A quadratic upper bound algorithm for regression analysis of credit risk under the proportional hazards model with case-cohort data

Abstract

A case-cohort design is a cost-effective biased-sampling scheme in studies on survival data. We study the regression analysis of credit risk by fitting the proportional hazards model to data collected via the case-cohort design. Using the minorization-maximization principle, we develop a new quadratic upper-bound algorithm for the calculation of estimators and obtain the convergence of the algorithm. The proposed algorithm involves the inversion of the derived upper-bound matrix only one time in the whole process and the upper-bound matrix is independent of parameters. These features make the proposed algorithm have simple update and low per-iterative cost, especially to large-dimensional problems. Rcpp is an R package which enables users to write R extensions with C++. In this paper, we write the program of the proposed algorithm via Rcpp and improve the efficiency of R program execution and realize the fast computing. We conduct simulation studies to illustrate the performance of the proposed algorithm. We analyze a real data example from a mortgage dataset for evaluating credit risk.

Appendix A: proofs the theorems:

Asymptotic properties of $\widehat{{\varvec{{\varvec{\beta }}}}}_P$ is established by (Self and Prentice 1988) for statistical inference. Before presenting the result of asymptotic properties, we introduce some notations first. Let ${\varvec{{\varvec{\beta }}}}_0$ denote the true value of ${\varvec{{\varvec{\beta }}}}$ and denote $\tau$ to be the time when the study ends or discontinues. Let $C = \{i: \xi _i = 1 \text { or } \Delta _i = 1\}$ and $\widetilde{C} = \{i:\xi _i = 1\}$. For $d = 0, 1, 2,$

$$\begin{aligned}{} & {} {} S^{(d)}({{\varvec{\beta }}}, t) = \frac{1}{n} \sum _{i \in {C}} Y_{i}(t) \exp \left( Z_{i}^{\prime } {{\varvec{\beta }}}\right) Z_{i}^{\otimes d},\nonumber \\{}{} & {} {} \widetilde{S}^{(d)}({{\varvec{\beta }}}, t) = \frac{1}{\widetilde{n}} \sum _{i \in \widetilde{{C}}} Y_{i}(t) \exp \left( Z_{i}^{\prime } {{\varvec{\beta }}}\right) Z_{i}^{\otimes d},\nonumber \\ {}{} & {} {} Q^{(d)}({{\varvec{\beta }}}, t, w) = \frac{1}{n} \sum _{i \in {C}} Y_{i}(t) Y_{i}(w) \exp \left( 2 Z_{i}^{\prime } {{\varvec{\beta }}}\right) Z_{i}^{\otimes d},\nonumber \\{}{} & {} {} \widetilde{Q}^{(d)}({{\varvec{\beta }}}, t, w)= \frac{1}{\widetilde{n}} \sum _{i \in \widetilde{{C}}} Y_{i}(t) Y_{i}(w) \exp \left( 2 Z_{i}^{\prime } {{\varvec{\beta }}}\right) Z_{i}^{\otimes d}, \end{aligned}$$

(A.1)

where $a^{\otimes 0} = 1$, $a^{\otimes 1} = a$, $a^{\otimes 2} = aa^{\top }$ and a is a vector. Define

$$\begin{aligned} E({\varvec{\beta }}, t)=\frac{S^{(1)}({\varvec{\beta }}, t)}{S^{(0)}({\varvec{\beta }}, t)}, \quad \widetilde{E}({\varvec{\beta }}, t)=\frac{\widetilde{S}^{(1)}({\varvec{\beta }}, t)}{\widetilde{S}^{(0)}({\varvec{\beta }}, t)}, \end{aligned}$$

(A.2)

Condition (C1)-(C7) ensure the asymptotic convergence of $\widehat{{\varvec{{\varvec{\beta }}}}}_P$:

• (C1) $\widetilde{n} / N \rightarrow \alpha$ for some $\alpha \in (0, 1)$.

• (C2) $\int _o^{\tau } \lambda _0(t)dt < \infty$.

• (C3) There exits $\delta >0$ such that $n^{-\frac{1}{2}} \sup _{i, t \in [0, \tau ]} Y_{i}(t)\left| Z_{i}\right| I\left\{ Z_{i}^{\prime } {\varvec{\beta }}_{0}>-\delta \left| Z_{i}\right| \right\} {\mathop {\rightarrow }\limits ^{P}} 0$.

