Noncentral t-distribution

Top

Home > Library > Miscellaneous > Wikipedia

Noncentral Student's `t`
Probability density function
parameters:	$\nu>0\,\!$ 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

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.

Contents

1 Characterization
- 1.1 Cumulative distribution function
- 1.2 Probability density function
2 Properties
3 Occurrences
- 3.1 Use in power analysis
4 Related distributions
- 4.1 Special cases
5 See also
6 References
7 External links

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 $k$ th 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 doubly noncentral t-distribution is zero, then it becomes a noncentral t-distribution.

Special cases

When $\mu=0\,\!$ , the noncentral t-distribution becomes the central (Student's) t-distribution with the same degrees of freedom.

References

^ ^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.
^ L. Scharf, Statistical Signal Processing, (Massachusetts: Addison-Wesley, 1991), p.177.
^ Hogben, D; Wilk, MB (1961). "The moments of the non-central t-distribution". Biometrika 48: 465–468. JSTOR 2332772.
^ 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. http://www.sciencedirect.com/science/article/B6TY8-47G44WX-V/2/7705d2642b1a384b13e0578898a22d48.
^ Helena Chmura Kraemer; Minja Paik (1979). "A Central t Approximation to the Noncentral t Distribution". Technometrics 21 (3): 357–360. JSTOR 1267759.

External links

Eric W. Weisstein. "Noncentral Student's t-Distribution." From MathWorld—A Wolfram Web Resource

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · Beta-binomial · binomial · categorical · hypergeometric · Poisson binomial · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial · Boltzmann · Conway–Maxwell–Poisson · discrete phase-type · extended negative binomial · Gauss–Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule–Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine · ARGUS · Balding-Nichols · Bates · Beta · Noncentral beta · Irwin–Hall · Kumaraswamy · logit-normal · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini · Benktander 1st kind · Benktander 2nd kind · Beta prime · Bose–Einstein · Burr · chi-squared · chi · Coxian · Dagum · Davis · Erlang · exponential · F · Fermi–Dirac · folded normal · Fréchet · Gamma · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-squared · hyper-exponential · hypoexponential · inverse chi-squared (scaled-inverse-chi-squared) · inverse Gaussian · inverse gamma · Kolmogorov · Lévy · log-Cauchy · log-Laplace · log-logistic · log-normal · Maxwell–Boltzmann · Maxwell speed · Mittag–Leffler · Nakagami · noncentral chi-squared · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy · exponential power · Fisher's z · generalized normal · generalized hyperbolic · geometric stable · Gumbel · Holtsmark · hyperbolic secant · Landau · Laplace · Linnik · logistic · noncentral t · normal (Gaussian) · normal-inverse Gaussian · skew normal · slash · stable · Student's t · type-1 Gumbel · variance-gamma · Voigt

Continuous univariate with support whose type varies

generalized extreme value · generalized Pareto · Tukey lambda · q-Gaussian · q-exponential · shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Pólya · negative multinomial Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · Multivariate stable · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional

Univariate (circular) directional: Circular uniform · univariate von Mises · wrapped normal · wrapped Cauchy · wrapped exponential · wrapped Lévy Bivariate (spherical): Kent Bivariate (toroidal): bivariate von Mises Multivariate: von Mises–Fisher · Bingham

Degenerate and singular

Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

Circular · compound Poisson · elliptical · exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · Tweedie · wrapped

v · d · eSome common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) · Median · Mode

Dispersion	Range · Standard deviation · Coefficient of variation · Percentile · Interquartile range

Shape	Variance · Skewness · Kurtosis · Moments · L-moments

Count data

Index of dispersion

Summary tables

Grouped data · Frequency distribution · Contingency table

Dependence

Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Partial correlation · Scatter plot

Statistical graphics

Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot · Radar chart

Data collection

Designing studies	Effect size · Standard error · Statistical power · Sample size determination

Survey methodology	Sampling · Stratified sampling · Opinion poll · Questionnaire

Controlled experiment	Design of experiments · Factorial experiment · Randomized experiment · Random assignment · Replication · Blocking · Optimal design

Uncontrolled studies	Natural experiment · Quasi-experiment · Observational study

Statistical inference

Correlation and regression analysis

Categorical, multivariate, time-series, or survival analysis

Applications

Category · Portal · Outline · Index

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Help us answer these

Degree of freedom in the t distribution?

Is the t-distribution symmetric?

Who helps t-mobile with distribution?

Is there a family of t distributions?

Post a question - any question - to the WikiAnswers community:

Copyrights:

Cite

Wikipedia on Answers.com This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Noncentral t-distribution. Read

Answer these

How does t statistic relate to t distribution?

Is a t distribution continuous?

What are some characteristics of the T distribution?

Which of the following is not a characteristic of t distribution?

» more

Featured guides

» More

Mentioned in

noncentral t distribution (statistics)

noncentral F distribution (statistics)

Noncentral parameter

Noncentral chi distribution

Fisher's noncentral hypergeometric distribution

» More

Company
About
Terms of Use
Privacy Policy
IP Issues
Disclaimer

Community
Guidelines
Reputation
Roles
Help

Updates
Email
Watchlist
RSS

International sites
English
|
Deutsch
|
Español
|
Français
|
Italiano
|
Tagalog

facebook
twitter
youtube

Noncentral t-distribution

Noncentral t-distribution