两次测量估两根杆长：能否优于各测一次

Doubling Your Accuracy

专题: Statistics / 统计
难度: L4
来源: QuantQuestion

题目详情

An unbiased instrument for measuring distances makes random errors whose distribution has standard deviation $\sigma$ . You are allowed two measurements all told to estimate the lengths of two cylindrical rods, one clearly longer than the other. Can you do better than to take one measurement on each rod? (An unbiased instrument is one that on the average gives the true measure.)

解析

可以。

设长杆真实长度 $A$ ，短杆为 $B$ 。用两次测量分别测“差”和“和”：

测 $D=(A-B)+\varepsilon_D$
测 $S=(A+B)+\varepsilon_S$

其中误差独立、均值 0、方差均为 $\sigma^2$ 。

则估计量

\hat A=\frac{D+S}{2}=A+\frac{\varepsilon_D+\varepsilon_S}{2},\qquad \hat B=\frac{S-D}{2}=B+\frac{\varepsilon_S-\varepsilon_D}{2}.

两者无偏。

方差：

\mathrm{Var}(\hat A)=\frac14(\sigma^2+\sigma^2)=\boxed{\frac12\sigma^2},

同理 $\mathrm{Var}(\hat B)=\frac12\sigma^2$ 。

相比“各测一次”得到的方差是 $\sigma^2$ ，该方案把方差减半。