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Student's t-distribution

From Wikipedia, the free encyclopedia
Probability distribution
This article is about the mathematics of Student'st-distribution. For its uses in statistics, seeStudent's t-test.
Student'st
Probability density function
Cumulative distribution function
Parametersν>0{\displaystyle \nu >0}degrees of freedom (real, almost always a positiveinteger)
Supportx∈(−∞,∞){\displaystyle x\in (-\infty ,\infty )}
PDFΓ(ν+12)πνΓ(ν2)(1+x2ν)−ν+12{\displaystyle {\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}}
CDF

12+xΓ(ν+12)πνΓ(ν2)×2F1(12,ν+12;32;−x2ν),{\displaystyle {\begin{aligned}&{\frac {1}{2}}+x{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}\times \\&\qquad {}_{2}F_{1}\!\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right),\end{aligned}}}

where2F1{\displaystyle {}_{2}F_{1}} is thehypergeometric function
Mean0{\displaystyle 0} forν>1,{\displaystyle \nu >1,} otherwiseundefined
Median0{\displaystyle 0}
Mode0{\displaystyle 0}
Varianceνν−2{\displaystyle {\frac {\nu }{\nu -2}}} forν>2,{\displaystyle \nu >2,}∞{\displaystyle \infty } for1<ν≤2,{\displaystyle 1<\nu \leq 2,} otherwiseundefined
Skewness0{\displaystyle 0} for ν>3 ,{\displaystyle \ \nu >3\ ,} otherwiseundefined
Excess kurtosis6ν−4{\displaystyle {\frac {6}{\nu -4}}} forν>4,{\displaystyle \nu >4,}∞{\displaystyle \infty } for2<ν≤4,{\displaystyle 2<\nu \leq 4,} otherwiseundefined
Entropy

ν+12[ψ(ν+12)−ψ(ν2)]+ln⁡[νB(ν2,12)] (nats),{\displaystyle {\begin{aligned}&{\frac {\nu +1}{2}}\left[\psi {\left({\frac {\nu +1}{2}}\right)}-\psi {\left({\frac {\nu }{2}}\right)}\right]\\&\quad +\ln \left[{\sqrt {\nu }}\,\mathrm {B} {\left({\frac {\nu }{2}},{\frac {1}{2}}\right)}\right]~{\text{(nats)}},\end{aligned}}}

whereψ{\displaystyle \psi } is thedigamma function andB{\displaystyle \mathrm {B} } is thebeta function
MGFundefined
CF

(ν|t|)ν/2Kν/2(ν|t|)Γ(ν/2)2ν/2−1{\displaystyle {\frac {{\big (}{\sqrt {\nu }}\,|t|{\big )}^{\nu /2}\,K_{\nu /2}{\big (}{\sqrt {\nu }}\,|t|{\big )}}{\Gamma (\nu /2)\,2^{\nu /2-1}}}} forν>0{\displaystyle \nu >0},

whereKν{\displaystyle K_{\nu }} is themodified Bessel function of the second kind[1]
Expected shortfall

μ+s((ν+[T−1(1−p)]2)×τ(T−1(1−p))(ν−1)(1−p)),{\displaystyle \mu +s\left({\frac {{\big (}\nu +[T^{-1}(1-p)]^{2}{\big )}\times \tau {\big (}T^{-1}(1-p){\big )}}{(\nu -1)(1-p)}}\right),}

whereT−1{\displaystyle T^{-1}} is the inverse standardized Student tCDF, andτ{\displaystyle \tau } is the standardized Student tPDF.[2]

Inprobability theory andstatistics,Student'st distribution (or simply thet distribution)tν{\displaystyle t_{\nu }} is a continuousprobability distribution that generalizes thestandard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.

However,tν{\displaystyle t_{\nu }} hasheavier tails, and the amount of probability mass in the tails is controlled by the parameterν{\displaystyle \nu }. Forν=1{\displaystyle \nu =1} the Student'st distributiontν{\displaystyle t_{\nu }} becomes the standardCauchy distribution, which has very"fat" tails; whereas forν→∞{\displaystyle \nu \to \infty } it becomes the standard normal distributionN(0,1),{\displaystyle {\mathcal {N}}(0,1),} which has very "thin" tails.

The name "Student" is a pseudonym used byWilliam Sealy Gosset in his scientific paper publications during his work at theGuinness Brewery inDublin, Ireland.

The Student'st distribution plays a role in a number of widely used statistical analyses, includingStudent'st-test for assessing thestatistical significance of the difference between two sample means, the construction ofconfidence intervals for the difference between two population means, and in linearregression analysis.

In the form of thelocation-scalet distributionℓst⁡(μ,τ2,ν){\displaystyle \operatorname {\ell st} (\mu ,\tau ^{2},\nu )} it generalizes thenormal distribution and also arises in theBayesian analysis of data from a normal family as acompound distribution when marginalizing over the variance parameter.

