0 引言

Lomax分布在寿命试验数据处理中起着重要的作用，很多统计学者对此分布进行了深入的探讨.文献[1]研究了Lomax分布参数极大似然估计的存在性和估计量的收敛性.文献[2]在完全样本下研究了两个参数及分位数的区间估计和假设检验.文献[3]研究了基于缺失数据样本下Lomax分布尺度参数的估计，并说明了确定最优置信区间的方法.文献[4-8]在不同损失函数下，当尺度参数已知时，讨论了形状参数的贝叶斯估计问题.在利用统计方法处理试验数据时，如何根据缺失数据进行统计推断是统计分析中的一个重要问题.文献[9-12]讨论了多种分布在缺失数据样本下的参数估计问题，而对Lomax分布在定时截尾数据缺失样本下的参数估计还没有人研究.本文假设尺度参数已知，在定时截尾数据缺失样本下给出了形状参数的极大似然估计，证明了估计量的相合性和渐近正态性，并给出了形状参数的置信区间和假设检验.

1 极大似然估计及其渐近性质

设样本观测数据来自Lomax分布总体，其密度函数为

$ f\left( {x;\theta, \lambda } \right) = \frac{\theta }{\lambda }{\left( {1 + \frac{x}{\lambda }} \right)^{ - \left( {\theta + 1} \right)}}, x, \theta, \lambda > 0, $

(1)

其中:θ为形状参数；λ为尺度参数；在本文中假设尺度参数已知.

现对上述Lomax分布总体进行n次独立观测，并到T₀时刻停止，每个样本观测值以概率1-p缺失, 以概率p被观测.用(Z_i, δ_i, α_i), i=1, 2, …，n表示总体观测值，其中：Z_i=min(T₀, X_i), X_i表示第i个样品的寿命；α_i=I_{Xi≤T0}-I_{Xi>T0}，即观测到具体的失效时间α_i=1，否则α_i=-1，并且第i个样品观测数据缺失时记δ_i=0，否则δ_i=1.

下面用极大似然估计方法对未知形状参数θ进行估计，基于上述样本观测值(Z_i, δ_i, α_i), i=1, 2, …, n, 可得到似然函数为L(θ)=$\prod\limits_{i = 1}^n {{{\left[{\frac{\theta }{\lambda }{{\left( {1 + \frac{{{Z_i}}}{\lambda }} \right)}^{-\left( {\theta + 1} \right)}}} \right]}^{{A_i}}}{{\left[{{{\left( {1 + \frac{{{Z_i}}}{\lambda }} \right)}^{-\theta }}} \right]}^{{B_i}}}} $，其中A_i=α_iδ_i(α_iδ_i+1)/2，B_i=α_iδ_i(α_iδ_i-1)/2，i=1, 2, …, n.对似然函数取对数，并由$ \frac{{\partial \ln L\left( \theta \right)}}{{\partial \theta }} = 0$，解得θ的极大似然估计为

$ \hat \theta = \frac{{\sum\limits_{i = 1}^n {{A_i}} }}{{\sum\limits_{i = 1}^n {\left( {{A_i} + {B_i}} \right)\ln \left( {1 + {\mathit{Z}_i}/\lambda } \right)} }} = \frac{{\sum\limits_{i = 1}^n {\left( {{\alpha _i}{\mathit{\delta }_i} + \alpha _i^2\delta _i^2} \right)} }}{{2\sum\limits_{i = 1}^n { {\alpha _i^2\delta _i^2} \ln \left( {1 + {\mathit{Z}_i}/\lambda } \right)} }}. $

(2)

由θ的极大似然估计可得到如下定理.

定理1  若(Z_i, δ_i, α_i), i=1, 2, …, n，是来自Lomax分布总体(1) 的样本观测值，则有$\hat \theta \to \theta $, a.s.

