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→Discrete and mixed null distribution 离散和混合零分布
From the right-continuity of <math>F(x)</math>, it follows that <math>F(F^{-1}(t)) \geq t</math> and <math>F^{-1}(F(x)) \leq x </math> and hence, the distribution of <math>D_{n}</math> depends on the null distribution <math>F(x)</math>, i.e., is no longer distribution-free as in the continuous case. Therefore, a fast and accurate method has been developed to compute the exact and asymptotic distribution of <math>D_{n}</math> when <math>F(x)</math> is purely discrete or mixed , implemented in C++ and in the KSgeneral package of the [[R (programming language)|R language]]. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> compute also the KS test statistic and p-values for purely discrete, mixed or continuous null distributions and arbitrary sample sizes. The KS test and its p-values for discrete null distributions and small sample sizes are also computed in as part of the dgof package of the R language. Major statistical packages among which [[SAS (software)|SAS]] <code>PROC NPAR1WAY</code> , [[Stata]] <code>ksmirnov</code> implement the KS test under the assumption that <math>F(x)</math> is continuous, which is more conservative if the null distribution is actually not continuous (see [15] [16] [17]).
From the right-continuity of <math>F(x)</math>, it follows that <math>F(F^{-1}(t)) \geq t</math> and <math>F^{-1}(F(x)) \leq x </math> and hence, the distribution of <math>D_{n}</math> depends on the null distribution <math>F(x)</math>, i.e., is no longer distribution-free as in the continuous case. Therefore, a fast and accurate method has been developed to compute the exact and asymptotic distribution of <math>D_{n}</math> when <math>F(x)</math> is purely discrete or mixed , implemented in C++ and in the KSgeneral package of the [[R (programming language)|R language]]. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> compute also the KS test statistic and p-values for purely discrete, mixed or continuous null distributions and arbitrary sample sizes. The KS test and its p-values for discrete null distributions and small sample sizes are also computed in as part of the dgof package of the R language. Major statistical packages among which [[SAS (software)|SAS]] <code>PROC NPAR1WAY</code> , [[Stata]] <code>ksmirnov</code> implement the KS test under the assumption that <math>F(x)</math> is continuous, which is more conservative if the null distribution is actually not continuous (see [15] [16] [17]).
从F（x）的右连续性，可以得出<math>F(F^{-1}(t)) \geq t</math>和<math>F^{-1}(F(x)) \leq x </math>，因此Dn的分布取决于零分布F（x），即在连续情况下不再无分布。目前已经开发出一种快速，准确的方法，以C ++和R语言的KSgeneral软件包来实现，当F（x）是纯离散或混合时，可以计算Dn的精确且渐近分布。函数disc_ks_test（），mixed_ks_test（）和cont_ks_test（）还针对纯离散，混合或连续的零分布和任意样本大小，计算KS检测统计量和p值。作为R语言的dgof软件包的一部分，还可以计算出KS检测及其用于离散零分布和小样本量的p值。另外主要统计软件包，其中SAS PROC NPAR1WAY和Stata ksmirnov是假设F（x）是连续的，因此执行KS检验时，如果零分布实际上不是连续的，则该检验更为保守。详情请见：
从F（x）的右连续性，可以得出<math>F(F^{-1}(t)) \geq t</math>和<math>F^{-1}(F(x)) \leq x </math>，因此Dn的分布取决于零分布F（x），即在连续情况下不再无分布。目前已经开发出一种快速，准确的方法，以C ++和R语言的KSgeneral软件包来实现，当F（x）是纯离散或混合时，可以计算出Dn的精确且渐近分布。函数disc_ks_test（），mixed_ks_test（）和cont_ks_test（）还可以针对纯离散，混合或连续的零分布和任意样本大小，计算出KS检测统计量和p值。另外作为R语言的dgof软件包的一部分，还可以计算出KS检测及其用于离散零分布和小样本量的p值。关于主要统计软件包，其中SAS PROC NPAR1WAY和Stata ksmirnov是假设F（x）是连续的，因此执行KS检验时，如果零分布实际上不是连续的，则该检验更为保守。详情请见：
1. 《关于离散案例中的Kolmogorov统计量的注释Note on the Kolmogorov Statistic in the Discrete Case》
1. 《关于离散案例中的Kolmogorov统计量的注释Note on the Kolmogorov Statistic in the Discrete Case》
2. 《皮尔逊卡方检验和Kolmogorov拟合优度检验在有效性方面的比较A Comparison of the Pearson Chi-Square and Kolmogorov Goodness-of-Fit Tests with Respect to Validity》
2. 《皮尔逊卡方检验和Kolmogorov拟合优度检验在有效性方面的比较A Comparison of the Pearson Chi-Square and Kolmogorov Goodness-of-Fit Tests with Respect to Validity》