Noncentral t-distribution

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. 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 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. The cumulative distribution function of noncentral t-distribution with ν degrees of freedom and noncentrality parameter μ can be expressed as where is the regularized incomplete beta function, and Φ is the cumulative distribution function of the standard normal distribution. Alternatively, the noncentral t-distribution CDF can be expressed as: where Γ is the gamma function and I is the regularized incomplete beta function. Although there are other forms of the cumulative distribution function, the first form presented above is very easy to evaluate through recursive computing. In statistical software R, the cumulative distribution function is implemented as pt. The probability density function (pdf) for the noncentral t-distribution with ν > 0 degrees of freedom and noncentrality parameter μ can be expressed in several forms. The confluent hypergeometric function form of the density function is where and where 1F1 is a confluent hypergeometric function. An alternative integral form is A third form of the density is obtained using its cumulative distribution functions, as follows. This is the approach implemented by the dt function in R.