Fisher's ''z''-distribution

Fisher's z
Probability density function
Parameters	$d_1>0,\ d_2>0$ deg. of freedom
Support	$x\in (-\infty ;+\infty )\!$
PDF	$\frac{2d_1^{d_1/2}d_2^{d_2/2}}{B(d_1/2,d_2/2)}\frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}\!$
Mode	$0$

Ronald Fisher

Fisher's z-distribution is the statistical distribution of half the logarithm of an F-distribution variate:

z={\frac {1}{2}}\log F

It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto.^[1] Nowadays one usually uses the F-distribution instead.

The probability density function and cumulative distribution function can be found by using the F-distribution at the value of $x' = e^{2x} \,$ . However, the mean and variance do not follow the same transformation.

The probability density function is^[2]^[3]

f(x; d_1, d_2) = \frac{2d_1^{d_1/2} d_2^{d_2/2}}{B(d_1/2, d_2/2)} \frac{e^{d_1 x}}{\left(d_1 e^{2 x} + d_2\right)^{(d_1+d_2)/2}},

where B is the beta function.

When the degrees of freedom becomes large ( $d_1, d_2 \rightarrow \infty$ ) the distribution approach normality with mean^[2]

{\bar {x}}={\frac {1}{2}}\left({\frac {1}{d_{2}}}-{\frac {1}{d_{1}}}\right)

and variance

\sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).

Related distribution

If $X \sim \operatorname{FisherZ}(n,m)$ then $e^{2X} \sim \operatorname{F}(n,m) \,$ (F-distribution)
If $X \sim \operatorname{F}(n,m)$ then ${\tfrac {\log X}{2}}\sim \operatorname {FisherZ} (n,m)$

References

↑ Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF). Proceedings of the International Congress of Mathematics, Toronto. 2: 805–813. Archived from the original (PDF) on April 12, 2011.
1 2 Leo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution". The Annals of Mathematical Statistics. 12 (4): 429–448. JSTOR 2235955. doi:10.1214/aoms/1177731681.
↑ Charles Ernest Weatherburn. A first course in mathematical statistics.

External links

MathWorld entry

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 Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace 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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