证明  由于{α_iδ_i, 1≤i≤n}为独立同分布随机变量，因此由强大数定律可得$\frac{1}{n}\sum\limits_{i = 1}^n {{\alpha _i}{\delta _i} \to E\left( {{\alpha _i}{\delta _i}} \right)} $, a.s. 其中E(α₁δ₁)=E(α₁)E(δ₁)=p[1-2(1+T₀/λ)^-θ]，因此$\frac{1}{n}\sum\limits_{i = 1}^n {{\alpha _i}{\delta _i} \to p\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right], a.s} $.又由于$E\left( {\alpha _1^2\delta _1^2} \right) = E\left( {\alpha _1^2} \right)E\left( {\delta _1^2} \right) = p, E\left( {\alpha _1^2\delta _1^2\ln \left( {1 + {Z_1}/\lambda } \right)} \right) = \frac{p}{\theta }\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right] $，则$\frac{1}{n}\sum\limits_{i = 1}^n {\alpha _1^2\delta _1^2 \to p, a.s.\frac{1}{n}\sum\limits_{i = 1}^n {\alpha _i^2\delta _i^2\ln} \left( {1 + {Z_i}/\lambda } \right) \to \frac{p}{\theta }\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right], a.s.} $由Slusky定理得到

$ \hat \theta = \frac{{\sum\limits_{i = 1}^n {\left( {{\alpha _i}{\mathit{\delta }_i} + \alpha _i^2\delta _i^2} \right)} }}{{2\sum\limits_{i = 1}^n { {\alpha _i^2\delta _i^2} \ln \left( {1 + {\mathit{Z}_i}/\lambda } \right)} }} = \frac{{\frac{1}{n}\sum\limits_{i = 1}^n {{\alpha _i}{\mathit{\delta }_i}} + \frac{1}{n}\sum\limits_{i = 1}^n {\alpha _i^2\delta _i^2} }}{{2\frac{1}{n}\sum\limits_{i = 1}^n {\alpha _i^2\delta _i^2\ln \left( {1 + {\mathit{Z}_i}/\lambda } \right)} }} \to \frac{{p\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right] + p}}{{\frac{{2p}}{\theta }\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right]}} = \theta, a.s. $

引理1^[9]  记T_n=(T_1n, …, T_kn)^T, β=(β₁, …, β_k)^T，设$ \sqrt n \left( {{\pmb{T}_n}-\pmb{\beta }} \right)\mathop \to \limits^L N\left( {0, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }} } \right)$其中Σ=(σ_ij)_k×k.又设g(t₁, …, t_n)对各t_i有连续偏导数，则当n→∞时，有$ \sqrt n \left[{g\left( {{T_{1n}}, \cdots, {T_{kn}}} \right)-g\left( {{\beta _1}, \cdots {\beta _k}} \right)} \right]\mathop \to \limits^L N\left( {0, {\sigma ^2}\left( \pmb{\beta } \right)} \right)$，其中${\sigma ^2}\left( \pmb{\beta } \right) = \sum {} \sum {} \left( {\frac{{\partial g}}{{\partial {\beta _i}}}\frac{{\partial g}}{{\partial {\beta _j}}}{\sigma _{ij}}} \right) $.

定理2  在前述记号下，$\sqrt n \left( {\hat \theta- \theta } \right)\mathop \to \limits^L N\left( {0, \frac{{{\theta ^2}}}{{p\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right]}}} \right) $.

证明  令${\pmb{W}_i} = {\left( {{\alpha _i}{\delta _i}, \alpha _i^2\delta _i^2, 2\alpha _i^2\delta _i^2\ln \left( {1 + {Z_i}/\lambda } \right)} \right)^{\rm{T}}} $，则{W_i, i≥1}为独立同分布随机变量序列，且E(W₁)=(p[1-2(1+T₀/λ)^-θ], $p, \frac{{2p}}{\theta }\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right] $)^T，令Σ=E(W₁-EW₁)(W₁-EW₁)^T，则由多元中心极限定理可得$ \sqrt n \left( {\frac{1}{n}\sum\limits_{i = 1}^n {{\pmb{W}_i}-\pmb{E}\left( {{W_1}} \right)} } \right)\mathop \to \limits^L N\left( {0, \mathit{\boldsymbol{ \boldsymbol{\varSigma} }} } \right)$, 记Σ=(σ_ij), (i, j=1, 2, 3).则

$ {\sigma _{11}} = E\left( {\alpha _1^2\mathit{\delta }_1^2} \right) - {\left( {E\left( {{\alpha _1}{\mathit{\delta }_1}} \right)} \right)^2} = p - {p^2}{\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right]^2}, $

