本文考虑以下线性回归模型:

$ \begin{equation} y = X\beta+\varepsilon , \varepsilon \sim N\left({0, {\sigma ^2}{I_n}} \right), \end{equation} $

(1.1)

其中y为$ n \times 1 $维观测向量, $ X = {\left( {{x_1}, {x_2}, \cdots {x_n}} \right)^\prime } $为$ n \times p $阶设计矩阵, $ rank\left( X \right) = p $, $ \beta $为$ p \times 1 $维未知参数向量, $ \varepsilon = {\left( {{\varepsilon _1}, {\varepsilon _2}, \cdots , {\varepsilon _n}} \right)^\prime } $为$ n \times 1 $维随机误差向量. 相应的最小二乘估计和岭估计分别为$ \hat \beta = {\left( {X'X} \right)^{ - 1}}X'y, {\hat \beta _k} = {\left( {X'X + kI} \right)^{ - 1}}X'y, $其中岭参数$ k > 0, I $为单位矩阵.

在回归分析中, 自变量之间存在多重共线性是一个常见的问题, 这对分析产生了严重的不良影响. 它对普通最小二乘法(OLS)的一个主要后果是, 估计产生了巨大的采样方差, 这可能导致模型中排除了重要系数. 为了处理这种不稳定性, 学者们提出了许多方法, 其中最著名的是Hoerl和Kennard^[1]提出的岭估计, 它是基于在$ X'X $对角线上添加少量正值, 这使得岭估计有偏差, 但确保了比OLS更小的均方误差(MSE). Farebrother^[2]利用广义逆讨论岭回归估计均方误差的进一步结果; Firinguetti^[3]通过大量的仿真实验研究了共线性和自相关干扰对几种岭回归估计的影响. 这些基于岭估计的文献由于其计算可行性和一些最优性质受到了相当大的关注, 但它们可能具有严重的偏差. 此后学者们为了减小偏差, 同时又尽可能地保留岭估计的优良性质, Quenouille^[4]提出将刀切方法应用于有偏估计以减小偏差. 刀切方法提供的估计不仅具有小的偏差, 而且具有所有理想的大样本特性. 文献[5–11]研究了利用刀切法减小岭估计的偏差以及刀切岭估计的性质. 另外, Chaubey^[13]指出刀切岭估计在处理更复杂的推理问题(如获得置信区间)时可提供精细的解决方案.

置信区间在应用统计领域已经非常成熟, 它是指由样本统计量所构造的总体参数的估计区间, 置信集则是置信区间推广到多维的形式. 先前已有学者对线性回归模型中参数的置信集做出大量研究, 例如: Vinod^[12]研究了通过Bootstrap和Stein方法构造岭回归的参数置信区间; Chaubey等人^[13]利用Bootstrap和Jackknife方法在岭估计基础上提出和比较回归模型中回归系数的不同置信区间; Firinguetti和Bobadilla^[14]在基于岭估计和Edgeworth展开建立回归系数的渐近置信区间; 张金^[15]提出了基于最小二乘估计的线性回归模型参数的最小体积置信集. 尽管学者们对于基于最小二乘估计的参数最小体积置信集的研究成果十分丰富, 但基于刀切岭估计的参数最小体积置信集的研究还未展开, 因此本文对此进行了研究.

为了下文的方便, 我们引入模型(1.1)的典则形式

$ \begin{equation} y = Z\gamma + \varepsilon , \varepsilon \sim N\left( {0, {\sigma ^2}{I_n}} \right), \end{equation} $

(1.2)

其中$ Z = XP, \gamma = P'\beta , Z'Z = P'X'XP = \Lambda = diag({\lambda _1}, {\lambda _2}, \ldots {\lambda _p}), \;{\rm{ }}{\lambda _i} \ge 0, {\rm{ }}i = 1, 2, \ldots , p.\ $ $ \gamma $的最小二乘和岭估计分别为$ \hat \gamma = {\left( {Z'Z} \right)^{ - 1}}Z'y, {\rm{ }}{\hat \gamma _k} = {\left( {Z'Z + kI} \right)^{ - 1}}Z'y.\ $另外, Hinkly在文献[11]中定义(加权)刀切岭估计为

$ \begin{equation} {\tilde \gamma _k} = \left[ {I + k{{\left( {Z'Z + kI} \right)}^{ - 1}}} \right]{\hat \gamma _k} = \left[ {I - {k^2}{{\left( {Z'Z + kI} \right)}^{ - 2}}} \right]\hat \gamma . \end{equation} $

(1.3)

于是原模型(1.1)的刀切岭估计可表示为

$ {\tilde \beta _k} = P{\tilde \gamma _k} = \left[ {P - P{k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right]\hat \gamma = \left[ {I - P{k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}P'} \right]\hat \beta . $

为了得到基于刀切岭估计的最小体积置信集, 我们先给出如下引理.

