961
个编辑
更改
跳到导航
跳到搜索
第82行：
第82行： + + + + + +
→Kolmogorov distribution Kolmogorov 分布
:<math>K=\sup_{t\in[0,1]}|B(t)|</math>
:<math>K=\sup_{t\in[0,1]}|B(t)|</math>
Kolmogorov分布是随机变量的分布
:<math>K=\sup_{t\in[0,1]}|B(t)|</math>
其中B（t）是布朗桥。K的累积分布函数为
\operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^\infty (-1)^{k-1} e^{-2k^2 x^2}=\frac{\sqrt{2\pi}}{x}\sum_{k=1}^\infty e^{-(2k-1)^2\pi^2/(8x^2)},
which can also be expressed by the Jacobi theta function \vartheta_{01}(z=0;\tau=2ix^2/\pi). Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available.
which can also be expressed by the Jacobi theta function \vartheta_{01}(z=0;\tau=2ix^2/\pi). Both the form of the Kolmogorov–Smirnov test statistic and its asymptotic distribution under the null hypothesis were published by Andrey Kolmogorov, while a table of the distribution was published by Nikolai Smirnov. Recurrence relations for the distribution of the test statistic in finite samples are available.