Generalized normal distribution

The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions 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 not a standard nomenclature. The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions 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 not a standard nomenclature. Known also as the exponential power distribution, or the generalized error distribution, this is a parametric family of symmetric distributions. It includes all normal and Laplace distributions, and as limiting cases it includes all continuous uniform distributions on bounded intervals of the real line. This family includes the normal distribution when β = 2 {displaystyle extstyle eta =2} (with mean μ {displaystyle extstyle mu } and variance α 2 2 {displaystyle extstyle {frac {alpha ^{2}}{2}}} ) and it includes the Laplace distribution when β = 1 {displaystyle extstyle eta =1} . As β → ∞ {displaystyle extstyle eta ightarrow infty } , the density converges pointwise to a uniform density on ( μ − α , μ + α ) {displaystyle extstyle (mu -alpha ,mu +alpha )} . This family allows for tails that are either heavier than normal (when β < 2 {displaystyle eta <2} ) or lighter than normal (when β > 2 {displaystyle eta >2} ). It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal ( β = 2 {displaystyle extstyle eta =2} ) to the uniform density ( β = ∞ {displaystyle extstyle eta =infty } ), and a continuum of symmetric, leptokurtic densities spanning from the Laplace ( β = 1 {displaystyle extstyle eta =1} ) to the normal density ( β = 2 {displaystyle extstyle eta =2} ). Parameter estimation via maximum likelihood and the method of moments has been studied. The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed. The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C∞ of smooth functions) only if β {displaystyle extstyle eta } is a positive, even integer. Otherwise, the function has ⌊ β ⌋ {displaystyle extstyle lfloor eta floor } continuous derivatives. As a result, the standard results for consistency and asymptotic normality of maximum likelihood estimates of β {displaystyle eta } only apply when β ≥ 2 {displaystyle extstyle eta geq 2} . It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method. With μ {displaystyle mu } initially set to the sample first moment m 1 {displaystyle m_{1}} , β {displaystyle extstyle eta } is estimated by using a Newton–Raphson iterative procedure, starting from an initial guess of β = β 0 {displaystyle extstyle eta = extstyle eta _{0}} ,