Generalized Error Distribution Function
Contents |
on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "version 1" and "version 2". However this is skewed generalized error distribution not a standard nomenclature. Contents 1 Version 1 1.1 Parameter estimation 1.1.1
Generalized Normal Distribution
Maximum likelihood estimator 1.2 Applications 1.3 Properties 2 Version 2 2.1 Parameter estimation 2.2 Applications 3 Other distributions related error distribution definition to the normal 4 See also 5 References Version 1[edit] Generalized Normal (version 1) Probability density function Cumulative distribution function Parameters μ {\displaystyle \mu \,} location (real) α {\displaystyle \alpha \,} generalized normal distribution matlab scale (positive, real) β {\displaystyle \beta \,} shape (positive, real) Support x ∈ ( − ∞ ; + ∞ ) {\displaystyle x\in (-\infty ;+\infty )\!} PDF β 2 α Γ ( 1 / β ) e − ( | x − μ | / α ) β {\displaystyle {\frac {\beta }{2\alpha \Gamma (1/\beta )}}\;e^{-(|x-\mu |/\alpha )^{\beta }}} Γ {\displaystyle \Gamma } denotes the
Exponential Power Distribution
gamma function CDF 1 2 + sgn ( x − μ ) γ [ 1 / β , ( | x − μ | α ) β ] 2 Γ ( 1 / β ) {\displaystyle {\frac {1}{2}}+\operatorname {sgn}(x-\mu ){\frac {\gamma \left[1/\beta ,\left({\frac {|x-\mu |}{\alpha }}\right)^{\beta }\right]}{2\Gamma (1/\beta )}}} γ {\displaystyle \gamma } denotes the lower incomplete gamma function Mean μ {\displaystyle \mu \,} Median μ {\displaystyle \mu \,} Mode μ {\displaystyle \mu \,} Variance α 2 Γ ( 3 / β ) Γ ( 1 / β ) {\displaystyle {\frac {\alpha ^{2}\Gamma (3/\beta )}{\Gamma (1/\beta )}}} Skewness 0 Ex. kurtosis Γ ( 5 / β ) Γ ( 1 / β ) Γ ( 3 / β ) 2 − 3 {\displaystyle {\frac {\Gamma (5/\beta )\Gamma (1/\beta )}{\Gamma (3/\beta )^{2}}}-3} Entropy 1 β − log [ β 2 α Γ ( 1 / β ) ] {\displaystyle {\frac {1}{\beta }}-\log \left[{\frac {\beta }{2\alpha \Gamma (1/\beta )}}\right]} [1] Known also as the exponential power distribution, or the generalized error distribution, this is a parametric family of symmetric distributions. It includes all Laplace distributions, and
0, sd = 1, nu = 2, log = FALSE) pged(q, mean = 0, sd = 1, nu generalized normal distribution r = 2) qged(p, mean = 0, sd = 1, generalized error distribution r nu = 2) rged(n, mean = 0, sd = 1, nu = 2) Arguments
Power Normal Distribution
mean, sd, nu location parameter mean, scale parameter sd, shape parameter nu. n the number of observations. p a numeric vector of probabilities. https://en.wikipedia.org/wiki/Generalized_normal_distribution x, q a numeric vector of quantiles. log a logical; if TRUE, densities are given as log densities. Value d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates, all values are numeric vectors. Author(s) Diethelm Wuertz http://finzi.psych.upenn.edu/library/fGarch/html/dist-ged.html for the Rmetrics R-port. References Nelson D.B. (1991); Conditional Heteroscedasticity in Asset Returns: A New Approach, Econometrica, 59, 347–370. Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages. Examples ## sged - par(mfrow = c(2, 2)) set.seed(1953) r = rsged(n = 1000) plot(r, type = "l", main = "sged", col = "steelblue") # Plot empirical density and compare with true density: hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue") box() x = seq(min(r), max(r), length = 201) lines(x, dsged(x), lwd = 2) # Plot df and compare with true df: plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue", ylab = "Probability") lines(x, psged(x), lwd = 2) # Compute quantiles: round(qsged(psged(q = seq(-1, 5, by = 1))), digits = 6) [Package fGarch version 3010.82.1 Index]
toolboxes, and other File Exchange content using Add-On Explorer in MATLAB. » Watch video Highlights from Generalized Error Distribution functions GEDcdf( X, alpha, beta )GED cumulative density function GEDinv(p,alpha,beta)GED inverse cumulative function GEDpdf( X, https://www.mathworks.com/matlabcentral/fileexchange/57283-generalized-error-distribution-functions alpha, beta )GED probabily density function View all files Join the 15-year community celebration. Play games and win prizes! » Learn more 4.0 4.0 | 1 rating Rate this file 6 Downloads (last 30 days) File Size: 1.06 KB File ID: #57283 Version: 1.0 Generalized Error Distribution functions by AV AV (view profile) 1 file 6 downloads 4.0 23 May 2016 (Updated 23 May 2016) error distribution GEDpdf, GEDcdf, GEDinv | Watch this File File Information Description Although I am completely aware they are simple functions I'd like to upload them because I noticed this distribution is not implemented by default and it could be useful for modelling logreturns distribution. MATLAB release MATLAB 8.6 (R2015b) MATLAB Search Path / Tags for This File Please login to tag files. egarchged Cancel Please generalized error distribution login to add a comment or rating. Comments and Ratings (5) 11 Oct 2016 Luca Attori Luca Attori (view profile) 0 files 0 downloads 0.0 14 Sep 2016 AV AV (view profile) 1 file 6 downloads 4.0 You're welcome! The scale parameter refers to the possibility of "extending the distribution to a larger area": the greater the scale parameter is, the more spread out the distribution will be. You could try different alpha in GEDinv for a given beta and U~uni(0,1); and finally plot them and compare the differences. Hope it will be useful. Comment only 13 Sep 2016 Bilal Arif Sheikh Bilal Arif Sheikh (view profile) 0 files 0 downloads 0.0 Thanks man. I understand the beta concept. but what does scale mean? Is it the degree of freedom? Comment only 11 Sep 2016 AV AV (view profile) 1 file 6 downloads 4.0 Sure, basically I used the same nomenclature of the wikipedia page: alpha is the scale (it has to be a real and positive value), while beta is the shape value (positive and real). Beta provides you the chance to obtain different distributions just modifying its value, for example if
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