Package 'DiscreteLaplace' reference manual

Title:	Discrete Laplace Distributions
Description:	Probability mass function, distribution function, quantile function, random generation and estimation for the skew discrete Laplace distributions.
Authors:	Alessandro Barbiero <[email protected]>, Riccardo Inchingolo <[email protected]>
Maintainer:	Alessandro Barbiero <[email protected]>
License:	GPL
Version:	1.1.1
Built:	2024-12-16 06:45:33 UTC
Source:	CRAN

Discrete Laplace Distributions

Description

Probability mass function, distribution function, quantile function, random generation and sample estimation for two discrete skew Laplace distributions on integers. The skew discrete Laplace distributions here considered are that proposed by Kozubowski and Inusah (2006), henceforth referred to as DSL, and the alternative one proposed by Barbiero (2014), henceforth ADSL.

Details

Package:	DiscreteLaplace
Type:	Package
Version:	1.1.1
Date:	2016-04-29
License:	GPL
LazyLoad:	yes

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

Maintainer: Alessandro Barbiero <[email protected]>

References

T. J. Kozubowski, S. Inusah (2006) A skew Laplace distribution on integers, Annals of the Institute of Statistical Mathematics, 58: 555-571, http://dx.doi.org/10.1007/s10463-005-0029-1

A. Barbiero (2014) An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67, http://dx.doi.org/10.1016/j.stamet.2013.07.002

Probability mass function, cumulative distribution function, quantile function and random generation of the DSL

Description

The function computes the probability mass function, the cumulative distribution function, the quantile function of the DSL and implements random generation.

Usage

ddlaplace(x, p, q)
pdlaplace(x, p, q)
qdlaplace(prob, p, q)
rdlaplace(n, p, q)
ddlaplace(x, p, q)
pdlaplace(x, p, q)
qdlaplace(prob, p, q)
rdlaplace(n, p, q)

Arguments

`x`	vector of quantiles
`p`	the first parameter $p$ in $(0,1)$ of the SDL
`q`	the second parameter $q$ in $(0,1)$ of the SDL
`prob`	vector of probabilities
`n`	number of observations

Details

The pmf of the SDL is given by

$P(X=x; p, q)=\frac{(1-p)(1-q)}{1-pq}p^x; x=0,1,2,3,\dots$

$P(X=x; p, q)=\frac{(1-p)(1-q)}{1-pq}q^{|x|}; x=0,-1,-2,-3,\dots$

whereas the cumulative distribution function is given by

$F(x; p, q)=P(X\leq x)=\frac{(1-p)q^{-\lfloor x\rfloor}}{1-pq},x<0$

$F(x; p, q)=P(X\leq x)=1-\frac{(1-q)p^{\lfloor x\rfloor+1}}{1-pq},x\geq 0$

Value

ddlaplace returns the probability of x; pdlaplace returns the cumulate probability of x; qdlaplace returns the prob- quantile; rdlaplace returns a random sample of size n from DSL.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

Tomasz J. Kozubowski, Seidu Inusah (2006) A skew Laplace distribution on integers, Annals of the Institute of Statistical Mathematics, 58: 555-571

Examples

# pmf
p<-0.7
q<-0.45
x<--10:10
prob<-ddlaplace(x, p, q)
plot(x, prob, type="h")
prob<-ddlaplace(x, q, p) # swap the parameters
plot(x, prob, type="h")
ddlaplace(-4:4, 1:9/10, 9:1/10) # letting p and q be vectors
# cdf
p<-0.2
q<-0.5
x<-c(-3, -1, pi)
pdlaplace(x, p, q)
# quantile function
p<-0.8
q<-0.4
prob<-c(0.2,0.5,0.8)
x<-qdlaplace(prob, p, q)
x # check
upper<-pdlaplace(x, p, q)
upper
lower<-pdlaplace(x-1, p, q)
lower
lower<=prob & prob<=upper
# random generation
n<-100
p<-0.3
q<-0.5
x<-rdlaplace(n, p, q)
x
t<-table(x)
t
plot(t)
# pmf
p<-0.7
q<-0.45
x<--10:10
prob<-ddlaplace(x, p, q)
plot(x, prob, type="h")
prob<-ddlaplace(x, q, p) # swap the parameters
plot(x, prob, type="h")
ddlaplace(-4:4, 1:9/10, 9:1/10) # letting p and q be vectors
# cdf
p<-0.2
q<-0.5
x<-c(-3, -1, pi)
pdlaplace(x, p, q)
# quantile function
p<-0.8
q<-0.4
prob<-c(0.2,0.5,0.8)
x<-qdlaplace(prob, p, q)
x # check
upper<-pdlaplace(x, p, q)
upper
lower<-pdlaplace(x-1, p, q)
lower
lower<=prob & prob<=upper
# random generation
n<-100
p<-0.3
q<-0.5
x<-rdlaplace(n, p, q)
x
t<-table(x)
t
plot(t)

