Penalized Lq-likelihood estimator and its influence function in generalized linear models

Abstract

Consider the following generalized linear model (GLM)

$$\begin{aligned} y_i=h(x_i^T\beta )+e_i,\quad i=1,2,\ldots ,n, \end{aligned}$$

where h(.) is a continuous differentiable function, $\{e_i\}$ are independent identically distributed (i.i.d.) random variables with zero mean and known variance $\sigma ^2$. Based on the penalized Lq-likelihood method of linear regression models, we apply the method to the GLM, and also investigate Oracle properties of the penalized Lq-likelihood estimator (PLqE). In order to show the robustness of the PLqE, we discuss influence function of the PLqE. Simulation results support the validity of our approach. Furthermore, it is shown that the PLqE is robust, while the penalized maximum likelihood estimator is not.

7. Appendix: Proofs of Lemmas and Theorems

In this section, we provide some proofs of technical lemmas and Theorem 3.1.

Proof of Lemma 3.1

We only prove that for any given $\varepsilon >0$, there exists a large constant C such that

$$\begin{aligned} P\left\{ \sup _{\Vert u\Vert =C}Q(\beta _0+\alpha _nu)<Q(\beta _0)\right\} \ge 1-\varepsilon . \end{aligned}$$

(7.1)

The case of $q_n=1$, the proof of (7.1) is similar to Theorem 2.1 of Wang and Tian (2019) or Wang and Wang (2014). In the following, we only consider the case of $q_n\ne 1$. Note that $p_{\lambda _n}(\cdot )\ge 0$ and $p_{\lambda _n}(0)=0$, and by Taylor expansion, we have

$$\begin{aligned} \begin{array}{lll} D_n(u)&{}=&{}Q(\beta _0+\alpha _nu)-Q(\beta _0)\\ &{}=&{}\sum \limits _{i=1}^n\left( L_{q_n}f(y_i-h(x_i^T(\beta _0+\alpha _nu)))-L_{q_n}f(y_i-h(x_i^T\beta _0)))\right) \\ &{}&{}-n\sum \limits _{j=1}^d\left( p_{\lambda _n}(\beta _{j0}+\alpha _nu_j)-p_{\lambda _n}(\beta _{j0})\right) \\ &{}\le &{}\frac{1}{1-q_n}\sum \limits _{i=1}^n\left( f^{1-q_n}(y_i-h(x_i^T(\beta _0+\alpha _nu)))-f^{1-q_n}(y_i-h(x_i^T\beta _0)))\right) \\ &{}&{}-n\sum \limits _{j=1}^s\left( p_{\lambda _n}(\beta _{j0}+\alpha _nu_j)-p_{\lambda _n}(\beta _{j0})\right) \\ &{}=&{}-\sum \limits _{i=1}^n\{-f^{-q_n}(y_i-h(x_i^T\beta _0))f'(y_i-h(x_i^T\beta _0))\left( h(x_i^T(\beta _0+\alpha _nu))-h(x_i^T\beta _0)\right) \\ &{}&{}-\frac{1}{2}\left( h(x_i^T(\beta _0+\alpha _nu))-h(x_i^T\beta _0)\right) ^2f^{-1-q_n}(y_i-{\tilde{h}}(x_i^T\beta _0))\\ &{}&{}\cdot [f(y_i-{\tilde{h}}(x_i^T\beta _0))f''(y_i-{\tilde{h}}(x_i^T\beta _0))-q_nf'^2(y_i-{\tilde{h}}(x_i^T\beta _0))]\}\\ &{}&{}-n\sum \limits _{j=1}^s\left( p'_{\lambda _n}(\beta _{j0})\alpha _nu_j+\frac{1}{2}p''_{\lambda _n}({\tilde{\beta }}_j)(\alpha _nu_j)^2\right) \\ &{}=&{}\sum \limits _{i=1}^n\{f^{-q_n}(y_i-h(x_i^T\beta _0))f'(y_i-h(x_i^T\beta _0))h'(x_i^T\beta ^*)x_i^T\alpha _nu\\ &{}&{}-\frac{1}{2}h'^2(x_i^T\beta ^*)(x_i^T\alpha _nu)^2f^{-1-q_n}(y_i-{\tilde{h}}(x_i^T\beta _0))\\ &{}&{}\cdot [f(y_i-{\tilde{h}}(x_i^T\beta _0))f''(y_i-{\tilde{h}}(x_i^T\beta _0))-q_nf'^2(y_i-{\tilde{h}}(x_i^T\beta _0))]\}\\ &{}&{}-n\sum \limits _{j=1}^s\left( p'_{\lambda _n}(\beta _{j0})\alpha _nu_j+\frac{1}{2}p''_{\lambda _n}({\tilde{\beta }}_j)(\alpha _nu_j)^2\right) \\ &{}=&{}\alpha _n\sum \limits _{i=1}^nf^{-q_n}(y_i-h(x_i^T\beta _0))f'(y_i-h(x_i^T\beta _0))h'(x_i^T\beta ^*)x_i^Tu\\ &{}&{}-\frac{1}{2}n\alpha ^2_nu^TI(\beta ^*)u(1+o_P(1))\\ &{}&{}-n\sum \limits _{j=1}^s\left( p'_{\lambda _n}(\beta _{j0})\alpha _nu_j+\frac{1}{2}p''_{\lambda _n}({\tilde{\beta }}_j)(\alpha _nu_j)^2\right) \\ &{}=&{}T_1+T_2+T_3, \end{array} \end{aligned}$$