引理2.1^[15]   假设

(1) $ s \in S\left( Z \right) $是$ \theta $的充分统计量, 其概率密度函数(pdf)为$ f\left( {s, \theta } \right) $, 其中$ S\left( Z \right) = \Theta $;

(2) 对于函数$ p\left( \theta \right) > 0 $, $ \tilde f\left( {S, \theta } \right) = {{f\left( {S, \theta } \right)} \mathord{\left/ {\vphantom {{f\left( {S, \theta } \right)} {p\left( \theta \right)}}} \right. } {p\left( \theta \right)}} $为一个枢轴量;

(3) 对任意$ \theta \in \Theta $, 置信集$ {C_m}\left( S \right) $定义为$ {C_m}\left( S \right) = \left\{ {\theta :\tilde f\left( {S, \theta } \right) \ge m} \right\} $, 其中$ m > 0 $是由$ P\left( {\theta \in {C_m}\left( S \right)} \right) = 1 - \alpha , \forall \alpha \in \left( {0, 1} \right) $所定义的临界值;

(4) 对函数$ p\left( \theta \right) > 0 $, 满足$ \left| {C_m^ * \left( \theta \right)} \right| = q\left( \theta \right)\left| {{C_m}\left( \theta \right)} \right| $, 以及对任意$ C\left( S \right) $和$ \theta \in \Theta $都有$ \left| {C_{}^ * \left( \theta \right)} \right| \le q\left( \theta \right)\left| {C\left( \theta \right)} \right|.\ $

若满足以上条件, 则$ {C_m}\left( S \right) $是$ \theta $的置信水平为$ 1 - \alpha $的最小体积置信集.

定理2.1    当$ {\sigma ^2} $已知, 对任意置信集$ C\left( S \right) $都有$ \left| {{C^ * }\left( \beta \right)} \right| \le \left| {C\left( \beta \right)} \right| $, 则$ \beta $的置信水平$ 1 - \alpha $的最小体积置信集为

$ \begin{equation} {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right). \end{equation} $

(2.1)

证    在模型的典则形式(1.2)式中, 令$ S = \hat \gamma = {\left( {{Z^\prime }Z} \right)^{ - 1}}Z'y $, 则它是$ \gamma $的充分统计量, S的pdf为

$ f\left( {s, \gamma } \right) \propto \exp \left( {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2{\sigma ^2}}}} \right. } {2{\sigma ^2}}}{{\left\| {Zs - Z\gamma } \right\|}^2}} \right). $

由引理2.1得$ \tilde f\left( {S, \gamma } \right) = f\left( {S, \gamma } \right) $并且

$ {C_m}\left( S \right) = \left\{ {\gamma :\tilde f\left( {S, \gamma } \right) \ge m} \right\} = \left\{ {\gamma :{{\left\| {\hat y - Z\gamma } \right\|}^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right)} \right\}, $

$ \left| {C_m^ * \left( \gamma \right)} \right| = \int\limits_{{{\left\| {\hat y - Z\gamma } \right\|}^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right)} {ds = \frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{{\left[ {{\sigma ^2}\chi _p^2\left( {1 - \alpha } \right)} \right]}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1} \right)}} = } \left| {{C_m}\left( \gamma \right)} \right|. $

于是在$ \left| {{C^ * }\left( \gamma \right)} \right| \le \left| {C\left( \gamma \right)} \right| $的限制条件下, $ \beta $的置信水平$ 1 - \alpha $的最小体积置信集为

$ {\left\| {\hat y - Z\gamma } \right\|^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right). $

因为$ {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|^2} = {\left\| {\hat y - Z\gamma } \right\|^2}, $所以结论成立.

定理2.2    当$ \beta $已知, 对任意置信集$ C\left( S \right) $都有$ \left| {{C^ * }\left( {{\sigma ^2}} \right)} \right| \le \left| {C\left( {{\sigma ^2}} \right)} \right| $, 则$ {\sigma ^2} $的置信水平$ 1 - \alpha $的最小体积置信区间为

$ \begin{equation} \left[ {\frac{{{{\left\| {y - X\beta } \right\|}^2}}}{{\chi _n^2\left( {1 - {{\alpha '}_1}} \right)}}\frac{{{{\left\| {y - X\beta } \right\|}^2}}}{{\chi _n^2\left( {\alpha - {{\alpha '}_1}} \right)}}} \right]. \end{equation} $

(2.2)

