The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.
Definitions If Z is a standard normal random variable, and V is a chi-squared distributed random variable with ν degrees of freedom that is independent of Z, then
T = Z + μ V / ν {\displaystyle T={\frac {Z+\mu }{\sqrt {V/\nu }}}}
is a noncentral t-distributed random variable with ν degrees of freedom and noncentrality parameter μ ≠ 0. Note that the noncentrality parameter may be negative.
Cumulative distribution function The cumulative distribution function of noncentral t-distribution with ν degrees of freedom and noncentrality parameter μ can be expressed as
F ν , μ ( x ) = { F ~ ν , μ ( x ) , if x ≥ 0 ; 1 − F ~ ν , − μ ( x ) , if x < 0 , {\displaystyle F_{\nu ,\mu }(x)={\begin{cases}{\tilde {F}}_{\nu ,\mu }(x),&{\mbox{if }}x\geq 0;\\1-{\tilde {F}}_{\nu ,-\mu }(x),&{\mbox{if }}x<0,\end{cases}}}
where
F ~ ν , μ ( x ) = Φ ( − μ ) + 1 2 ∑ j = 0 ∞ [ p j I y ( j + 1 2 , ν 2 ) + q j I y ( j + 1 , ν 2 ) ] , {\displaystyle {\tilde {F}}_{\nu ,\mu }(x)=\Phi (-\mu )+{\frac {1}{2}}\sum _{j=0}^{\infty }\left[p_{j}I_{y}\left(j+{\frac {1}{2}},{\frac {\nu }{2}}\right)+q_{j}I_{y}\left(j+1,{\frac {\nu }{2}}\right)\right],}
I y ( a , b ) {\displaystyle I_{y}\,\!(a,b)} is the regularized incomplete beta function,
y = x 2 x 2 + ν , {\displaystyle y={\frac {x^{2}}{x^{2}+\nu }},}
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