We present here a unit-log-symmetric model based on the bivariate log-symmetric distribution.
It is a flexible family of distributions over the interval $(0, 1)$. We then discuss its mathematical properties such as stochastic representation, symmetry, modality, moments,  quantile function, entropy and maximum likelihood estimators, paying particular attention to the special cases of unit-log-normal,  unit-log-Student-$t$ and unit-log-Laplace distributions. Finally, some empirical results and practical illustrations are presented.

Keywords. Unit-log-symmetric distribution $\cdot$ Log-symmetric distribution $\cdot$ Bivariate model $\cdot$ Bounded distributions $\cdot$  MCMC.

Mathematics Subject Classification (2010). MSC 60E05 $\cdot$ MSC 62Exx $\cdot$ MSC 62Fxx.

§ INTRODUCTION

Distribution theory has mostly focused on unbounded distributions. However, in the recent years, bounded distributions have also become a subject of interest. In many practical applications, such as in medicine, biology, economics and finance, bounded data arise naturally. Some of the most common models used in this case are the beta distribution and its generalizations, Kumaraswamy and Topp-Leone distribution.

In order to have including flexible models for real world problems, it is often desirable to have a whole family of distributions available. For this purpose, transformations have been developed in the statistical literature which construct a bounded distribution for any given distribution over the real line $(-\infty, \infty)$ or over $(0, \infty)$, respectively, such as logistic transformation.

In this work, we study a transformation that contains the latter as a special case (for $\mu=0$ and $\sigma=1$). However, our transformation is derived from the bivariate log-symmetric distribution
which is defined for the first time in this paper.

We introduce  the unit-log-symmetric (ULS) distribution in Section <ref>. We pay special attention to the cases of unit-log-normal, unit-log-Student-$t$ and unit-log-Laplace distributions. In Section <ref>, mathematical properties of these models are studied. Stochastic representations of the ULS are given which are used to simplify the density considerably. The symmetry of the ULS density is established and then the above-mentioned special cases are explicitly formulated, and graphically illustrated. The modality is also discussed in detail.
The unit-log-normal density may be unimodal or bimodal; the unit-log-Student-$t$ density may have a bathtub shape or possess two minimum points; the unit-log-Laplace density may be unimodal or possess two minimum points. The quantile function is studied in Subsection <ref> while the moments
and the Shannon entropy in Subsections <ref> and <ref>, respectively. Finally, the maximum likelihood (ML) estimators are studied in Subsection <ref>. In Section <ref>, the Monte Carlo simulation results are presented for demonstrating the performance of the estimators and for assessing the empirical distribution of the residuals. The paper ends with the application of the unit-log-normal, unit-log-Student-$t$ and unit-log-Laplace distributions for modeling an internet access data. Specifically, the three models are fitted to the proportion of
world population using Internet (Section <ref>). Finally, in Section <ref>, some concluding remarks are made.

§ UNIT-LOG-SYMMETRIC DISTRIBUTION

We say that a continuous random variable $W$, with support
$(0, 1)$, follows a unit-log-symmetric  (ULS) distribution with parameter $\sigma_\rho=\sigma\sqrt{2(1-\rho)}$, denoted by $W\sim {\rm ULS}(\sigma_\rho,g_c)$, if its probability density function (PDF) is given by

\begin{eqnarray}\label{ULS-PDF}
{1\over w(1-w) \sigma^2\sqrt{1-\rho^2} Z_{g_c}}\,
\int_{0}^{\infty}
{1\over t}\,
\over
\Biggr)
{\rm d}t,
\quad
\\[0,3cm]
\text{where} \quad
t_w= \bigg({w\over 1-w}\bigg) t,
\
\widetilde{t}_w
\log\biggl[\Bigl({t_w\over \eta}\Bigr)^{1/\sigma}\biggr],
\
\widetilde{t}
\log\biggl[\Bigl({t\over \eta}\Bigr)^{1/\sigma}\biggr],
\ \eta=\exp(\mu),
\nonumber
\end{eqnarray}

with $\mu\in\mathbb{R}$, $\sigma>0$ and $\rho\in(-1,1)$. Moreover, $Z_{g_c}$ is a positive constant defined as

\begin{align}\label{partition function}
\int_{0}^{\infty}\int_{0}^{\infty}
{1\over t_1t_2\sigma^2\sqrt{1-\rho^2}}\,
\over
\Biggr)\, {\rm d}t_1{\rm d}t_2
\pi \int_{0}^{\infty}  g_c(u)\,{\rm d}u,
%\, {\rm d}{z_1}{\rm d}{z_2},
\end{align}

with $g_c$ being a scalar function, referred to as the density generator, which may or may not depend on extra parameters considered known

We have defined the PDF in (<ref>) as a uniparameter function for two reasons: first, to avoid identification problems (see Proposition <ref>) and second, because the shape of this PDF depends only on the parameter $\sigma_\rho$ (see Subsection <ref>).

