1.3.6.6.12. Double Exponential Distribution

The equation for the standard double exponential distribution is ... The following is the plot of the double exponential probability density function. 1. ExploratoryDataAnalysis 1.3. EDATechniques 1.3.6. ProbabilityDistributions 1.3.6.6. GalleryofDistributions 1.3.6.6.12. DoubleExponentialDistribution ProbabilityDensityFunction Thegeneralformulafortheprobability densityfunctionofthedoubleexponentialdistributionis \(f(x)=\frac{e^{-\left|\frac{x-\mu}{\beta}\right|}}{2\beta}\) whereμisthelocationparameterand βisthescaleparameter.Thecase whereμ=0andβ=1iscalledthestandard doubleexponentialdistribution.Theequationforthestandarddouble exponentialdistributionis \(f(x)=\frac{e^{-|x|}}{2}\) Sincethegeneralformofprobabilityfunctionscanbe expressedintermsofthestandard distribution,allsubsequentformulasinthissectionare givenforthestandardformofthefunction. Notethatthedoubleexponentialdistributionisalsocommonly referredtoastheLaplacedistribution. Thefollowingistheplotofthedoubleexponentialprobability densityfunction. CumulativeDistributionFunction Theformulaforthecumulativedistribution functionofthedoubleexponentialdistributionis \(F(x)=\begin{array}{ll}\frac{e^{x}}{2}&\mbox{for$x<0$}\\ 1-\frac{e^{-x}}{2}&\mbox{for$x\ge0$}\end{array}\) Thefollowingistheplotofthedoubleexponentialcumulative distributionfunction. PercentPointFunction Theformulaforthepercentpoint functionofthedoubleexponentialdistributionis \(G(P)=\begin{array}{ll}\log(2p)&\mbox{for$p\le0.5$}\\ -\log(2(1-p))&\mbox{for$p>0.5$}\end{array}\) Thefollowingistheplotofthedoubleexponentialpercentpoint function. HazardFunction Theformulaforthehazardfunctionof thedoubleexponentialdistributionis \(h(x)=\begin{array}{ll}\frac{e^{x}}{2-e^{x}}& \mbox{for$x<0$}\\1&\mbox{for$x\ge0$}\end{array}\) Thefollowingistheplotofthedoubleexponentialhazardfunction. CumulativeHazardFunction Theformulaforthecumulativehazard functionofthedoubleexponentialdistributionis \(H(x)=\begin{array}{ll}-log{(1-\frac{e^{x}}{2})}& \mbox{for$x<0$}\\x+\log{(2)}&\mbox{for$x\ge0$}\end{array}\) Thefollowingistheplotofthedoubleexponentialcumulativehazard function. SurvivalFunction Theformulaforthesurvivalfunction ofthedoubleexponentialdistributionis \(S(x)=\begin{array}{ll}1-\frac{e^{x}}{2}&\mbox{for$x<0$}\\ \frac{e^{-x}}{2}&\mbox{for$x\ge0$}\end{array}\) Thefollowingistheplotofthedoubleexponentialsurvivalfunction. InverseSurvivalFunction Theformulafortheinverse survivalfunctionofthedoubleexponentialdistributionis \(Z(P)=\begin{array}{ll}\log(2(1-p))&\mbox{for$p\le0.5$}\\ -\log(2p)&\mbox{for$p>0.5$}\end{array}\) Thefollowingistheplotofthedoubleexponentialinversesurvival function. CommonStatistics Mean μ Median μ Mode μ Range \(-\infty\mbox{to}\infty\) StandardDeviation \(\sqrt{2}\beta\) Skewness 0 Kurtosis 6 CoefficientofVariation \(\sqrt{2}(\frac{\beta}{\mu})\) ParameterEstimation Themaximumlikelihoodestimatorsofthelocationandscale parametersofthedoubleexponentialdistributionare \(\hat{\mu}=\tilde{X}\) \(\hat{\beta}=\frac{\sum_{i=1}^{N}|X_{i}-\tilde{X}|}{N}\) where \(\tilde{X}\)isthesamplemedian. Software Somegeneralpurposestatisticalsoftwareprogramssupportatleast someoftheprobabilityfunctionsforthedoubleexponential distribution.