证    因为$ S = {1 \mathord{\left/ {\vphantom {1 {{{\left\| {y - Z\gamma } \right\|}^2}}}} \right. } {{{\left\| {y - Z\gamma } \right\|}^2}}} $是$ {\sigma ^2} $的充分统计量, 其pdf为$ \tilde g\left( {s, {\sigma ^2}} \right) = {g_n}\left( {{1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}s}}} \right. } {{\sigma ^2}s}}} \right){\left( {{1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}s}}} \right. } {{\sigma ^2}s}}} \right)^2}, t > 0, $其中$ {g_n} $为$ \chi _n^2 $的pdf. 由引理2.1得

$ {C_m}\left( S \right) = \left\{ {{\sigma ^2}:\tilde g\left( {S, {\sigma ^2}} \right) \ge m} \right\} = \left\{ {{\sigma ^2}:{m_1} \le {1 \mathord{\left/ {\vphantom {1 {\left( {{\sigma ^2}S} \right)}}} \right. } {\left( {{\sigma ^2}S} \right)}} \le {m_2}} \right\}, $

其中$ {m_1} = \chi _n^2\left( {\alpha - {\alpha _1}} \right), {m_2} = \chi _n^2\left( {1 - {\alpha _1}} \right).\ $令$ {\alpha '_1} = \arg \mathop {\min }\limits_{0 < {\alpha _1} < \alpha } \left\{ {m_1^{ - 1} - m_2^{ - 1}} \right\} $, 即

$ \left( {{d \mathord{\left/ {\vphantom {d {d{\alpha _1}}}} \right. } {d{\alpha _1}}}} \right)\left( {m_1^{ - 1} - m_2^{ - 1}} \right) = {m_1}^{ - 2}g_n^{ - 1}\left( {{m_1}} \right) - {m_2}^{ - 2}g_n^{ - 1}\left( {{m_2}} \right) = 0, {m_1}^2{g_n}\left( {{m_1}} \right) = {m_2}^2{g_n}\left( {{m_2}} \right) \ne 0. $

$ \left| {C_m^ * \left( {{\sigma ^2}} \right)} \right| = \frac{1}{{{\sigma ^2}}}\left( {\frac{1}{{\chi _n^2\left( {\alpha - {{\alpha '}_1}} \right)}} - \frac{1}{{\chi _n^2\left( {1 - {{\alpha '}_1}} \right)}}} \right) = \left| {{C_m}\left( {{\sigma ^2}} \right)} \right|. $

因此$ \left| {{C^ * }\left( {{\sigma ^2}} \right)} \right| \le \left| {C\left( {{\sigma ^2}} \right)} \right| $的限制条件下, $ {\sigma ^2} $的置信水平为$ 1 - \alpha $的最小体积置信区间为

$ {C_m}\left( S \right) = \left[ {{{{{\left\| {y - Z\gamma } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {y - Z\gamma } \right\|}^2}} {\chi _n^2\left( {1 - {{\alpha '}_1}} \right)}}} \right. } {\chi _n^2\left( {1 - {{\alpha '}_1}} \right)}}, {{{{\left\| {y - Z\gamma } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {y - Z\gamma } \right\|}^2}} {\chi _n^2\left( {\alpha - {{\alpha '}_1}} \right)}}} \right. } {\chi _n^2\left( {\alpha - {{\alpha '}_1}} \right)}}} \right]. $

由于$ {\left\| {y - Z\gamma } \right\|^2} = {\left\| {y - X\beta } \right\|^2}, $故结论成立.

定理2.3    当$ \beta , {\sigma ^2} $未知, 对任意置信集$ C\left( S \right) $都有$ \left| {{C^*}\left( {\beta , {\sigma ^2}} \right)} \right| \le {\sigma ^p}\left| {C\left( {\beta , {\sigma ^2}} \right)} \right| $, 则$ \left( {\beta , {\sigma ^2}} \right) $的置信水平$ 1 - \alpha $的最小体积置信集为

$ {C_m}\left( {{S_1}, {S_2}} \right) := \left\{ {\begin{array}{*{20}{c}} {}\\ {\left( {\beta , {\sigma ^2}} \right):}\\ {} \end{array}\begin{array}{*{20}{c}} {g\left( {\frac{1}{{n{\sigma ^2}}}{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}} \right)\begin{array}{*{20}{c}} {}&{} \end{array}}\\ { + \frac{1}{{n{\sigma ^2}}}{{\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|}^2} \le {m_\alpha }} \end{array}} \right\}, $

等价形式

$ {C_m}\left( {{S_1}, {S_2}} \right) := \left\{ \begin{array}{l} {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|^2}\\ \begin{array}{*{20}{c}} {}&{}&{}&{} \end{array} \le n{\sigma ^2}\left[ {{m_\alpha } - g\left( {\frac{1}{{n{\sigma ^2}}}{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}} \right)} \right], \\ \frac{1}{{n{m_2}}}{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + K} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|^2} \le {\sigma ^2} \\ \begin{array}{*{20}{c}} {}&{}&{}&{} \end{array} \le \frac{1}{{n{m_1}}}{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + K} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|^2}, \end{array} \right. $

其中$ {m_\alpha }, {m_1}, {m_2} $见下文(2.5)式.

