estdlaplace {DiscreteLaplace}R Documentation

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 pp and qq coincide with the maximum likelihood estimates.

Usage

estdlaplace(x)

Arguments

x

a vector of observations from the DSL

Details

See the reference. If xˉ+=1ni=1nxi+\bar{x}^{+}=\frac{1}{n}\sum_{i=1}^n x_i^{+}, xˉ=1ni=1nxi\bar{x}^{-}=\frac{1}{n}\sum_{i=1}^n x_i^{-} where x+x^{+} and xx^{-} are the positive and the negative parts of xx, respectively: x+=xx^{+}=x if x0x\geq 0 and zero otherwise, x=(x)+x^{-}=(-x)^{+}, then

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

when xˉ0\bar{x}\geq 0 and

p^=2xˉ+(1xˉ)12xˉ+xˉ+1+4xˉxˉ+\hat{p}=\frac{2\bar{x}^{+}(1-\bar{x})}{1-2\bar{x}^{+}\bar{x}+\sqrt{1+4\bar{x}^{-}\bar{x}^{+}}}, q^=p^xˉ(1p^)1xˉ(1p^)\hat{q}=\frac{\hat{p}-\bar{x}(1-\hat{p})}{1-\bar{x}(1-\hat{p})}

when xˉ0\bar{x}\leq 0.

Value

A list comprising

hatp

estimate of pp

hatq

estimate of qq

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

See Also

ddlaplace

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]

[Package DiscreteLaplace version 1.1.1 Index]