Definitions

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Probability density function

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Student'st distribution has theprobability density function (PDF) given byf(t)=Γ(ν+12)πνΓ(ν2)(1+t2ν)−(ν+1)/2,{\displaystyle f(t)={\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}whereν{\displaystyle \nu } is the number ofdegrees of freedom, andΓ{\displaystyle \Gamma } is thegamma function. This may also be written asf(t)=1νB(12,ν2)(1+t2ν)−(ν+1)/2,{\displaystyle f(t)={\frac {1}{{\sqrt {\nu }}\,\mathrm {B} {\left({\frac {1}{2}},{\frac {\nu }{2}}\right)}}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2},}whereB{\displaystyle \mathrm {B} } is thebeta function. In particular, for positive integer-valued degrees of freedomν > 1 we have:

ν∈N≥2k={2,ν evenπ,ν oddΓ(ν+12)πνΓ(ν2)=(ν−1)!!kν(ν−2)!!{\displaystyle {\begin{aligned}\nu &\in \mathbb {N} \geq 2\\k&={\begin{cases}2,&\nu {\text{ even}}\\\pi ,&\nu {\text{ odd}}\\\end{cases}}\\{\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{{\sqrt {\pi \nu }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}&={\frac {(\nu -1)!!}{k{\sqrt {\nu }}(\nu -2)!!}}\\\end{aligned}}}

The probability density function issymmetric, and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, thet distribution approaches the normal distribution with mean 0 and variance 1. For this reasonν{\displaystyle {\nu }} is also known as the normality parameter.[3]

The following images show the density of thet distribution for increasing values ofν.{\displaystyle \nu .} The normal distribution is shown as a blue line for comparison. Note that thet distribution (red line) becomes closer to the normal distribution asν{\displaystyle \nu } increases.

Density of thet distribution (red) for 1, 2, 3, 5, 10, and 30 degrees of freedom compared to the standard normal distribution (blue).
Previous plots shown in green.
1 degree of freedom
2 degrees of freedom
3 degrees of freedom
5 degrees of freedom
10 degrees of freedom
30 degrees of freedom

Cumulative distribution function

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Thecumulative distribution function (CDF) can be written in terms ofI, the regularizedincomplete beta function. Fort > 0 ,

F(t)=∫−∞tf(u)du = 1−12Ix(t)(ν2,12),{\displaystyle F(t)=\int _{-\infty }^{t}f(u)\,du~=~1-{\frac {1}{2}}I_{x(t)}{\left({\frac {\nu }{2}},\,{\frac {1}{2}}\right)},}

where

x(t)=νt2+ν.{\displaystyle x(t)={\frac {\nu }{t^{2}+\nu }}\,.}

Other values would be obtained by symmetry. An alternative formula, valid fort2<ν,{\displaystyle t^{2}<\nu \,,} is

∫−∞tf(u)du=12+tΓ(ν+12)πνΓ(ν 2 )2F1(12,ν+12;32;−t2ν),{\displaystyle \int _{-\infty }^{t}f(u)\,du={\frac {1}{2}}+t\,{\frac {\Gamma \!\left({\frac {\nu +1}{2}}\right)}{{\sqrt {\pi \nu }}\,\Gamma \!\left({\frac {\nu }{\ 2\ }}\right)}}\;{}_{2}F_{1}\!\left({\frac {1}{2}},{\frac {\nu +1}{2}};\,{\frac {3}{2}};\,-{\frac {t^{2}}{\nu }}\right),}

where2F1( , ; ; ){\displaystyle {}_{2}F_{1}(\ ,\ ;\ ;\ )} is a particular instance of thehypergeometric function.

For information on its inverse cumulative distribution function, seequantile function § Student's t-distribution.

Special cases

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Certain values of ν {\displaystyle \ \nu \ } give a simple form for Student's t-distribution.

 ν {\displaystyle \ \nu \ }PDFCDFnotes
11π(1+t2){\displaystyle {\frac {1}{\pi (1+t^{2})}}}12+1πarctan⁡(t){\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\arctan(t)}SeeCauchy distribution
2122(1+t22)3/2{\displaystyle {\frac {1}{2\,{\sqrt {2}}\,\left(1+{\frac {t^{2}}{2}}\right)^{3/2}}}}12+t221+t22{\displaystyle {\frac {1}{2}}+{\frac {t}{2{\sqrt {2}}\,{\sqrt {1+{\frac {t^{2}}{2}}}}}}}
32π3(1+t23)2{\displaystyle {\frac {2}{\pi {\sqrt {3}}\,\left(1+{\frac {t^{2}}{3}}\right)^{2}}}}12+1π[t31+t23+arctan⁡t3]{\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\left[{\frac {\frac {t}{\sqrt {3}}}{1+{\frac {t^{2}}{3}}}}+\arctan {\frac {t}{\sqrt {3}}}\right]}
438(1+t24)5/2{\displaystyle {\frac {3}{8\left(1+{\frac {t^{2}}{4}}\right)^{5/2}}}}12+38[t1+t24][1−t212(1+t24)]{\displaystyle {\frac {1}{2}}+{\frac {3}{8}}\left[{\frac {t}{\sqrt {1+{\frac {t^{2}}{4}}}}}\right]\left[1-{\frac {t^{2}}{12\left(1+{\frac {t^{2}}{4}}\right)}}\right]}
583π5(1+t25)3{\displaystyle {\frac {8}{3\pi {\sqrt {5}}\,\left(1+{\frac {t^{2}}{5}}\right)^{3}}}}12+1π[t5(1+t25)(1+23(1+t25))+arctan⁡t5]{\displaystyle {\frac {1}{2}}+{\frac {1}{\pi }}\left[{\frac {t}{{\sqrt {5}}\left(1+{\frac {t^{2}}{5}}\right)}}\left(1+{\frac {2}{3\left(1+{\frac {t^{2}}{5}}\right)}}\right)+\arctan {\frac {t}{\sqrt {5}}}\right]}
 ∞ {\displaystyle \ \infty \ }12πe−t2/2{\displaystyle {\frac {1}{\sqrt {2\pi }}}\,e^{-t^{2}/2}}12[1+erf⁡(t2)]{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {t}{\sqrt {2}}}\right)\right]}SeeNormal distribution,Error function

