Noncentral Student'st
Probability density function
Parameters	ν > 0 degrees of freedom ${\displaystyle \mu \in \mathrm{\Re }}$ noncentrality parameter
Support	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$
PDF	see text
CDF	see text
Mean	see text
Mode	see text
Variance	see text
Skewness	see text
Excess kurtosis	see text

Thenoncentralt-distribution generalizesStudent'st-distribution using anoncentrality parameter. Whereas the centralprobability distribution describes how a test statistict is distributed when the difference tested is null, the noncentral distribution describes howt is distributed when the null is false. This leads to its use in statistics, especially calculatingstatistical power. The noncentralt-distribution is also known as the singly noncentralt-distribution, and in addition to its primary use instatistical inference, is also used inrobust modeling fordata.

IfZ is a standardnormal random variable, andV is achi-squared distributed random variable with νdegrees of freedom that is independent ofZ, then

${\displaystyle T=\frac{Z+\mu }{\sqrt{V/\nu }}}$

is a noncentralt-distributed random variable with ν degrees of freedom andnoncentrality parameter μ ≠ 0. Note that the noncentrality parameter may be negative.

Thecumulative distribution function of noncentralt-distribution with ν degrees of freedom and noncentrality parameter μ can be expressed as^[1]

where

${\displaystyle {\tilde{F}}_{\nu ,\mu }(x)=\mathrm{\Phi }(-\mu )+\frac{1}{2}\sum _{j=0}^{\mathrm{\infty }}\left[p_j{I}_y\left(j+\frac{1}{2},\frac{\nu }{2}\right)+q_j{I}_y\left(j+1,\frac{\nu }{2}\right)\right],}$

${\displaystyle I_y(a,b)}$ is theregularized incomplete beta function,

${\displaystyle y=\frac{x^2}{x^2+\nu },}$

${\displaystyle p_j=\frac{1}{j!}\exp \left\{-\frac{{\mu }^2}{2}\right\}{\left(\frac{{\mu }^2}{2}\right)}^j,}$

${\displaystyle q_j=\frac{\mu }{\sqrt{2}\mathrm{\Gamma }(j+3/2)}\exp \left\{-\frac{{\mu }^2}{2}\right\}{\left(\frac{{\mu }^2}{2}\right)}^j,}$

and Φ is the cumulative distribution function of thestandard normal distribution.

Alternatively, the noncentralt-distribution CDF can be expressed as^{[citation needed]}:

where Γ is thegamma function andI is theregularized incomplete beta function.

Although there are other forms of the cumulative distribution function, the first form presented above is very easy to evaluate throughrecursive computing.^[1] In statistical softwareR, the cumulative distribution function is implemented aspt.

Theprobability density function (pdf) for the noncentralt-distribution with ν > 0 degrees of freedom and noncentrality parameter μ can be expressed in several forms.

Theconfluent hypergeometric function form of the density function is

${\displaystyle f(x)=\underset{\text{StudentT}(x;\mu =0)}{\underbrace{\frac{\mathrm{\Gamma }(\frac{\nu +1}{2})}{\sqrt{\nu \pi }\mathrm{\Gamma }(\frac{\nu }{2})}{\left(1+\frac{x^2}{\nu }\right)}^{-\frac{\nu +1}{2}}}}\exp (-\frac{{\mu }^2}{2})\{A_{\nu }(x;\mu )+B_{\nu }(x;\mu )\},}$

where

and where₁F₁ is aconfluent hypergeometric function.

An alternative integral form is^[2]

${\displaystyle f(x)=\frac{{\nu }^{\frac{\nu }{2}}\exp \left(-\frac{\nu {\mu }^2}{2(x^2+\nu )}\right)}{\sqrt{\pi }\mathrm{\Gamma }(\frac{\nu }{2})2^{\frac{\nu -1}{2}}(x^2+\nu {)}^{\frac{\nu +1}{2}}}{\int }_0^{\mathrm{\infty }}{y}^{\nu }\exp \left(-\frac{1}{2}{\left(y-\frac{\mu x}{\sqrt{x^2+\nu }}\right)}^2\right)dy.}$

A third form of the density is obtained using its cumulative distribution functions, as follows.

This is the approach implemented by thedt function inR.