Probability mass function of the ADSL

Description

The function computes the probability mass function, the cumulative distribution function, the quantile function of the ADSL and provides random generation of samples from the same model

Usage

ddlaplace2(x, p, q)
palaplace2(x, p, q)
pdlaplace2(x, p, q)
qdlaplace2(prob, p, q)
rdlaplace2(n, p, q)
ddlaplace2(x, p, q)
palaplace2(x, p, q)
pdlaplace2(x, p, q)
qdlaplace2(prob, p, q)
rdlaplace2(n, p, q)

Arguments

`x`	vector of quantiles
`p`	the first parameter $p$ in $(0,1)$ of the ADSL
`q`	the second parameter $q$ in $(0,1)$ of the ADSL
`prob`	vector of probabilities
`n`	number of observations

Details

The probability mass funtion of the ADSL distribution is given by:

$P(X=x;p,q)=\frac{\log p}{\log (pq)}q^{-(x+1)}(1-q)$ for $x=\dots, -2, -1$

and

$P(X=x;p,q)=\frac{\log q}{\log (pq)}p^{x}(1-p)$ for $x=0, 1, 2, \dots$

Its cumulative distribution function is:

$F(x;p,q)=\frac{\log p}{\log (pq)}q^{-(\lfloor x \rfloor+1)}$ for $x<0$

and

$F(x;p,q)=1-\frac{\log q}{\log (pq)}p^{(\lfloor x \rfloor+1)}$ for $x\geq 0$

Value

ddlaplace2 returns the probability of x; pdlaplace2 returns the cumulate probability of x; qdlaplace2 returns the prob- quantile; rdlaplace2 returns a random sample of size n from ADSL.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

# pmf
p <- 0.7
q <- 0.45
x <- -10:10
prob <- ddlaplace2(x, p, q)
plot(x, prob, type="h")
# swap the parameters
prob <- ddlaplace2(x, q, p)
plot(x, prob, type="h")
# letting p and q be vectors...
ddlaplace2(-4:4, 1:9/10, 9:1/10)
# cdf
pdlaplace2(x, p, q)
pdlaplace2(pi, p, q)
pdlaplace2(floor(pi), p, q)
# quantile function
qdlaplace(1:9/10, p, q)
# random generation
y <- rdlaplace2(n=1000, p, q)
plot(table(y))
# pmf
p <- 0.7
q <- 0.45
x <- -10:10
prob <- ddlaplace2(x, p, q)
plot(x, prob, type="h")
# swap the parameters
prob <- ddlaplace2(x, q, p)
plot(x, prob, type="h")
# letting p and q be vectors...
ddlaplace2(-4:4, 1:9/10, 9:1/10)
# cdf
pdlaplace2(x, p, q)
pdlaplace2(pi, p, q)
pdlaplace2(floor(pi), p, q)
# quantile function
qdlaplace(1:9/10, p, q)
# random generation
y <- rdlaplace2(n=1000, p, q)
plot(table(y))

Log-likelihood function for the ADSL distribution

Description

Log-likelihood function (changed in sign) for the ADSL distribution.

