Noncentral t-distribution

In probability and statistics, the noncentral t-distribution (also known as the singly noncentral t-distribution) generalizes Student's t-distribution. 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 ${\displaystyle t}$-distribution arises in power analysis.

Noncentral Student's t
Probability density function
Parameters	${\displaystyle \nu >0}$ degrees of freedom (real) ${\displaystyle \mu \in \Re }$ noncentrality parameter (real)
Support	${\displaystyle x\in (-\infty ;+\infty )\!}$
PDF	see text
CDF	see text
Mean	see text
Variance	see text
Skewness	see text
Excess kurtosis	see text

Characterization

If ${\displaystyle Z}$  is a normally distributed random variable with unit variance and zero mean, and ${\displaystyle V}$  is a Chi-square distributed random variable with ${\displaystyle \nu }$  degrees of freedom that is statistically independent of ${\displaystyle Z}$ , then

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

is a noncentral ${\displaystyle t}$ -distributed random variable with ${\displaystyle \nu }$  degrees of freedom and noncentrality parameter ${\displaystyle \mu }$ . Note that the noncentrality parameter may be negative.

Cumulative distribution function

The cumulative distribution function can be expressed as ^[1]

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

where ${\displaystyle \mu }$  is the noncentrality parameter, ${\displaystyle \nu }$  is the degrees of freedom, ${\displaystyle I_{y}\,(a,b)}$  is the regularized incomplete beta function,

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

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

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

and ${\displaystyle \Phi }$  is the cumulative distribution function of standard normal distribution.

This form of the cumulative distribution function is 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 for the noncentral ${\displaystyle t}$ -distribution with ${\displaystyle \nu }$ ${\displaystyle >0}$  degrees of freedom and noncentrality parameter ${\displaystyle \mu }$  can be expressed in several forms.

The confluent hypergeometric function form of the density function is

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

${\displaystyle \left.+{\frac {\,_{1}F_{1}((\nu +1)/2;\,1/2;\,\mu ^{2}x^{2}/(2(\nu +x^{2})))}{{\sqrt {\nu +x^{2}}}\Gamma (\nu /2+1)}}\right\}}$ 

where ${\displaystyle \,_{1}F_{1}}$  is a confluent hypergeometric function.

An alternative integral form is ^[2]

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

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

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

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

Properties

Moments of the Noncentral t-distribution

In general, the ${\displaystyle k}$ th raw moment of the non-central ${\displaystyle t}$ -distribution is ^[3].

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

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

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

and

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

Occurences

Use in power analysis

Suppose we have an independent and identically distributed sample ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ , each of which is normally distributed with mean ${\displaystyle \theta }$  and variance ${\displaystyle \sigma }$ ${\displaystyle ^{2}}$ , and we are interested in testing the null hypothesis ${\displaystyle \theta }$ ${\displaystyle =0}$  vs. the alternative hypothesis ${\displaystyle \theta }$ ${\displaystyle \neq 0}$ . We can perform a one sample ${\displaystyle t}$ -test using the test statistic

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

where ${\displaystyle {\bar {X}}}$  is the sample mean and ${\displaystyle {\hat {\sigma }}^{2}}$  is the unbiased sample variance. Since the right hand side of the second equality exactly matches the characterization of a noncentral ${\displaystyle t}$ -distribution as described above, ${\displaystyle T}$  has a noncentral ${\displaystyle t}$ -distribution with ${\displaystyle n-1}$  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 ${\displaystyle \alpha /2}$  quantile of the (central) Student's t-distribution for a pre-specified ${\displaystyle \alpha \in (0,1)}$ , 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 noncentral t-distribution can be found in the power analysis of the general normal-theory linear models, which includes the above one sample ${\displaystyle t}$ -test as a special case.

Related distributions

• Central t approximation: 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 noncentral t-distribution.

• If ${\displaystyle T}$  is noncentral t-distributed with ${\displaystyle \nu }$  degrees of freedom and noncentrality parameter ${\displaystyle \mu }$  and ${\displaystyle F=T^{2}}$ , then ${\displaystyle Z}$  has a noncentral F-distribution with 1 numerator degree of freedom, ${\displaystyle \nu }$  denominator degrees of freedom, and noncentrality parameter ${\displaystyle \mu }$ ${\displaystyle ^{2}}$ .

• If ${\displaystyle T}$  is noncentral t-distributed with ${\displaystyle \nu }$  degrees of freedom and noncentrality parameter ${\displaystyle \mu }$  and ${\displaystyle Z=\lim _{\nu \rightarrow \infty }T}$ , then ${\displaystyle Z}$  has a normal distribution with mean ${\displaystyle \mu }$  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 ${\displaystyle \mu }$ ${\displaystyle =0}$ , the noncentral ${\displaystyle t}$ -distribution becomes the central (Student's) t-distribution with the same degrees of freedom.

See also

• Chi-square distribution
• Noncentral F-distribution
• Normal distribution
• Student's t-distribution
• doubly noncentral t-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 (Applied Statistics). 38: 185–189.
2. L. Scharf, Statistical Signal Processing, (Massachusetts: Addison-Wesley, 1991), p.177.
3. Hogben, D (1961). "The moments of the non-central t-distribution". Biometrika. 48: 465–468. {{cite journal}}: Unknown parameter |coauthor= ignored (|author= suggested) (help)

External links

• Comparison of noncentral and central t-distributions Density plot, critical value, cumulative probability, etc., noncentral t-distribution online calculator, based on R platform embedded in Mediawiki
• Eric W. Weisstein. "Noncentral Student's t-Distribution." From MathWorld--A Wolfram Web Resource