Inference for the Generalized Normal Laplace Distribution

Letter to the editor comments on Groparu-cojocaru and Doray (2013)

“…In a paper recently published in this journal, Groparu-cojocaru and Doray (2013) minimize an L 2 distance between the empirical and true characteristic functions to estimate $\theta =(\mu ,\sigma ,\alpha ,\beta ,\rho )\in \mathbb{ℝ}\times {(0,\infty )}^4$, the five-dimensional parameter of a Generalized Normal Laplace (GNL) distribution. We recall that a random variable X follows such a distribution if there exist $Z~N(0,1),Y_1~\Gamma (\rho ,1)\text{and}{Y}_2~\Gamma (\rho ,1)$, which are mutually independent such that $$X=\rho \mu +\sqrt{\rho \sigma }Z+\frac{1}{\alpha }{Y}_1-\frac{1}{\beta }{Y}_2.$$ Unless ρ = 1, in which case the distribution specializes to the Normal-Laplace distribution; see e.g.…”

“…Also, let U n et V n be the real and imaginary parts of Φ n , that is $U_n\left(t\right)=n^{-1}{\sum }_{j=1}^n\text{cos}\left(X_{j}t\right)\text{and}{V}_n\left(t\right)=n^{-1}{\sum }_{j=1}^n\text{sin}\left(X_{j}t\right)$. It is known that $$\upsilon (t,\theta )=\text{exp}true(-\frac{\rho {\sigma }^2{t}^2}{2}true){\left[\frac{{\alpha }^2{\beta }^2}{({\alpha }^2+t^2)({\beta }^2+t^2)}\right]}^{\rho ∕2}\text{cos}true[\rho true(\mu t+\text{arctan}true(\frac{t}{\alpha }true)-\text{arctan}true(\frac{t}{\beta }true)true)true]$$ and $$v(t,\theta )=\text{exp}true(-\frac{\rho {\sigma }^2{t}^2}{2}true){\left[\frac{{\alpha }^2+{\beta }^2}{({\alpha }^2+t^2)({\beta }^2+t^2)}\right]}^{\rho ∕2}\text{sin}true[\rho true(\mu t+\text{arctan}true(\frac{t}{\alpha }true)-\text{arctan}true(\frac{t}{\beta }true)true)true]$$ for t ∈ ℝ; see Groparu-cojocaru and Doray (2013), page 1990.…”

“…For given real points { t 1 , … , t k }, Groparu-cojocaru and Doray (2013) define the vectors $$Z_n=(U_n\left(t\right),\dots ,U_n\left(t_k\right),V_n\left(t_1\right),\dots ,{\upsilon }_n\left(t_k\right))$$ and $$Z\left(\theta \right)=(u(t_1,\theta ),\dots ,u(t_k,\theta ),v(t_1,\theta ),\dots v(t_k,\theta ))$$ and consider minimizing the L 2 distance $$d_n\left(\theta \right)={[Z_{\text{n}}-Z\left(\theta \right)]}^{T}Q\left(\theta \right)(Z_n-Z\left(\theta \right)).$$…”

“…To choose the numbers t i , i ∈ {1, … , k }, Groparu-cojocaru and Doray (2013) compute the respective variances of $U_n\left(t\right)=\text{Re}\left({\Phi }_n\left(t\right)\right)\text{and}{V}_n\left(t\right)=\text{Im}\left({\Phi }_n\left(t\right)\right)$ and their covariance. The authors write on page 1993…”

“…In case the matrix Q ( θ ) is taken to be the identity, the obtained estimator is denoted by $\hat{\theta }$ in Groparu-cojocaru and Doray (2013). In this case, the estimator $\hat{\theta }$ simply minimizes $$d_n\left(\theta \right)\sum _{j=1}^{k}true\{{\left[(U_n\left(t_j\right)-u(t_j,\theta ))\right]}^2+{\left[V_n\left(t_j\right)-\upsilon (t_j,\theta )\right]}^{2}true\}.$$…”