证    由因子分解定理, $ S = \left( {{{\hat \gamma } \mathord{\left/ {\vphantom {{\hat \gamma } {\left\| {\hat \varepsilon } \right\|}}} \right. } {\left\| {\hat \varepsilon } \right\|}}, {1 \mathord{\left/ {\vphantom {1 {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right. } {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right) $是$ \left( {\gamma , {\sigma ^2}} \right) $的充分统计量, 其中$ {\left\| {\hat \varepsilon } \right\|^2} = {\left\| y \right\|^2} - {\left\| {Z\hat \gamma } \right\|^2}.\ $令$ \left( {{N_1}, {N_2}} \right) = \left( {\hat \gamma , {{\left\| {\hat \varepsilon } \right\|}^2}} \right) $, 则其pdf为

$ {\left| {{{Z'Z} \mathord{\left/ {\vphantom {{Z'Z} {\left( {2\pi {\sigma ^2}} \right)}}} \right. } {\left( {2\pi {\sigma ^2}} \right)}}} \right|^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}\exp \left\{ {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2{\sigma ^2}{{\left\| {Z{n_1} - Z\gamma } \right\|}^2}}}} \right. } {2{\sigma ^2}{{\left\| {Z{n_1} - Z\gamma } \right\|}^2}}}} \right\}{f_{n - p}}\left( {{{{n_2}} \mathord{\left/ {\vphantom {{{n_2}} {{\sigma ^2}}}} \right. } {{\sigma ^2}}}} \right){1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}}}} \right. } {{\sigma ^2}}}. $

于是$ S = \left( {{S_1}, {S_2}} \right) $的pdf为

$ f\left( {{s_1}, {s_2}, \gamma , {\sigma ^2}} \right) = \frac{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}}}{{{{\left( {2\pi {\sigma ^2}} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}\exp \left\{ {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2{\sigma ^2}{{\left\| {Z{s_1}/\sqrt {{s_2}} - Z\gamma } \right\|}^2}}}} \right. } {2{\sigma ^2}{{\left\| {Z{s_1}/\sqrt {{s_2}} - Z\gamma } \right\|}^2}}}} \right\}{f_{n - p}}\left( {{1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}{s_2}}}} \right. } {{\sigma ^2}{s_2}}}} \right)\frac{1}{{{\sigma ^2}}}\left| {\frac{{\partial \left( {{n_1}, {n_2}} \right)}}{{\partial \left( {{s_1}, {s_2}} \right)}}} \right|, {s_2} > 0, $

其中$ \left( {{N_1}, {N_2}} \right) = \left( {{{{S_1}} \mathord{\left/ {\vphantom {{{S_1}} {\sqrt {{S_2}} }}} \right. } {\sqrt {{S_2}} }}, {1 \mathord{\left/ {\vphantom {1 {{S_2}}}} \right. } {{S_2}}}} \right) $, 雅可比行列式$ \left| {{{\partial \left( {{n_1}, {n_2}} \right)} \mathord{\left/ {\vphantom {{\partial \left( {{n_1}, {n_2}} \right)} {\partial \left( {{s_1}, {s_2}} \right)}}} \right. } {\partial \left( {{s_1}, {s_2}} \right)}}} \right| = 1/{s_2}^{2 + p/2}. $由引理2.1得

$ \tilde f\left( {{s_1}, {s_2}, \gamma , {\sigma ^2}} \right) = \exp \left\{ {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {2{\sigma ^2}{{\left\| {Z{s_1}/\sqrt {{s_2}} - Z\gamma } \right\|}^2}}}} \right. } {2{\sigma ^2}{{\left\| {Z{s_1}/\sqrt {{s_2}} - Z\gamma } \right\|}^2}}}} \right\}{f_{n - p}}\left( {{1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}{s_2}}}} \right. } {{\sigma ^2}{s_2}}}} \right){\left( {{1 \mathord{\left/ {\vphantom {1 {{\sigma ^2}{s_2}}}} \right. } {{\sigma ^2}{s_2}}}} \right)^{2 + {p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}, {s_2} > 0, $

$ {C_m}\left( {{s_1}, {s_2}} \right) = \left\{ {\left( {\gamma , {\sigma ^2}} \right):\tilde f\left( {{s_1}, {s_2}, \gamma , {\sigma ^2}} \right) \ge m} \right\}, $