Usage

dlaplacelike2(par, x)
dlaplacelike2(par, x)

Arguments

`par`	the vector of parameters $(p,q)$
`x`	a vector of observations from ADSL

Value

The log-likelihood function with changed sign.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

p <- 0.25
q <- 0.7
x <- rdlaplace2(n=100, p, q)
par <- estdlaplace2(x, "ML")
-dlaplacelike2(par, x) # greater than...
-dlaplacelike2(c(p, q), x)
p <- 0.25
q <- 0.7
x <- rdlaplace2(n=100, p, q)
par <- estdlaplace2(x, "ML")
-dlaplacelike2(par, x) # greater than...
-dlaplacelike2(c(p, q), x)

Moments of the discrete Laplace distribution

Description

The function provides the expected value and the variance of the SDL, and the expectation of its absolute value.

Usage

Edlaplace(p, q)
Edlaplace(p, q)

Arguments

`p`	the first parameter, in $(0,1)$ , of the DSL
`q`	the second parameter, in $(0,1)$ , of the DSL

Details

$E(X;p,q)=\frac{1}{1-p}-\frac{1}{1-q}=\frac{p}{1-p}-\frac{q}{1-q}$ ,

$E(|X|;p,q)=\frac{q(1-p)^2+p(1-q)^2}{(1-qp)(1-q)(1-p)}$ ,

$V(X;p,q)=\frac{1}{(1-p)^2(1-q)^2}[\frac{q(1-p)^3(1+q)+p(1-q)^3(1+p)}{1-pq}-(p-q)^2]$

Value

A list of three items:

`E1`	expected value
`E1a`	expectation of the absolute value
`V`	variance

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

T. J. Kozubowski, S. Inusah (2006) A skew Laplace distribution on integers, Annals of the Institute of Statistical Mathematics, 58: 555-571

Examples

# ex.1
p<-0.5
q<-0.4
Edlaplace(p, q)
# ex.2
p<-0.1
q<-0.9
Edlaplace(p, q)
# ex.1
p<-0.5
q<-0.4
Edlaplace(p, q)
# ex.2
p<-0.1
q<-0.9
Edlaplace(p, q)

First- and second-order moments of ADSL

Description

First- and second-order moment of the ADSL distribution.

Usage

Edlaplace2(p, q)
Edlaplace2(p, q)

Arguments

`p`	the first parameter $p$ , in $(0,1)$ , of the ADSL
`q`	the first parameter $q$ , in $(0,1)$ , of the ADSL

Details

For the ADSL distribution,

$E(X;p,q)=\frac{\log q}{\log(pq)}\frac{p}{1-p}-\frac{\log p}{\log(pq)}\frac{1}{1-q}$

and

$E(X^2;p,q)=\frac{\log q}{\log(pq)}\frac{p(1+p)}{(1-p)^2}+\frac{\log p}{\log(pq)}\frac{1+q}{(1-q)^2}$

Value

A list containing the first- and the second-order moments of the ADSL distribution, E1 and E2.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

Edlaplace2(p=0.3, q=0.3)
Edlaplace2(p=0.3, q=0.6)
Edlaplace2(p=0.6, q=0.3)
Edlaplace2(p=0.6, q=0.6)
Edlaplace2(p=0.3, q=0.3)
Edlaplace2(p=0.3, q=0.6)
Edlaplace2(p=0.6, q=0.3)
Edlaplace2(p=0.6, q=0.6)

Sample estimation for the DSL

Description

The function provides the maximum likelihood estimates for the parameters of the DSL and the estimate of the inverse of the Fisher information matrix. The method of moments estimates of $p$ and $q$ coincide with the maximum likelihood estimates.

Usage

estdlaplace(x)
estdlaplace(x)

Arguments

`x`	a vector of observations from the DSL

Details

See the reference. If $\bar{x}^{+}=\frac{1}{n}\sum_{i=1}^n x_i^{+}$ , $\bar{x}^{-}=\frac{1}{n}\sum_{i=1}^n x_i^{-}$ where $x^{+}$ and $x^{-}$ are the positive and the negative parts of $x$ , respectively: $x^{+}=x$ if $x\geq 0$ and zero otherwise, $x^{-}=(-x)^{+}$ , then

$\hat{q}=\frac{2\bar{x}^{-}(1+\bar{x})}{1+2\bar{x}^{-}\bar{x}+\sqrt{1+4\bar{x}^{-}\bar{x}^{+}}}$ , $\hat{p}=\frac{\hat{q}+\bar{x}(1-\hat{q})}{1+\bar{x}(1-\hat{q})}$

when $\bar{x}\geq 0$ and

$\hat{p}=\frac{2\bar{x}^{+}(1-\bar{x})}{1-2\bar{x}^{+}\bar{x}+\sqrt{1+4\bar{x}^{-}\bar{x}^{+}}}$ , $\hat{q}=\frac{\hat{p}-\bar{x}(1-\hat{p})}{1-\bar{x}(1-\hat{p})}$

when $\bar{x}\leq 0$ .

