Laplace function

The Laplace Distribution.

The Laplace Distribution.

Density, distribution, quantile, random number generation and parameter estimation functions for the Laplace distribution with location parameter μ\mu and scale parameter bb. 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\mu and scale=bb has probability density function

f(x)=(1/2b)exp(xμ/b) f(x) = (1/2b) exp(-|x-\mu|/b)

where <x<-\infty < x < \infty and b>0b > 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\mu, is estimated using the sample median and bb by the mean of xb|x-b|.

Johnson et.al (p.172) also provides the log-likelihood function for the Laplace distribution

l(μ,bx)=nln(2b)b1xiμ. l(\mu, b | x) = -n ln(2b) - b^{-1} \sum |xi-\mu|.

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 parameters X <- rLaplace(n=500, mu=1, b=2) est.par <- eLaplace(X, method="analytic.MLE"); est.par plot(est.par) # Fitted density curve and histogram den.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 parameters est.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=3.36 flood <- 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 function lLaplace(flood,param=est.par) # Evaluating the precision by the Hessian matrix H <- 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

See Also

ExtDist for other standard distributions.

Author(s)

A. Jonathan R. Godfrey and Haizhen Wu.

Updates and bug fixes by Sarah Pirikahu

  • Maintainer: Oleksii Nikolaienko
  • License: GPL (>= 2)
  • Last published: 2023-08-21

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