Noncentral t-distribution

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Noncentral t-distribution
parameters: Probability density function $\nu>0\,\!$ degrees of freedom $\mu \in \Re \,\!$ noncentrality parameter $x \in (-\infty; +\infty)\,\!$ see text see text see text see text see text see text see text

In probability and statistics, the noncentral t-distribution (also known as the singly noncentral t-distribution) generalizes Student's t-distribution using a noncentrality parameter. Like the central t-distribution, the noncentral t-distribution is primarily used in statistical inference, although it may also be used in robust modeling for data. In particular, the noncentral t-distribution arises in power analysis.

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 $\nu\,\!$ 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 $\nu\,\!$ degrees of freedom and noncentrality parameter $\mu\,\!$. Note that the noncentrality parameter may be negative.

Cumulative distribution function

The cumulative distribution function of noncentral t-distribution with $\nu\,\!$ degrees of freedom and noncentrality parameter $\mu\,\!$ can be expressed as [1]

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

where

$\tilde{F}_{\nu,\mu}(x)= \Phi(-\mu)+\frac{1}{2}\sum_{j=0}^\infty\left[p_jI_y\left(j+\frac{1}{2},\frac{\nu}{2}\right)+q_jI_y\left(j+1,\frac{\nu}{2}\right)\right],$
$I_y\,\!(a,b)$ 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

$\Phi\,\!$ is the cumulative distribution function of the standard normal distribution.

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

$F_{v,\mu}(x)= \begin{cases}\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\mu\sqrt{2})^je^{\frac{-\mu^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\Gamma(1/2)}I\left (\frac{v}{v+x^2};\frac{v}{2},\frac{j+1}{2}\right ), & x\ge 0 \\ 1-\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\mu\sqrt{2})^je^{\frac{-\mu^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\Gamma(1/2)}I\left (\frac{v}{v+x^2};\frac{v}{2},\frac{j+1}{2}\right ), & x < 0 \end{cases}$

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 for the noncentral t-distribution with $\nu>0\,\!$ degrees of freedom and noncentrality parameter $\mu\,\!$ can be expressed in several forms.

The confluent hypergeometric function form of the density function is

$f(x)=\frac{\nu^{\nu/2}\Gamma(\nu+1)\exp(-\mu^2/2)}{2^\nu(\nu+x^2)^{\nu/2}\Gamma(\nu/2)} \left\{\frac{\sqrt{2}\mu x\,_1F_1(\nu/2+1;\, 3/2;\, \mu^2x^2/(2(\nu+x^2)))}{(\nu+x^2)\Gamma((\nu+1)/2)} \right.$
$\left. + \frac{\,_1F_1((\nu+1)/2;\, 1/2;\, \mu^2x^2/(2(\nu+x^2)))}{\sqrt{\nu+x^2}\Gamma(\nu/2+1)}\right\}$

where $\,_1F_1$ is a confluent hypergeometric function.

An alternative integral form is [2]

$f(x) =\frac{\nu^{\nu/2} \exp\left\{-\frac{\nu\mu^2}{2(x^2+\nu)} \right\} } {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(x^2+\nu)^{(\nu+1)/2}} \int_0^\infty y^\nu\exp\left\{-\frac{1}{2}\left(y-\frac{\mu x}{\sqrt{x^2+\nu}}\right)^2\right\}\,dy\,.$

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

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

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

Properties

Moments of the Noncentral t-distribution

In general, the kth raw moment of the non-central t-distribution is [3]

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

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

$\mbox{E}\left[T\right]= \begin{cases} \mu\sqrt{\frac{\nu}{2}}\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}, &\mbox{if }\nu>1 ;\\ \mbox{Does not exist}, &\mbox{if }\nu\le1 ,\\ \end{cases}$

and

$\mbox{Var}\left[T\right]= \begin{cases} \frac{\nu(1+\mu^2)}{\nu-2} -\frac{\mu^2\nu}{2} \left(\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}\right)^2 , &\mbox{if }\nu>2 ;\\ \mbox{Does not exist}, &\mbox{if }\nu\le2 .\\ \end{cases}$

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 it always lies in the interval[4]

$\left( \sqrt{\frac{2\nu}{2\nu+5}}\mu,\,\sqrt{\frac{\nu}{\nu+1}}\mu \right)$ when $\mu>0,\,\!$ and
$\left( \sqrt{\frac{\nu}{\nu+1}}\mu,\,\sqrt{\frac{2\nu}{2\nu+5}}\mu \right)$ when $\mu<0.\,\!$

Moreover, the mode always has the same sign as the noncentrality parameter $\mu,\,\!$ and the negative of the mode is exactly the mode for a noncentral t-distribution with the same number of degrees of freedom $\nu\,\!$ but noncentrality parameter $-\mu.\,\!$

The mode is strictly increasing with $\mu\,\!$ when $\mu>0,\,\!$ and strictly decreasing with $\mu\,\!$ when $\mu<0.\,\!$ In the limit, when $\mu\,\!$ approaches zero, 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 $\mu\,\!$ approaches infinity, 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 $X_1,X_2,\ldots,X_n$, each of which is normally distributed with mean $\theta\,\!$ and variance $\sigma^2\,\!$, and we are interested in testing the null hypothesis $\theta=0\,\!$ vs. the alternative hypothesis $\theta\neq0\,\!$. We can perform a one sample t-test using the test statistic

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

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 $\alpha/2\,\!$ quantile of the (central) Student's t-distribution for a pre-specified $\alpha\in(0,1)\,\!$, 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.

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 non-central t-distribution.[5]
• If $T\,\!$ is noncentral t-distributed with $\nu\,\!$ degrees of freedom and noncentrality parameter $\mu\,\!$ and $F=T^2\,\!$, then $F\,\!$ has a noncentral F-distribution with 1 numerator degree of freedom, $\nu\,\!$ denominator degrees of freedom, and noncentrality parameter $\mu^2\,\!$.
• If $T\,\!$ is noncentral t-distributed with $\nu\,\!$ degrees of freedom and noncentrality parameter $\mu\,\!$ and $Z=\lim_{\nu\rightarrow\infty} T$, then $Z\,\!$ has a normal distribution with mean $\mu\,\!$ and unit variance.
• When the denominator noncentrality parameter of a edit] Special cases

References

1. ^ a b Lenth, Russell V (1989). "Algorithm AS 243: Cumulative Distribution Function of the Non-central t Distribution". Journal of the Royal Statistical Society. Series C (Applied Statistics) 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. 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. ^ Helena Chmura Kraemer; Minja Paik (1979). "A Central t Approximation to the Noncentral t Distribution". Technometrics 21 (3): 357–360. JSTOR 1267759.

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