Properties

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Moments

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Forν>1{\displaystyle \nu >1}, theraw moments of thet distribution are

E⁡{Tk}={0k odd ,0<k<ν,1πΓ(ν2)[Γ(k+12)Γ(ν−k2)νk2]k even ,0<k<ν.{\displaystyle \operatorname {\mathbb {E} } \left\{T^{k}\right\}={\begin{cases}\quad 0&k{\text{ odd }},\quad 0<k<\nu \,,\\[2ex]{\frac {1}{{\sqrt {\pi }}\,\Gamma {\left({\frac {\nu }{2}}\right)}}}\left[\Gamma \!\left({\frac {k+1}{2}}\right)\,\Gamma \!\left({\frac {\nu -k}{2}}\right)\,\nu ^{\frac {k}{2}}\right]&k{\text{ even }},\quad 0<k<\nu \,.\end{cases}}}

Moments of order ν {\displaystyle \ \nu \ } or higher do not exist.[4]

The term for0<k<ν{\displaystyle 0<k<\nu },k even, may be simplified using the properties of thegamma function to

E⁡{Tk}=νk2∏j=1k/22j−1ν−2jk even,0<k<ν .{\displaystyle \operatorname {\mathbb {E} } \left\{T^{k}\right\}=\nu ^{\frac {k}{2}}\,\prod _{j=1}^{k/2}{\frac {2j-1}{\nu -2j}}\qquad k{\text{ even}},\quad 0<k<\nu ~.}

For at distribution withν{\displaystyle \nu } degrees of freedom, theexpected value is0{\displaystyle 0} ifν>1,{\displaystyle \nu >1\,,} and itsvariance isνν−2{\displaystyle {\frac {\nu }{\nu -2}}} ifν>2.{\displaystyle \nu >2\,.} Theskewness is 0 ifν>3{\displaystyle \nu >3} and theexcess kurtosis is6ν−4{\displaystyle {\frac {6}{\nu -4}}} ifν>4.{\displaystyle \nu >4\,.}

How thet distribution arises (characterization)

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As the distribution of a test statistic

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Student'st-distribution withν{\displaystyle \nu } degrees of freedom can be defined as the distribution of therandom variableT with[5][6]

T=ZV/ν=ZνV,{\displaystyle T={\frac {Z}{\sqrt {V/\nu }}}=Z{\sqrt {\frac {\nu }{V}}},}

where

A different distribution is defined as that of the random variable defined, for a given constant μ, by(Z+μ)νV.{\displaystyle (Z+\mu ){\sqrt {\frac {\nu }{V}}}.}This random variable has anoncentralt-distribution withnoncentrality parameterμ. This distribution is important in studies of thepower of Student'st-test.

Derivation
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SupposeX1, ...,Xn areindependent realizations of the normally-distributed, random variableX, which has an expected valueμ andvarianceσ2. Let

X¯n=1n(X1+⋯+Xn){\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}

be the sample mean, and

s2=1n−1∑i=1n(Xi−X¯n)2{\displaystyle s^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}\left(X_{i}-{\overline {X}}_{n}\right)^{2}}

be an unbiased estimate of the variance from the sample. It can be shown that the random variable

V=(n−1)s2σ2{\displaystyle V=(n-1){\frac {s^{2}}{\sigma ^{2}}}}

has a chi-squared distribution withν=n−1{\displaystyle \nu =n-1} degrees of freedom (byCochran's theorem).[7] It is readily shown that the quantity

Z=(X¯n−μ)nσ{\displaystyle Z=\left({\overline {X}}_{n}-\mu \right){\frac {\sqrt {n}}{\sigma }}}

is normally distributed with mean 0 and variance 1, since the sample meanX¯n{\displaystyle {\overline {X}}_{n}} is normally distributed with meanμ and varianceσ2/n. Moreover, it is possible to show that these two random variables (the normally distributed oneZ and the chi-squared-distributed oneV) are independent. Consequently[clarification needed] thepivotal quantity

T≡ZV/ν=(X¯n−μ)ns,{\displaystyle T\equiv {\frac {Z}{\sqrt {V/\nu }}}=\left({\overline {X}}_{n}-\mu \right){\frac {\sqrt {n}}{s}},}

which differs fromZ in that the exact standard deviationσ is replaced by the sample standard errors, has a Student'st-distribution as defined above. Notice that the unknown population varianceσ2 does not appear inT, since it was in both the numerator and the denominator, so it canceled. Gosset intuitively obtained the probability density function stated above, withν{\displaystyle \nu } equal ton − 1, and Fisher proved it in 1925.[8]

The distribution of the test statisticT depends onν{\displaystyle \nu }, but notμ orσ; the lack of dependence onμ andσ is what makes thet-distribution important in both theory and practice.

Sampling distribution of t-statistic

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Thet distribution arises as thesampling distributionof thet statistic. Below the one-samplet statistic is discussed, for the corresponding two-samplet statistic seeStudent's t-test.