$ \left| {C_m^*\left( {\gamma , {\sigma ^2}} \right)} \right| = \iint\limits_{\tilde f\left( {{s_1}, {s_2}, \gamma , {\sigma ^2}} \right) \ge m}{d{s_1}d{s_2}} = \iint\limits_{\tilde f\left( {\gamma , {\sigma ^2}, {{\tilde s}_1}, {{\tilde s}_2}} \right) \ge m}{{\sigma ^p}}{d{{\tilde s}_1}d{{\tilde s}_2}} = {\sigma ^p}\left| {{C_m}\left( {\gamma , {\sigma ^2}} \right)} \right|. $

其中$ {\tilde s_1}, {\tilde s_2} $由$ {s_1} = \sigma {\tilde s_1} + \gamma \sqrt {{{\tilde s}_2}} - \gamma , {s_2} = {\tilde s_2} $决定.

因此, 在$ \left| {{C^*}\left( {\gamma , {\sigma ^2}} \right)} \right| \le {\sigma ^p}\left| {C\left( {\gamma , {\sigma ^2}} \right)} \right| $的限制条件下, $ \left( {\beta , {\sigma ^2}} \right) $的置信水平$ 1 - \alpha $的最小体积置信集为

$ \begin{equation} {C_m}\left( {{S_1}, {S_2}} \right) = \left\{ {\left( {\gamma , {\sigma ^2}} \right):g\left( {\frac{{{{\left\| {\hat \varepsilon } \right\|}^2}}}{{n{\sigma ^2}}}} \right) + \frac{{{{\left\| {\hat y - Z\gamma } \right\|}^2}}}{{n{\sigma ^2}}} \le {m_\alpha }} \right\}, \end{equation} $

(2.3)

其中$ g\left( x \right) = x - \left( {1 + {2 \mathord{\left/ {\vphantom {2 n}} \right. } n}} \right)\log x $是凸函数和$ {m_\alpha } $是被确定的临界值. 因为

$ {\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|^2} = {\left\| {\hat \varepsilon } \right\|^2}, $

$ {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|^2} = {\left\| {\hat y - Z\gamma } \right\|^2}, $

所以结论成立.

又因为

$ {C_m}\left( {{S_1}, {S_2}} \right):=\left\{ \begin{array}{l} {\left\| {\hat y - Z\gamma } \right\|^2} \le n{\sigma ^2}\left[ {{m_\alpha } - g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right)} \right], \\ g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right) \le {m_\alpha }. \end{array} \right. $

并且, $ g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right) \le {m_\alpha } $等价于$ {m_1} \le {{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}} \le {m_2}, g\left( {{m_1}} \right) = g\left( {{m_2}} \right) = {m_\alpha }.\ $

因此, (2.3)式的最小体积置信集的等价形式为

$ \begin{equation} {C_m}\left( {{S_1}, {S_2}} \right):=\left\{ \begin{array}{l} {\left\| {\hat y - Z\gamma } \right\|^2} \le n{\sigma ^2}\left[ {{m_\alpha } - g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right)} \right], \\ {{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{m_2}}}} \right. } {n{m_2}}} \le {\sigma ^2} \le {{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{m_1}}}} \right. } {n{m_1}}}, \end{array} \right. \end{equation} $

(2.4)

其中$ {m_\alpha } $取决于

$ \begin{equation} 1 - \alpha = n\int_{{m_1}}^{{m_2}} {{f_{n - {\rm{p}}}}\left( {nx} \right)} {F_p}\left( {n{m_\alpha } - ng\left( x \right)} \right)dx, \end{equation} $

$ {F_{n - p}} $为$ \chi _{n - p}^2 $的分布函数.

首先, 参数$ \beta , {\sigma ^2} $的经典置信集如下, 分以下三种情况

情况1: 当$ {\sigma ^2} $已知时, $ \beta $的置信水平为$ 1 - \alpha $置信集为

$ \begin{equation} {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right), \end{equation} $

(3.1)

情况2: 当$ \beta $已知时, $ {\sigma ^2} $的置信水平为$ 1 - \alpha $置信区间为

$ \begin{equation} \left[ {{{{{\left\| {y - X\beta } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {y - X\beta } \right\|}^2}} {\chi _n^2\left( {1 - {\alpha _1}} \right)}}} \right. } {\chi _n^2\left( {1 - {\alpha _1}} \right)}}{{{{\left\| {y - X\beta } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {y - X\beta } \right\|}^2}} {\chi _n^2\left( {{\alpha _2}} \right)}}} \right. } {\chi _n^2\left( {{\alpha _2}} \right)}}} \right], \end{equation} $