(7.2)

where ${\tilde{h}}(x_i^T\beta _0)$ is between $h(x_i^T(\beta _0+\alpha _nu))$ and $h(x_i^T\beta _0)$, ${\tilde{\beta }}$ and $\beta ^*$ is between $\beta _0$ and $\beta _0+\alpha _nu$.

Note that $E(f^{-q_n}(e_i)f'(e_i))^2<\infty$. By the condition (A) and weak large number theorem, we easily obtain

$$\begin{aligned} \begin{array}{lll} T_1&{}=&{}-\alpha _n\sum \limits _{i=1}^nf^{-q_n}(e_i)f'(e_i)h'(x_i^T{\tilde{\beta }}_0)x_i^Tu\\ &{}&{} \rightarrow -\alpha _nE\left( f^{-q_n}(e_i)f'(e_i)\right) \sum \limits _{i=1}^nh'(x_i^T{\tilde{\beta }}_0)x_i^Tu=0. \end{array} \end{aligned}$$

(7.3)

Since f, $f'$, $f''$ and h are continuous, we have

$$\begin{aligned} T_2= & {} -\frac{1}{2}n\alpha _n^2u^TI(\beta _0)u(1+o_P(1))=O_P(n\alpha _n^2\Vert u\Vert ^2). \end{aligned}$$

(7.4)

$$\begin{aligned} T_3= & {} -\left( O(n\alpha _n\Vert u\Vert )+o(n\alpha _n^2\Vert u\Vert ^2)\right) . \end{aligned}$$

(7.5)

Thus, (7.1) follows from (7.2)–(7.5). $\square$

Proof of Lemma 3.2

It is sufficient to show, for $j=s+1,\ldots ,d$,

$$\begin{aligned} \frac{\partial Q(\beta )}{\partial \beta _j}:\left\{ \begin{array}{cc}<0,&{} 0<\beta _j<cn^{-\frac{1}{2}},\\ >0,&{} -cn^{-\frac{1}{2}}<\beta _j<0. \end{array} \right. \end{aligned}$$

(7.6)

By Taylor!‘$^{-}$s expansion, we have

$$\begin{aligned} \frac{\partial Q(\beta )}{\partial \beta _j}= & {} -\sum \limits _{i=1}^nf^{-q_n}(e_i)f'(e_i)h'(x_i^T\beta )x_{ij}-np'_{\lambda _n}(\beta _j) \nonumber \\= & {} -\sum \limits _{i=1}^nh'(x_i^T\beta _0)x_{ij}f^{-q_n}(e_{i0})f'(e_{i0})-np'_{\lambda _n}(\beta _j) \nonumber \\{} & {} +\sum \limits _{i=1}^nx_{ij}f^{-1-q_n}(e_{i0})\nonumber \\{} & {} \cdot \left( h'^2(x_i^T\beta _0)(f(e_{i0})f''(e_{i0})-q_nf'^2(e_{i0}))-h''^2(x_i^T\beta _0)f(e_{i0})f'(e_{i0})\right) \nonumber \\{} & {} \cdot \sum \limits _{l=1}^dx_{il}(\beta _l-\beta _{l0})\nonumber \\{} & {} -\sum \limits _{i=1}^nx_{ij}h''(x_i^T\beta _0)f^{-q_n}(e_{i0})f'(e_{i0})\sum \limits _{l=1}^dx_{il}(\beta _l-\beta _{l0})\nonumber \\{} & {} +\frac{1}{2}\sum \limits _{i=1}^n\frac{\partial ^3L_q(f(e_i))}{\partial \beta _j\partial \beta _l\partial \beta _k}|_{\beta =\beta ^*}x_{ij} \sum \limits _{l=1}^d\sum \limits _{k=1}^dx_{il}x_{ik}(\beta _l-\beta _{l0})(\beta _k-\beta _{k0})\nonumber \\= & {} T_1+T_2+T_3+T_4+T_5, \end{aligned}$$