For simplicity in presentation, we focus our attention specifically on the density generators of unit-log-normal, unit-log-Student-$t$ and unit-log-Laplace (Table <ref>), though other forms of density generators can be handled in an analogous manner.

Normalization constants $(Z_{g_c})$ and density generators $(g_c)$ for some ULS distributions.

$Z_{g_c}$     $g_c$     Parameter

height 1.0pt
$2\pi$     $\exp(-x/2)$     $-$

${{\Gamma({\nu/ 2})}\nu\pi\over{\Gamma({(\nu+2)/ 2})}}$
$(1+{x\over\nu})^{-(\nu+2)/ 2}$       $\nu>0$

$\pi$      $K_0(\sqrt{2x})$     $-$

Here, in Table <ref>,

$K_0(u)=\int_0^\infty t^{-1} \exp(-t-{u^2\over 4t}) \,{\rm d}t/2$, $u>0$, is the Bessel function of the third kind

§ SOME BASIC PROPERTIES

We now discuss some mathematical properties of the unit-log-symmetric distribution stated in the last section.
For this purpose, the following definition of bivariate log-symmetric  distribution

becomes essential.

We say that a continuous random vector $(T_1,T_2)$, with $T_1$ and $T_2$ being identically distributed, follows a bivariate log-symmetric (BLS) distribution  if its joint PDF is given by

\begin{eqnarray*}%\label{PDF}
{1\over t_1t_2\sigma^2\sqrt{1-\rho^2}Z_{g_c}}\,
\over
\Biggr),
\quad
\\[0,3cm]
\text{where}\quad
\widetilde{t_i}
\log\biggl[\Bigl({t_i\over \eta}\Bigr)^{1/\sigma}\biggr], \ \eta=\exp(\mu), \ i=1,2, \nonumber
\end{eqnarray*}

with $\boldsymbol{\theta}=(\eta,\sigma,\rho)$ being the parameter vector, $\mu\in\mathbb{R}$, $\sigma>0$, $\rho\in(-1,1)$,
and $g_c$ and $Z_{g_c}$ are as given in (<ref>) and (<ref>), respectively.
Observe that $Z_{g_c}$ in (<ref>) is
the normalization constant for $f_{T_1,T_2}$ and that the joint PDF of $(T_1, T_2)$ can be obtained as the PDF of an  exponential transformation of an elliptically symmetric vector <cit.>.

§.§ Stochastic representation

A simple observation shows that if $(T_1,T_2)\sim {\rm BLS}(\boldsymbol{\theta},g_c)$, then

\begin{align}\label{CDF-general}
\mathbb{P}\bigg({T_1\over T_1+T_2}\leqslant w\bigg)=\mathbb{P}\bigg({T_1\over T_2}\leqslant {w\over 1-w}\bigg),
\quad
\end{align}

and so,

\begin{align*}
f_{{T_1\over T_1+T_2}}(w)
{1\over (1-w)^2}\, f_{\frac{T_1}{T_2}}\bigg({w\over 1-w}\bigg)
{1\over (1-w)^2}\,
\int_{0}^{\infty}
t f_{T_1, T_2}(t_w, t;\boldsymbol{\theta})\,
{\rm d}t
%{1\over (1-w)^2}\,
%y f_{T_2}(y \, \vert \, T_1=y_w)\,
%{\rm d}y
\end{align*}

This means that if $(T_1,T_2)\sim {\rm BLS}(\boldsymbol{\theta},g_c)$,

the ULS random variable $W$ admits the stochastic representation

\begin{align}\label{stochastic representation}
W={T_1\over T_1+T_2}.
\end{align}

§.§ Characterizations

From the classic stochastic representation of a bivariate
elliptical vector <cit.>, it follows that $(T_1,T_2)\sim {\rm BLS}(\boldsymbol{\theta},g_c)$ has the following representation:

\begin{align} \label{rep-stoch-biv-student-t}
\begin{array}{lllll}
&T_1=\eta \exp(\sigma Z_1),
\\[0,3cm]
\exp(\sigma[ {\rho} Z_1+\sqrt{1-\rho^2} Z_2]),
\end{array}
\end{align}

where $Z_1=RDU_1$ and $Z_2=R\sqrt{1-D^2}U_2$ with
$U_1$, $U_2$,$R$ and $D$ being mutually independent, $\rho\in(-1,1)$, $\eta=\exp(\mu)$, $\mathbb{P}(U_i = -1) = \mathbb{P}(U_i = 1) = 1/2$, $i=1,2$, and the variables $D$ and $R$ have PDFs $f_D(d)={2/(\pi\sqrt{1-d^2})}, \ d\in(0,1)$, and
$	f_R(r)={2r g_c(r^2)/[\int_{0}^{\infty}
g_c(u) \, {\rm d}{u}]}, \ r>0$, respectively.