Unbiased variance estimate
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Let x1,…,xn∼N(μ,σ2) {\displaystyle \ x_{1},\ldots ,x_{n}\sim {\mathcal {N}}(\mu ,\sigma ^{2})\ } be independent and identically distributed samples from a normal distribution with meanμ{\displaystyle \mu } and variance σ2 .{\displaystyle \ \sigma ^{2}~.} The sample mean and unbiasedsample variance are given by:

x¯= x1+⋯+xn n ,s2=1 n−1  ∑i=1n(xi−x¯)2 .{\displaystyle {\begin{aligned}{\bar {x}}&={\frac {\ x_{1}+\cdots +x_{n}\ }{n}}\ ,\\[5pt]s^{2}&={\frac {1}{\ n-1\ }}\ \sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}~.\end{aligned}}}

The resulting (one sample)t statistic is given by

t=x¯−μ s/n  ∼tn−1 .{\displaystyle t={\frac {{\bar {x}}-\mu }{\ s/{\sqrt {n\ }}\ }}\sim t_{n-1}~.}

and is distributed according to a Student'st distribution with n−1 {\displaystyle \ n-1\ } degrees of freedom.

Thus for inference purposes thet statistic is a useful "pivotal quantity" in the case when the mean and variance(μ,σ2){\displaystyle (\mu ,\sigma ^{2})} are unknown population parameters, in the sense that thet statistic has then a probability distribution that depends on neitherμ{\displaystyle \mu } nor σ2 .{\displaystyle \ \sigma ^{2}~.}

ML variance estimate
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Instead of the unbiased estimate s2 {\displaystyle \ s^{2}\ } we may also use the maximum likelihood estimate sML2= 1 n ∑i=1n(xi−x¯)2 {\displaystyle \ s_{\mathsf {ML}}^{2}={\frac {\ 1\ }{n}}\ \sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}\ }yielding the statistic tML=x¯−μsML2/n =nn−1  t .{\displaystyle \ t_{\mathsf {ML}}={\frac {{\bar {x}}-\mu }{\sqrt {s_{\mathsf {ML}}^{2}/n\ }}}={\sqrt {{\frac {n}{n-1}}\ }}\ t~.}This is distributed according to the location-scalet distribution:tML∼ℓst⁡(0, τ2=n/(n−1), n−1) .{\displaystyle t_{\mathsf {ML}}\sim \operatorname {\ell st} (0,\ \tau ^{2}=n/(n-1),\ n-1)~.}

Compound distribution of normal with inverse gamma distribution

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The location-scalet distribution results fromcompounding aGaussian distribution (normal distribution) withmean μ {\displaystyle \ \mu \ } and unknownvariance, with aninverse gamma distribution placed over the variance with parametersa=ν2{\textstyle a={\frac {\nu }{2}}} andb=ντ22.{\textstyle b={\frac {\nu \tau ^{2}}{2}}\,.} In other words, therandom variableX is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance ismarginalized out (integrated out).

Equivalently, this distribution results from compounding a Gaussian distribution with ascaled-inverse-chi-squared distribution with parametersν{\displaystyle \nu } and τ2 .{\displaystyle \ \tau ^{2}~.} The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e.ν=2a,τ2=ba.{\displaystyle \nu =2a,\;\tau ^{2}={\frac {b}{a}}\,.}

The reason for the usefulness of this characterization is that inBayesian statistics the inverse gamma distribution is theconjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scalet distribution arises naturally in many Bayesian inference problems.[9]

Maximum entropy distribution

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Student'st distribution is themaximum entropy probability distribution for a random variateX having a certain value ofE⁡{ln⁡(ν+X2)}{\textstyle \operatorname {\mathbb {E} } \left\{\ln(\nu +X^{2})\right\}}.[10][clarification needed][better source needed]This follows immediately from the observation that the pdf can be written inexponential family form withν+X2{\displaystyle \nu +X^{2}} as sufficient statistic.

Integral of Student's probability density function andp-value

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The functionA(t |ν) is the integral of Student's probability density function,f(t) between  -t andt, fort ≥ 0 . It thus gives the probability that a value oft less than that calculated from observed data would occur by chance. Therefore, the functionA(t |ν) can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value oft and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly int tests. For the statistict, withν degrees of freedom,A(t |ν) is the probability thatt would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so thatt ≥ 0 ). It can be easily calculated from thecumulative distribution functionFν(t) of thet distribution:

A(t∣ν)=Fν(t)−Fν(−t)=1−Iνν+t2(ν2,12),{\displaystyle A(t\mid \nu )=F_{\nu }(t)-F_{\nu }(-t)=1-I_{\frac {\nu }{\nu +t^{2}}}\!\left({\frac {\nu }{2}},{\frac {1}{2}}\right),}

whereIx(a,b) is the regularizedincomplete beta function.

For statistical hypothesis testing this function is used to construct thep-value.

Related distributions

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In general

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Location-scalet-distribution

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Location-scale transformation

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Student'st distribution generalizes to the three parameterlocation-scalet distributionℓst⁡(μ, τ2, ν) {\displaystyle \operatorname {\ell st} (\mu ,\ \tau ^{2},\ \nu )\ } by introducing alocation parameter μ {\displaystyle \ \mu \ } and ascale parameter τ .{\displaystyle \ \tau ~.} With T∼tν {\displaystyle \ T\sim t_{\nu }\ }andlocation-scale family transformation X=μ+τ T {\displaystyle \ X=\mu +\tau \ T\ }we getX∼ℓst⁡(μ, τ2, ν) .{\displaystyle X\sim \operatorname {\ell st} (\mu ,\ \tau ^{2},\ \nu )~.}

The resulting distribution is also called thenon-standardized Student'st distribution.

