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LogLaplace( \alpha ,\beta .\delta) - no Crystal Ball function

LogLaplace equations



Examples of the LogLaplace distribution are given below. \delta  is just a scaling factor, giving the location of the point of inflection of the density function, so \delta  = 1 is used for these graphs. The LogLaplace distribution takes a variety of shapes, depending on the value of \beta. For example, where \beta = 1, the LogLaplace distribution is uniform for x < \delta .




Kozubowski TJ and Podgórski K review many uses of the LogLaplace distribution. The most commonly quoted use (for the symmetric LogLaplace) has been for modeling "moral fortune', a state of well-being that is the logarithm of income, based on a formula by Daniel Bernoulli.


The asymmetric LogLaplace distribution has been fit to pharmacokinetic and particle size data (particle size studies often show the log size to follow a tent'shaped distribution like the Laplace). It has been used to model growth rates, stock prices, annual gross domestic production, interest and forex rates. Some explanation for the goodness of fit of the LogLaplace has been suggested because of its relationship to Brownian motion stopped at a random exponential time.


If log(X) takes a Laplace( \mu ,\sigma) distribution, then the variable X takes the symmetric LogLaplace( \alpha ,\alpha ,\delta) distribution, where \alpha = SQRT(2)/ \sigma and \delta = Exp( \mu). This is analogous to the relationship between the Normal and lognormal distributions


The LogLaplace distribution described here is the asymmetric form of the distribution which offers a greater variety of shapes. If X is a LogLaplace( \alpha ,\beta .\delta) distribution, then log(X) is an asymmetric-Laplace distributed variable with probability density given by:



f(x)=\frac{\alpha \beta}{\alpha +\beta}exp[-\alpha(x-\ln \delta)] \quad for~x\geq \ln \delta
f(x)=\frac{\alpha \beta}{\alpha +\beta}exp[\beta(x-\ln \delta)] \quad for~x < \ln \delta



The symmetric form is the special case where \alpha = \beta.


If a variable X is LogLaplace(t,t, \delta) distributed (i.e. \alpha = \beta) then X and 1/X take the same distribution.


Other names: skew log-Laplace, double Pareto.



Kozubowski TJ and Podgórski K give an excellent review of the Log-Laplace distribution.





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