(7.7)

where $e_{i0}=y_i-h(x_i^T\beta _0)$, $e^*_i=y_i-h(x_i^T\beta ^*)$, $\beta ^*$ is between $\beta _0$ and $\beta$. Observe that we have

$$\begin{aligned} T_1\rightarrow -E(f^{-q_n}(e_{i0})f'(e_{i0}))\sum _{i=1}^nh'(x_i^T\beta _0)x_{ij}=0, \end{aligned}$$

(7.8)

$$\begin{aligned} T_2=-n\lambda _n\cdot \frac{p'_{\lambda _n}(\beta _j)}{\lambda _n}, \end{aligned}$$

(7.9)

$$\begin{aligned} T_3\rightarrow I_{jl}\sum _{i=1}^nx_{ij}h'^2(x_i^T\beta _0)\sum _{l=1}^dx_{il}(\beta _l-\beta _{l0})=O_P(n^{\frac{1}{2}}), \end{aligned}$$

(7.10)

$$\begin{aligned} T_4\rightarrow E\left( f^{-q_n}(e_{i0})f'(e_{i0})\right) \sum _{i=1}^nx_{ij}h''(x_i^T\beta _0)\sum _{l=1}^dx_{il}(\beta _l-\beta _{l0})=0, \end{aligned}$$

(7.11)

$$\begin{aligned} T_5\le C\sum _{i=1}^nx_{ij}\sum _{l=1}^d\sum _{k=1}^dx_{il}x_{ik}(\beta _l-\beta _{l0})(\beta _k-\beta _{k0})=O(1). \end{aligned}$$

(7.12)

By $n^{-\frac{1}{2}}\lambda _n\rightarrow 0$, we have

$$\begin{aligned} \begin{array}{lll} \frac{\partial Q(\beta )}{\partial \beta _j}&{}=&{}n\lambda _n\left( -\frac{p'_{\lambda _n}(\beta _j)}{\lambda _n}-n^{-\frac{1}{2}}\lambda ^{-1}_n- O(1)(n\lambda _n)^{-1}+o_P(1)\right) \\ &{}=&{}n\lambda _n\left( -\frac{p'_{\lambda _n}(\beta _j)}{\lambda _n}+o_P(1)\right) . \end{array} \end{aligned}$$

(7.13)

Note that $\liminf _{n\rightarrow \infty }\liminf _{\theta \rightarrow 0^+}\frac{p'_{\lambda _n}(\theta )}{\lambda _n}>0$, and $\liminf _{n\rightarrow \infty }\liminf _{\theta \rightarrow 0^-}\frac{p'_{\lambda _n}(\theta )}{\lambda _n}<0$. Hence, (7.6) follows from (7.13). $\square$

Proof of Theorem 3.1

By Lemma 3.2, we only prove the asymptotic normality.

For $j=1,2,\ldots ,s$, we have

$$\begin{aligned} 0= & {} \frac{\partial Q(\beta )}{\partial \beta _j}|_{\beta =({\hat{\beta }}_1^T,0)^T} \nonumber \\= & {} -\sum \limits _{i=1}^nf^{-q_n}({\hat{e}}_i)f'({\hat{e}}_i)h'(x_i^T{\hat{\beta }})x_{ij}-np'_{\lambda _n}({\hat{\beta }}_j)\nonumber \\= & {} -\sum \limits _{i=1}^nh'(x_i^T\beta _0)x_{ij}f^{-q_n}(e_{i0})f'(e_{i0})\nonumber \\{} & {} +\sum \limits _{i=1}^nx_{ij}\{h'^2(x_i^T\beta ^*)f^{-1-q_n}(e^*_i)\left( f(e^*_i)f''(e^*_i)-q_nf'^2(e^*_i)\right) \nonumber \\{} & {} -h''(x_i^T\beta ^*)f^{-q_n}(e^*_i)f'(e^*_i)\}\sum \limits _{l=1}^dx_{il}({\hat{\beta }}_l-\beta _{l0})\nonumber \\{} & {} -n\left( p'_{\lambda _n}(\beta _{j0})+p''_{\lambda _n}(\beta ^*_j)({\hat{\beta }}_j-\beta _{j0})\right) . \end{aligned}$$

