0–1膨胀几何分布的客观贝叶斯分析

肖翔; 古晞

0–1膨胀几何分布的客观贝叶斯分析

肖翔^1,,
古晞²

1.
上海工程技术大学数理与统计学院，上海 201620
2.
同济大学数学科学学院，上海 200092

基金项目: 全国统计科学研究资助项目（2020LY080）

详细信息

作者简介:
肖翔：肖　翔（1980−），男，讲师，硕士，研究方向为统计学. E-mail：xiaoxiang@sues.edu.cn

中图分类号: O212.1
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- 被引次数: 3
出版历程
- 收稿日期: 2021-05-06
- 刊出日期: 2021-09-30

Objective Bayesian analysis of zero-and-one-inflated geometric distribution

XIAO Xiang^1
,,
GU Xi²

1.
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
2.
School of Mathematical Science, Tongji University, Shanghai 200092, China

摘要

摘要: 在医疗卫生、金融证券等应用领域，经常会同时出现零观测值、一观测值较多的情况. 为更好地拟合这类数据，提出0–1膨胀几何分布模型并进行客观贝叶斯分析. 通过参数变换，得到Jeffreys先验和reference先验. 设计后验分布的抽样算法，设置不同的样本量和参数真值，采用数值模拟方法对不同客观先验下的估计效果进行评估.
- 0–1膨胀几何分布 /
- Jeffreys先验 /
- reference先验 /
- 客观贝叶斯分析
Abstract: Count datas with excess zeros and ones arise frequently in various fields such as medical health, finance and securities. In order to fit these datas better, a zero-and-one-inflated geometric distribution model was proposed and the objective Bayesian analysis was carried out. Based on parameter transformation, the Jeffreys prior and the reference prior were derived for this model. For different sample size and different true values of the parameters, simulations were adopted to assess the performance under the different objective priors through the sampling algorithm.
- zero-and-one-inflated geometric distribution /
- Jeffreys prior /
- reference prior /
- objective Bayesian analysis

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在医疗卫生、保险金融及可靠性等许多实际应用领域，样本数据不仅会出现零过多的情况，也会出现一过多的情况. 例如，在新型冠状病毒（COVID–19）大流行时，个体感染COVID–19后，自身就会产生抗体，使其感染次数最多可能一次. 又如，在商场买衣服时，出于货比三家的心理，很多顾客没有购买衣服或者只购买一件衣服.

近年来，国内外很多文献对0–1膨胀泊松分布模型进行了深入研究，取得丰富的研究成果. 田震^[1]基于数据删失和加权扰动模型对0–1膨胀泊松分布模型进行统计推断. Tang等^[2]构造0–1膨胀泊松分布模型的等价表达式，采用极大似然估计与贝叶斯方法对新加坡军团菌感染病例数据进行研究. Liu等^[3]利用广义最大期望（EM）算法对0–1膨胀泊松分布回归模型中的参数进行估计，对美国底特律城市交通事故死亡数据进行拟合. 夏丽丽等^[4]使用局部多项式核回归法对0–1膨胀泊松分布模型进行参数估计，通过对北京市糖尿病患者数据的分析，验证了局部多项式核回归方法的有效性.

对于0–1膨胀泊松分布模型，当数据存在较大变异时，即样本均值与样本方差不相等时，如果仍然用模型进行拟合，效果往往不好. 而0–1膨胀几何分布模型，不仅可以用于处理样本数据的变异，也适应于样本尾部数据退化较慢的情形. 对于0–1膨胀几何分布及其回归模型，目前研究文献较少，肖翔^[5]利用贝叶斯方法对0–1膨胀几何分布回归模型进行参数估计，Xiao等^[6]基于Polya-Gamma潜变量设计0–1膨胀几何分布回归模型中后验样本的抽样机制. 本研究对0–1膨胀几何分布模型进行参数变换，计算出客观贝叶斯先验，以期得到更好的拟合效果.

1. 0–1膨胀几何分布模型

本研究提出0–1膨胀几何分布(简称为ZOIGE）模型，即一个非负的0-1膨胀几何分布的随机变量 $Y$ ，可以表示为 $Y = V(1 - {B_1}) + {B_1}(1 - {B_2})$ . 其中， ${B_1}、{B_2}、V$ 相互独立， ${B_1}$ 为一个试验成功概率为 ${p_1}$ 的伯努利随机变量； ${B_2}$ 为一个试验成功概率为 ${p_2}$ 的伯努利随机变量； $V$ 为一个服从于试验成功概率为 $\theta$ 的几何分布随机变量，即 $P(V = k) = {\theta ^k}(1 - \theta )$ ， $k = 0,1, \cdots$ . 随机变量 $Y$ 的分布律为

