The Laplace distribution, also known as the double exponential distribution, is a continuous probability distribution that is often used to model data with sharp peaks and heavy tails. It is parameterized by a location parameter \(\mu\) and a scale parameter \(b > 0\).
Usage
Laplace(mu = 0, sigma = 1)
dlaplace(x, mu, sigma, log = FALSE)
plaplace(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qlaplace(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rlaplace(n, mu, sigma)
# S4 method for class 'Laplace,numeric'
d(distr, x, log = FALSE)
# S4 method for class 'Laplace,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)
# S4 method for class 'Laplace,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)
# S4 method for class 'Laplace,numeric'
r(distr, n)
# S4 method for class 'Laplace'
mean(x)
# S4 method for class 'Laplace'
median(x)
# S4 method for class 'Laplace'
mode(x)
# S4 method for class 'Laplace'
var(x)
# S4 method for class 'Laplace'
sd(x)
# S4 method for class 'Laplace'
skew(x)
# S4 method for class 'Laplace'
kurt(x)
# S4 method for class 'Laplace'
entro(x)
# S4 method for class 'Laplace'
finf(x)
lllaplace(x, mu, sigma)
# S4 method for class 'Laplace,numeric'
ll(distr, x)
elaplace(x, type = "mle", ...)
# S4 method for class 'Laplace,numeric'
mle(distr, x, na.rm = FALSE)
# S4 method for class 'Laplace,numeric'
me(distr, x, na.rm = FALSE)
vlaplace(mu, sigma, type = "mle")
# S4 method for class 'Laplace'
avar_mle(distr)
# S4 method for class 'Laplace'
avar_me(distr)Arguments
- mu, sigma
numeric. The distribution parameters.
- x
For the density function,
xis a numeric vector of quantiles. For the moments functions,xis an object of classLaplace. For the log-likelihood and the estimation functions,xis the sample of observations.- log, log.p
logical. Should the logarithm of the probability be returned?
- q
numeric. Vector of quantiles.
- lower.tail
logical. If TRUE (default), probabilities are \(P(X \leq x)\), otherwise \(P(X > x)\).
- p
numeric. Vector of probabilities.
- n
number of observations. If
length(n) > 1, the length is taken to be the number required.- distr
an object of class
Laplace.- type
character, case ignored. The estimator type (mle or me).
- ...
extra arguments.
- na.rm
logical. Should the
NAvalues be removed?
Value
Each type of function returns a different type of object:
Distribution Functions: When supplied with one argument (
distr), thed(),p(),q(),r(),ll()functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distrandx), they evaluate the aforementioned functions directly.Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The
moments()function returns a list with all the available methods.Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.
Details
The probability density function (PDF) of the Laplace distribution is: $$ f(x; \mu, b) = \frac{1}{2b} \exp\left(-\frac{|x - \mu|}{b}\right) .$$
Examples
# -----------------------------------------------------
# Laplace Distribution Example
# -----------------------------------------------------
# Create the distribution
m <- 3 ; s <- 5
D <- Laplace(m, s)
# ------------------
# dpqr Functions
# ------------------
d(D, c(0.3, 2, 10)) # density function
#> [1] 0.05827483 0.08187308 0.02465970
p(D, c(0.3, 2, 10)) # distribution function
#> [1] 0.2913741 0.4093654 0.8767015
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 1.884282 7.581454
x <- r(D, 100) # random generator function
# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#> [1] 0.0255049011 0.0411834040 0.0462429105 0.0465894662 0.0415778861
#> [6] 0.0568282165 0.0681336696 0.0926202989 0.0971288137 0.0039095983
#> [11] 0.0122019484 0.0969991081 0.0364322510 0.0830665678 0.0050211357
#> [16] 0.0500123026 0.0663597172 0.0716630141 0.0643986045 0.0450556273
#> [21] 0.0664329885 0.0381311629 0.0798682086 0.0306833740 0.0112360638
#> [26] 0.0111551880 0.0900457788 0.0189437140 0.0939240431 0.0843399497
#> [31] 0.0154120931 0.0970061188 0.0414377373 0.0358009313 0.0643595795
#> [36] 0.0426303986 0.0065052702 0.0041909611 0.0533525518 0.0426843733
#> [41] 0.0154930544 0.0568691757 0.0656494853 0.0721713971 0.0102965415
#> [46] 0.0772819964 0.0209844087 0.0292521537 0.0088187437 0.0619484873
#> [51] 0.0692220467 0.0524187332 0.0413714186 0.0832574055 0.0315657900
#> [56] 0.0940235447 0.0001178064 0.0857673395 0.0039464330 0.0395695716
#> [61] 0.0661753114 0.0056440283 0.0464413826 0.0823896556 0.0868047631
#> [66] 0.0410684344 0.0553180110 0.0412130329 0.0653067930 0.0272090108
#> [71] 0.0768721399 0.0870514842 0.0151375011 0.0270272243 0.0156571540
#> [76] 0.0542485144 0.0222024417 0.0543644589 0.0216261976 0.0109751517
#> [81] 0.0123335455 0.0873977414 0.0317832441 0.0721013897 0.0445756467
#> [86] 0.0492795968 0.0139500877 0.0492574904 0.0641078407 0.0913747413
#> [91] 0.0609337283 0.0519808882 0.0925293788 0.0399075269 0.0768249620
#> [96] 0.0303843228 0.0987945568 0.0730543641 0.0534809090 0.0792518551
# ------------------
# Moments
# ------------------
mean(D) # Expectation
#> [1] 3
median(D) # Median
#> [1] 3
mode(D) # Mode
#> [1] 3
var(D) # Variance
#> [1] 50
sd(D) # Standard Deviation
#> [1] 7.071068
skew(D) # Skewness
#> [1] 0
kurt(D) # Excess Kurtosis
#> [1] 6
entro(D) # Entropy
#> [1] 3.302585
finf(D) # Fisher Information Matrix
#> mu sigma
#> mu 1 0.0
#> sigma 0 0.2
# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 3
# ------------------
# Point Estimation
# ------------------
elaplace(x, type = "mle")
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
elaplace(x, type = "me")
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
mle(D, x)
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
me(D, x)
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
e(D, x, type = "mle")
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
mle("laplace", x) # the distr argument can be a character
#> $mu
#> [1] 2.954329
#>
#> $sigma
#> [1] 4.948933
#>
# ------------------
# Estimator Variance
# ------------------
vlaplace(m, s, type = "mle")
#> mu sigma
#> mu 1 0
#> sigma 0 5
vlaplace(m, s, type = "me")
#> mu sigma
#> mu 1 0
#> sigma 0 5
avar_mle(D)
#> mu sigma
#> mu 1 0
#> sigma 0 5
avar_me(D)
#> mu sigma
#> mu 1 0
#> sigma 0 5
v(D, type = "mle")
#> mu sigma
#> mu 1 0
#> sigma 0 5