Density and first two moments

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The location-scalet distribution has a density defined by:[13]

p(x∣ν,μ,τ)=Γ(ν+12)Γ(ν2)τπν(1+1ν(x−μτ)2)−(ν+1)/2{\displaystyle p(x\mid \nu ,\mu ,\tau )={\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{\Gamma {\left({\frac {\nu }{2}}\right)}\tau {\sqrt {\pi \nu }}}}\left(1+{\frac {1}{\nu }}\left({\frac {x-\mu }{\tau }}\right)^{2}\right)^{-(\nu +1)/2}}

Equivalently, the density can be written in terms ofτ2{\displaystyle \tau ^{2}}:

p(x∣ν,μ,τ2)=Γ(ν+12)Γ(ν2)πντ2(1+1ν(x−μ)2τ2)−(ν+1)/2{\displaystyle p(x\mid \nu ,\mu ,\tau ^{2})={\frac {\Gamma {\left({\frac {\nu +1}{2}}\right)}}{\Gamma {\left({\frac {\nu }{2}}\right)}{\sqrt {\pi \nu \tau ^{2}}}}}\left(1+{\frac {1}{\nu }}{\frac {(x-\mu )^{2}}{\tau ^{2}}}\right)^{-(\nu +1)/2}}

Other properties of this version of the distribution are:[13]

E⁡{ X }=μ for ν>1 ,var⁡{ X }=τ2νν−2 for ν>2 ,mode⁡{ X }=μ .{\displaystyle {\begin{aligned}\operatorname {\mathbb {E} } \{\ X\ \}&=\mu &{\text{ for }}\nu >1\ ,\\\operatorname {var} \{\ X\ \}&=\tau ^{2}{\frac {\nu }{\nu -2}}&{\text{ for }}\nu >2\ ,\\\operatorname {mode} \{\ X\ \}&=\mu ~.\end{aligned}}}

Special cases

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Occurrence and applications

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In frequentist statistical inference

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Student'st distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additiveerrors. If (as in nearly all practical statistical work) the populationstandard deviation of these errors is unknown and has to be estimated from the data, thet distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of thet distribution.

Confidence intervals andhypothesis tests are two statistical procedures in which thequantiles of the sampling distribution of a particular statistic (e.g. thestandard score) are required. In any situation where this statistic is alinear function of thedata, divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student'st distribution. Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.

Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student'st distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of thevariance as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

Hypothesis testing

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A number of statistics can be shown to havet distributions for samples of moderate size undernull hypotheses that are of interest, so that thet distribution forms the basis for significance tests. For example, the distribution ofSpearman's rank correlation coefficientρ, in the null case (zero correlation) is well approximated by thet distribution for sample sizes above about 20.[citation needed]

Confidence intervals

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Suppose the numberA is so chosen that

 P⁡{ −A<T<A }=0.9 ,{\displaystyle \ \operatorname {\mathbb {P} } \left\{\ {-A}<T<A\ \right\}=0.9\ ,}

whenT has at distribution withn − 1   degrees of freedom. By symmetry, this is the same as saying thatA satisfies

 P⁡{ T<A }=0.95 ,{\displaystyle \ \operatorname {\mathbb {P} } \left\{\ T<A\ \right\}=0.95\ ,}

soA is the "95th percentile" of this probability distribution, or A=t(0.05,n−1) .{\displaystyle \ A=t_{(0.05,n-1)}~.} Then

 P⁡{ −A< X¯n−μ Sn/n <A }=0.9 ,{\displaystyle \ \operatorname {\mathbb {P} } \left\{\ {-A}<{\frac {\ {\overline {X}}_{n}-\mu \ }{S_{n}/{\sqrt {n\ }}}}<A\ \right\}=0.9\ ,}

whereSn is the sample standard deviation of the observed values. This is equivalent to

 P⁡{ X¯n−ASn n  <μ<X¯n+A Sn n   }=0.9.{\displaystyle \ \operatorname {\mathbb {P} } \left\{\ {\overline {X}}_{n}-A{\frac {S_{n}}{\ {\sqrt {n\ }}\ }}<\mu <{\overline {X}}_{n}+A\ {\frac {S_{n}}{\ {\sqrt {n\ }}\ }}\ \right\}=0.9.}

Therefore, the interval whose endpoints are

 X¯n ±A Sn n   {\displaystyle \ {\overline {X}}_{n}\ \pm A\ {\frac {S_{n}}{\ {\sqrt {n\ }}\ }}\ }

is a 90%confidence interval for μ. Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use thet distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on anull hypothesis.

It is this result that is used in theStudent'st tests: since the difference between the means of samples from two normal distributions is itself distributed normally, thet distribution can be used to examine whether that difference can reasonably be supposed to be zero.

If the data are normally distributed, the one-sided(1 −α) upper confidence limit (UCL) of the mean, can be calculated using the following equation:

UCL1−α=X¯n+tα,n−1 Sn n   .{\displaystyle {\mathsf {UCL}}_{1-\alpha }={\overline {X}}_{n}+t_{\alpha ,n-1}\ {\frac {S_{n}}{\ {\sqrt {n\ }}\ }}~.}

The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words,X¯n{\displaystyle {\overline {X}}_{n}} being the mean of the set of observations, the probability that the mean of the distribution is inferior toUCL1 −α is equal to the confidencelevel1 −α .