$\begin{split} & P(Y = k) =\\& \left\{ {\begin{array}{*{20}{l}} {{p_1}{p_2} + (1 - {p_1})(1 - \theta ) }\quad {k = 0} \\ {{p_1}(1 - {p_2}) + (1 - {p_1})\theta (1 - \theta ) } \quad{k = 1} \\ {(1 - {p_1}){\theta ^k}(1 - \theta ) }\quad{k = 2,3, \cdots } \end{array}} \right. \end{split}$

(1)

式中： $0 \leqslant {p_1} \leqslant 1$ ， $0 \leqslant {p_2} \leqslant 1$ ， $0 \leqslant \theta \leqslant 1$ . 可以看出，0–1膨胀几何分布是由伯努利分布与几何分布按照比例 ${p_1}$ 和 $1 - {p_1}$ 组成的混合分布. 当 ${p_2} = 1$ 时，ZOIGE变成零膨胀几何分布(ZIGE)^[7-8]，当 ${p_1} = 0$ 时，ZOIGE退化成几何分布.

进行参数变换，令

$\left\{ \begin{array}{l} {q_1} = {p_1}{p_2} + (1 - {p_1})(1 - \theta ) \hfill \\ {q_2} = {p_1}(1 - {p_2}) + (1 - {p_1})\theta (1 - \theta ) \hfill \end{array} \right.$

(2)

可得

$1 - {p_1} = \frac{{1 - {q_1} - {q_2}}}{{{\theta ^2}}}$

(3)

通过上述重参数化，式(1)变为

$P(Y = k)\left\{ {\begin{array}{*{20}{l}} {{q_1}}\;\;\;{k = 0} \\ {{q_2}}\;\;\;{k = 1} \\ {(1 - {q_1} - {q_2}){\theta ^{k - 2}}(1 - \theta )}\\\;\;\;\;\;\;\;\; {k = 2,3, \cdots } \end{array}} \right.$

(4)

式中： ${q_1} \geqslant 0$ ， ${q_2} \geqslant 0$ ， ${q_1} + {q_2} \leqslant 1$ ， $0 \leqslant \theta \leqslant 1$ .

设 $Y = ({Y_1},{Y_2}, \cdots ,{Y_n})$ 为取自0–1膨胀几何分布的观测值，由式(4)得出似然函数公式为

$\begin{split} L({p_1},{p_2},\theta |Y) =& q_1^{{S_0}}q_2^{{S_1}}{(1 - {q_1} - {q_2})^{n - {S_0} - {S_1}}}\times\\& {\theta ^{S - 2(n - {S_0} - {S_1})}}{(1 - \theta )^{n - {S_0} - {S_1}}} \end{split}$

(5)

式中： ${S_0} = \# \{ i:{Y_i} = 0\}$ 为集合 $\{ i:{Y_i} = 0\}$ 中包含元素的个数； ${S_1} = \# \{ i:{Y_i} = 1\}$ 为集合 $\{ i:{Y_i} = 1\}$ 中包含元素的个数； ${S_2} = \displaystyle\sum\limits_{{Y_i} \geqslant 2} {{Y_i}}$ .

式(5)两边取对数，得到对数似然函数为

$\begin{split} \ln L({p_1},{p_2},\theta |Y) &= {S_0}\ln {q_1} + {S_1}\ln {q_2} + \\&(n - {S_0} - {S_1}) \ln (1 - {q_1} - {q_2}) + \\& [S - 2(n - {S_0} - {S_1})]\ln \theta + \\& (n - {S_0} - {S_1})\ln (1 - \theta ) \end{split}$

(6)

2. 客观先验

2.1 Fisher信息矩阵

计算随机变量 $Y,{S_0},{S_1},S$ 的期望为

$\begin{split} E(Y) =& {q_2} + \sum\limits_{k = 2}^\infty {(1 - {q_1} - {q_2})k{\theta ^{k - 2}}(1 - \theta )} =\\[-4pt]& {q_2} +(1 - {q_1} - {q_2})\frac{{2 - \theta }}{{1 - \theta }} \end{split}$

$\begin{split}& E({S_0}) = n{q_1}\\& E({S_1}) = n{q_2}\\[-2pt] & E(S) = n(E(Y) - {q_2}) = n(1 - {q_1} - {q_2})\frac{{2 - \theta }}{{1 - \theta }} \end{split}$