• (C4) There exits $s^{(d)}({\varvec{\beta }}, t)$ defined in $\mathbb {B} \times [0, \tau ]$ satisfying:

• (1) $\sup _{{\varvec{\beta }} \in \mathbb {B}, t \in [0, \tau ]}\left\| S^{(d)}({\varvec{\beta }}, t)-s^{(d)}({\varvec{\beta }}, t)\right\| {\mathop {\longrightarrow }\limits ^{P}} 0;$

• (2) $s^{(d)}({\varvec{\beta }}, t)$ are continuous functions of ${\varvec{\beta }}$ uniformly in $t \in [0, \tau ]$; $s^{(d)}({\varvec{\beta }}, t)$ are bounded in $\mathbb {B} \times [0, \tau ]$, $s^{(0)}({\varvec{\beta }}, t)$ is bounded above 0; $s^{(1)}({\varvec{\beta }}, t)=\nabla _{{\varvec{\beta }}} s^{(0)}({\varvec{\beta }}, t), s^{(2)}({\varvec{\beta }}, t)=\nabla _{{\varvec{\beta }}}^{2} s^{(0)}({\varvec{\beta }}, t)$ for ${\varvec{\beta }} \in \mathbb {B}, t \in [0, \tau ]$,

• (3) The matrix

  $$\begin{aligned} \Sigma =\int _{0}^{\tau } v\left( {\varvec{\beta }}_{0}, t\right) s^{(0)}\left( {\varvec{\beta }}_{0}, t\right) \lambda _{0}(t) \textrm{d} t \end{aligned}$$

  (A.3)

  is positive definite, where $v({\varvec{\beta }}, t)=s^{(2)}({\varvec{\beta }}, t) / s^{(0)}({\varvec{\beta }}, t) -\left( s^{(1)}({\varvec{\beta }}, t) / s^{(0)}({\varvec{\beta }}, t)\right) ^{\otimes 2}$.

• (C5) The sequence of distributions of $n^{\frac{1}{2}}(\widetilde{E}({\varvec{\beta }}_{0}, t)-E ({\varvec{\beta }}_{0}, t))$ is tight on the space of càdlàg functions equipped with the product Skorohod topology.

• (C6) There exist $q^{(d)}({\varvec{\beta }}, t, w)$ defined on $\mathbb {B} \times [0, \tau ]^{2}$, satisfying:

• (1) $\sup _{{\varvec{\beta }} \in \mathcal {B},(t, w) \in [0,\tau ]^{2}}\Vert Q^{(d)}({\varvec{\beta }}, t, w)-q^{(d)}({\varvec{\beta }}, t, w)\Vert {\mathop {\longrightarrow }\limits ^{P}} 0;$

• (2) $q^{(d)}({\varvec{\beta }}, t, w), d = 0,1,2$ are continuous function of ${\varvec{\beta }}$ uniformly in $(t, w) \in [0, \tau ]^{2}$; and $q^{(d)}({\varvec{\beta }}, t, w)$ are bounded on $\mathcal {B} \times [0, \tau ]^{2}$;

• (3) $\sup _{n \ge 1} E\left( Q^{(d)}({\varvec{\beta }}, t, w)\right)$ is bounded sequence.

• (C7) for $d=0,1,2$,

  $$\begin{aligned} \sup _{{\varvec{\beta }} \in \mathbb {B}, t \in [0, \tau ]}\left\| \tilde{S}^{(d)}({\varvec{\beta }}, t)-s^{(d)}({\varvec{\beta }}, t)\right\| {\mathop {\longrightarrow }\limits ^{P}} 0, \end{aligned}$$

  (A.4)

  and

  $$\begin{aligned} \sup _{{\varvec{\beta }} \in \mathbb {B},(t, w) \in [0, \tau ]^{2}}\left\| \tilde{Q}^{(d)}({\varvec{\beta }}, t, w)-q^{(d)}({\varvec{\beta }}, t, w)\right\| {\mathop {\longrightarrow }\limits ^{P}} 0.\nonumber \\ \end{aligned}$$

  (A.5)