(7.14)

By (7.10), we have

$$\begin{aligned}{} & {} \sqrt{n}({\hat{\beta }}_1-\beta _{10})\nonumber \\{} & {} \quad =-\left\{ (b+o_P(1))\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_i(x_{i1},x_{i2},\ldots ,x_{is})+ p''_{\lambda _n}(\beta ^*_1)\right\} ^{-1}\nonumber \\{} & {} \quad \cdot \left\{ n^{-\frac{1}{2}}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_if^{-q_n}(e_{i0})f'(e_{i0})+\sqrt{n}p'_{\lambda _n}(\beta _1)\right\} \nonumber \\{} & {} \quad =-\left\{ b\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_i(x_{i1},x_{i2},\ldots ,x_{is})+ p''_{\lambda _n}(\beta ^*_1)\right\} ^{-1}\nonumber \\{} & {} \quad \cdot \left\{ n^{-\frac{1}{2}}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_if^{-q_n}(e_{i0})f'(e_{i0})+\sqrt{n}p'_{\lambda _n}(\beta _1)\right\} +o_P(1).\nonumber \\ \end{aligned}$$

(7.15)

Taking $\forall u=(u_1,u_2,\ldots ,u_s)$ such that $\Vert u\Vert =1$, we will prove that

$$\begin{aligned} n^{-\frac{1}{2}}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_if^{-q_n}(e_{i0})f'(e_{i0}) \rightarrow N\left( 0,Var(f^{-q_n}(e_{i0})f'(e_{i0}))C_{11}\right) .\nonumber \\ \end{aligned}$$

(7.16)

Since $\left\{ \sum _{j=1}^su_jx_{ij}f^{-q_n}(e_{i0})f'(e_{i0}),\quad i=1,2,\ldots ,n \right\}$ are independent random variables, we have

$$\begin{aligned} Var\left( \sum \limits _{j=1}^su_jx_{ij}f^{-q_n}(e_{i0})f'(e_{i0})\right) =Var\left( f^{-q_n}(e_{i0})f'(e_{i0})\right) \sum \limits _{j=1}^s(u_jx_{ij})^2 \end{aligned}$$

and

$$\begin{aligned} \begin{array}{lll} B_n^2&{}=&{}\sum \limits _{i=1}^nh'^2(x_i^T\beta _0)Var\left( \sum \limits _{j=1}^su_jx_{ij}f^{-q_n}(e_{i0})f'(e_{i0})\right) \\ &{}=&{}Var\left( f^{-q_n}(e_{i0})f'(e_{i0})\right) \sum \limits _{i=1}^nh'^2(x_i^T\beta _0)\sum \limits _{j=1}^s(u_jx_{ij})^2 =O(n). \end{array} \end{aligned}$$

(7.17)

Note that

$$\begin{aligned} \begin{array}{lll} &{}&{}\sum \limits _{i=1}^nE\left| h'(x_i^T\beta _0)\sum \limits _{j=1}^su_jx_{ij}f^{-q_n}(e_{i0})f'(e_{i0})\right| ^v\\ &{}&{}\quad \le C\sum \limits _{i=1}^n|h'(x_i^T\beta _0)|^v\left| \sum \limits _{j=1}^su_jx_{ij}\right| ^vE\left| f^{-q_n}(e_{i0})f'(e_{i0})\right| ^v\\ &{}&{} \quad =CE\left| f^{-q_n}(e_{i0})f'(e_{i0})\right| ^v\sum \limits _{i=1}^n|h'(x_i^T\beta _0)|^v \left| \sum \limits _{j=1}^su_jx_{ij}\right| ^v\\ &{}&{}\quad \le Cn=o(B_n^v). \end{array} \nonumber \\ \end{aligned}$$

(7.18)

