StatsCMSSDCDF

StatsCMSSDCDF (C, n)

The StatsCMSSDCDF function returns the cumulative distribution function of the C distribution (mean square successive difference), which is

$$\displaystyle f(C, n)=\frac{\Gamma(2 m+2)}{a 2^{2 m+1}[\Gamma(m+1)]^{2}}\left(1-\frac{C^{2}}{a^{2}}\right)^{m},$$

where

$$\displaystyle a^{2}=\frac{\left(n^{2}+2 n-12\right)(n-2)}{\left(n^{3}-13 n+24\right)},$$ $$\displaystyle m=\frac{\left(n^{4}-n^{3}-13 n^{2}+37 n-60\right)}{2\left(n^{3}-13 n+24\right)} .$$

The distribution (for C >0) can then be expressed as

$$\displaystyle F(C, n)=\frac{\Gamma(2 m+2)}{a 2^{2 m+1}[\Gamma(m+1)]^{2}} C_{2} F_{1}\left(\frac{1}{2},-m, \frac{3}{2}, \frac{C^{2}}{a^{2}}\right),$$

where

$$\displaystyle { }_{2} F_{1}$$

is the Hypergeometric function HyperG2F1.

References

Young, L.C., On randomness in ordered sequences, Annals of Mathematical Statistics, 12, 153-162, 1941.

See Also

Statistical Analysis, StatsCMSSDCDF, StatsSRTest