Lemma A.1

Under the regularity conditions (C1)-(C7), as $n\rightarrow \infty$, $\widehat{{\varvec{\beta }}}_P$ converges to ${\varvec{\beta }}_0$ in probability and

$$\begin{aligned} \sqrt{n}\left( \widehat{{\varvec{\beta }}}_{P}-{\varvec{\beta }}_{0}\right) {\mathop {\longrightarrow }\limits ^{d}} N\left( 0, \Sigma ^{-1}+\Sigma ^{-1} \Omega \Sigma ^{-1}\right) , \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \Omega =&\int _{0}^{\tau } \int _{0}^{\tau } G\left( {\varvec{\beta }}_{0}, t, w\right) s^{(0)}\left( {\varvec{\beta }}_{0}, t\right) s^{(0)}\left( {\varvec{\beta }}_{0}, w\right) \\&\lambda _{0}(t) \lambda _{0}(w) \textrm{d} t \mathrm {~d} w,\\ G({\varvec{\beta }}, t, w)=&(1-\alpha ) \alpha ^{-1}\\&\left( \left( s^{(0)}({\varvec{\beta }}, t) s^{(0)}({\varvec{\beta }}, w)\right) ^{-1} h^{(1)}({\varvec{\beta }}, t, w)\right. \\&+\left( s^{(0)}({\varvec{\beta }}, t) s^{(0)}({\varvec{\beta }}, w)\right) ^{-2} s^{(1)}({\varvec{\beta }}, t)\\&s^{(1)}({\varvec{\beta }}, w)^{\top } h^{(0)}({\varvec{\beta }}, t, w) \\&-s^{(0)}({\varvec{\beta }}, t)^{-1} s^{(0)}({\varvec{\beta }}, w)^{-2}\\&s^{(1)}({\varvec{\beta }}, w) h^{(2)}({\varvec{\beta }}, w, t) \\&-s^{(0)}({\varvec{\beta }}, w)^{-1} s^{(0)}({\varvec{\beta }}, t)^{-2}\\&\left. s^{(1)}({\varvec{\beta }}, t) h^{(2)}({\varvec{\beta }}, t, w)\right) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&h^{(i)}({\varvec{\beta }}, t, w)=q^{(i)}({\varvec{\beta }}, t, w)-s^{(i)}({\varvec{\beta }}, t) s^{(i)}({\varvec{\beta }}, w), i = 0,1,2. \end{aligned} \end{aligned}$$

Proof of Theorem 3.1:

The observed information matrix is

$$\begin{aligned}&H_P({\varvec{\beta }})=\sum _{i=1}^{N} \Delta _{i} \nonumber \\&\quad \left[ \frac{\sum _{l \in \widetilde{R}\left( T_{i}\right) } {\varvec{Z}}_{l}^{\otimes 2} e^{{\varvec{Z}}_{l}^{\prime } {\varvec{\beta }}}}{\sum _{l \in \widetilde{R}\left( T_{i}\right) } e^{{\varvec{Z}}_{l}^{\prime } {\varvec{\beta }}}}- \left\{ \frac{\sum _{l \in \widetilde{R}\left( T_{i}\right) } {\varvec{Z}}_{l} e^{{\varvec{Z}}_{l}^{\prime } {\varvec{\beta }}}}{\sum _{l \in \widetilde{R}\left( T_{i}\right) } e^{{\varvec{Z}}_{l}^{\prime } {\varvec{\beta }}}}\right\} ^{\otimes 2} \right] . \end{aligned}$$

(A.6)

Define

$$\begin{aligned} p_k^i = \frac{e^{{\varvec{Z}}_k'{\varvec{\beta }}}}{\sum _{l \in \widetilde{R}\left( T_{i}\right) }e^{{\varvec{Z}}_l'{\varvec{\beta }}}},\; k \in \widetilde{R}(T_i). \end{aligned}$$

(A.7)

$H_p({\varvec{\beta }})$ can be written as

$$\begin{aligned}{} & {} H_p({\varvec{\beta }}) = \sum _{i=1}^N\Delta _i\left[ \sum _{j\in \widetilde{R}(T_i)}{\varvec{Z}}_j{\varvec{Z}}_j^\top p_j^i-\left\{ \sum _{j\in \widetilde{R}(T_i)}{\varvec{Z}}_jp_{j}^i\right\} ^{\otimes 2}\right] \nonumber \\{} & {} \quad \triangleq \sum _{i=1}^N\Delta _i M_i \;. \end{aligned}$$

(A.8)

For any $x \ne 0$, we have

$$\begin{aligned} x^\top M_i x= & {} \sum _{j \in \widetilde{R}(T_i)}(x^\top {\varvec{Z}}_j)^2p_j^i \nonumber \\{} & {} - \left[ \sum _{j\in \widetilde{R}(T_i)}x^\top {\varvec{Z}}_jp_{j}^i\right] ^2 \nonumber \\= & {} \textbf{E}_i[W^2] - \textbf{E}_i[W]^2, \end{aligned}$$

(A.9)

where W is a discrete random variable with support $\{W_j|W_j = x^\top {\varvec{Z}}_j, j \in \widetilde{R}(T_i)\}$ and the $\textbf{E}_i[\cdot ]$ is with respect to probability $\{p_k^i,\; k \in \widetilde{R}_i(T_i)\}$.