{T_1/ T_2}
\exp(\sigma[(1 -{\rho}) Z_1-\sqrt{1-\rho^2} Z_2]).
and so we obtain

\begin{align*}
\mathbb{P}\bigg({T_1\over T_2}\leqslant {w\over 1-w}\bigg)
\mathbb{P}(
\leqslant
\end{align*}

\begin{align}\label{sigma-and-Z-rho}
Z_\rho={1\over \sqrt{2(1-\rho)}}\,[(1-{\rho}) Z_1-\sqrt{1-\rho^2} Z_2]
\end{align}

\begin{align}\label{def-A}
A(w)=\log\bigg[\Big({w\over 1-w}\Big)^{1/\sigma_\rho}\bigg].
\end{align}

Thus, from (<ref>) and (<ref>), the cumulative distribution function (CDF) of  $W\sim {\rm ULS}(\sigma_\rho,g_c)$,  denoted by $F_W(w;\sigma_\rho)$, can be expressed as

\begin{align}\label{general cdf formula}
%\mathbb{P}\bigg({T_1\over T_2}\leqslant {w\over 1-w}\bigg)
\mathbb{P}(
\leqslant
\quad
\end{align}

From (<ref>), the following result follows immediately.

If the distribution of $Z_\rho$ depends (or not) only on the parameters of the density generator $g_c$, then the ULS distribution is identifiable.

The inverse function of $A$, denoted by $A^{-1}$, is given by

\begin{align}\label{inverse-A}
A^{-1}(w)={1\over 1+\exp(-\sigma_\rho w)}.
\end{align}

So, using this notation, the following result follows immediately from (<ref>).

Given the distribution of $Z_\rho$, we have $A^{-1}(Z_\rho)\sim {\rm ULS}(\sigma_\rho,g_c)$.  On the other hand, if $W\sim {\rm ULS}(\sigma_\rho,g_c)$, then $A(W)$ and $Z_\rho$ have the same distribution.

Differentiating (<ref>) with respect to $w$, the ULS PDF in (<ref>) of $W$ is characterized as

\begin{align}\label{general pdf formula}
{1\over w(1-w) \sigma_\rho}\,
\quad
\end{align}

where $f_{Z_\rho}$ denotes the PDF of $Z_\rho$.

This means that, the distribution of $W$ is completely determined by the distribution of $Z_\rho$.

The ULS PDF in (<ref>) is symmetric around $w_0 = 1/2$ provided that the PDF of  $Z_\rho$ is symmetric around $z_0=0$.
Furthermore, in this case, the median and the mean of a ULS distribution both occur at $w_0 = 1/2$.

A simple algebraic manipulation shows that

f_W\left({1\over 2}-w;\sigma_\rho\right)
\quad
\forall 0<w<1,

provided that $f_{Z_\rho}(-z)=f_{Z_\rho}(z)$, $\forall z\in\mathbb{R}$.
The required result then follows.

Since the PDFs of ${Z_1}$ and ${Z_2}$ are even functions <cit.>,
and provided that $Z_1$ and $Z_2$ are independent, we have that $f_{Z_\rho}(-z)=f_{Z_\rho}(z)$, $\forall z\in\mathbb{R}$.

§.§.§ Unit-log-normal

It is well-known that the bivariate log-normal distribution admits a stochastic representation as in (<ref>),  where $Z_1$ and $Z_2$ are
independent and identically distributed (i.i.d.)  standard normal  random variables <cit.>. Consequently, $Z_\rho$ in (<ref>) is standard normally distributed. So, by using (<ref>) and (<ref>), we obtain

\begin{align}\label{ULS-PDF-normal}
\Phi(A(w)),
\quad
{1\over w(1-w) \sigma_\rho}\,
\phi(A(w)),
\quad
\end{align}

where $\phi$ and $\Phi$ denote the PDF and CDF of a standard normal distribution, respectively.