计算对数似然函数式(6)的一阶偏导数为

$\begin{split}& \frac{{\partial \ln L}}{{\partial {q_1}}} = \frac{{{S_0}}}{{{q_1}}} - \frac{{n - {S_0} - {S_1}}}{{1 - {q_1} - {q_2}}}\\& \frac{{\partial \ln L}}{{\partial {q_2}}} = \frac{{{S_1}}}{{{q_2}}} - \frac{{n - {S_0} - {S_1}}}{{1 - {q_1} - {q_2}}}\\& \frac{{\partial \ln L}}{{\partial \theta }} = \frac{{S - 2(n - {S_0} - {S_1})}}{\theta } - \frac{{n - {S_0} - {S_1}}}{{1 - \theta }} \end{split}$

计算对数似然函数式(6)的二阶偏导数为

$\begin{split}& \frac{{{\partial ^2}\ln L}}{{\partial q_1^2}} = - \frac{{{S_0}}}{{q_1^2}} - \frac{{n - {S_0} - {S_1}}}{{{{(1 - {q_1} - {q_2})}^2}}}\\& \frac{{{\partial ^2}\ln L}}{{\partial q_2^2}} = - \frac{{{S_1}}}{{q_2^2}} - \frac{{n - {S_0} - {S_1}}}{{{{(1 - {q_1} - {q_2})}^2}}} \end{split}$

$\frac{{{\partial ^2}\ln L}}{{\partial {\theta ^2}}} = - \frac{{S - 2(n - {S_0} - {S_1})}}{{{\theta ^2}}} - \frac{{n - {S_0} - {S_1}}}{{{{(1 - \theta )}^2}}}$

$\frac{{{\partial ^2}\ln L}}{{\partial {q_1}\partial {q_2}}} = \frac{{{\partial ^2}\ln L}}{{\partial {q_2}\partial {q_1}}} = - \frac{{n - {S_0} - {S_1}}}{{{{(1 - {q_1} - {q_2})}^2}}}$

$\begin{split} & \frac{{{\partial ^2}\ln L}}{{\partial {q_1}\partial \theta }} = \frac{{{\partial ^2}\ln L}}{{\partial \theta \partial {q_1}}} = 0\\ &\frac{{{\partial ^2}\ln L}}{{\partial {q_2}\partial \theta }} = \frac{{{\partial ^2}\ln L}}{{\partial \theta \partial {q_2}}} = 0 \end{split}$

进一步计算二阶偏导数期望的相反数，它们是Fisher信息阵的组成元素. 表达式为

$\begin{split}& - E\Bigg(\frac{{{\partial ^2}\ln L}}{{\partial q_1^2}}\Bigg) = \frac{{n(1 - {q_2})}}{{{q_1}(1 - {q_1} - {q_2})}}\\& - E\Bigg(\frac{{{\partial ^2}\ln L}}{{\partial q_2^2}}\Bigg) = \frac{{n(1 - {q_1})}}{{{q_2}(1 - {q_1} - {q_2})}}\end{split}$

$\begin{split} &- E\Bigg(\frac{{{\partial ^2}\ln L}}{{\partial {q_1}\partial {q_2}}}\Bigg) = - E\Bigg(\frac{{{\partial ^2}\ln L}}{{\partial {q_2}\partial {q_1}}}\Bigg) = \frac{n}{{1 - {q_1} - {q_2}}} \\&- E\Bigg(\frac{{{\partial ^2}\ln L}}{{\partial {\theta ^2}}}\Bigg) = \frac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}} \end{split}$

因此， $({q_1},{q_2},\theta )$ 的Fisher信息阵为

$\qquad\qquad\qquad\qquad\displaystyle\sum ({q_1},{q_2},\theta ) = \left( {\begin{array}{*{20}{c}} {\dfrac{{n(1 - {q_2})}}{{{q_1}(1 - {q_1} - {q_2})}}}&{\dfrac{n}{{1 - {q_1} - {q_2}}}}&0 \\ {\dfrac{n}{{1 - {q_1} - {q_2}}}}&{\dfrac{{n(1 - {q_1})}}{{{q_2}(1 - {q_1} - {q_2})}}}&0 \\ 0&0&{\dfrac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}}} \end{array}} \right)$

(7)

2.2 Jeffreys先验

与Laplace先验比较，Jeffreys先验能够在参数变换下保持不变性，比Laplace先验具有更广泛的应用场合^[9]. 参数 $({q_1},{q_2},\theta )$ 的Jeffreys先验与Fisher信息矩阵行列式的平方根成正比，通过式(7)可以计算 $({q_1},{q_2},\theta )$ 的Jeffreys先验，公式为

${\pi _J} \propto \det {\left(\displaystyle\sum \right)^{1/2}} \propto q_1^{ - 1/2}{q_2}^{ - 1/2}{\theta ^{ - 1/2}}{(1 - \theta )^{ - 1}}$