(3.2)

其中$ 0 < \alpha < 1 $, 且$ {\alpha _1} + {\alpha _2} = \alpha .\ $

情况3: 当$ \beta $和$ {\sigma ^2} $未知时, $ \left( {\beta , {\sigma ^2}} \right) $的置信水平为$ 1 - \alpha $的经典置信集为

$ \begin{equation} \begin{aligned} & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}} P'{{\tilde \beta }_k} - X\beta } \right\|^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right), \\ &\frac{{{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}}}{{\chi _{n - p}^2\left( {1 - {\alpha _1}^{\prime \prime }} \right)}} \le {\sigma ^2} \le \frac{{{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}}}{{\chi _{n - p}^2\left( {{\alpha _2}^{\prime \prime }} \right)}}\\ \end{aligned} \end{equation} $

(3.3)

其中$ \left( {1 - \alpha '} \right)\left( {1 - \alpha ''} \right) = 1 - \alpha , {\alpha _1}^{\prime \prime } + {\alpha _2}^{\prime \prime } = \alpha '' $. 上式结果由以下两式可得

$ P\left[ {{{\left\| {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k} - X\beta } \right\|}^2} \le {\sigma ^2}\chi _p^2\left( {1 - \alpha } \right)} \right] = 1 - \alpha ', $

$ P\left[ {\chi _{n - p}^2\left( {{\alpha _2}^{\prime \prime }} \right) \le {{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {{\sigma ^2}}}} \right. } {{\sigma ^2}}} \le \chi _{n - p}^2\left( {1 - {\alpha _1}^{\prime \prime }} \right)} \right] = 1 - \alpha '', $

其中$ Z\hat \gamma $与$ \hat \varepsilon $相互独立.

下面将以上经典置信集和基于刀切岭估计的最小体积置信集进行比较.

情况1: 当$ {\sigma ^2} $已知，$ \beta $未知时. 在$ \left| {{C^ * }\left( \gamma \right)} \right| \le \left| {C\left( \gamma \right)} \right| $的限制条件下, 置信水平为$ 1 - \alpha $的$ \beta $经典置信集(3.1)式和最小体积置信集(2.1)式相同.

情况2: 当$ \beta $已知，$ {\sigma ^2} $未知时. 在置信水平为$ 1 - \alpha $中$ {\sigma ^2} $的经典置信区间(3.2)式, 其常用区间“等尾”$ \left( {{\alpha _1} = {\alpha _2} = {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}} \right) $, 在$ \left| {{C^ * }\left( {{\sigma ^2}} \right)} \right| \le \left| {C\left( {{\sigma ^2}} \right)} \right| $的限制条件下, 最优经典置信区间与最小体积置信区间(2.2)式相同, 是所有置信区间中最优.

情况3: 当$ \beta , {\sigma ^2} $未知时. (3.3)式中常用的经典置信集是“等尾”$ \left( {{\alpha _1}^{\prime \prime } = {\alpha _2}^{\prime \prime } = {{\alpha ''} \mathord{\left/ {\vphantom {{\alpha ''} 2}} \right. } 2}} \right) $“等概率”$ 1 - \alpha ' = 1 - \alpha '' = \sqrt {1 - \alpha } $. (3.3)式的经典置信集有体积

$ \begin{equation} \begin{split} \left| {C\left( S \right)} \right| &= \int_{{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {x_{n - p}^2\left( {1 - {\alpha _1}^{\prime \prime }} \right)}}} \right. } {x_{n - p}^2\left( {1 - {\alpha _1}^{\prime \prime }} \right)}}}^{{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {\chi _{n - p}^2\left( {{\alpha _2}^{\prime \prime }} \right)}}} \right. } {\chi _{n - p}^2\left( {{\alpha _2}^{\prime \prime }} \right)}}} {\frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{{\left[ {{\sigma ^2}\chi _p^2\left( {1 - \alpha '} \right)} \right]}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1} \right)}}} d{\sigma ^2}\nonumber\\ &= \frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{{\left\| {\hat \varepsilon } \right\|}^{p + 2}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 2} \right)}}\left[ {\frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {{\alpha _2}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}} - \frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {1 - {\alpha _1}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}}} \right], \nonumber \end{split} \end{equation} $