Prediction intervals

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Thet distribution can be used to construct aprediction interval for an unobserved sample from a normal distribution with unknown mean and variance.

In Bayesian statistics

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The Student'st distribution, especially in its three-parameter (location-scale) version, arises frequently inBayesian statistics as a result of its connection with the normal distribution. Whenever thevariance of a normally distributedrandom variable is unknown and aconjugate prior placed over it that follows aninverse gamma distribution, the resultingmarginal distribution of the variable will follow a Student'st distribution. Equivalent constructions with the same results involve a conjugatescaled-inverse-chi-squared distribution over the variance, or a conjugate gamma distribution over theprecision. If animproper prior proportional to⁠1/ σ² ⁠ is placed over the variance, thet distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to aconjugate normally distributed prior, or is unknown distributed according to an improper constant prior.

Related situations that also produce at distribution are:

Robust parametric modeling

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Thet distribution is often used as an alternative to the normal distribution as a model for data, which often has heavier tails than the normal distribution allows for; see e.g. Lange et al.[14] The classical approach was to identifyoutliers (e.g., usingGrubbs's test) and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially inhigh dimensions), and thet distribution is a natural choice of model for such data and provides a parametric approach torobust statistics.

A Bayesian account can be found in Gelman et al.[15] The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors[citation needed] report that values between 3 and 9 are often good choices. Venables and Ripley[citation needed] suggest that a value of 5 is often a good choice.

Student'st process

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For practicalregression andprediction needs, Student'st processes were introduced, that are generalisations of the Studentt distributions for functions. A Student'st process is constructed from the Studentt distributions like aGaussian process is constructed from theGaussian distributions. For aGaussian process, all sets of values have a multidimensional Gaussian distribution. Analogously,X(t){\displaystyle X(t)} is a Studentt process on an intervalI=[a,b]{\displaystyle I=[a,b]} if the correspondent values of the process X(t1), … ,X(tn) {\displaystyle \ X(t_{1}),\ \ldots \ ,X(t_{n})\ } (ti∈I{\displaystyle t_{i}\in I}) have a jointmultivariate Studentt distribution.[16] These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi-output prediction, the multivariate Studentt processes are introduced and used.[17]

Table of selected values

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The following table lists values fort distributions withν degrees of freedom for a range of one-sided or two-sided critical regions. The first column isν, the percentages along the top are confidence levels α ,{\displaystyle \ \alpha \ ,} and the numbers in the body of the table are thetα,n−1{\displaystyle t_{\alpha ,n-1}} factors described in the section onconfidence intervals.

The last row with infiniteν gives critical points for a normal distribution since at distribution with infinitely many degrees of freedom is a normal distribution. (SeeRelated distributions above).

One-sided75%80%85%90%95%97.5%99%99.5%99.75%99.9%99.95%
Two-sided50%60%70%80%90%95%98%99%99.5%99.8%99.9%
11.0001.3761.9633.0786.31412.70631.82163.657127.321318.309636.619
20.8161.0611.3861.8862.9204.3036.9659.92514.08922.32731.599
30.7650.9781.2501.6382.3533.1824.5415.8417.45310.21512.924
40.7410.9411.1901.5332.1322.7763.7474.6045.5987.1738.610
50.7270.9201.1561.4762.0152.5713.3654.0324.7735.8936.869
60.7180.9061.1341.4401.9432.4473.1433.7074.3175.2085.959
70.7110.8961.1191.4151.8952.3652.9983.4994.0294.7855.408
80.7060.8891.1081.3971.8602.3062.8963.3553.8334.5015.041
90.7030.8831.1001.3831.8332.2622.8213.2503.6904.2974.781
100.7000.8791.0931.3721.8122.2282.7643.1693.5814.1444.587
110.6970.8761.0881.3631.7962.2012.7183.1063.4974.0254.437
120.6950.8731.0831.3561.7822.1792.6813.0553.4283.9304.318
130.6940.8701.0791.3501.7712.1602.6503.0123.3723.8524.221
140.6920.8681.0761.3451.7612.1452.6242.9773.3263.7874.140
150.6910.8661.0741.3411.7532.1312.6022.9473.2863.7334.073
160.6900.8651.0711.3371.7462.1202.5832.9213.2523.6864.015
170.6890.8631.0691.3331.7402.1102.5672.8983.2223.6463.965
180.6880.8621.0671.3301.7342.1012.5522.8783.1973.6103.922
190.6880.8611.0661.3281.7292.0932.5392.8613.1743.5793.883
200.6870.8601.0641.3251.7252.0862.5282.8453.1533.5523.850
210.6860.8591.0631.3231.7212.0802.5182.8313.1353.5273.819
220.6860.8581.0611.3211.7172.0742.5082.8193.1193.5053.792
230.6850.8581.0601.3191.7142.0692.5002.8073.1043.4853.767
240.6850.8571.0591.3181.7112.0642.4922.7973.0913.4673.745
250.6840.8561.0581.3161.7082.0602.4852.7873.0783.4503.725
260.6840.8561.0581.3151.7062.0562.4792.7793.0673.4353.707
270.6840.8551.0571.3141.7032.0522.4732.7713.0573.4213.690
280.6830.8551.0561.3131.7012.0482.4672.7633.0473.4083.674
290.6830.8541.0551.3111.6992.0452.4622.7563.0383.3963.659
300.6830.8541.0551.3101.6972.0422.4572.7503.0303.3853.646
400.6810.8511.0501.3031.6842.0212.4232.7042.9713.3073.551
500.6790.8491.0471.2991.6762.0092.4032.6782.9373.2613.496
600.6790.8481.0451.2961.6712.0002.3902.6602.9153.2323.460
800.6780.8461.0431.2921.6641.9902.3742.6392.8873.1953.416
1000.6770.8451.0421.2901.6601.9842.3642.6262.8713.1743.390
1200.6770.8451.0411.2891.6581.9802.3582.6172.8603.1603.373
∞0.6740.8421.0361.2821.6451.9602.3262.5762.8073.0903.291
One-sided75%80%85%90%95%97.5%99%99.5%99.75%99.9%99.95%
Two-sided50%60%70%80%90%95%98%99%99.5%99.8%99.9%
Calculating the confidence interval