2.3 Reference先验

对于参数组合 $\{ ({q_1},{q_2}),\theta \}$ ， $({q_1},{q_2})$ 为感兴趣的参数，Fisher信息矩阵 $\displaystyle\sum ({q_1},{q_2},\theta )$ 可写成

${\displaystyle\sum\nolimits_{1}} = \left[ {\begin{array}{*{20}{c}} {{\sum\nolimits _{11}}}&0 \\ 0&{{I_{33}}} \end{array}} \right]$

其中

$\begin{split}&{\displaystyle\sum \nolimits_{11}} = \left( {\begin{array}{*{20}{c}} {\dfrac{{n(1 - {q_2})}}{{{q_1}(1 - {q_1} - {q_2})}}}&{\dfrac{n}{{1 - {q_1} - {q_2}}}} \\ {\dfrac{n}{{1 - {q_1} - {q_2}}}}&{\dfrac{{n(1 - {q_1})}}{{{q_2}(1 - {q_1} - {q_2})}}} \end{array}} \right) \\&{I_{33}} = \dfrac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}} \end{split}$

根据文献[10]，reference先验求解过程中，先求出 ${h_1}$ 和 ${h_2}$ ，公式为

$\begin{split}& {h_1} = \det \left({\displaystyle\sum\nolimits _{11}}\right) = \frac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}} \\& {h_2} = {I_{33}} = \frac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}} \end{split}$

再完成以下4个步骤.

步骤1 选取参数空间的一组紧子集为 ${\Omega _i} = {\Omega _{12}} \times {\Omega _{3i}} =$ $\{ ({q_1},{q_2})|0 < {q_1} < 1, 0 < {q_2} < 1 - {q_1}\} \times \{ \theta |{a_i} < \theta < {b_i}\}$ .使得当 $i \to \infty$ 时，有 ${a_i} \to 0$ 和 ${b_i} \to 1$ .通过换元法计算区域 ${\Omega _{12}}$ 上的两个二重积分，公式为

$\left\{\begin{array}{l} \displaystyle\iint_{{\Omega _{12}}} {\dfrac{1}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }}}{\rm{d}}{q_1}{\rm{d}}{q_2} = 2{\text{π}} \\ \displaystyle\iint_{{\Omega _{12}}} {\dfrac{{\log (1 - {q_1} - {q_2})}}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }}}{\rm{d}}{q_1}{\rm{d}}{q_2} = - 4{\text{π}} \end{array}\right.$

(8)

步骤2 当 $({q_1},{q_2})$ 给定时， $\theta$ 的条件先验为

$\begin{split} \pi _{R1}^i(\theta |{q_1},{q_2}) =& \frac{{\sqrt {{h_2}} {\Omega _{3i}}}}{{\displaystyle\int_{{\Omega _{3i}}} {\sqrt {{h_2}} {\rm{d}}\theta } }} =\\ &\frac{{\sqrt {n(1 - {q_1} - {q_2})} {\theta ^{ - 1/2}}{{(1 - \theta )}^{ - 1}}{\Omega _{3i}}}}{{\sqrt {n(1 - {q_1} - {q_2})} \displaystyle\int_{{a_i}}^{{b_i}} {{\theta ^{ - 1/2}}{{(1 - \theta )}^{ - 1}}{\rm{d}}\theta } }}= \\ & \dfrac{{{\theta ^{ - 1/2}}{{(1 - \theta )}^{ - 1}}{\Omega _{3i}}}}{{\log \dfrac{{(1 + {b_i})(1 - {a_i})}}{{(1 + {a_i})(1 - {b_i})}}}} \propto {\theta ^{ - 1/2}}{(1 - \theta )^{ - 1}}{\Omega _{3i}} \end{split}$

