(m+1)}}{\mathop {\longrightarrow }\limits ^{L}}N(0,1). \end{aligned}$$

(18)

According to Lemmas 4 and 5, we have

$$\begin{aligned} \Vert T_n^*-T_n\Vert =o_p(1). \end{aligned}$$

Combining this with (18), we can conclude that

$$\begin{aligned} \frac{T_n-\frac{m(m+1)}{2}}{\sqrt{m(m+1)}}{\mathop {\longrightarrow }\limits ^{L}}N(0,1). \end{aligned}$$

$\square$