Behrens–Fisher distribution

Statistician and biologist Ronald Fisher

In statistics, the Behrens–Fisher distribution, named after Ronald Fisher and W. V. Behrens, is a parameterized family of probability distributions arising from the solution of the Behrens–Fisher problem proposed first by Behrens and several years later by Fisher. The Behrens–Fisher problem is that of statistical inference concerning the difference between the means of two normally distributed populations when the ratio of their variances is not known (and in particular, it is not known that their variances are equal).

Definition

The Behrens–Fisher distribution is the distribution of a random variable of the form

T_{2}\cos \theta -T_{1}\sin \theta \,

where T₁ and T₂ are independent random variables each with a Student's t-distribution, with respective degrees of freedom ν₁ = n₁ − 1 and ν₂ = n₂ − 1, and θ is a constant. Thus the family of Behrens–Fisher distributions is parametrized by ν₁, ν₂, and θ.

Derivation

Suppose it were known that the two population variances are equal, and samples of sizes n₁ and n₂ are taken from the two populations:

{\begin{aligned}X_{{1,1}},\ldots ,X_{{1,n_{1}}}&\sim \operatorname {i.i.d.}N(\mu _{1},\sigma ^{2}),\\[6pt]X_{{2,1}},\ldots ,X_{{2,n_{2}}}&\sim \operatorname {i.i.d.}N(\mu _{2},\sigma ^{2}).\end{aligned}}

where "i.i.d" are independent and identically distributed random variables and N denotes the normal distribution. The two sample means are

{\begin{aligned}{\bar {X}}_{1}&=(X_{{1,1}}+\cdots +X_{{1,n_{1}}})/n_{1}\\[6pt]{\bar {X}}_{2}&=(X_{{2,1}}+\cdots +X_{{2,n_{2}}})/n_{2}\end{aligned}}

The usual "pooled" unbiased estimate of the common variance σ² is then

S_{{\mathrm {pooled}}}^{2}={\frac {\sum _{{k=1}}^{{n_{1}}}(X_{{1,k}}-{\bar X}_{1})^{2}+\sum _{{k=1}}^{{n_{2}}}(X_{{2,k}}-{\bar X}_{2})^{2}}{n_{1}+n_{2}-2}}={\frac {(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}}

where S₁² and S₂² are the usual unbiased (Bessel-corrected) estimates of the two population variances.

Under these assumptions, the pivotal quantity

{\frac {(\mu _{2}-\mu _{1})-({\bar X}_{2}-{\bar X}_{1})}{\displaystyle {\sqrt {{\frac {S_{{\mathrm {pooled}}}^{2}}{n_{1}}}+{\frac {S_{{\mathrm {pooled}}}^{2}}{n_{2}}}}}}}

has a t-distribution with n₁ + n₂ − 2 degrees of freedom. Accordingly, one can find a confidence interval for μ₂ − μ₁ whose endpoints are

{\bar {X}}_{2}-{\bar {X_{1}}}\pm A\cdot S_{{\mathrm {pooled}}}{\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}},

where A is an appropriate percentage point of the t-distribution.

However, in the Behrens–Fisher problem, the two population variances are not known to be equal, nor is their ratio known. Fisher considered the pivotal quantity

{\frac {(\mu _{2}-\mu _{1})-({\bar X}_{2}-{\bar X}_{1})}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}.

This can be written as

T_{2}\cos \theta -T_{1}\sin \theta ,\,

where

T_{i}={\frac {\mu _{i}-{\bar {X}}_{i}}{S_{i}/{\sqrt {n_{i}}}}}{\text{ for }}i=1,2\,

are the usual one-sample t-statistics and

\tan \theta ={\frac {S_{1}/{\sqrt {n_{1}}}}{S_{2}/{\sqrt {n_{2}}}}}

and one takes θ to be in the first quadrant. The algebraic details are as follows:

{\begin{aligned}{\frac {(\mu _{2}-\mu _{1})-({\bar X}_{2}-{\bar X}_{1})}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}&={\frac {\mu _{2}-{\bar {X}}_{2}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}-{\frac {\mu _{1}-{\bar {X}}_{1}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\\[10pt]&=\underbrace {{\frac {\mu _{2}-{\bar {X}}_{2}}{S_{2}/{\sqrt {n_{2}}}}}}_{{{\text{This is }}T_{2}}}\cdot \underbrace {\left({\frac {S_{2}/{\sqrt {n_{2}}}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\right)}_{{{\text{This is }}\cos \theta }}-\underbrace {{\frac {\mu _{1}-{\bar {X}}_{1}}{S_{1}/{\sqrt {n_{1}}}}}}_{{{\text{This is }}T_{1}}}\cdot \underbrace {\left({\frac {S_{1}/{\sqrt {n_{1}}}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\right)}_{{{\text{This is }}\sin \theta }}.\qquad \qquad \qquad (1)\end{aligned}}

The fact that the sum of the squares of the expressions in parentheses above is 1 implies that they are the cosine and sine of some angle.

The Behren–Fisher distribution is actually the conditional distribution of the quantity (1) above, given the values of the quantities labeled cos θ and sin θ. In effect, Fisher conditions on ancillary information.

Fisher then found the "fiducial interval" whose endpoints are

{\bar {X}}_{2}-{\bar {X}}_{1}\pm A{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}

where A is the appropriate percentage point of the Behrens–Fisher distribution. Fisher claimed that the probability that μ₂ − μ₁ is in this interval, given the data (ultimately the Xs) is the probability that a Behrens–Fisher-distributed random variable is between −A and A.

Fiducial intervals versus confidence intervals

Bartlett showed that this "fiducial interval" is not a confidence interval because it does not have a constant coverage rate. Fisher did not consider that a cogent objection to the use of the fiducial interval.

Probability distributions

List

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

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

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

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

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

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Behrens–Fisher distribution

Definition

Derivation

Fiducial intervals versus confidence intervals

Further reading