步骤3 结合式(8)， $({q_1},{q_2})$ 的边缘先验为

$\qquad\qquad\qquad\begin{split} \pi _{R1}^i({q_1},{q_2}) =& \dfrac{{\exp \Bigg\{ \dfrac{1}{2}\displaystyle\int_{{\Omega _{3i}}} {\pi _{R1}^i(\theta\, |\,{q_1},\,{q_2}\,)\log ({h_1}){\rm{d}}\theta } \Bigg\} {\Omega _{12}}}}{{\displaystyle\iint_{{\Omega _{12}}} {\exp \Bigg\{ \dfrac{1}{2}\displaystyle\int_{{\Omega _{3i}}} {\pi _{R1}^i(\theta \, |\, {q_1},\, {q_2}\, )\log ({h_1}){\rm{d}}\theta } \Bigg\} {\rm{d}}{q_1}{\rm{d}}{q_2}}}} \propto\\& \dfrac{{\exp \Bigg\{ \dfrac{1}{2}\displaystyle\int_{{\Omega _{3i}}} {{\theta ^{ - 1/2}}{{(1 - \theta )}^{ - 1}}\log \Bigg(\dfrac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}}\Bigg){\rm{d}}\theta } \Bigg\} {\Omega _{12}}}}{{\displaystyle\iint_{{\Omega _{12}}} {\exp \Bigg\{ \dfrac{1}{2}\displaystyle\int_{{\Omega _{3i}}} {{\theta ^{ - 1/2}}{{(1 - \theta )}^{ - 1}}\log \Bigg(\dfrac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}}\Bigg)} {\rm{d}}\theta \Bigg\} {\rm{d}}{q_1}{\rm{d}}{q_2}}}} \propto\\& \dfrac{{{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}\dfrac{{(1 + {b_i})(1 - {a_i})}}{{(1 + {a_i})(1 - {b_i})}}{\Omega _{12}}}}{{\dfrac{{(1 + {b_i})(1 - {a_i})}}{{(1 + {a_i})(1 - {b_i})}}\displaystyle\iint_{{\Omega _{12}}} {{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}{\rm{d}}{q_1}{\rm{d}}{q_2}}}} \propto\\& \dfrac{{{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}{\Omega _{12}}}}{{2{\text{π}} }} \propto {q_1}^{ - 1/2}q_2^{ - 1/2}{(1 - {q_1} - {q_2})^{ - 1/2}}{\Omega _{12}} \end{split}$

步骤4 ${\varPhi}$ 的reference先验为

$\begin{split}& {\pi _{R1}} = \mathop {\lim }\limits_{i \to \infty } \frac{{\pi _{R1}^i({q_1},{q_2})\pi _{R1}^i(\theta \, |\,{q_1},\,{q_2}\,)}}{{\pi _{R1}^i(q_1^*,q_2^*)\pi _{R1}^i({\theta \, ^*}\,|\,q_1^*,\,q_2^*)\,}} \propto \\&\quad {q_1}^{ - 1/2}q_2^{ - 1/2}{(1 - {q_1} - {q_2})^{ - 1/2}}{\theta ^{ - 1/2}}{(1 - \theta )^{ - 1}} \end{split}$

式中， $q_1^*$ 、 $q_2^*$ 和 ${\theta ^*}$ 为参数空间中事先给定的值.

对于参数组合 $\{ \theta \,,\,({q_1},\,{q_2}\,)\}$ ， $\theta$ 为感兴趣的参数，Fisher信息矩阵 $\displaystyle\sum ({q_1},{q_2},\theta )$ 可写成

${\displaystyle\sum\nolimits _2} = \left[ {\begin{array}{*{20}{c}} {{I_{33}}}&0 \\ 0&{{\sum\nolimits _{11}}} \end{array}} \right]$

得到

$\begin{split} & {h_1} = \dfrac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}}\\ & {h_2} = \dfrac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}} \end{split}$

步骤1 选取与 $\{\, ({q_1},\,{q_2}),\,\theta \,\}$ 参数空间中相同的一组紧子集.

步骤2 当 $\theta$ 给定时，结合式(8)， $({q_1},{q_2})$ 的条件先验为

$\begin{split} \pi _{R2}^i({q_1},{q_2}|\theta ) = &\dfrac{{\sqrt {{h_2}} {\Omega _{12}}}}{{\displaystyle\iint_{{\Omega _{12}}} {\sqrt {{h_2}} {\rm{d}}{q_1}{\rm{d}}{q_2}}}} =\\&\dfrac{{\sqrt {\dfrac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}}} {\Omega _{12}}}}{{\displaystyle\iint_{{\Omega _{12}}} {\sqrt {\dfrac{{{n^2}}}{{{q_1}{q_2}(1 - {q_1} - {q_2})}}} {\rm{d}}{q_1}{\rm{d}}{q_2}}}} =\\& \dfrac{{{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}{\Omega _{12}}}}{{2{\text{π}} }}\propto \\& {q_1}^{ - 1/2}q_2^{ - 1/2}{(1 - {q_1} - {q_2})^{ - 1/2}}{\Omega _{12}}\end{split}$

