# Laplace: The Laplace Distribution. In ExtDist: Extending the Range of Functions for Probability Distributions

## Description

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.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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.

## 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) = -n ln(2b) - b^{-1} ∑ |xi-μ|.

## Value

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.

## 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.

## Author(s)

A. Jonathan R. Godfrey and Haizhen Wu.
Updates and bug fixes by Sarah Pirikahu

## 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

ExtDist for other standard distributions.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```# 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 ```

### Example output   ```Attaching package: 'ExtDist'

The following object is masked from 'package:stats':

BIC

Parameters for the Laplace distribution.
(found using the  analytic.MLE method.)

Parameter     Type Estimate
mu location 1.056785
b    scale 1.993565

\$mu
 1.056785

\$b
 1.993565

Parameters for the Laplace distribution.
(found using the  numerical.MLE method.)

Parameter     Type Estimate      S.E.
mu location 10.13000 0.4149289
b    scale  3.36091 0.5850647

 -95.87684
mu         b
0.4149289 0.5850647
```

ExtDist documentation built on Jan. 8, 2021, 2:39 a.m.