Let $w_{\textrm{min}}$ and $w_{\textrm{max}}$ represent minimum and maximum value of $W_{j}.$ Easily note that this variance of W can be maximized when the $w_{\textrm{min}}$ and $w_{\textrm{max}}$ are both taken on with the probability 1/2. So equation (A.9) can be dominated by:

$$\begin{aligned} \begin{aligned} x^{\top } M_{i} x&\le \frac{w_{\textrm{min}}^{2}+w_{\textrm{max}}^{2}}{2}-\left[ \frac{w_{\textrm{min}}+w_{\textrm{max}}}{2}\right] ^{2} \\&\le \frac{w_{\textrm{min}}^{2}+w_{\textrm{max}}^{2}}{2} \le \frac{\sum _{j \in \widetilde{R}(T_i)} W_{j}^{2}}{2}\\&=\frac{\sum _{j \in \widetilde{R}(T_i)} x^{\top } {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top } x}{2}. \end{aligned} \end{aligned}$$

Hence, $M_i \le \sum _{j \in \widetilde{R}_{i}} {\varvec{Z}}_jZ_j^\top$ and we can deduce that:

$$\begin{aligned} H_P({\varvec{\beta }})=\sum _{i =1}^N \Delta _{i} M_{i} \le \sum _{i = 1}^N \Delta _{i} \sum _{j \in \widetilde{R}(t_{i})} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top } / 2. \end{aligned}$$

(A.10)

Let p be a n-dimensional vector and define D(p) be a diagonal matrix, with the k-th diagonal element being the k-element of p and other elements being 0. Due to (A.8), this quadratic form can be rewritten as:

$$\begin{aligned} \begin{aligned}&x^{\top } M_{i} x =\sum _{j \in \widetilde{R}(T_i)}\left\{ x^{\top } {\varvec{Z}}_{j}\right\} ^{2} p_j^i-\left[ \sum _{j \in \widetilde{R}(T_i)} x^{\top } {\varvec{Z}}_{j} p_j^i\right] ^{2} \\&\quad =\left\{ {\varvec{Z}}^{\top } x\right\} ^{\top } D\left( p^i\right) \left\{ {\varvec{Z}}^{\top } x\right\} -\left\{ {\varvec{Z}}^{\top } x\right\} ^{\top } \\&\quad p^i\left[ \left\{ {\varvec{Z}}^{\top } x\right\} ^{\top } p^i\right] ^{\top } \\&\quad =\left\{ {\varvec{Z}}^{\top } x\right\} ^{\top }\left[ D\left( p^i\right) -p^i\left( p^i\right) ^{\top }\right] \left\{ {\varvec{Z}}^{\top } x\right\} , \end{aligned} \end{aligned}$$

(A.11)

where ${\varvec{Z}} = [{\varvec{Z}}_1, \ldots , {\varvec{Z}}_{n_i}]^{\top }$ and $p^i = [p_1^i, \ldots , p_{n_i}^i]^{\top }$. $n_i$ is the number of the elements of $\widetilde{R}(T_i)$. Let $w = [W_1,\ldots , W_{n_i}]^{\top } = {\varvec{Z}}^{\top } x$ and $G(p) = D(p) - pp^{\top }$. Then (A.11) can be rewritten as

$$\begin{aligned} x^{\top } M_i x = w^{\top } G(p^i) w. \end{aligned}$$

(A.12)

We hope to find an upper bound of $M_i$, which is independent of $\varvec{\beta }$. Note that in (A.12), ${\varvec{\beta }}$ is only in $G(p^i)$, so we want to seek an upper bound of $G(p^i)$. Consider a proportional expression as follow:

$$\begin{aligned} \left( w^{\top } G(p^i) w\right) / \left( w^{\top } G(p^{*}) w\right) , \end{aligned}$$