步骤3 结合式(8)， $\theta$ 的边缘先验为

$\qquad\qquad\begin{split} \pi _{R2}^i(\theta ) =& \dfrac{{\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {\pi _{R2}^i(\,{q_1},\,{q_2}\,|\,\theta \,)\log ({h_1}){\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\Omega _{3i}}}}{{\displaystyle\int_{{\Omega _{3i}}} {\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {\pi _{R2}^i(\,{q_1},\,{q_2}\,|\,\theta \,)\log ({h_1}){\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\rm{d}}\theta } }} =\\& \dfrac{{\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}\log \dfrac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}}{\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\Omega _{3i}}}}{{\displaystyle\int_{{\Omega _{3i}}} {\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {{q_1}^{ - 1/2}q_2^{ - 1/2}{{(1 - {q_1} - {q_2})}^{ - 1/2}}\log \dfrac{{n(1 - {q_1} - {q_2})}}{{\theta {{(1 - \theta )}^2}}}{\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\rm{d}}\theta } }}= \\& \dfrac{{\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {\left[ {\dfrac{{\log (1 - {q_1} - {q_2})}}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }} + \dfrac{1}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }}\log \dfrac{n}{{\theta {{(1 - \theta )}^2}}}} \right]{\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\Omega _{3i}}}}{{\displaystyle\int_{{\Omega _{3i}}} {\exp \left\{ \dfrac{1}{2}\displaystyle\iint_{{\Omega _{12}}} {\left[ {\dfrac{{\log (1 - {q_1} - {q_2})}}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }} + \dfrac{1}{{\sqrt {{q_1}{q_2}(1 - {q_1} - {q_2})} }}\log \dfrac{n}{{\theta {{(1 - \theta )}^2}}}} \right]{\rm{d}}{q_1}{\rm{d}}{q_2}}\right\} {\rm{d}}\theta } }} = \end{split}$

$\qquad\qquad\qquad \dfrac{{\exp \left\{ - 2{\text{π}} + {\text{π}} \log \dfrac{n}{{\theta {{(1 - \theta )}^2}}}\right\} {\Omega _{3i}}}}{{\displaystyle\int_{{\Omega _{3i}}} {\exp \left\{ - 2{\text{π}} + {\text{π}} \log \dfrac{n}{{\theta {{(1 - \theta )}^2}}}\right\} {\rm{d}}\theta } }} = \dfrac{{{{\rm{e}}^{ - 2{\text{π}} }}{{\left( {\dfrac{n}{{\theta {{(1 - \theta )}^2}}}} \right)}^{\text{π}} }{\Omega _{3i}}}}{{{{\rm{e}}^{ - 2{\text{π}} }}\displaystyle\int_{{a_i}}^{{b_i}} {{{\left( {\dfrac{n}{{\theta {{(1 - \theta )}^2}}}} \right)}^{\text{π}} }{\rm{d}}\theta } }} \propto {\theta ^{ - {\text{π}} }}{(1 - \theta )^{ - 2{\text{π}} }}{\Omega _{3i}}$

步骤4 $\varPhi$ 的reference先验为

$\begin{split} {\pi _{R2}} =& \mathop {\lim }\limits_{i \to \infty } \frac{{\pi _{R2}^i(\theta )\pi _{R2}^i({q_1},{q_2}|\theta )}}{{\pi _{R2}^i({\theta ^*})\pi _{R2}^i(q_1^*,q_2^*|{\theta ^*})}} \propto \\& {q_1}^{ - 1/2} q_2^{ - 1/2}{(1 - {q_1} - {q_2})^{ - 1/2}}{\theta ^{ - {\text{π}} }}{(1 - \theta )^{ - 2{\text{π}} }} \end{split}$

式中： $q_1^*$ 、 $q_2^*$ 和 ${\theta ^*}$ 为参数空间中事先给定的值.

3. 后验分析

基于先验分布 ${\pi _J}$ ， ${\pi _{R1}}$ 和 ${\pi _{R2}}$ ，分别得到它们的后验分布，通过R软件进行抽样，获取后验样本. 以 ${\pi _{R1}}$ 为例， $({q_1},{q_2},\theta )$ 的后验分布为

${\pi _{R1}}({q_1},{q_2},\theta \,|\,Y) \propto L({q_1},{q_2},\theta \,|\,Y){\pi _{R1}}$

(9)

式中： $Y = ({Y_1},{Y_2}, \cdots ,{Y_n})$ 为观测数据. 式(9)的具体形式为

$\begin{split}& {\pi _{R1}}({q_1},{q_2},\theta\, |\,Y) \propto \\&q_1^{{S_0} - \frac{1}{2}}q_2^{{S_1} - \frac{1}{2}}{(1 - {q_1} - {q_2})^{n - {S_0} - {S_1} - \frac{1}{2}}} \times \\&{\theta ^{S - 2(n - {S_0} - {S_1}) - \frac{1}{2}}}{(1 - \theta )^{n - {S_0} - {S_1} - 1}}\\[-13pt] \end{split}$

(10)

从式(10)可以看出， $({q_1},{q_2})$ 的后验边缘分布为

$\begin{split} & {\pi _{R1}}({q_1},{q_2}\,|\,Y) \propto\\ &\quad q_1^{{S_0} - \frac{1}{2}}q_2^{{S_1} - \frac{1}{2}}{(1 - {q_1} - {q_2})^{n - {S_0} - {S_1} - \frac{1}{2}}} \end{split}$

