Noncentral t-distribution

Noncentral Student's t

Probability density function
Parameters	ν > 0 degrees of freedom $\mu \in \Re \,\!$ noncentrality parameter
Support	$x\in (-\infty ;+\infty )\,\!$
PDF	see text
CDF	see text
Mean	see text
Mode	see text
Variance	see text
Skewness	see text
Ex. kurtosis	see text

As with other noncentrality parameters, the noncentral t-distribution generalizes a probability distribution – Student's t-distribution – using a noncentrality parameter. Whereas the central 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.

Characterization

If Z is a normally distributed random variable with unit variance and zero mean, and V is a Chi-squared distributed random variable with ν degrees of freedom that is statistically independent of Z, then

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

is a noncentral t-distributed random variable with ν degrees of freedom and noncentrality parameter μ. 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^[1]

$F_{{\nu ,\mu }}(x)={\begin{cases}{\tilde {F}}_{{\nu ,\mu }}(x),&{\mbox{if }}x\geq 0;\\1-{\tilde {F}}_{{\nu ,-\mu }}(x),&{\mbox{if }}x<0,\end{cases}}$

where

${\tilde {F}}_{{\nu ,\mu }}(x)=\Phi (-\mu )+{\frac {1}{2}}\sum _{{j=0}}^{\infty }\left[p_{j}I_{y}\left(j+{\frac {1}{2}},{\frac {\nu }{2}}\right)+q_{j}I_{y}\left(j+1,{\frac {\nu }{2}}\right)\right],$

$I_{y}\,\!(a,b)$ is the regularized incomplete beta function,

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

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

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

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

Alternatively, the noncentral t-distribution CDF can be expressed as:

${\displaystyle F_{v,\mu }(x)={\begin{cases}{\frac {1}{2}}\sum _{j=0}^{\infty }{\frac {1}{j!}}(-\mu {\sqrt {2}})^{j}e^{\frac {-\mu ^{2}}{2}}{\frac {\Gamma ({\frac {j+1}{2}})}{\sqrt {\pi }}}I\left({\frac {v}{v+x^{2}}};{\frac {v}{2}},{\frac {j+1}{2}}\right),&x\geq 0\\1-{\frac {1}{2}}\sum _{j=0}^{\infty }{\frac {1}{j!}}(-\mu {\sqrt {2}})^{j}e^{\frac {-\mu ^{2}}{2}}{\frac {\Gamma ({\frac {j+1}{2}})}{\sqrt {\pi }}}I\left({\frac {v}{v+x^{2}}};{\frac {v}{2}},{\frac {j+1}{2}}\right),&x<0\end{cases}}}$

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.^[1] 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

$f(x)={\frac {\nu ^{{{\frac {\nu }{2}}}}\Gamma (\nu +1)\exp \left(-{\frac {\mu ^{2}}{2}}\right)}{2^{\nu }(\nu +x^{2})^{{{\frac {\nu }{2}}}}\Gamma ({\frac {\nu }{2}})}}\left\{{\sqrt {2}}\mu x{\frac {{}_{1}F_{1}\left({\frac {\nu }{2}}+1;\,{\frac {3}{2}};\,{\frac {\mu ^{2}x^{2}}{2(\nu +x^{2})}}\right)}{(\nu +x^{2})\Gamma ({\frac {\nu +1}{2}})}}+{\frac {{}_{1}F_{1}\left({\frac {\nu +1}{2}};\,{\frac {1}{2}};\,{\frac {\mu ^{2}x^{2}}{2(\nu +x^{2})}}\right)}{{\sqrt {\nu +x^{2}}}\Gamma ({\frac {\nu }{2}}+1)}}\right\}$

where ₁F₁ is a confluent hypergeometric function.

An alternative integral form is^[2]

$f(x)={\frac {\nu ^{{{\frac {\nu }{2}}}}\exp \left(-{\frac {\nu \mu ^{2}}{2(x^{2}+\nu )}}\right)}{{\sqrt {\pi }}\Gamma ({\frac {\nu }{2}})2^{{{\frac {\nu -1}{2}}}}(x^{2}+\nu )^{{{\frac {\nu +1}{2}}}}}}\int _{0}^{\infty }y^{\nu }\exp \left(-{\frac {1}{2}}\left(y-{\frac {\mu x}{{\sqrt {x^{2}+\nu }}}}\right)^{2}\right)dy.$

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

$f(x)={\begin{cases}{\frac {\nu }{x}}\left\{F_{{\nu +2,\mu }}\left(x{\sqrt {1+{\frac {2}{\nu }}}}\right)-F_{{\nu ,\mu }}(x)\right\},&{\mbox{if }}x\neq 0;\\{\frac {\Gamma ({\frac {\nu +1}{2}})}{{\sqrt {\pi \nu }}\Gamma ({\frac {\nu }{2}})}}\exp \left(-{\frac {\mu ^{2}}{2}}\right),&{\mbox{if }}x=0.\end{cases}}$

This is the approach implemented by the dt function in R.

