# estdlaplace: Sample estimation for the DSL In DiscreteLaplace: Discrete Laplace Distributions

## 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

 1 estdlaplace(x) 

## Arguments

 x a vector of observations from the DSL

## Details

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

\hat{q}=\frac{2\bar{x}^{-}(1+\bar{x})}{1+2\bar{x}^{-}\bar{x}+√{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}≥q 0 and

\hat{p}=\frac{2\bar{x}^{+}(1-\bar{x})}{1-2\bar{x}^{+}\bar{x}+√{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}≤q 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

ddlaplace
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 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]