(2.3)式或(2.4)式最小体积置信集的体积为

$ \begin{equation} \begin{split} \left| {{C_m}\left( S \right)} \right| &= \iint_{g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right) + {{{{\left\| {\hat y - Z\gamma } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat y - Z\gamma } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}} \le {m_\alpha }}{d{\sigma ^2}d\gamma } = \iint_{g\left( {{{{{\left\| {\hat \varepsilon } \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {\hat \varepsilon } \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}}} \right) + {{{{\left\| {{{\left. {\left( {Z'Z} \right.} \right)}^{^{1/2}}}\left( {\hat \gamma - \gamma } \right)} \right\|}^2}} \mathord{\left/ {\vphantom {{{{\left\| {{{\left. {\left( {Z'Z} \right.} \right)}^{^{1/2}}}\left( {\hat \gamma - \gamma } \right)} \right\|}^2}} {n{\sigma ^2}}}} \right. } {n{\sigma ^2}}} \le {m_\alpha }}{d{\sigma ^2}d\gamma }\nonumber\\ &= \frac{{{{\left\| {\hat \varepsilon } \right\|}^{p + 2}}}}{{n{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}}}\iint_{g\left( x \right) + {{\left\| y \right\|}^2} \le {m_\alpha }}{{x^{ - {p \mathord{\left/ {\vphantom {p 2}} \right. } 2} - 2}}}{dxdy} = \frac{{{\pi ^{p/2}}{{\left\| {\hat \varepsilon } \right\|}^{p + 2}}}}{{n{{\left| {Z'Z} \right|}^{1/2}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1} \right)}}\int_{{m_1}}^{{m_2}} {\frac{{{{\left[ {{m_\alpha } - g\left( x \right)} \right]}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{{x^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 2}}}}dx} .\nonumber \end{split} \end{equation} $

假设$ C\left( S \right) $是(3.3)式的置信集, 其中$ S = \left( {{S_1}, {S_2}} \right) = \left( {{{\hat \gamma } \mathord{\left/ {\vphantom {{\hat \gamma } {\left\| {\hat \varepsilon } \right\|}}} \right. } {\left\| {\hat \varepsilon } \right\|}}, {1 \mathord{\left/ {\vphantom {1 {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right. } {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right) $则$ C\left( S \right) $满足$ \left| {{C^ * }\left( \theta \right)} \right| = {\sigma ^p}\left| {C\left( \theta \right)} \right|, \theta \in \Theta $. 事实上, 注意到

$ \left| {C\left( S \right)} \right| = \left| {C\left( {{{\hat \gamma } \mathord{\left/ {\vphantom {{\hat \gamma } {\left\| {\hat \varepsilon } \right\|}}} \right. } {\left\| {\hat \varepsilon } \right\|}}, {1 \mathord{\left/ {\vphantom {1 {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right. } {{{\left\| {\hat \varepsilon } \right\|}^2}}}} \right)} \right| = \frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{{\left\| {\hat \varepsilon } \right\|}^{p + 2}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 2} \right)}}\left[ {\frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - {\rm{p}}}^2{{\left( {{\alpha _2}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}} - \frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {1 - {\alpha _1}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}}} \right], $

$ \left| {C\left( \theta \right)} \right| = \left| {C\left( {\gamma , {\sigma ^2}} \right)} \right| = \frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{\sigma ^{ - p - 2}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 2} \right)}}\left[ {\frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {{\alpha _2}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}} - \frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {1 - {\alpha _1}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}}} \right], $

于是

$ \left| {{C^ * }\left( \theta \right)} \right| = \int_{{{\left[ {{\sigma ^2}\chi _{n - p}^2\left( {1 - {\alpha _1}^{\prime \prime }} \right)} \right]}^{ - 1}}}^{{{\left[ {{\sigma ^2}\chi _{n - p}^2\left( {{\alpha _2}^{\prime \prime }} \right)} \right]}^{ - 1}}} {\frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{{\left[ {{\sigma ^2}\chi _p^2\left( {{\rm{1}} - \alpha '} \right){s_2}} \right]}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + {\rm{1}}} \right)}}d{s_2}} , $

$ \left| {{C^ * }\left( \theta \right)} \right| = \frac{{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}{\sigma ^{ - {\rm{2}}}}}}{{{{\left| {Z'Z} \right|}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + {\rm{2}}} \right)}}\left[ {\frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {{\alpha _2}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}} - \frac{{\chi _p^2{{\left( {1 - \alpha '} \right)}^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2}}}}}{{\chi _{n - p}^2{{\left( {1 - {\alpha _1}^{\prime \prime }} \right)}^{^{{p \mathord{\left/ {\vphantom {p 2}} \right. } 2} + 1}}}}}} \right] = {\sigma ^p}\left| {C\left( \theta \right)} \right|. $

因此由引理2.1得$ \left| {{C_m}\left( S \right)} \right| \le \left| {C\left( S \right)} \right|.\ $

在本节中, 我们利用文献[16]给出的数据来说明上文置信集的可靠性. 该数据集来自对不同组成的波兰水泥在凝固和硬化过程中产生的热量y与生产水泥的四种化学成分$ {x_1}, {x_2}, {x_3}, {x_4} $含量之间的关系的实验研究. 该数据样本量包括$ n = 13 $个观测值, 一个响应变量和四个解释变量.