(11)

式(11)是形状参数为 $\Bigg({S_0} - \dfrac{1}{2}，{S_1} - \dfrac{1}{2}，n - {S_0} -$ ${S_1} - \dfrac{1}{2}\Bigg)$ 的Dirichlet分布，可以调用R软件包中rdirichlet(N, alpha)函数进行抽样. 另一方面， $\theta$ 的后验边缘分布为

${\pi _{R1}}(\theta \,|\,Y) \propto {\theta ^{S - 2(n - {S_0} - {S_1}) - \frac{1}{2}}}{(1 - \theta )^{n - {S_0} - {S_1} - 1}}$

(12)

从贝塔分布 $B(S - 2(n - {S_0} - {S_1}) + \dfrac{1}{2},$ ${\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n - {S_0} - {S_1})$ 中可以抽样得到式(12)的样本. 通过参数变换表达式式(2)和式(3)， $({q_1},{q_2},\theta )$ 的后验样本可转化为 $({p_1},{p_2},\theta )$ 的后验样本.

4. 数值模拟

本节基于Jeffreys先验和reference先验，通过数值模拟对ZOIGE模型的参数进行估计. 样本容量分别设为 $n = 20$ 和 $n = 50$ ， $\theta$ 值设为 $0.8$ ， ${q_1}$ 的值分别设为 $0.3$ 和 $0.7$ ， ${q_2}$ 值分别设为 $0.4$ 和 $0.6$ ，所有模拟重复2000次，计算出参数估计量的均值和均方误差见表1和表2. 从表中可以看出，随着样本容量的增加，3种客观贝叶斯先验下的估计值越来越接近真值，均方误差也越来越小. 对于 ${q_1}$ 和 ${q_2}$ 的估计， ${\pi _{R1}}$ 、 ${\pi _{R2}}$ 比 ${\pi _J}$ 表现更好，这是因为在 ${\pi _{R1}}$ 、 ${\pi _{R2}}$ 中包含 ${q_1}$ 和 ${q_2}$ 的信息更加丰富. 对于 $\theta$ 的估计， ${\pi _{R2}}$ 比 ${\pi _{R1}}$ 、 ${\pi _J}$ 表现更好，这是因为在 ${\pi _{R2}}$ 中 $\theta$ 为感兴趣参数，集中了更多的样本信息.

表 1

$\theta = 0.8$ 下参数估计量的均值

Table 1. Mean of parameter estimators when

$\theta = 0.8$

${q_1}$	${q_2}$	$n$	${\theta _J}$	${q_{1J}}$	${q_{2J}}$	${\theta _{R1}}$	${q_{1R1}}$	${q_{2R1}}$	${\theta _{R2}}$	${q_{1R2}}$	${q_{2R2}}$
0.3	0.4	20	0.768	0.283	0.357	0.783	0.289	0.361	0.792	0.288	0.362
	0.4	50	0.779	0.284	0.374	0.787	0.291	0.388	0.793	0.289	0.387
	0.6	20	0.776	0.282	0.504	0.785	0.290	0.556	0.788	0.290	0.555
	0.6	50	0.781	0.287	0.584	0.791	0.302	0.592	0.792	0.298	0.594
0.7	0.4	20	0.772	0.669	0.347	0.786	0.672	0.373	0.790	0.671	0.376
	0.4	50	0.787	0.685	0.376	0.792	0.694	0.388	0.794	0.693	0.386
	0.6	20	0.775	0.724	0.545	0.784	0.724	0.575	0.789	0.723	0.581
	0.6	50	0.791	0.721	0.569	0.793	0.716	0.594	0.795	0.715	0.593

下载: 导出CSV

| 显示表格

表 2

$\theta = 0.8$ 下参数估计量的均方误差

Table 2. Mean squared error of parameter estimators when

$\theta = 0.8$

${q_1}$	${q_2}$	$n$	${\theta _J}$	${q_{1J}}$	${q_{2J}}$	${\theta _{R1}}$	${q_{1R1}}$	${q_{2R1}}$	${\theta _{R2}}$	${q_{1R2}}$	${q_{2R2}}$
0.3	0.4	20	0.083	0.074	0.098	0.081	0.072	0.088	0.079	0.072	0.087
	0.4	50	0.065	0.037	0.085	0.062	0.033	0.075	0.061	0.034	0.075
	0.6	20	0.076	0.072	0.093	0.078	0.062	0.092	0.075	0.063	0.091
	0.6	50	0.061	0.036	0.072	0.058	0.024	0.063	0.055	0.026	0.062
0.7	0.4	20	0.087	0.045	0.096	0.086	0.041	0.086	0.084	0.042	0.084
	0.4	50	0.056	0.038	0.074	0.066	0.032	0.073	0.061	0.035	0.072
	0.6	20	0.065	0.037	0.083	0.072	0.031	0.082	0.069	0.032	0.081
	0.6	50	0.057	0.035	0.078	0.056	0.026	0.072	0.045	0.028	0.072