By the central limit theorem, we have

$$\begin{aligned} \begin{array}{lll} &{}&{}\sum \limits _{i=1}^nh'(x_i^T\beta _0)\sum \limits _{j=1}^su_jx_{ij}f^{-q_n}(e_{i0})f'(e_{i0})\\ &{}&{}\quad \rightarrow N(E(f^{-q_n}(e_{i0})f'(e_{i0}))\sum \limits _{i=1}^nh'(x_i^T\beta _0)\sum \limits _{j=1}^su_jx_{ij},\\ &{}&{}\quad Var\left( f^{-q_n}(e_{i0})f'(e_{i0})\right) \sum \limits _{i=1}^nh'^2(x_i^T\beta _0)\sum \limits _{j=1}^s(u_jx_{ij})^2) \end{array} \end{aligned}$$

(7.19)

Thus (7.12) follows from $E(f^{-q_n}(e_{i0})f'(e_{i0}))=0$ and (7.15). Note that

$$\begin{aligned} \begin{array}{lll} &{}&{}E\left( \sqrt{n}({\hat{\beta }}_1-\beta _{10})\right) \\ &{}&{}\quad =-\left\{ bn^{-1}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_i(x_{i1},x_{i2},\ldots ,x_{is})+ p''_{\lambda _n}(\beta ^*_1)\right\} ^{-1}\\ &{}&{}\quad \sqrt{n}p'_{\lambda _n}(\beta ^*_1) \end{array} \end{aligned}$$

(7.20)

and

$$\begin{aligned} \begin{array}{lll} &{}&{}Var\left( \sqrt{n}({\hat{\beta }}_1-\beta _{10})\right) \\ &{}&{}\quad =\left\{ bn^{-1}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_i(x_{i1},x_{i2},\ldots ,x_{is})+ p''_{\lambda _n}(\beta ^*_1)\right\} ^{-1}\\ &{}&{}\quad \cdot Var\left\{ f^{-q_n}(e_{i0})f'(e_{i0})\right\} C_{11}\\ &{}&{}\quad \cdot \left\{ bn^{-1}\sum \limits _{i=1}^nh'(x_i^T\beta _0)x^{(s)}_i(x_{i1},x_{i2},\ldots ,x_{is})+ p''_{\lambda _n}(\beta ^*_1)\right\} ^{-T}. \end{array} \nonumber \\ \end{aligned}$$

(7.21)

By (7.11)(7.12) and (7.16)(7.17), we complete the proof. $\square$

Proof of Theorem 4.1

For the objective function of $PL_qE$, corresponding to the population level is given by

$$\begin{aligned} E_{F_\varepsilon }(L_qf(y-h(x^T\beta )))-\sum \limits _{j=1}^d p_\lambda (\beta _j)=E_{F_\varepsilon }(L_qf(e))-\sum \limits _{j=1}^d p_\lambda (\beta _j). \end{aligned}$$

(7.22)

After differentiating with respect to $\beta$ and setting its derivative equal to zero (the solution is denoted as $\beta _{PL_qE}(F_\varepsilon )$), we get

$$\begin{aligned} \begin{array}{lll} &{}&{}-(1-\varepsilon )E_{F_0}\left( f^{-q}(e)f'(e)h'(x^T\beta _{PL_qE}(F_\varepsilon ))x\right) \\ &{}&{}-\varepsilon f^{-q}(e_0)f'(e_0)h'(x_0^T\beta _{PL_qE}(F_\varepsilon ))x_0-p'_\lambda (\beta _{PL_qE}(F_\varepsilon ))=0. \end{array} \end{aligned}$$

(7.23)

Derivation formula (7.23) with respect to $\varepsilon$, and taking the limit as $\varepsilon \rightarrow 0$, we yield

