Density, distribution, quantile, random number generation and parameter estimation functions for the Laplace distribution with location parameter μ and scale parameter b. Parameter estimation can for the Laplace distribution can be carried out numerically or analytically but may only be based on an unweighted i.i.d. sample.
dLaplace(x, mu =0, b =1, params = list(mu, b),...)pLaplace(q, mu =0, b =1, params = list(mu, b),...)qLaplace(p, mu =0, b =1, params = list(mu, b),...)rLaplace(n, mu =0, b =1, params = list(mu, b),...)eLaplace(X, w, method = c("analytic.MLE","numerical.MLE"),...)lLaplace(x, w =1, mu =0, b =1, params = list(mu, b), logL =TRUE,...)
Arguments
x, q: A vector of quantiles.
mu: Location parameter.
b: Scale parameter.
params: A list that includes all named parameters
...: Additional parameters.
p: A vector of probabilities.
n: Number of observations.
X: Sample observations.
w: Optional vector of sample weights.
method: Parameter estimation method.
logL: logical; if TRUE, lLaplace gives the log-likelihood, otherwise the likelihood is given.
Returns
dLaplace gives the density, pLaplace the distribution function, qLaplace the quantile function, rLaplace generates random deviates, and eLaplace estimates the distribution parameters. lLaplace provides the log-likelihood function, sLaplace the score function, and iLaplace the observed information matrix.
Details
The dLaplace(), pLaplace(), qLaplace(),and rLaplace() functions allow for the parameters to be declared not only as individual numerical values, but also as a list so parameter estimation can be carried out.
The Laplace distribution with parameters location = μ and scale=b has probability density function
f(x)=(1/2b)exp(−∣x−μ∣/b)
where −∞<x<∞ and b>0. The cumulative distribution function for pLaplace is defined by Johnson et.al (p.166).
Parameter estimation can be carried out analytically via maximum likelihood estimation, see Johnson et.al (p.172). Where the population mean, μ, is estimated using the sample median and b by the mean of ∣x−b∣.
Johnson et.al (p.172) also provides the log-likelihood function for the Laplace distribution
l(μ,b∣x)=−nln(2b)−b−1∑∣xi−μ∣.
Note
The estimation of the population mean is done using the median of the sample. Unweighted samples are not yet catered for in the eLaplace() function.
Examples
# Parameter estimation for a distribution with known shape parametersX <- rLaplace(n=500, mu=1, b=2)est.par <- eLaplace(X, method="analytic.MLE"); est.par
plot(est.par)# Fitted density curve and histogramden.x <- seq(min(X),max(X),length=100)den.y <- dLaplace(den.x, location = est.par$location, scale= est.par$scale)hist(X, breaks=10, probability=TRUE, ylim = c(0,1.1*max(den.y)))lines(den.x, den.y, col="blue")lines(density(X), lty=2)# Extracting location or scale parametersest.par[attributes(est.par)$par.type=="location"]est.par[attributes(est.par)$par.type=="scale"]# Parameter estimation for a distribution with unknown shape parameters# Example from Best et.al (2008). Original source of flood data from Gumbel & Mustafi.# Parameter estimates as given by Best et.al mu=10.13 and b=3.36flood <- c(1.96,1.96,3.60,3.80,4.79,5.66,5.76,5.78,6.27,6.30,6.76,7.65,7.84,7.99,8.51,9.18,10.13,10.24,10.25,10.43,11.45,11.48,11.75,11.81,12.34,12.78,13.06,13.29,13.98,14.18,14.40,16.22,17.06)est.par <- eLaplace(flood, method="numerical.MLE"); est.par
plot(est.par)#log-likelihood functionlLaplace(flood,param=est.par)# Evaluating the precision by the Hessian matrixH <- attributes(est.par)$nll.hessian
var <- solve(H)se <- sqrt(diag(var));se
References
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapter 24, Wiley, New York.
Best, D.J., Rayner, J.C.W. and Thas O. (2008) Comparison of some tests of fit for the Laplace distribution, Computational Statistics and Data Analysis, Vol. 52, pp.5338-5343.
Gumbel, E.J., Mustafi, C.K., 1967. Some analytical properties of bivariate extremal distributions. J. Am. Stat. Assoc. 62, 569-588