🙅🏿 👯 ⛸️ 简单快速地近似统计函数 🛤️ 🚣 🏆

任务。有一个计算器，但手头没有统计表。例如，您需要学生分布的关键点表来计算置信区间。使用Excel获取计算机？不运动。

不需要很高的精度，您可以使用近似公式。下列公式的想法是通过变换参数，可以将所有分布以某种方式简化为正态分布。近似值应既提供累积分布函数的计算，又提供其反函数的计算。

让我们从正态分布开始。

Φ (z) = P = \frac{1}{2} [1 + e r f (\frac{z}{\sqrt{2}})]

$\Phi(z)=P=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{z}{\sqrt{2}}\right)\right]$

z = Φ^{- 1} (P) = \sqrt{2} \cdot {e r f}^{- 1} (2 P - 1)

$z=\Phi^{-1}(P)=\sqrt{2}\cdot\mathrm{erf}^{-1}(2P-1)$

它需要计算功能 $\mathrm{erf}(x)$ 和相反。我使用了近似值[1]：

e r f (x) = s i g n (x) \cdot \sqrt{1 - \exp (- x^{2} \cdot \frac{\frac{4}{π} + a x^{2}}{1 + a x^{2}})}

$\mathrm{erf}(x)=\mathrm{sign}(x)\cdot\sqrt{1-\exp\left(-x^{2}\cdot\frac{\frac{4}{\pi}+ax^{2}}{1+ax^{2}}\right)}$

{e r f}^{- 1} (x) = s i g n (x) \cdot \sqrt{- t_{2} + \sqrt{t_{2}^{2} - \frac{1}{a} \cdot \ln t_{1}}}

$\mathrm{erf}^{-1}(x)=\mathrm{sign}(x)\cdot\sqrt{-t_2 + \sqrt{t_2^{2}-\frac{1}{a}\cdot \ln t_1}}$

哪里 $t_1$ 和 $t_2$ -辅助变量：

t_{1} = 1 - x^{2}, t_{2} = \frac{2}{π a} + \frac{\ln t_{1}}{2}

$t_1=1-x^{2},\:t_2=\frac{2}{\pi a}+\frac{\ln t_1}{2}$

和常数 $a=0.147$ ... 下面是八度语言的代码。

function y = erfa(x)
  a  = 0.147;
  x2 = x**2; t = x2*(4/pi + a*x2)/(1 + a*x2);
  y  = sign(x)*sqrt(1 - exp(-t));
endfunction

function y = erfinva(x)
  a  = 0.147; 
  t1 = 1 - x**2; t2 = 2/pi/a + log(t1)/2;
  y  = sign(x)*sqrt(-t2 + sqrt(t2**2 - log(t1)/a));
endfunction

function y = normcdfa(x)
  y = 1/2*(1 + erfa(x/sqrt(2)));
endfunction

function y = norminva(x)
  y = sqrt(2)*erfinva(2*x - 1);
endfunction

现在我们有了正态分布函数，我们给出一个参数并计算学生的t分布[2]：

F_{t} (x, n) = Φ (\sqrt{\frac{1}{t_{1}} \cdot \ln (1 + \frac{x^{2}}{n})})

$F_t(x,n)=\Phi\left(\sqrt{\frac{1}{t_1}\cdot\ln(1+\frac{x^{2}}{n})}\right)$

t = F_{t}^{- 1} (P, n) = \sqrt{n \cdot \exp (Φ^{- 1} (P)^{2} \cdot t_{1}) - n}

$t=F_t^{-1}(P,n)=\sqrt{n\cdot\exp\left(\Phi^{-1}(P)^{2}\cdot t_1\right)-n}$

辅助变量 $t_1$ 有

t_{1} = \frac{n - 1.5}{(n - 1)^{2}}

$t_1=\frac{n-1.5}{(n-1)^{2}}$

function y = tcdfa(x,n)
  t1 = (n - 1.5)/(n - 1)**2;
 y = normcdfa(sqrt(1/t1*log(1 + x**2/n)));
endfunction

function y = tinva(x,n)
  t1 = (n - 1.5)/(n - 1)**2;
  y  = sqrt(n*exp(t1*norminva(x)**2) - n);
endfunction

