estdlaplace {DiscreteLaplace}

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)

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

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]