卡方分布密度函数与分布函数的渐近展开

Asymptotic Expansions of the Probability Density Function and the Distribution Function of Chi-Square Distribution

通过对χ2分布概率密度函数的自变量进行标准化变换,将其展开成如下形式:(1/2)nχ2(x;n)=1+r1(t)n+r2(t)n+r3(t)n n+r4(t)n2[]φ(t)+o1n2(),其中n为自由度,φ(t)为标准正态分布的密度函数,ri(t)(1≤i≤4)均为关于t的多项式.从该展开式得到χ2分布密度函数的一个近似计算公式.进一步建立φ(t)的幂系数积分递推关系,得到χ2分布函数的渐近展开式.最后通过数值计算验证了这些结果在实际应用中的有效性.

Through the transformation of the independent variable of χ^2distribution probability density function,degree of freedom of which is n,the equation can be expanded as follows： √2nχ^2（x;n）=[1＋r1（t）/√n＋r2（t）/√n＋r3（t）/n√n＋r4（t）/n^2]φ（t）＋o（1/n^2）,here,φ（ t） is a density function of standard normal distribution;ri（ t） is a 3i order polynomial of t（ 1≤i≤4）. An approximate formula can be obtained from the expansion of the distribution density function. We further establish the integral recurrence relations of the power coefficients of the standard normal density function and obtain the asymptotic expansion of the distribution function of χ^2. Finally,the effectiveness of these results in practical application was verified by the numerical calculations.