Distributions Derived from the Continuous Iteration of the Hyperbolic Sine Function

Abstract

Families of distributions built from the fractional or continuous iteration of exponential-type functions are characterized by a wide range of tail-heaviness. The present paper aims to define classes of distributions supported on the whole real line based on the continuous iteration of the hyperbolic sine function sinh. This function has already been commonly employed in univariate transformations such as the Johnson’s $S_{U}$ and sinh–arcsinh transforms. The tail versatility generated by a transformation based on the continuous iteration of sinh is highlighted based on an initial logistic distribution. It leads to the Hyperbolic Tetration distribution. The Double Hyperbolic Tetration distribution, defined from two successive hyperbolic transformations, is also introduced. It is among the first class of distributions with potential distinct tetration indices at plus and minus infinity. The distributions are applied to multiple data sets in hydrology.

Appendices

APPENDIX

A. PROOF OF PROPOSITION 1

The proof of Proposition 1 can be first illustrated by the retrieval of the coefficients of $\widehat{Q_{1}}$. The first coefficients of the power of expansion for $\{\psi^{[n]}\}_{1\leq n\leq 2}$ are presented in (28)

$$\psi^{[1]}(z)=\mathrm{sinh}(z)=z+\displaystyle\frac{z^{3}}{6}+\displaystyle\frac{z^{5}}{120}+O(z^{7}),$$

$$\psi^{[2]}(z)=\mathrm{sinh}(\mathrm{sinh}(z))=z+\displaystyle\frac{z^{3}}{3}+\displaystyle\frac{z^{5}}{10}+O(z^{7}).$$

(28)

The coefficients in $z^{5}$ from (28) can be coincided with $\delta\widehat{Q_{1}}$ from (5) for $\delta\in\{1,2\}$. The corresponding linear system can be written as in (29)

$$\begin{bmatrix}1&1\\ 1&2\end{bmatrix}\begin{bmatrix}\mathcal{Y}_{1,0}\\ \mathcal{Y}_{1,1}\end{bmatrix}\begin{bmatrix}\displaystyle\frac{1}{1}\times\frac{1}{120}\\ \displaystyle\frac{1}{2}\times\frac{1}{10}\end{bmatrix}$$

(29)

The solution of this system is $\begin{bmatrix}\frac{-1}{30},\frac{1}{24}\end{bmatrix}$ and corresponds to the second row of $\mathcal{Y}$. This method can be generalized by noticing that the matrix in (29) can be extended to a Vandermonde matrix while the constant terms in (29) can be obtained from the powers of the iteration matrix $M[\psi]$. Let $V(n)$ denote a Vandermonde matrix of order $n+1$ with entries $\{(i+1)^{j}\}_{\begin{subarray}{c}0\leq i\leq n\\ 0\leq j\leq n\end{subarray}}$. The coefficients in ascending order of $\widehat{Q_{n}}$ can be obtained by solving the matrix equation (30). $V(n)$ is invertible, which leads to Eq. (11)

$$V(n)\times\begin{bmatrix}\mathcal{Y}_{n,0}\\ \vdots\\ \mathcal{Y}_{n,p}\\ \vdots\\ \mathcal{Y}_{n,n}\end{bmatrix}\begin{bmatrix}\displaystyle\frac{\mathcal{M}[\psi]_{2n+3,1}}{1}\\ \vdots\\ \displaystyle\frac{\mathcal{M}^{p}[\psi]_{2n+3,1}}{p}\\ \vdots\\ \displaystyle\frac{\mathcal{M}^{n+1}[\psi]_{2n+3,1}}{n+1}\end{bmatrix}$$

(30)

The computation of the inverse of $V$ is not easily tractable. Therefore, as in Dijoux [1] for the continuous iteration of the exponential-minus-one function, the Vandermonde matrix is augmented. Let $\tilde{V}(n)$ denote a Vandermonde matrix of order $n+2$ with entries $\left\{\tilde{V}(n)_{ij}=i^{j}\right\}_{\begin{subarray}{c}0\leq i\leq n+1\\ 0\leq j\leq n+1\end{subarray}}$. First, Eq. (30) can be rewritten as in (31)

$$\tilde{V}(n)\times\begin{bmatrix}\mathcal{Y}_{n,0}\\ \vdots\\ \mathcal{Y}_{n,p}\\ \vdots\\ \mathcal{Y}_{n,n}\\ 0\end{bmatrix}\begin{bmatrix}\mathfrak{y}_{n}\\ \displaystyle\frac{\mathcal{M}[\psi]_{2n+3,1}}{1}\\ \vdots\\ \displaystyle\frac{\mathcal{M}^{p}[\psi]_{2n+3,1}}{p}\\ \vdots\\ \displaystyle\frac{\mathcal{M}^{n+1}[\psi]_{2n+3,1}}{n+1}\end{bmatrix}.$$

(31)

Second, the entries of the inverse of $\tilde{V}(n)$ can be simply expressed as in (32) [1], where $s(.,.)$ denotes the Stirling numbers of the first kind, see Comtet [31]. Equation (12) can be finally obtained by a matrix product from the inversion of (31), which ends the proof

$$\left(\tilde{V}(n)\right)^{-1}_{i,j}=\displaystyle\sum_{\textrm{max}(i,j)}^{n+1}\frac{(-1)^{j+k}}{k!}C_{k}^{j}s(k,i).$$

(32)

B. INFORMATION ON THE METEOROLOGICAL SUBDIVISIONS

The observations used in Section 7.1 consist of the annual precipitations in 30 Indian meteorological subdivisions from 1871 to 2016 [46]. The data compiled are either obtained either from a network of 306 rain gauge stations or from the India Meteorological Department. The name of each of the meteorological subdivision is provided in Table 5 along with the number of stations involved in the data compilation.

Table 5 Information on the subdivisions