Value

A list comprising

`hatp`	estimate of $p$
`hatq`	estimate of $q$
`hatSigma`	estimate of the inverse of the Fisher information matrix

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

T. J. Kozubowski, S. Inusah (2006) A skew Laplace distribution on integers, Annals of the Institute of Statistical Mathematics, 58: 555-571

Examples

p<-0.6
q<-0.3
n<-20
x<-rdlaplace(n, p, q)
est<-estdlaplace(x)
est[1]
est[2]
est[3]
# increase n
n<-100
x<-rdlaplace(n, p, q)
est<-estdlaplace(x)
est[1]
est[2]
est[3]
# swap the parameters
x<-rdlaplace(n, q, p)
est<-estdlaplace(x)
est[1]
est[2]
est[3]
p<-0.6
q<-0.3
n<-20
x<-rdlaplace(n, p, q)
est<-estdlaplace(x)
est[1]
est[2]
est[3]
# increase n
n<-100
x<-rdlaplace(n, p, q)
est<-estdlaplace(x)
est[1]
est[2]
est[3]
# swap the parameters
x<-rdlaplace(n, q, p)
est<-estdlaplace(x)
est[1]
est[2]
est[3]

Sample estimation for the ADSL

Description

The function provides the point estimates for the parameters of the ASDL, resorting to four possible methods: method of moments, maximum likelihood method, method of proportion, modified method of moments. For details, please take a look at the references.

Usage

estdlaplace2(x, method = "M", err = 0.001, parml = c(exp(-1), exp(-1)))
estdlaplace2(x, method = "M", err = 0.001, parml = c(exp(-1), exp(-1)))

Arguments

`x`	a vector of observations from the ADSL
`method`	`M` for the method of moments, `ML` for the maximum likelihood methods, `P` for the method of proportion, `MM` for the modified method of moments
`err`	a positive tolerance value, as small as possible, used in the definition of lower and upper bounds of the parameters $p$ and $q$ in the minimization algorithm utilized by the method of moments
`parml`	starting values for $p$ and $q$ in the optimization process for the maximum likelihood method

Value

a vector with the parameter estimates of $p$ and $q$ .

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

p <- 0.4
q <- 0.6
x <- rdlaplace2(n=100, p, q)
est <- matrix(0, 5, 2)
est[1,] <- c(p,q)
est[2,] <- estdlaplace2(x, method="M")
est[3,] <- estdlaplace2(x, method="ML")
est[4,] <- estdlaplace2(x, method="P")
est[5,] <- estdlaplace2(x, method="MM")
dimnames(est)[[1]]<-c("true","M","ML","P","MM")
dimnames(est)[[2]]<-c("p","q")
xlim <- c(min(est[,1])*.98,max(est[,1])*1.02)
ylim <- c(min(est[,2])*.98,max(est[,2])*1.02)
plot(est, pch=19, col=1:5, xlim=xlim, ylim=ylim)
text(est, dimnames(est)[[1]], pos=3, col=1:5, cex= .75)
p <- 0.4
q <- 0.6
x <- rdlaplace2(n=100, p, q)
est <- matrix(0, 5, 2)
est[1,] <- c(p,q)
est[2,] <- estdlaplace2(x, method="M")
est[3,] <- estdlaplace2(x, method="ML")
est[4,] <- estdlaplace2(x, method="P")
est[5,] <- estdlaplace2(x, method="MM")
dimnames(est)[[1]]<-c("true","M","ML","P","MM")
dimnames(est)[[2]]<-c("p","q")
xlim <- c(min(est[,1])*.98,max(est[,1])*1.02)
ylim <- c(min(est[,2])*.98,max(est[,2])*1.02)
plot(est, pch=19, col=1:5, xlim=xlim, ylim=ylim)
text(est, dimnames(est)[[1]], pos=3, col=1:5, cex= .75)

Inverse of Fisher Information matrix

Description

Inverse of Fisher Information matrix for the DSL.