(A.13)

where $p^{*} = [1/n_i, \ldots , 1/n_i]^{\top }$. If the above ratio is controlled by a proper constant independent of ${\varvec{\beta }}$, then we get an upper bound. Since in the proof of Theorem 2 we have found that

$$\begin{aligned} w^{\top } G(p^i) w\le & {} \frac{w_{\textrm{min}}^{2}+w_{\textrm{max}}^{2}}{2}-\left[ \frac{w_{\textrm{min}}+w_{\textrm{max}}}{2}\right] ^{2} \\= & {} \frac{(w_{\max } - w_{\min })^2}{4}, \end{aligned}$$

we need only to find a lower bound of $w^{\top } G(p^{*}) w$.

Rewriting $w^{\top } G(p^{*}) w$ as:

$$\begin{aligned}{} & {} w^{\top } G\left( p^{*}\right) w =w^{\top } D\left( p^{*}\right) w-\left( w^{\top } p^{*}\right) ^{2} \\{} & {} \quad =\sum _{j \in \widetilde{R}(T_i)} w_{j}^{2} / n_{i}-\left( \sum _{j \in \widetilde{R}(T_i)} w_{j} / n_{i}\right) ^{2}. \end{aligned}$$

We can think of it as the variance of a random variable with probability $1/ n_i$ at $w_i, i= 1, 2,\ldots , n_i$, so our goal is equivalently to find a lower bound of the variance. Keep the maximum and the minimum of W fixed while consider all other components of the vector w as unknown variables of the variance function. In this case, the variance is minimized when the all remaining position variables take the value $(w_{\min } + w_{\max }) / 2$.

$$\begin{aligned} \begin{aligned} w^{\top } G\left( p^{*}\right) w&\ge \left( w_{\max }-\frac{w_{\max }+w_{\min }}{2}\right) ^2 \frac{1}{n_{i}}\\&+\left( w_{\min }-\frac{w_{\max }+w_{\min }}{2}\right) ^2 \frac{1}{n_{i}} \\&=\frac{\left( w_{\max }-w_{\min }\right) ^{2}}{2 n_{i}}, \end{aligned} \end{aligned}$$

Hence we bound the ratio:

$$\begin{aligned} \frac{w^{\top } G\left( p^{(i)}\right) w}{w^{\top } G\left( p^{*}\right) w} \le \frac{\left( w_{\max }-w_{\min }\right) ^{2}}{4} / \frac{\left( w_{\max }-w_{\min }\right) ^{2}}{2 n_{i}}=\frac{n_{i}}{2}. \end{aligned}$$

According to this, we can obtain that

$$\begin{aligned} \begin{aligned} x^{\top } M_{i} x&=w^{\top } G\left( p^{(i)}\right) w \le w^{\top } \frac{n_{i}}{2} G\left( p^{*}\right) w \\&=\frac{n_{i}}{2}\left\{ {\varvec{Z}}^{\top } x\right\} ^{\top }\left[ D\left( p^{*}\right) -p^{*}\left( p^{*}\right) ^{\top }\right] \left\{ {\varvec{Z}}^{\top } x\right\} \\&=\frac{n_{i}}{2} \sum _{j \in \widetilde{R}(T_i)}\left\{ x^{\top } {\varvec{Z}}_{j}\right\} ^{2} \frac{1}{n_{i}}-\frac{n_{i}}{2}\left[ \sum _{j \in \widetilde{R}(T_i)} x^{\top } {\varvec{Z}}_{j} \frac{1}{n_{i}}\right] ^{2} \\&=x^{\top }\left[ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top } / 2-\left\{ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j}\right\} ^{\otimes 2} /\left( 2 n_{i}\right) \right] x. \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} M_{i} \le \left[ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top }-\left\{ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j}\right\} ^{\otimes 2} / n_{i}\right] / 2, \end{aligned}$$

and we can deduce that

$$\begin{aligned}{} & {} H_P({\varvec{\beta }})=\sum _{i=1}^N\Delta _{i} M_{i} \le \sum _{i = 1}^N \Delta _{i}\\{} & {} \quad \left[ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top }-\left\{ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j}\right\} ^{\otimes 2} / n_{i}\right] / 2. \end{aligned}$$

$\square$

Proof of Theorem 3.2:

Let ${\varvec{\beta }}^{(m+1)} = {\varvec{\beta }}^{(m)} + B^{-1}U_P({\varvec{\beta }})$, we first prove that $l_P({\varvec{\beta }}^{(m)})$ is an ascending sequence. consider the quadratic approximation of $l_P({\varvec{\beta }}^{(m+1)}) - l_P({\varvec{\beta }}^{(m)})$:

$$\begin{aligned} \begin{aligned}&Q({\varvec{\beta }}^{(m+1)}|{\varvec{\beta }}^{(m)})- l_P({\varvec{\beta }}^{(m)})= U_P({\varvec{\beta }}^{(m)})^{\top }({\varvec{\beta }}^{(m+1)} - {\varvec{\beta }}^{(m)}) \\&\quad - \frac{1}{2}({\varvec{\beta }}^{(m+1)} - {\varvec{\beta }}^{(m)}) ^{\top } \textbf{B} ({\varvec{\beta }}^{(m+1)} - {\varvec{\beta }}^{(m)})\\&\quad = U_P({\varvec{\beta }}^{(m)})^{\top }\textbf{B}^{-1}U_P({\varvec{\beta }}^{(m)}) - \frac{1}{2}U_P({\varvec{\beta }}^{(m)})^{\top }\textbf{B}^{-1}\textbf{B}\textbf{B}^{-1}U_P({\varvec{\beta }}^{(m)})\\&\quad = \frac{1}{2}U_P({\varvec{\beta }}^{(m)})^{\top }\textbf{B}^{-1}U_P({\varvec{\beta }}^{(m)}) \ge 0, \end{aligned} \end{aligned}$$

where the inequality is strict if $U_P({\varvec{\beta }}^{(m)})\ne 0$. Since $l_P({\varvec{\beta }}^{(m+1)})\ge Q({\varvec{\beta }}^{(m+1)}|{\varvec{\beta }}^{(m)})$ for the reason $H_P({\varvec{\beta }})\le \textbf{B}$, we have $l_P({\varvec{\beta }}^{(m + 1)}) \ge l_P({\varvec{\beta }}^{(m)})$.

Suppose for purpose of contradiction that $\Vert U_P({\varvec{\beta }}^{(m)})\Vert$ is bounded below 0, then

$$\begin{aligned} l_P({\varvec{\beta }}^{(m+1)}) - l_P({\varvec{\beta }}^{(m)}) \ge Q({\varvec{\beta }}^{(m+1)}|{\varvec{\beta }}^{(m)})- l_P({\varvec{\beta }}^{(m)}) > 0, \end{aligned}$$

which means the increments of $l_P({\varvec{\beta }}^{(m)})$ are positive, contradicting to boundedness of $l_P({\varvec{\beta }})$. Therefore, $U_P({\varvec{\beta }}^{(m)})$ converges to 0, deducing that ${\varvec{\beta }}^{(m)}$ converges to $\widehat{{\varvec{\beta }}}_P$. $\square$

Proof of Theorem 3.3

From the proof of Theorem 1 we can learn that

$$\begin{aligned}{} & {} Q_P\left( \varvec{\beta }^{(m+1)} \mid \varvec{\beta }^{(m)}\right) -l_P\left( \varvec{\beta }^{(m)}\right) \\{} & {} \quad =\frac{1}{2} U_P\left( \varvec{\beta }^{(m)}\right) ^{\top } \textbf{B}^{-1} U_P\left( \varvec{\beta }^{(m)}\right) . \end{aligned}$$

Note that the rate of convergence, measured by degree of improvement at each iterative step is monotone decreasing with the least bound matrix $\textbf{B}$ from (A.19). And

$$\begin{aligned} \begin{aligned} \textbf{B}_{1}-\textbf{B}_{2}&=\sum _{i=1}^{N} \Delta _{i} \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top } / 2-\sum _{i=1}^{N} \Delta _{i}\\&\left[ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j} {\varvec{Z}}_{j}^{\top }-\left\{ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j}\right\} ^{\otimes 2} / n_{i}\right] / 2 \\&=\sum _{i=1}^{N} \Delta _{i}\left[ \left\{ \sum _{j \in \widetilde{R}(T_i)} {\varvec{Z}}_{j}\right\} ^{\otimes 2} / n_{i}\right] / 2 \ge 0. \end{aligned} \end{aligned}$$

As a consequence that $\textbf{B}_{1} \ge \textbf{B}_{2}$, we can deduce that QUB2 converges faster than QUB1 algorithm. $\square$