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5. 结　语

本研究对0–1膨胀几何分布模型进行客观贝叶斯分析，巧妙地进行重参数化，写出具有分块对角形式的Fisher信息矩阵. 因而，较容易推导出参数的Jeffreys先验和reference先验，这种方法和技巧可以推广到其他形式的0–1膨胀分布模型中去.

表 1 $\theta = 0.8$ 下参数估计量的均值

Table 1. Mean of parameter estimators when $\theta = 0.8$

${q_1}$	${q_2}$	$n$	${\theta _J}$	${q_{1J}}$	${q_{2J}}$	${\theta _{R1}}$	${q_{1R1}}$	${q_{2R1}}$	${\theta _{R2}}$	${q_{1R2}}$	${q_{2R2}}$
0.3	0.4	20	0.768	0.283	0.357	0.783	0.289	0.361	0.792	0.288	0.362
	0.4	50	0.779	0.284	0.374	0.787	0.291	0.388	0.793	0.289	0.387
	0.6	20	0.776	0.282	0.504	0.785	0.290	0.556	0.788	0.290	0.555
	0.6	50	0.781	0.287	0.584	0.791	0.302	0.592	0.792	0.298	0.594
0.7	0.4	20	0.772	0.669	0.347	0.786	0.672	0.373	0.790	0.671	0.376
	0.4	50	0.787	0.685	0.376	0.792	0.694	0.388	0.794	0.693	0.386
	0.6	20	0.775	0.724	0.545	0.784	0.724	0.575	0.789	0.723	0.581
	0.6	50	0.791	0.721	0.569	0.793	0.716	0.594	0.795	0.715	0.593

下载: 导出CSV

表 2 $\theta = 0.8$ 下参数估计量的均方误差

Table 2. Mean squared error of parameter estimators when $\theta = 0.8$

${q_1}$	${q_2}$	$n$	${\theta _J}$	${q_{1J}}$	${q_{2J}}$	${\theta _{R1}}$	${q_{1R1}}$	${q_{2R1}}$	${\theta _{R2}}$	${q_{1R2}}$	${q_{2R2}}$
0.3	0.4	20	0.083	0.074	0.098	0.081	0.072	0.088	0.079	0.072	0.087
	0.4	50	0.065	0.037	0.085	0.062	0.033	0.075	0.061	0.034	0.075
	0.6	20	0.076	0.072	0.093	0.078	0.062	0.092	0.075	0.063	0.091
	0.6	50	0.061	0.036	0.072	0.058	0.024	0.063	0.055	0.026	0.062
0.7	0.4	20	0.087	0.045	0.096	0.086	0.041	0.086	0.084	0.042	0.084
	0.4	50	0.056	0.038	0.074	0.066	0.032	0.073	0.061	0.035	0.072
	0.6	20	0.065	0.037	0.083	0.072	0.031	0.082	0.069	0.032	0.081
	0.6	50	0.057	0.035	0.078	0.056	0.026	0.072	0.045	0.028	0.072

下载: 导出CSV

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${q_1}$	${q_2}$	$n$	${\theta _J}$	${q_{1J}}$	${q_{2J}}$	${\theta _{R1}}$	${q_{1R1}}$	${q_{2R1}}$	${\theta _{R2}}$	${q_{1R2}}$	${q_{2R2}}$
0.3	0.4	20	0.768	0.283	0.357	0.783	0.289	0.361	0.792	0.288	0.362
	0.4	50	0.779	0.284	0.374	0.787	0.291	0.388	0.793	0.289	0.387
	0.6	20	0.776	0.282	0.504	0.785	0.290	0.556	0.788	0.290	0.555
	0.6	50	0.781	0.287	0.584	0.791	0.302	0.592	0.792	0.298	0.594
0.7	0.4	20	0.772	0.669	0.347	0.786	0.672	0.373	0.790	0.671	0.376
	0.4	50	0.787	0.685	0.376	0.792	0.694	0.388	0.794	0.693	0.386
	0.6	20	0.775	0.724	0.545	0.784	0.724	0.575	0.789	0.723	0.581
	0.6	50	0.791	0.721	0.569	0.793	0.716	0.594	0.795	0.715	0.593