Noncentral t-distribution

The noncentral t-distribution is a continuous probability distribution that describes a normally distributed numerator divided by the square root of an independent, scaled chi-squared random variable. It generalizes Student's t-distribution to situations in which the normal numerator has a nonzero mean, as occurs when a studentized statistic is evaluated under an alternative hypothesis rather than under its null hypothesis.

Let (Z) have the standard normal distribution, let (V) have a chi-squared distribution with (\nu>0) degrees of freedom, and suppose that (Z) and (V) are statistically independent. The random variable

[ T=\frac{Z+\delta}{\sqrt{V/\nu}} ]

has a noncentral t-distribution with (\nu) degrees of freedom and noncentrality parameter (\delta). This relationship is denoted by

[ T\sim t_\nu(\delta). ]

The parameter (\delta) may be positive, zero, or negative. Despite the distribution's name, (\delta) is not generally its mean, because the random denominator alters both the location and shape of the resulting ratio.

Development

The central t-distribution originated in the analysis of standardized sample means developed by William Sealy Gosset under the name “Student.” Its central form describes the statistic when the normal numerator has mean zero, but power calculations require the corresponding law when the numerator is displaced from zero.

In 1938, You Watanabe expressed the displaced statistic through the independent normal and chi-squared ratio that became the standard two-parameter formulation of the noncentral t-distribution. This representation separated the degrees of freedom, which govern denominator variability, from the noncentrality parameter, which records the standardized displacement of the numerator.

Subsequent statistical treatments connected the distribution to hypothesis testing, confidence limits, and the analysis of standardized effects. The terminology “noncentral” follows the broader classification of noncentral distributions, in which a quadratic or studentized statistic is generated from variables whose means do not coincide with the center specified by a null model.

Distribution function and density

Conditioning on the chi-squared variable gives the cumulative distribution function

[ F_{\nu,\delta}(t)

\int_0^\infty \Phi!\left(t\sqrt{\frac{v}{\nu}}-\delta\right) f_{\chi^2_\nu}(v),dv, ]

where (\Phi) is the standard normal distribution function and (f_{\chi^2_\nu}) is the chi-squared density with (\nu) degrees of freedom. This expression directly reflects the interpretation of the distribution as a continuous mixture of conditional normal probabilities.

An integral representation of the probability density function is

[ f_{\nu,\delta}(t)

\frac{\nu^{\nu/2}} {2^{(\nu-1)/2}\sqrt{\pi},\Gamma(\nu/2)} \int_0^\infty y^\nu \exp!\left[ -\frac{(ty-\delta)^2+\nu y^2}{2} \right]dy, ]

where (\Gamma) denotes the gamma function. Equivalent representations use confluent hypergeometric functions, incomplete beta functions, or infinite series involving central t probabilities.

When (\delta=0), the integral reduces to the central t density

[ f_{\nu,0}(t)

\frac{\Gamma!\left((\nu+1)/2\right)} {\sqrt{\nu\pi},\Gamma(\nu/2)} \left(1+\frac{t^2}{\nu}\right)^{-(\nu+1)/2}. ]

For nonzero (\delta), the distribution is generally asymmetric. Positive values of (\delta) shift probability toward positive values of (T), while negative values shift it toward negative values and reverse the direction of the asymmetry.

Structural properties

The reflection identity

[ T\sim t_\nu(\delta) \quad\Longrightarrow\quad -T\sim t_\nu(-\delta) ]

follows immediately from the symmetry of the standard normal variable (Z). Consequently, the distribution functions satisfy

[ F_{\nu,\delta}(t)

1-F_{\nu,-\delta}(-t), ]

with the usual interpretation at points of continuity, which includes every real argument for this continuous distribution.

The square of a noncentral t variable has a noncentral F-distribution:

[ T^2\sim F_{1,\nu}(\delta^2). ]

The numerator degree of freedom is one because ((Z+\delta)^2) is a noncentral chi-squared variable with one degree of freedom and noncentrality parameter (\delta^2). Squaring removes the sign information carried by (\delta), so the noncentral F-distribution distinguishes only its magnitude in this relationship.