首先, 通过文献[16]可得到$ X'X $的特征值为$ {\lambda _1} = 105.406, {\lambda _2} = 809.952, {\lambda _3} = 5965.339, {\lambda _4} = 44663.303, $其条件数为

$ k = \frac{{{\lambda _{\max }}}}{{{\lambda _{\min }}}} = \frac{{44663.303}}{{105.406}} = 423.726, $

该信息表明回归向量之间存在严重的多重共线性. 我们得到了$ \beta $和$ \sigma ^2 $的最小二乘估计

$ \hat \beta = (2.193, 1.1533, 0.7582, 0.4863)', {\rm{ }}\hat \sigma _{OLS}^2 = 5.8455. $

然后, 考虑随机约束$ r = R\beta + e, e \sim N(0, {\sigma ^2}), $其中$ R = (1, - 1, 1, 0), r = 0. $由文献[16]可知, 随机约束Liu-型估计(SRLTE)为

$ {\hat \beta _{SRLTE}}\left( {k, d} \right) = {\left( {X'X + kI} \right)^{ - 1}}\left( {X'X - dI} \right){\left( {X'X + R'R} \right)^{ - 1}}X'y, k > 0, - \infty < d < + \infty . $

令$ k = 0.05, d = - 0.5, $则$ {\hat \beta _{SRLTE}}\left( {0.05, - 0.5} \right) = (2.17985, 1.15688, 0.7466, 0.48857)', $在$ k = 0.05, d = - 0.5 $时, $ MSE\left( {{{\hat \beta }_{SRLTE}}} \right) = 0.0641. $

经计算得$ \beta $的刀切岭估计及其均方误差分别为

$ {\tilde \beta _k} = ({\rm{2}}{\rm{.1930457}}, {\rm{1}}{\rm{.153326}}, {\rm{0}}{\rm{.7585}}, {\rm{0}}{\rm{.48632}})' $

$ MSE\left( {{{\tilde \beta }_k}} \right) = {\sigma ^2}\sum\limits_{i = 1}^4 {\frac{{{\lambda _i}{{\left( {{\lambda _i} + 2k} \right)}^2}}}{{{{\left( {{\lambda _i} + k} \right)}^4}}}} + {k^4}\sum\limits_{i = 1}^4 {\frac{{\gamma _i^2}}{{{{\left( {{\lambda _i} + k} \right)}^4}}}} = 0.0637, $

其中$ {\sigma ^2} $取$ \hat \sigma _{OLS}^2 $, $ {\gamma _i} $取$ {\hat \gamma _i} = P'\hat \beta .\ $

因此, 当$ k = 0.05, d = - 0.5 $时, $ MSE\left( {{{\tilde \beta }_k}} \right) < MSE\left( {{{\hat \beta }_{SRLTE}}} \right) $, 此时我们提出的刀切岭估计$ {\tilde \beta _k} $更优势.

此外, 置信水平为$ 95\% \left( {\beta , {\sigma ^2}} \right) $的最小体积置信集为

$ \begin{array}{l} \ \ \ \ \ \ \ \ \frac{{{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}}}{{13{\sigma ^2}}} - 1.1538\log \frac{{{{\left\| {y - XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P'{{\tilde \beta }_k}} \right\|}^2}}}{{13{\sigma ^2}}} + \\ \ \ \ \ \frac{{{{\left\| {{{\left( {{{\tilde \beta }_k} - \beta } \right)}^\prime }{{\left[ {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P' - X} \right]}^\prime }\left[ {XP{{\left( {I - {k^2}{{\left( {\Lambda + kI} \right)}^{ - 2}}} \right)}^{ - 1}}P' - X} \right]\left( {{{\tilde \beta }_k} - \beta } \right)} \right\|}^2}}}{{13{\sigma ^2}}} \\ \le {m_{0.05}}, \end{array} $

其中$ {m_{0.05}} = 2.249 $, P为$ X'X $的特征向量, $ \Lambda = diag({\lambda _1}, {\lambda _2}, \ldots {\lambda _p}) $, $ {\lambda _i} $为$ X'X $的特征根.

令上述最小体积置信集中的$ \beta $为$ {\hat \beta _{SRLTE}}\left( {0.05, - 0.5} \right) $, 通过Matlab计算, 得到左边等式为$ 1.1166 $. 由于$ 1.1166 < 2.249 $, 说明$ {\hat \beta _{SRLTE}} $包含在基于刀切岭估计的最小体积置信集中.