Differential equation

The pdf of the noncentral t-distribution is a solution of the following differential equation:

$\left\{\begin{array}{l} \left(\nu+x^2\right)^2 f''(x)+x f'(x) \left(\nu \left(-\mu^2+2\nu+5\right)+(2\nu+5) x^2\right)+(\nu+1) f(x) \left(-\mu^2 \nu+\nu+(\nu+3) x^2\right)=0, \\[12pt] f(0)=\frac{\exp\left(-\frac{\mu^2}{2}\right) \Gamma\left(\frac{\nu +1}{2}\right)} {\sqrt{\pi} \sqrt{\nu} \Gamma\left(\frac{\nu}{2}\right)}, \\[12pt] f'(0)=-\frac{\exp\left(-\frac{\mu ^2}{2}\right) \mu}{\sqrt{2 \pi}} \end{array}\right\}$

Properties

Moments of the Noncentral t-distribution

In general, the kth raw moment of the noncentral t-distribution is^[3]

${\mbox{E}}\left[T^{k}\right]={\begin{cases}\left({\frac {\nu }{2}}\right)^{{{\frac {k}{2}}}}{\frac {\Gamma \left({\frac {\nu -k}{2}}\right)}{\Gamma \left({\frac {\nu }{2}}\right)}}{\mbox{exp}}\left(-{\frac {\mu ^{2}}{2}}\right){\frac {d^{k}}{d\mu ^{k}}}{\mbox{exp}}\left({\frac {\mu ^{2}}{2}}\right),&{\mbox{if }}\nu >k;\\{\mbox{Does not exist}},&{\mbox{if }}\nu \leq k.\\\end{cases}}$

In particular, the mean and variance of the noncentral t-distribution are

${\begin{aligned}{\mbox{E}}\left[T\right]&={\begin{cases}\mu {\sqrt {{\frac {\nu }{2}}}}{\frac {\Gamma ((\nu -1)/2)}{\Gamma (\nu /2)}},&{\mbox{if }}\nu >1;\\{\mbox{Does not exist}},&{\mbox{if }}\nu \leq 1,\\\end{cases}}\\{\mbox{Var}}\left[T\right]&={\begin{cases}{\frac {\nu (1+\mu ^{2})}{\nu -2}}-{\frac {\mu ^{2}\nu }{2}}\left({\frac {\Gamma ((\nu -1)/2)}{\Gamma (\nu /2)}}\right)^{2},&{\mbox{if }}\nu >2;\\{\mbox{Does not exist}},&{\mbox{if }}\nu \leq 2.\\\end{cases}}\end{aligned}}$

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

Asymmetry

The noncentral t-distribution is asymmetric unless μ is zero, i.e., a central t-distribution. 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^[4]

${\displaystyle {\sqrt {\frac {\nu }{\nu +(5/2)}}}<{\frac {\mathrm {mode} }{\mu }}<{\sqrt {\frac {\nu }{\nu +1}}}}$

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 when μ > 0 and strictly decreasing with μ when μ < 0. In the limit, when μ → 0, the mode is approximated by

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

and when μ → ∞, the mode is approximated by

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

Occurrences

Use in power analysis

Suppose we have an independent and identically distributed sample X₁, ..., X_n each of which is normally distributed with mean θ and variance σ², 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

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

where ${\bar {X}}$ is the sample mean and ${\hat {\sigma }}^{2}\,\!$ 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 ${\sqrt {n}}\theta /\sigma \,\!$.

If the test procedure rejects the null hypothesis whenever $|T|>t_{{1-\alpha /2}}\,\!$, where $t_{{1-\alpha /2}}\,\!$ 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

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

Similar applications of the 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.^[5] This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.

Related distributions

• Central t-distribution: the central t-distribution can be converted into a location/scale family. This family of distributions is used in data modeling to capture various tail behaviors. The location/scale generalization of the central t-distribution is a different distribution from the noncentral t-distribution discussed in this article. In particular, this approximation does not respect the asymmetry of the noncentral t-distribution. However, the central t-distribution can be used as an approximation to the noncentral t-distribution.^[6]
• If T is noncentral t-distributed with ν degrees of freedom and noncentrality parameter μ and F = T², then F has a noncentral F-distribution with 1 numerator degree of freedom, ν denominator degrees of freedom, and noncentrality parameter μ².
• If T is noncentral t-distributed with ν degrees of freedom and noncentrality parameter μ and $Z=\lim _{{\nu \rightarrow \infty }}T$, then Z has a normal distribution with mean μ and unit variance.
• When the denominator noncentrality parameter of a doubly noncentral t-distribution is zero, then it becomes a noncentral t-distribution.

Special cases

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

See also

• Noncentral F-distribution

References

1. Lenth, Russell V (1989). "Algorithm AS 243: Cumulative Distribution Function of the Non-central t Distribution". Journal of the Royal Statistical Society, Series C. 38: 185–189. JSTOR 2347693.
2. L. Scharf, Statistical Signal Processing, (Massachusetts: Addison-Wesley, 1991), p.177.
3. Hogben, D; Wilk, MB (1961). "The moments of the non-central t-distribution". Biometrika. 48: 465–468. doi:10.1093/biomet/48.3-4.465. JSTOR 2332772.
4. van Aubel, A; Gawronski, W (2003). "Analytic properties of noncentral distributions". Applied Mathematics and Computation. 141: 3–12. doi:10.1016/S0096-3003(02)00316-8.
5. Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software. 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013., p.23
6. Helena Chmura Kraemer; Minja Paik (1979). "A Central t Approximation to the Noncentral t Distribution". Technometrics. 21 (3): 357–360. doi:10.1080/00401706.1979.10489781. JSTOR 1267759.

External links

• Eric W. Weisstein. "Noncentral Student's t-Distribution." From MathWorld—A Wolfram Web Resource
• High accuracy calculation for life or science.: Noncentral t-distribution From Casio company.