In general, thekth raw moment of the noncentralt-distribution is^[3]

In particular, the mean and variance of the noncentralt-distribution are

An excellent approximation to${\displaystyle \sqrt{\frac{\nu }{2}}\frac{\mathrm{\Gamma }((\nu -1)/2)}{\mathrm{\Gamma }(\nu /2)}}$ is${\displaystyle {\left(1-\frac{3}{4\nu -1}\right)}^{-1}}$, which can be used in both formulas.^[4][5]

The non-centralt-distribution is asymmetric unless μ is zero, i.e., a centralt-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.

The noncentralt-distribution is always unimodal and bell shaped, but the mode is not analytically available, although for μ ≠ 0 we have^[6]

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 noncentralt-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

${\displaystyle \sqrt{\frac{\nu }{2}}\frac{\mathrm{\Gamma }\left(\frac{\nu +2}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu +3}{2}\right)}\mu ;}$

and when μ → ∞, the mode is approximated by

${\displaystyle \sqrt{\frac{\nu }{\nu +1}}\mu .}$

• Centralt-distribution: the centralt-distribution can be converted into alocation/scale family. This family of distributions is used in data modeling to capture various tail behaviors. The location/scale generalization of the centralt-distribution is a different distribution from the noncentralt-distribution discussed in this article. In particular, this approximation does not respect the asymmetry of the noncentralt-distribution. However, the centralt-distribution can be used as an approximation to the noncentralt-distribution.^[7]
• IfT is noncentralt-distributed with ν degrees of freedom and noncentrality parameter μ andF =T², thenF has anoncentralF-distribution with 1 numerator degree of freedom, ν denominator degrees of freedom, and noncentrality parameter μ².
• IfT is noncentralt-distributed with ν degrees of freedom and noncentrality parameter μ and${\displaystyle Z=\underset{\nu \to \mathrm{\infty }}{lim}T}$, thenZ has a normal distribution with mean μ and unit variance.
• When thedenominator noncentrality parameter of adoubly noncentralt-distribution is zero, then it becomes a noncentralt-distribution.

• When μ = 0, the noncentralt-distribution becomes thecentral (Student's)t-distribution with the same degrees of freedom.

Suppose we have an independent and identically distributed sampleX₁, ...,X_n each of which is normally distributed with mean θ and variance σ², and we are interested in testing thenull hypothesis θ = 0 vs. thealternative hypothesis θ ≠ 0. We can perform aone samplet-test using thetest statistic

${\displaystyle T=\frac{\overline{X}}{\hat{\sigma }/\sqrt{n}}=\frac{\frac{\overline{X}-\theta }{(\sigma /\sqrt{n})}+\frac{\theta }{(\sigma /\sqrt{n})}}{\sqrt{\left(\frac{{\hat{\sigma }}^2}{{\sigma }^2/(n-1)}\right)/(n-1)}}}$

where${\displaystyle \overline{X}}$ is the sample mean and${\displaystyle {\hat{\sigma }}^2}$ is the unbiasedsample variance. Since the right hand side of the second equality exactly matches the characterization of a noncentralt-distribution as described above,T has a noncentralt-distribution withn−1 degrees of freedom and noncentrality parameter${\displaystyle \sqrt{n}\theta /\sigma }$.

If the test procedure rejects the null hypothesis whenever${\displaystyle |T|>t_{1-\alpha /2}}$, where${\displaystyle t_{1-\alpha /2}}$ is the upper α/2 quantile of the(central) Student'st-distribution for a pre-specified α ∈ (0, 1), then the power of this test is given by

${\displaystyle 1-F_{n-1,\sqrt{n}\theta /\sigma }(t_{1-\alpha /2})+F_{n-1,\sqrt{n}\theta /\sigma }(-t_{1-\alpha /2}).}$

Similar applications of the noncentralt-distribution can be found in thepower analysis of the general normal-theorylinear models, which includes the aboveone samplet-test as a special case.

One-sided normaltolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentralt-distribution.^[8] This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.

• NoncentralF-distribution

• Eric W. Weisstein. "Noncentral Student'st-Distribution." From MathWorld—A Wolfram Web Resource
• High accuracy calculation for life or science.: Noncentralt-distribution From Casio company.

• Continuous distributions

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