$$\begin{aligned} \begin{array}{lll} &{}&{}E_{F_0}\left( f^{-q}(e)f'(e)h'(x^T\beta _{PL_qE}(F_\varepsilon ))x\right) \\ &{}&{}\quad -(1-\varepsilon )E_{F_0}\{[qf^{-q-1}(e)f'^2(e)h'^2(x^T\beta _{PL_qE}(F_\varepsilon ))\\ &{}&{}\quad -f^{-q}(e)f''(e)h'^2(x^T\beta _{PL_qE}(F_\varepsilon )) +f^{-q}(e)f'(e)h''(x^T\beta _{PL_qE}(F_\varepsilon ))]xx^T\}\\ &{}&{}\qquad \frac{\partial }{\partial \varepsilon }\beta _{PL_qE}(F_\varepsilon )\\ &{}&{}\quad -f^{-q}(e_0)f'(e_0)h'(x_0^T\beta _{PL_qE}(F_\varepsilon ))x_0\\ &{}&{}\quad -\varepsilon [qf^{-q-1}(e_0)f'^2(e_0)h'^2(x_0^T\beta _{PL_qE}(F_\varepsilon )) -f^{-q}(e_0)f''(e_0)h'^2(x_0^T\beta _{PL_qE}(F_\varepsilon ))\\ &{}&{}\quad +f^{-q}(e_0)f'(e_0)h''(x_0^T\beta _{PL_qE}(F_\varepsilon ))]x_0x_0^T]\frac{\partial }{\partial \varepsilon }\beta _{PL_qE}(F_\varepsilon )\\ &{}&{}\qquad -diag(p''_\lambda (\beta _{PL_qE}(F_\varepsilon )))=0. \end{array} \nonumber \\ \end{aligned}$$

(7.24)

Note that

$$\begin{aligned} \frac{\partial }{\partial \varepsilon }(\beta _{PL_qE}(F_\varepsilon ))|_{\varepsilon =0}=IF((x_0,y_0),\beta _{PL_qE},F_0), \end{aligned}$$

we easily obtain the Theorem 4.1.

To verify Corollary 4.2, we first give a lemma (see Xia 2003). $\square$

Lemma 7.1

Let A and B be real symmetric matrices of order n, and A is a positive definite matrix, then A and B can be contract diagonalized at the same time.

Proof of Corollary 4.2

Let $B=xx^T-b^{-1}U, A=bB^{-1}=(a_{ij})_{d\times d}$, $l(e_0)=f^{-q}(e_0)f'(e_0)$, $l(e)=f^{-q}(e)f'(e)$, $D=l(e_0)x_0-E_{F_0}l(e)x$. Then

$$\begin{aligned} IF((x_0,y_0),\beta _{PL_qE},F_0)=AD. \end{aligned}$$

(7.25)

Note that

$$\begin{aligned} \begin{array}{lll} b&{}=&{}E_{F_0}{\tilde{e}}(y,\beta _{PL_qE}(F_0))\\ &{}=&{}E_{F_0}\left( (2\pi )^{\frac{q-1}{2}}((1-q)e^2-1)\exp \{\frac{q-1}{2}e^2\}\right) \\ &{}=&{}\int _{-\infty }^{+\infty }(2\pi )^{\frac{q-1}{2}}((1-q)v^2-1)\exp \{\frac{q-1}{2}v^2\}(2\pi )^{\frac{1}{2}}\exp \{-\frac{1}{2}v^2\}dv\\ &{}=&{}-(2\pi )^{\frac{q-1}{2}}(2-q)^{-\frac{3}{2}}. \end{array} \end{aligned}$$

(7.26)

By lemma 4.1, there exists an invertible matrix Q such that

$$\begin{aligned} Q(xx^T)Q^T=E{} & {} =diag(1,1,\ldots ,1),Q(b^{-1}U)Q^T\\{} & {} =diag(b^{-1}w_1,b^{-1}w_2,\ldots ,b^{-1}w_d). \end{aligned}$$

Thus

$$\begin{aligned} Q(xx^T)Q^T-Q(b^{-1}U)Q^T=diag(1-b^{-1}w_1,1-b^{-1}w_2,\ldots ,1-b^{-1}w_d). \end{aligned}$$

Since $b\ne w_i$, the matrix B is invertible. Therefore,

$$\begin{aligned} \Vert A\Vert _1=\max _{1\le j\le d}\sum \limits _{i=1}^d|a_{ij}|<\infty . \end{aligned}$$

(7.27)

Obviously,

$$\begin{aligned} |l(e_0)|\le \frac{(2\pi )^{\frac{q-1}{2}}}{\sqrt{(1-q)e_0}}. \end{aligned}$$

(7.28)

Since $l(e)=(2\pi )^{\frac{q-1}{2}}\exp \{\frac{q-1}{2}e^2\}e$ is an odd function for e,

$$\begin{aligned} E_{F_0}(l(e))x=0. \end{aligned}$$

(7.29)

Thus, by (7.28)(7.29), we have

$$\begin{aligned} \Vert D\Vert _1=|l(e_0)|\Vert x_0\Vert _1<\infty . \end{aligned}$$

(7.30)

The corollary follows from (7.25)(7.27) and (7.30). $\square$