$ \begin{array}{l} {\sigma _{12}} = {\sigma _{21}} = E\left( {\alpha _1^3\mathit{\delta }_1^3} \right) - E\left( {{\alpha _1}{\mathit{\delta }_1}} \right)E\left( {\alpha _1^2\mathit{\delta }_1^2} \right) = \\p\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right] - {p^2}\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right] = \left( {p - {p^2}} \right)\left[{1-2{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right], \end{array} $

$ \begin{array}{l} {\sigma _{13}} = {\sigma _{31}} = 2E\left( {\alpha _1^3\mathit{\delta }_1^3\ln \left( {1 + {Z_1}/\lambda } \right)} \right) - 2E\left( {{\alpha _1}{\mathit{\delta }_1}} \right)E\left( {\alpha _1^2\mathit{\delta }_1^2\ln \left( {1 + {Z_1}/\lambda } \right)} \right) =\\ \frac{{2p}}{\theta }\left[{1-{{\left( {1 + {T_0}/\lambda } \right)}^{^{-\theta }}}} \right] - 4p{\left( {1 + {T_0}/\lambda } \right)^{^{ - \theta }}}\ln \left( {1 + {T_0}/\lambda } \right) - \frac{{2{p^2}}}{\theta }\\\left( {1 - 2{{\left( {1 + {T_0}/\lambda } \right)}^{^{ - \theta }}}} \right)\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{^{ - \theta }}}} \right), \end{array} $

$ {\sigma _{22}} = E\left( {\alpha _1^4\mathit{\delta }_1^4} \right) - {\left( {E\left( {\alpha _1^2\mathit{\delta }_1^2} \right)} \right)^2} = p - {p^2}, $

$ \begin{array}{l} {\sigma _{23}} = {\sigma _{32}} = 2E\left( {\alpha _1^4\mathit{\delta }_1^4\ln \left( {1 + {Z_1}/\lambda } \right)} \right) - \\ 2E\left( {\alpha _1^2\mathit{\delta }_1^2} \right)E\left( {\alpha _1^2\mathit{\delta }_1^2\ln \left( {1 + {Z_1}/\lambda } \right)} \right) = \frac{{2p\left( {1 - p} \right)}}{\theta }\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{^{ - \theta }}}} \right), \end{array} $

$ \begin{array}{l} {\sigma _{33}} = 4E\left( {\alpha _1^4\mathit{\delta }_1^4{{\left[{\ln \left( {1 + {Z_1}/\lambda } \right)} \right]}^2}} \right) - 4{\left[{E\left( {\alpha _1^2\mathit{\delta }_1^2\ln \left( {1 + {Z_1}/\lambda } \right)} \right)} \right]^2} = \\\frac{{8p}}{{{\theta ^2}}}\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{^{ - \theta }}}} \right) - \frac{{8p}}{\theta }{\left( {1 + {T_0}/\lambda } \right)^{^{ - \theta }}}\ln \left( {1 + {T_0}/\lambda } \right) - \frac{{4{p^2}}}{{{\theta ^2}}}{\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{^{ - \theta }}}} \right)^2}, \end{array} $

$ g\left( {{t_1}, {t_2}, {t_3}} \right) = \frac{{{t_1} + {t_2}}}{{{t_3}}}, {T_{1n}} = \frac{1}{n}\sum\limits_{i = 1}^n {{\alpha _i}} {\mathit{\delta }_i}, {\mathit{\beta }_1} = p\left( {1 - 2{{\left( {1 + {T_0}/\lambda } \right)}^{ - \theta }}} \right), $

$ {\mathit{T}_{2n}} = \frac{1}{n}\sum\limits_{i = 1}^n {\alpha _i^2} \delta _i^2, {\mathit{\beta }_2} = p, {T_{3n}} = \frac{2}{n}\sum\limits_{i = 1}^n {\alpha _i^2\delta _i^2\ln } \left( {1 + {Z_i}/\lambda } \right), {\mathit{\beta }_3} =\\ \frac{{2p}}{\theta }\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \theta }}} \right), $

可得$g\left( {{T_{1n}}, {T_{2n}}, {T_{3n}}} \right) = \hat \theta, g\left( {{\beta _1}, {\beta _2}, {\beta _3}} \right) = \frac{{{\beta _1} + {\beta _2}}}{{{\beta _3}}} = \theta $.