粗略计算分布的想法 $\chi^{2}$ 由公式[3]清楚地表示：

σ^{2} = \frac{2}{9 n}, μ = 1 - σ^{2}

$\sigma^{2}=\frac{2}{9n},\:\mu=1-\sigma^{2}$

F_{χ^{2}} (x, n) = Φ (\frac{{(\frac{x}{n})}^{1 / 3} - μ}{σ})

$F_{\chi^{2}}(x,n)=\Phi\left(\frac{\left(\frac{x}{n}\right)^{1/3}-\mu}{\sigma}\right)$

χ^{2} = F_{χ^{2}}^{- 1} (P, n) = n \cdot {(Φ^{- 1} (P) \cdot σ + μ)}^{3}

$\chi^2=F_{\chi^2}^{-1}(P,n)=n\cdot\left(\Phi^{-1}(P)\cdot\sigma + \mu\right)^3$

function y = chi2cdfa(x,n)
  s2 = 2/9/n; mu = 1 - s2;
  y  = normcdfa(((x/n)**(1/3) - mu)/sqrt(s2));
endfunction

function y = chi2inva(x,n)
 s2 = 2/9/n; mu = 1 - s2;
  y = n*(norminva(x)*sqrt(s2) + mu)**3;
endfunction

费舍尔分布（用于 $n/k\geq3$ 和 $n\geq3$ ) . $\chi^2$ [4], , .

σ^{2} = \frac{2}{9 n}, μ = 1 - σ^{2}

$\sigma^2=\frac{2}{9n},\:\mu=1-\sigma^2$

λ = \frac{2 n + k \cdot x / 3 + (k - 2)}{2 n + 4 k \cdot x / 3}

$\lambda=\frac{2n+k\cdot x/3+(k-2)}{2n+4k\cdot x/3}$

F_{f} (x; k, n) = Φ (\frac{{(λ \cdot x)}^{1 / 3} - μ}{σ})

$F_f(x;k,n)=\Phi\left(\frac{\left(\lambda\cdot x\right)^{1/3}-\mu}{\sigma}\right)$

, .

q = {(Φ^{- 1} (P) \cdot σ + μ)}^{3}

$q=\left(\Phi^{-1}(P)\cdot\sigma+\mu\right)^3$

b = 2 n + k - 2 - 4 / 3 \cdot k q

$b=2n+k-2-4/3\cdot kq$

D = b^{2} + 8 / 3 \cdot k n q

$D=b^2+8/3\cdot knq$

x = F_{f}^{- 1} (P; k, n) = \frac{- b + \sqrt{D}}{2 k / 3}

$x=F_f^{-1}(P;k,n)=\frac{-b+\sqrt{D}}{2k/3}$

function y = fcdfa(x,k,n)
  mu = 1-2/9/k; s = sqrt(2/9/k);
  lambda = (2*n + k*x/3 + k-2)/(2*n + 4*k*x/3);
  normcdfa(((lambda*x)**(1/3)-mu)/s)
endfunction

function y = finva(x,k,n)
  mu = 1-2/9/k; s = sqrt(2/9/k);
  q = (norminva(x)*s + mu)**3;
  b = 2*n + k-2 -4/3*k*q;
  d = b**2 + 8/3*k*n*q;
  y = (sqrt(d) - b)/(2*k/3);
endfunction

Sergei Winitzki. A handy approximation for the error function and its inverse. February 6, 2008.
Gleason J.R. A note on a proposed Student t approximation // Computational statistics & data analysis. – 2000. – Vol. 34. – №. 1. – Pp. 63-66.
Wilson E.B., Hilferty M.M. The distribution of chi-square // Proceedings of the National Academy of Sciences. – 1931. – Vol. 17. – №. 12. – Pp. 684-688.
Li B. and Martin E.B. An approximation to the F-distribution using the chi-square distribution. Computational statistics & data analysis. – 2002. Vol. 40. – №. 1. pp. 21-26.

简单快速地近似统计函数

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