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. Definitions. If "Z" is a normally distributed random variable with unit variance and mean  μ ≠ 0, 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. Cumulative distribution function. The cumulative distribution function of noncentral "t"-distribution with ν degrees of freedom and noncentrality parameter μ can be expressed as where 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. Probability density function. 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 "F" 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. Properties. Moments of the noncentral "t"-distribution. In general, the "k"th raw moment of the noncentral "t"-distribution is In particular, the mean and variance of the noncentral "t"-distribution are An excellent approximation to formula_15 is formula_16, which can be used in both formulas. Asymmetry. The non-central "t"-distribution is asymmetric unless μ is zero, i.e., a central "t"-distribution. In addition, the asymmetry becomes smaller the larger degree of freedom. The right tail will be heavier than the left when μ  > 0, and vice versa. However, the usual skewness is not generally a good measure of asymmetry for this distribution, because if the degrees of freedom is not larger than 3, the third moment does not exist at all. Even if the degrees of freedom is greater than 3, the sample estimate of the skewness is still very unstable unless the sample size is very large. Mode. The noncentral "t"-distribution is always unimodal and bell shaped, but the mode is not analytically available, although for μ ≠ 0 we have In particular, the mode always has the same sign as  the noncentrality parameter μ. Moreover, the negative of the mode is exactly the mode for a noncentral "t"-distribution with the same number of degrees of freedom ν but noncentrality parameter −μ. The mode is strictly increasing with μ (it always moves in the same direction as μ is adjusted in). In the limit, when μ → 0, the mode is approximated by and when μ → ∞, the mode is approximated by Occurrence and applications. Use in power analysis. Suppose we have an independent and identically distributed sample "X", ..., "X" each of which is normally distributed with mean θ and  variance σ, and we are interested in testing the null hypothesis θ = 0 vs. the alternative hypothesis θ ≠ 0. We can perform a one sample "t"-test using the test statistic where formula_22 is the sample mean and formula_23 is the unbiased sample variance. Since the right hand side of the second equality exactly matches the characterization of a noncentral "t"-distribution as described above, "T" has a noncentral "t"-distribution with "n"−1 degrees of freedom and noncentrality parameter formula_24. If the test procedure rejects the null hypothesis whenever formula_25, where formula_26 is the upper α/2 quantile of the (central) Student 's "t"-distribution for a pre-specified α ∈ (0, 1), then the power of this test is given by Similar applications of the noncentral "t"-distribution can be found in the power analysis of the general normal-theory linear models, which includes the above one sample "t"-test as a special case. Use in tolerance intervals. One-sided normal tolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentral "t"-distribution. This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls. Noncentral