$ \frac{{\partial g}}{{\partial {\mathit{\beta }_1}}} = \frac{{\partial g}}{{\partial {\mathit{\beta }_2}}} = \frac{1}{{{\mathit{\beta }_3}}} = \frac{\theta }{{2p\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \theta }}} \right)}}\frac{{\partial g}}{{\partial {\mathit{\beta }_3}}} = - \frac{{{\mathit{\beta }_{\rm{1}}}\mathit{ + }{\mathit{\beta }_{\rm{2}}}}}{{\mathit{\beta }_3^2}} = - \frac{{{\theta _1}}}{{{\mathit{\beta }_3}}} = - \frac{{{\theta ^2}}}{{2p\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \theta }}} \right)}}. $

因此由引理1可得到$ \sqrt n \left( {\hat \theta-\theta } \right) = \sqrt n \left( {g\left( {{T_{1n}}, {T_{2n}}, {T_{3n}}} \right)-g\left( {{\beta _1}, {\beta _2}, {\beta _3}} \right)} \right)\mathop \to \limits^L N\left( {0, {\sigma ^2}} \right)$，其中，

$ {\sigma ^2} = \left( {\frac{{\partial g}}{{\partial {\mathit{\beta }_1}}}, \frac{{\partial g}}{{\partial {\mathit{\beta }_2}}}, \frac{{\partial g}}{{\partial {\mathit{\beta }_3}}}} \right)\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}{\left( {\frac{{\partial g}}{{\partial {\mathit{\beta }_1}}}, \frac{{\partial g}}{{\partial {\mathit{\beta }_2}}}, \frac{{\partial g}}{{\partial {\mathit{\beta }_3}}}} \right)^{\rm{T}}} = \frac{{{\theta ^2}}}{{p\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \theta }}} \right)}}, $

即        $\sqrt n \left( {\hat \theta-\theta } \right)\mathop \to \limits^L N\left( {0, \frac{{{\theta ^2}}}{{p\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\theta }}} \right)}}} \right). $

2 区间估计及假设检验

在实际中，对参数真值范围的研究，可以归结到参数的置信区间问题.对于本文中讨论的问题可得到如下定理.

定理3  在前面的记号下，如果${\hat \theta } $是(2) 式所给出的形状参数θ的极大似然估计，若0 < γ < 1，则θ的置信水平为1-γ的近似置信区间为

$ \left( {\hat \theta - {u_{1 - \mathit{\gamma }/2}}\sqrt {\frac{{{{\hat \theta }^2}}}{{np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)}}, } \hat \theta + {u_{1 - \mathit{\gamma }/2}}\sqrt {\frac{{{{\hat \theta }^2}}}{{np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)}}} } \right), $

其中u_γ为标准正态分布的γ下分位数.

证明  由于$ \hat \theta \to \theta, a.s.$因此$\frac{{\sqrt {np\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\hat \theta }}} \right)} \left( {\hat \theta-\theta } \right)}}{{\hat \theta }}\mathop \to \limits^L N\left( {0, 1} \right) $.

$ p\left\{ { - {u_{1 - \mathit{\gamma }/2}} < \frac{{\sqrt {np\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\hat \theta }}} \right)} \left( {\hat \theta-\theta } \right)}}{{\hat \theta }} < {u_{1 - \mathit{\gamma }/2}}} \right\} \approx 1 - \mathit{\gamma, } $

$ \mathit{p}\left\{ {\hat \theta - {u_{1 - \mathit{\gamma }/2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }} < \theta < \hat \theta + {u_{1 - \mathit{\gamma }/2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }}} \right\} \approx 1 - \mathit{\gamma, } $

因此θ的置信水平为1-γ的近似置信区间为

$ \left( {\hat \theta - {u_{1 - \mathit{\gamma }/2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }}, \hat \theta + {u_{1 - \mathit{\gamma }/2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }}} \right). $

由于Lomax的分布函数为$ F\left( x \right) = 1-{\left( {1 + \frac{x}{\lambda }} \right)^{-\theta }}$，对任意p∈(0, 1)，当F(x_p)=p时，解得Lomax分布的p分位数为$ {x_p} = \lambda \left[{{{\left( {1-p} \right)}^{-\frac{1}{\theta }}}-1} \right]$.