Usage

iFI(p, q)
iFI(p, q)

Arguments

`p`	first parameter $p$ , in $(0,1)$ , of the DSL
`q`	second parameter $q$ , in $(0,1)$ , of the DSL

Value

The inverse of Fisher Information matrix. Take a look at the references for more details.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

T. J. Kozubowski, S. Inusah (2006) A skew Laplace distribution on integers, Annals of the Institute of Statistical Mathematics, 58: 555-571

Examples

p <- 0.2
q <- 0.8
iFI(p, q)
p <- 0.2
q <- 0.8
iFI(p, q)

Inverse of Fisher Information matrix

Description

Inverse of Fisher Information matrix for the ADSL.

Usage

iFI2(p, q)
iFI2(p, q)

Arguments

`p`	first parameter $p$ , in $(0,1)$ , of the ADSL
`q`	second parameter $q$ , in $(0,1)$ , of the ADSL

Value

The inverse of Fisher Information matrix. Take a look at the references for more details.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

p <- 0.2
q <- 0.8
iFI2(p, q)
p <- 0.2
q <- 0.8
iFI2(p, q)

Inverse of the observed Fisher Information matrix

Description

Inverse of the observed Fisher Information matrix computed on a random sample of ADSL values.

Usage

ioFI2(x)
ioFI2(x)

Arguments

`x`	a vector of observations from the ADSL

Value

The inverse of the observed Fisher Information matrix.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

n <- 100
p <- 0.4
q <- 0.7
x <- rdlaplace2(n, p, q)
M <- ioFI2(x)
par <- estdlaplace2(x, "ML")
se <- diag(sqrt(M))
par # MLEs
se # standard errors
M # compare with  the inverse of Fisher Information matrix
iFI2(par[1], par[2])/n # with MLEs plugged in
iFI2(p, q)/n # or the true values
n <- 100
p <- 0.4
q <- 0.7
x <- rdlaplace2(n, p, q)
M <- ioFI2(x)
par <- estdlaplace2(x, "ML")
se <- diag(sqrt(M))
par # MLEs
se # standard errors
M # compare with  the inverse of Fisher Information matrix
iFI2(par[1], par[2])/n # with MLEs plugged in
iFI2(p, q)/n # or the true values

Loss function for the method of moments

Description

A loss function used for the implementation of the method of moments (for the ADSL).

Usage

loss(par, x)
loss(par, x)

Arguments

`par`	the vector of parameters, `p` and `q`, of the ADSL
`x`	a vector of sample values from the ADSL

Value

The value $L=[E(X)-m_1(x)]^2+[E(X^2)-m_2(x)]^2$ , where $m_1$ and $m_2$ are the first- and second-order sample moments.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

A. Barbiero, An alternative discrete Laplace distribution, Statistical Methodology, 16: 47-67

Examples

p <- 0.3
q <- 0.7
x <- rdlaplace2(n=100, p, q)
par <- estdlaplace2(x, "M")
loss(par, x) # should be near zero
loss(c(p,q), x) # may be far greater than zero
p <- 0.3
q <- 0.7
x <- rdlaplace2(n=100, p, q)
par <- estdlaplace2(x, "M")
loss(par, x) # should be near zero
loss(c(p,q), x) # may be far greater than zero

Package 'DiscreteLaplace'

Help Index

Discrete Laplace Distributions

Description

Details

Author(s)

References

Probability mass function, cumulative distribution function, quantile function and random generation of the DSL

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Probability mass function of the ADSL

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Log-likelihood function for the ADSL distribution

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Moments of the discrete Laplace distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

First- and second-order moments of ADSL

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Sample estimation for the DSL

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Sample estimation for the ADSL

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Inverse of Fisher Information matrix

Description

Usage

Arguments

Value