Let's say we have a sample with size 11, sample mean 10, and sample variance 2. For 90% confidence with 10 degrees of freedom, the one-sidedt value from the table is 1.372 . Then with confidence interval calculated from

X¯n±tα,νSnn,{\displaystyle {\overline {X}}_{n}\pm t_{\alpha ,\nu }\,{\frac {S_{n}}{\sqrt {n}}}\,,}

we determine that with 90% confidence we have a true mean lying below

10+1.372211=10.585.{\displaystyle 10+1.372\,{\frac {\sqrt {2}}{\sqrt {11}}}=10.585\,.}

In other words, 90% of the times that an upper threshold is calculated by this method from particular samples, this upper threshold exceeds the true mean.

And with 90% confidence we have a true mean lying above

 10−1.372 2  11  =9.414 .{\displaystyle \ 10-1.372\ {\frac {\sqrt {2\ }}{\ {\sqrt {11\ }}\ }}=9.414~.}

In other words, 90% of the times that a lower threshold is calculated by this method from particular samples, this lower threshold lies below the true mean.

So that at 80% confidence (calculated from 100% − 2 × (1 − 90%) = 80%), we have a true mean lying within the interval

(10−1.372211,10+1.372211)=(9.414,10.585).{\displaystyle \left(10-1.372\,{\frac {\sqrt {2}}{\sqrt {11}}},\,10+1.372\,{\frac {\sqrt {2}}{\sqrt {11}}}\right)=(9.414,\,10.585)\,.}

Saying that 80% of the times that upper and lower thresholds are calculated by this method from a given sample, the true mean is both below the upper threshold and above the lower threshold is not the same as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method; seeconfidence interval andprosecutor's fallacy.

Nowadays, statistical software, such as theR programming language, and functions available in manyspreadsheet programs compute values of thet distribution and its inverse without tables.

Computational methods

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Monte Carlo sampling

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There are various approaches to constructing random samples from the Student'st distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of aquantile function touniform samples; e.g., in the multi-dimensional applications basis ofcopula-dependency.[citation needed] In the case of stand-alone sampling, an extension of theBox–Muller method and itspolar form is easily deployed.[18] It has the merit that it applies equally well to all real positivedegrees of freedom,ν, while many other candidate methods fail ifν is close to zero.[18]

History

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StatisticianWilliam Sealy Gosset, known as "Student"

In statistics, thet distribution was first derived as aposterior distribution in 1876 byHelmert[19][20][21] andLüroth.[22][23][24] As such, Student's t-distribution is an example ofStigler's Law of Eponymy. Thet distribution also appeared in a more general form asPearson type IV distribution inKarl Pearson's 1895 paper.[25]

In the English-language literature, the distribution takes its name fromWilliam Sealy Gosset's 1908 paper inBiometrika under the pseudonym "Student" during his work at theGuinness Brewery inDublin, Ireland.[26] One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Another version is that Guinness did not want their competitors to know that they were using thet test to determine the quality of raw material.[27][28]

Gosset worked at Guinness and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became well known through the work ofRonald Fisher, who called the distribution "Student's distribution" and represented the test value with the lettert.[8][29]