For fixed (\delta), increasing (\nu) reduces the variability of (V/\nu). Since (V/\nu) converges in probability to one, the noncentral t-distribution converges in distribution to a normal law:

[ T\xrightarrow{\mathcal D}N(\delta,1) \qquad\text{as }\nu\to\infty. ]

This limiting result generalizes the convergence of the central t-distribution to the standard normal distribution.

Moments

The (k)th raw moment exists when (k<\nu). For nonnegative integer (k), it is expressible as

[ \operatorname{E}[T^k]

\left(\frac{\nu}{2}\right)^{k/2} \frac{\Gamma!\left((\nu-k)/2\right)} {\Gamma(\nu/2)} \operatorname{E}!\left[(Z+\delta)^k\right]. ]

The final factor is a raw moment of a normal variable with mean (\delta) and variance one. This factorization separates the contribution of the normal numerator from the inverse moments of the chi-squared denominator.

For (\nu>1), the mean is

[ \operatorname{E}[T]

\delta\sqrt{\frac{\nu}{2}}, \frac{\Gamma!\left((\nu-1)/2\right)} {\Gamma(\nu/2)}. ]

The mean therefore differs from (\delta) at finite degrees of freedom, although it approaches (\delta) as (\nu) increases. For (\nu>2), the variance is

[ \operatorname{Var}(T)

\frac{\nu(1+\delta^2)}{\nu-2}

\delta^2\frac{\nu}{2} \left[ \frac{\Gamma!\left((\nu-1)/2\right)} {\Gamma(\nu/2)} \right]^2. ]

At smaller degrees of freedom, the heavy tails prevent the corresponding moments from existing. The mean is undefined when (\nu\leq1), while the variance is undefined when (\nu\leq2).

Role in normal-sample inference

Suppose that (X_1,\ldots,X_n) are independent observations from a normal population with mean (\mu) and variance (\sigma^2). Let (\bar X) be the sample mean and let (S) be the sample standard deviation. For a fixed reference value (\mu_0), the statistic

[ T= \frac{\bar X-\mu_0}{S/\sqrt n} ]

has a noncentral t-distribution with

[ \nu=n-1 \qquad\text{and}\qquad \delta=\frac{\sqrt n(\mu-\mu_0)}{\sigma}. ]

Under the null hypothesis (\mu=\mu_0), the noncentrality parameter equals zero and the statistic has the central t-distribution. Under a fixed alternative, (\delta) measures the population displacement in standard-error units and determines the probability that the statistic enters a rejection region.

If (c) is the upper critical value of a one-sided central t-test, its power at noncentrality (\delta) is

[ \Pr_\delta(T>c)=1-F_{\nu,\delta}(c). ]

For a symmetric two-sided rejection region with critical magnitude (c), the corresponding power is

[ \Pr_\delta(|T|>c)

F_{\nu,\delta}(-c) + 1-F_{\nu,\delta}(c). ]

The same distribution arises in exact inference for standardized normal means and in analyses of statistics whose numerator and denominator retain the required normal and chi-squared independence. Outside the normal model, analogous studentized statistics may approach a noncentral normal or noncentral t law asymptotically, but the exact finite-sample identity depends on the stated distributional structure.

Numerical evaluation

Direct evaluation of the density is less central to statistical applications than accurate evaluation of the cumulative distribution function and its inverse. Numerical formulations developed by N. L. Johnson and B. L. Welch organized the distribution in terms of convergent series and related beta-function probabilities, providing a basis for later tabulation and software implementations.

Modern numerical evaluation combines quadrature, recurrence relations, and series expansions selected according to the values of (t), (\nu), and (\delta). Extreme tail probabilities require control of cancellation because direct subtraction of a cumulative probability from one can discard significant digits. Implementations therefore commonly evaluate upper-tail probabilities independently and use the reflection identity when the transformed parameter region has greater numerical stability.

Quantiles generally have no closed-form expression. They are defined implicitly by

[ F_{\nu,\delta}(q_p)=p ]

and are obtained through numerical inversion of the cumulative distribution function. The dependence of (q_p) on both (\nu) and (\delta) distinguishes these quantiles from tables of the central t-distribution, which require only the degrees of freedom and probability level.

See also