因为x_p是θ的单调递减函数，因此当λ已知时，p分位数x_p的置信水平为1-γ的近似置信区间为

$ \left( {\lambda \left( {{{\left( {1 - p} \right)}^{ - \frac{1}{{{{\hat \theta }_U}}}}} - 1} \right), \lambda \left( {{{\left( {1 - p} \right)}^{ - \frac{1}{{{{\hat \theta }_L}}}}} - 1} \right)} \right), $

其中: ${{\hat \theta }_L} = \hat \theta-{u_{1-\gamma /2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }};{{\hat \theta }_U} = \hat \theta + {u_{1 - \gamma /2}}\frac{{\hat \theta }}{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} }} $.

由文献[4]知，Lomax分布的失效率函数为$H\left( x \right) = \frac{\theta }{{\lambda + x}} $，可靠度函数为$ R\left( x \right) = {\left( {1 + \frac{x}{\lambda }} \right)^{-\theta }}$.且$\frac{\theta }{{\lambda + x}} $是θ的单调递增函数，$ {\left( {1 + \frac{x}{\lambda }} \right)^{-\theta }}$是θ的单调递减函数.由形状参数θ的置信区间可得到失效率的置信水平为1-γ的近似置信区间为$\left( {\frac{{{{\hat \theta }_L}}}{{\lambda + x}}, \frac{{{{\hat \theta }_U}}}{{\lambda + x}}} \right) $.

同样可得到可靠度的置信水平为1-γ的近似置信区间为$\left( {{{\left( {1 + \frac{x}{\lambda }} \right)}^{-{{\hat \theta }_U}}}, {{\left( {1 + \frac{x}{\lambda }} \right)}^{-{{\hat \theta }_L}}}} \right) $.

1) 对于假设检验问题H₀:θ=θ₀↔H₁:θ≠θ₀，其中θ₀已知，当H₀成立时$ \frac{{\sqrt {np\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\hat \theta }}} \right)} \left( {\hat \theta-{\theta _0}} \right)}}{{\hat \theta }}\mathop \to \limits^L N\left( {0, 1} \right)$，对于给定的显著性水平γ(0 < γ < 1)，检验的拒绝域为

$ {\mathit{\boldsymbol{W}}_1} = \left\{ {\left( {{\mathit{Z}_i}, {\delta _i}, {\alpha _i}} \right), i = 1, 2, \cdots, n} ||\frac{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} \left( {\hat \theta - {\theta _0}} \right)}}{{\hat \theta }}| \ge {u_{1 - \mathit{\gamma }/2}} \right\}. $

2) 对于假设检验问题H₀:θ>θ₀↔H₁:θ≤θ₀.同理可得，对于给定的显著性水平γ(0 < γ < 1)，检验的拒绝域为${\pmb{W}_2} = \left\{ {\left( {{Z_i}, {\delta _i}, {\alpha _i}} \right), i = 1, 2, \cdots, n|\frac{{\sqrt {np\left( {1-{{\left( {1 + {T_0}/\lambda } \right)}^{-\hat \theta }}} \right)} \left( {\hat \theta-{\theta _0}} \right)}}{{\hat \theta }} \le {u_\gamma }} \right\} $.

3) 对于假设检验问题H₀:θ < θ₀↔H₁:θ≥θ₀.同理可得，对于给定的显著性水平γ(0 < γ < 1)，检验的拒绝域为

$ {\mathit{\boldsymbol{W}}_3} = \left\{ {\left( {{\mathit{Z}_i}, {\delta _i}, {\alpha _i}} \right), i = 1, 2, \cdots, n} | {\frac{{\sqrt {np\left( {1 - {{\left( {1 + {T_0}/\lambda } \right)}^{ - \hat \theta }}} \right)} \left( {\hat \theta - {\theta _0}} \right)}}{{\hat \theta }}} \ge {u_{1 - \mathit{\gamma }}} \right\}. $

根据定理2可以得到两个独立Lomax分布总体形状参数之差的区间估计和假设检验问题.

3 随机模拟

当λ分别取1.2和2时，在形状参数θ取不同真值的情况下，通过随机模拟的方法，产生一个服从Lomax分布(1) 的样本，且样本容量n=100.取缺失概率1-P=0.1，置信水平1-γ=0.95，对于给定截尾时间T₀，利用上述样本可以得到参数θ的估计，以上过程重复1 000次，可以得到参数估计的均值、均方误差、置信区间的上下限均值及覆盖率，模拟结果见表 1.对形状参数的估计都很接近参数真值，并且均方误差较小.θ的真值介于下限均值与上限均值之间，覆盖率很接近近似置信水平0.95.