See also

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Notes

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  1. ^Hurst, Simon."The characteristic function of the Studentt distribution". Financial Mathematics Research Report. Statistics Research Report No. SRR044-95. Archived fromthe original on February 18, 2010.
  2. ^Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019)."Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"(PDF).Annals of Operations Research.299 (1–2). Springer:1281–1315.arXiv:1811.11301.doi:10.1007/s10479-019-03373-1.S2CID 254231768. Retrieved2023-02-27.
  3. ^Kruschke, J. K. (2015).Doing Bayesian Data Analysis (2nd ed.). Academic Press.ISBN 9780124058880.OCLC 959632184.
  4. ^Casella G, Berger RL (1990).Statistical Inference. Duxbury Resource Center. p. 56.ISBN 9780534119584.
  5. ^Johnson NL, Kotz S, Balakrishnan N (1995). "Chapter 28".Continuous Univariate Distributions. Vol. 2 (2nd ed.). Wiley.ISBN 9780471584940.
  6. ^Hogg RV, Craig AT (1978).Introduction to Mathematical Statistics (4th ed.). New York: Macmillan.ASIN B010WFO0SA. Sections 4.4 and 4.8{{cite book}}: CS1 maint: postscript (link)
  7. ^Cochran, W. G. (1934). "The distribution of quadratic forms in a normal system, with applications to the analysis of covariance".Mathematical Proceedings of the Cambridge Philosophical Society.30 (2):178–191.Bibcode:1934PCPS...30..178C.doi:10.1017/S0305004100016595.S2CID 122547084.
  8. ^abFisher RA (1925)."Applications of 'Student's' distribution"(PDF).Metron.5:90–104. Archived fromthe original(PDF) on 5 March 2016.
  9. ^Gelman AB, Carlin JS, Rubin DB, Stern HS (1997).Bayesian Data Analysis (2nd ed.). Boca Raton, FL: Chapman & Hal l. p. 68.ISBN 9780412039911.
  10. ^Park SY, Bera AK (2009). "Maximum entropy autoregressive conditional heteroskedasticity model".Journal of Econometrics.150 (2):219–230.doi:10.1016/j.jeconom.2008.12.014.
  11. ^Ord JK (1972).Families of Frequency Distributions. London, UK: Griffin. Table 5.1.ISBN 9780852641378.
  12. ^Ord JK (1972).Families of frequency distributions. London, UK: Griffin. Chapter 5.ISBN 9780852641378.
  13. ^abJackman, S. (2009).Bayesian Analysis for the Social Sciences. Wiley Series in Probability and Statistics. Wiley. p. 507.doi:10.1002/9780470686621.ISBN 9780470011546.
  14. ^Lange KL, Little RJ, Taylor JM (1989)."Robust Statistical Modeling Using thet Distribution"(PDF).Journal of the American Statistical Association.84 (408):881–896.doi:10.1080/01621459.1989.10478852.JSTOR 2290063.
  15. ^Gelman AB, Carlin JB, Stern HS, et al. (2014). "Computationally efficient Markov chain simulation".Bayesian Data Analysis. Boca Raton, Florida: CRC Press. p. 293.ISBN 9781439898208.
  16. ^Shah, Amar; Wilson, Andrew Gordon; Ghahramani, Zoubin (2014)."Student t processes as alternatives to Gaussian processes"(PDF).JMLR.33 (Proceedings of the 17th International Conference on Artificial Intelligence and Statistics (AISTATS) 2014, Reykjavik, Iceland):877–885.arXiv:1402.4306.
  17. ^Chen, Zexun; Wang, Bo; Gorban, Alexander N. (2019)."Multivariate Gaussian and Student t process regression for multi-output prediction".Neural Computing and Applications.32 (8):3005–3028.arXiv:1703.04455.doi:10.1007/s00521-019-04687-8.
  18. ^abBailey RW (1994). "Polar generation of random variates with thet distribution".Mathematics of Computation.62 (206):779–781.Bibcode:1994MaCom..62..779B.doi:10.2307/2153537.JSTOR 2153537.S2CID 120459654.
  19. ^Helmert FR (1875). "Über die Berechnung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler".Zeitschrift für Angewandte Mathematik und Physik (in German).20:300–303.
  20. ^Helmert FR (1876). "Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und uber einige damit in Zusammenhang stehende Fragen".Zeitschrift für Angewandte Mathematik und Physik (in German).21:192–218.
  21. ^Helmert FR (1876)."Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit" [The accuracy of Peters' formula for calculating the probable observation error of direct observations of the same accuracy].Astronomische Nachrichten (in German).88 (8–9):113–132.Bibcode:1876AN.....88..113H.doi:10.1002/asna.18760880802.
  22. ^Lüroth J (1876)."Vergleichung von zwei Werten des wahrscheinlichen Fehlers".Astronomische Nachrichten (in German).87 (14):209–220.Bibcode:1876AN.....87..209L.doi:10.1002/asna.18760871402.
  23. ^Pfanzagl J, Sheynin O (1996). "Studies in the history of probability and statistics. XLIV. A forerunner of thet distribution".Biometrika.83 (4):891–898.doi:10.1093/biomet/83.4.891.MR 1766040.
  24. ^Sheynin O (1995). "Helmert's work in the theory of errors".Archive for History of Exact Sciences.49 (1):73–104.doi:10.1007/BF00374700.S2CID 121241599.
  25. ^Pearson, K. (1895)."Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material".Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.186 (374):343–414.Bibcode:1895RSPTA.186..343P.doi:10.1098/rsta.1895.0010.ISSN 1364-503X.
  26. ^"Student" [pseu.William Sealy Gosset] (1908)."The probable error of a mean"(PDF).Biometrika.6 (1):1–25.doi:10.1093/biomet/6.1.1.hdl:10338.dmlcz/143545.JSTOR 2331554.{{cite journal}}: CS1 maint: numeric names: authors list (link)
  27. ^Wendl MC (2016). "Pseudonymous fame".Science.351 (6280): 1406.Bibcode:2016Sci...351.1406W.doi:10.1126/science.351.6280.1406.PMID 27013722.
  28. ^Mortimer RG (2005).Mathematics for Physical Chemistry (3rd ed.). Burlington, MA: Elsevier. pp. 326.ISBN 9780080492889.OCLC 156200058.
  29. ^Walpole RE, Myers R, Myers S, Ye K (2006).Probability & Statistics for Engineers & Scientists (7th ed.). New Delhi, IN: Pearson. p. 237.ISBN 9788177584042.OCLC 818811849.
  30. ^Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021)."The Modified-Half-Normal distribution: Properties and an efficient sampling scheme".Communications in Statistics - Theory and Methods.52 (5):1591–1613.doi:10.1080/03610926.2021.1934700.ISSN 0361-0926.S2CID 237919587.

References

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