|
|
| Line 1: |
Line 1: |
| {{DISPLAYTITLE:Noncentral ''t''-distribution}}
| | Andrew Simcox is the title his parents gave him and he completely enjoys this name. North Carolina is exactly where we've been living for many years and will never move. My day occupation is a journey agent. One of the issues she loves most is canoeing and she's been doing it for fairly a whilst.<br><br>Stop by my homepage - [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review clairvoyant psychic] |
| {{Probability distribution |
| |
| name =Noncentral Student's ''t''|
| |
| type =density|
| |
| pdf_image =[[Image:nc student t pdf.svg|325px]]|
| |
| cdf_image =|
| |
| parameters = ν > 0 degrees of freedom<br/><math>\mu \in \Re \,\!</math> noncentrality parameter |
| |
| support =<math>x \in (-\infty; +\infty)\,\!</math>|
| |
| pdf =see text|
| |
| <!--For "<math>":
| |
| -- Split formulas by "\begin{matrix}" with "\\[0.5em]" split --
| |
| -- as 0.5em interline spacing; end with "\end{matrix}". --
| |
| -- Fractions have 2 brace-pairs. A centered dot is "\cdot". -->
| |
| cdf =see text|
| |
| mean =see text|
| |
| median =|
| |
| mode =see text|
| |
| variance =see text|
| |
| skewness =see text|
| |
| kurtosis =see text|
| |
| entropy =|
| |
| mgf =|
| |
| char =
| |
| }}
| |
| | |
| As with other [[noncentrality parameter]]s, the '''noncentral ''t''-distribution''' generalizes a [[probability distribution]] – [[Student's t-distribution|Student's ''t''-distribution]] – using a noncentrality parameter. Whereas the central distribution describes how a test statistic is distributed when the difference tested is null, the noncentral distribution also 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 statistics|robust modeling]] for [[data]].
| |
| | |
| ==Characterization==
| |
| If ''Z'' is a [[normal distribution|normally distributed]] random variable with unit variance and zero mean, and ''V'' is a [[Chi-squared distribution|Chi-squared distributed]] random variable with ν [[Degrees of freedom (statistics)|degrees of freedom]] that is [[Statistical independence|statistically independent]] of ''Z'', then
| |
| | |
| :<math>T=\frac{Z+\mu}{\sqrt{V/\nu}} </math>
| |
| | |
| 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 <ref name=lenth>
| |
| {{cite journal | last=Lenth | first= Russell V | title=Algorithm AS 243: Cumulative Distribution Function of the Non-central ''t'' Distribution | journal=Journal of the Royal Statistical Society, Series C | year=1989| volume=38 | pages=185–189 | jstor=2347693}}</ref>
| |
| | |
| :<math>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}</math>
| |
| | |
| where
| |
| :<math>\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],</math>
| |
| :<math>I_y\,\!(a,b)</math> is the [[Beta function|regularized incomplete beta function]],
| |
| :<math>y=\frac{x^2}{x^2+\nu},</math>
| |
| :<math>p_j=\frac{1}{j!}\exp\left\{-\frac{\mu^2}{2}\right\}\left(\frac{\mu^2}{2}\right)^j,</math>
| |
| :<math>q_j=\frac{\mu}{\sqrt{2}\Gamma(j+3/2)}\exp\left\{-\frac{\mu^2}{2}\right\}\left(\frac{\mu^2}{2}\right)^j,</math>
| |
| and | |
| :Φ is the cumulative distribution function of the [[standard normal distribution]].
| |
| | |
| Alternatively, the noncentral t-distribution CDF can be expressed as:
| |
| :<math>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}</math>
| |
| 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 [[Recursion|recursive computing]].<ref name=lenth/> In statistical software [[R (programming language)|R]], the cumulative distribution function is implemented as '''pt'''.
| |
| | |
| ===Probability density function===
| |
| The [[probability density function]] 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
| |
| | |
| :<math>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{{}_1F_1\left(\frac{\nu}{2}+1;\, \frac{3}{2};\, \frac{\mu^2x^2}{2(\nu+x^2)} \right )}{(\nu+x^2)\Gamma(\frac{\nu+1}{2})} + \frac{{}_1F_1\left(\frac{\nu+1}{2};\, \frac{1}{2};\, \frac{\mu^2x^2}{2(\nu+x^2)} \right )}{\sqrt{\nu+x^2}\Gamma(\frac{\nu}{2}+1)}\right \}</math>
| |
| | |
| where <sub>1</sub>''F''<sub>1</sub> is a [[confluent hypergeometric function]]. | |
| | |
| An alternative integral form is <ref>L. Scharf, Statistical Signal Processing, (Massachusetts: Addison-Wesley, 1991), p.177.</ref>
| |
| | |
| :<math> 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.</math>
| |
| | |
| A third form of the density is obtained using its cumulative distribution functions, as follows.
| |
| :<math>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}</math>
| |
| | |
| This is the approach implemented by the '''dt''' function in [[R (programming language)|R]].
| |
| | |
| ==Properties==
| |
| ===Moments of the Noncentral ''t''-distribution===
| |
| In general, the ''k''th raw moment of the noncentral ''t''-distribution is <ref>{{cite journal | last=Hogben | first=D | coauthor=Wilk, MB | title=The moments of the non-central ''t''-distribution | journal=Biometrika | year=1961 | volume=48 | pages=465–468 | jstor=2332772}}</ref>
| |
| | |
| :<math>\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}</math>
| |
| | |
| In particular, the mean and variance of the noncentral ''t''-distribution are
| |
| | |
| :<math>\begin{align}
| |
| \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} \\
| |
| \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}
| |
| \end{align}</math>
| |
| | |
| An excellent approximation to <math>\sqrt{\frac{\nu}{2}}\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} </math> is <math>\left(1-\frac{3}{4\nu-1}\right)^{-1}</math>, 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 it always lies in the interval<ref>{{cite journal | last=van Aubel| first=A| coauthor=Gawronski, W | title=Analytic properties of noncentral distributions | journal=Applied Mathematics and Computation| year=2003| volume=141 | pages=3–12 | url=http://www.sciencedirect.com/science/article/B6TY8-47G44WX-V/2/7705d2642b1a384b13e0578898a22d48 | doi=10.1016/S0096-3003(02)00316-8}}</ref>
| |
| :<math> \left( \sqrt{\frac{2\nu}{2\nu+5}}\mu,\,\sqrt{\frac{\nu}{\nu+1}}\mu \right)</math> when μ > 0, and
| |
| :<math> \left( \sqrt{\frac{\nu}{\nu+1}}\mu,\,\sqrt{\frac{2\nu}{2\nu+5}}\mu \right)</math> when μ < 0.
| |
| | |
| Moreover, the mode always has the same sign as the noncentrality parameter μ and 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
| |
| :<math>\sqrt{\frac{\nu}{2}}\frac{\Gamma\left(\frac{\nu+2}{2}\right)}{\Gamma\left(\frac{\nu+3}{2}\right)}\mu;\,</math>
| |
| and when μ → ∞, the mode is approximated by
| |
| :<math>\sqrt{\frac{\nu}{\nu+1}}\mu.</math>
| |
| | |
| ==Occurrences==
| |
| ===Use in power analysis===
| |
| Suppose we have an independent and identically distributed sample ''X''<sub>1</sub>, ..., ''X<sub>n</sub>'' each of which is normally distributed with mean θ and variance σ<sup>2</sup>, and we are interested in testing the [[null hypothesis]] θ = 0 vs. the [[alternative hypothesis]] θ ≠ 0. We can perform a [[Student's t-test|one sample ''t''-test]] using the [[test statistic]]
| |
| | |
| :<math>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} }}</math>
| |
| | |
| where <math>\bar{X}</math> is the sample mean and <math>\hat{\sigma}^2\,\!</math> 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 <math>\sqrt{n}\theta/\sigma\,\!</math>.
| |
| | |
| If the test procedure rejects the null hypothesis whenever <math>|T|>t_{1-\alpha/2}\,\!</math>, where <math>t_{1-\alpha/2}\,\!</math> is the upper α/2 quantile of the [[Student's t-distribution|(central) Student's ''t''-distribution]] for a pre-specified α ∈ (0, 1), then the power of this test is given by
| |
| | |
| :<math>1-F_{n-1,\sqrt{n}\theta/\sigma}(t_{1-\alpha/2})+F_{n-1,\sqrt{n}\theta/\sigma}(-t_{1-\alpha/2}) .</math>
| |
| | |
| Similar applications of the noncentral ''t''-distribution can be found in the [[Statistical power|power analysis]] of the general normal-theory [[general linear model|linear models]], which includes the above [[Student's t-test|one sample ''t''-test]] as a special case.
| |
| | |
| ==Related distributions==
| |
| *Central ''t'' distribution: The central ''t''-distribution can be converted into a [[location parameter|location]]/[[scale parameter|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.<ref>{{cite journal | title=A Central t Approximation to the Noncentral t Distribution | author1=Helena Chmura Kraemer | author2=Minja Paik | journal=Technometrics | volume=21 | number=3 | year=1979 | pages=357–360 | jstor=1267759}}</ref>
| |
| | |
| *If ''T'' is noncentral ''t''-distributed with ν degrees of freedom and noncentrality parameter μ and ''F'' = ''T''<sup>2</sup>, then ''F'' has a [[noncentral F-distribution|noncentral ''F''-distribution]] with 1 numerator degree of freedom, ν denominator degrees of freedom, and noncentrality parameter μ<sup>2</sup>.
| |
| | |
| *If ''T'' is noncentral ''t''-distributed with ν degrees of freedom and noncentrality parameter μ and <math> Z=\lim_{\nu\rightarrow\infty} T </math>, then ''Z'' has a normal distribution with mean μ and unit variance.
| |
| | |
| *When the ''denominator'' noncentrality parameter of a [[doubly noncentral t-distribution|doubly noncentral ''t''-distribution]] is zero, then it becomes a noncentral ''t''-distribution.
| |
| | |
| ===Special cases===
| |
| *When μ = 0, the noncentral ''t''-distribution becomes the [[Student's t-distribution|central (Student's) ''t''-distribution]] with the same degrees of freedom.
| |
| | |
| ==See also==
| |
| * [[Noncentral F-distribution|Noncentral <var>F</var>-distribution]]
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| ==External links==
| |
| * [http://mathworld.wolfram.com/NoncentralStudentst-Distribution.html Eric W. Weisstein. "Noncentral Student's ''t''-Distribution."] From MathWorld—A Wolfram Web Resource
| |
| | |
| {{ProbDistributions|continuous-infinite}}
| |
| {{Common univariate probability distributions|state=collapsed}}
| |
| {{Statistics|state=collapsed}}
| |
| | |
| [[Category:Continuous distributions]]
| |
| [[Category:Probability distributions]]
| |