§.§.§ Unit-log-Student-$t$

It is well-known that the bivariate log-Student-$t$ distribution has a stochastic representation as in (<ref>), where $Z_1=Z_1^* \sqrt{\nu/Q}\sim t_\nu$ and $Z_2= Z_2^* \sqrt{\nu/Q}\sim t_\nu$ <cit.>. Here, $Q\sim\chi^2_\nu$ (chi-square with $\nu$ degrees of freedom) is independent by of $Z_1^*$ and ${\rho} Z_1^* +\sqrt{1-\rho^2} Z_2^*$,
whereas $Z_1^*$ and $Z_2^*$ are i.i.d.  standard normal  random variables.

$[1/\sqrt{2(1-\rho)}][(1-{\rho}) Z_1^*-\sqrt{1-\rho^2} Z_2^*]\sim
we have

\begin{align}\label{distributed-t}
{1\over \sqrt{2(1-\rho)}}\,[(1-{\rho}) Z_1^*-\sqrt{1-\rho^2} Z_2^*]
\sqrt{\nu\over Q}\sim t_\nu.
\end{align}

By combining (<ref>) and (<ref>) with (<ref>), we obtain

\begin{align*}
\quad
{1\over w(1-w)\sigma_\rho}\,
\quad
\end{align*}

where $f_\nu$ and $F_\nu$ denote the PDF and CDF of a Student-$t$ distribution with $\nu$ degrees of freedom, respectively.

§.§.§ Unit-log-Laplace

By using the general algorithm for simulation of symmetric
bivariate Laplace variables <cit.>, it follows that the bivariate log-Laplace distribution has a stochastic representation as in (<ref>), where $Z_1=Z_1^* \sqrt{B}\sim {\rm Laplace}(0,1/\sqrt{2})$ and $Z_2= Z_2^* \sqrt{B}\sim {\rm Laplace}(0,1/\sqrt{2})$. Here, $B$ is a standard exponential variable independent by of $Z_1^*$ and ${\rho} Z_1^* +\sqrt{1-\rho^2} Z_2^*$,
whereas $Z_1^*$ and $Z_2^*$ are i.i.d.  standard normal  random variables.

$[1/\sqrt{2(1-\rho)}][(1-{\rho}) Z_1^*-\sqrt{1-\rho^2} Z_2^*]\sim
we have

\begin{align}\label{distributed-Laplace}
{1\over \sqrt{2(1-\rho)}}\,[(1-{\rho}) Z_1^*-\sqrt{1-\rho^2} Z_2^*]
\sqrt{B}\sim {\rm Laplace}(0,1/\sqrt{2}).
\end{align}

Then, by (<ref>), (<ref>) and (<ref>) we obtain

\begin{align}\label{ULS-PDF-Laplace}
\quad
{1\over w(1-w)\sigma_\rho}\,
\quad
\end{align}

where $F_\ell$ and $f_\ell$ denote the CDF and PDF of Laplace distribution with location parameter $0$ and scale parameter $1/\sqrt{2}$, respectively.

Figure <ref> displays different shapes of the unit-log-symmetric PDFs for different choices of parameters. From this figure, we observe that the parameter $\sigma_{\rho}$ controls the shape of the unit-log-normal, unit-log-Laplace and unit-log-Student-$t$ densities. We also note that the PDFs of these three models are symmetric. Finally, from Figure <ref>(d), we see the clear effect the kurtosis parameter $\nu$ has on the unit-log-Student-$t$ density.

Unit-log-symmetric PDFs for some choices of parameters.

§.§ Modality

Differentiating $A$ in (<ref>) with respect to $w$, we get

\begin{align}\label{a-derivatives}
A'(w)={1\over w(1-w)\sigma_\rho}
\quad \text{and}\quad
A''(w)=(2w-1)[A'(w)]^2 \sigma_\rho.
\end{align}

Suppose the derivative of $f_W(w;\sigma_\rho)$ with respect to $w$  exists at every point in its domain. In the case there is a countable number of points on its domain where $f_W$ is not differentiable, by definition, these are considered to be critical points of $f_W$.

Thus, from (<ref>),
we have $f_W(w;\sigma_\rho)=A'(w) f_{Z_\rho}(A(w))$ so that

\begin{align*}
A''(w) f_{Z_\rho}(A(w))+[A'(w)]^2f'_{Z_\rho}(A(w)).
\end{align*}

If $f'_{Z_\rho}(z)=-r(z)f_{Z_\rho}(z)$, for some real-valued function $r$,  then

\begin{align*}
\{A''(w)-[A'(w)]^2 r(A(w))\} f_{Z_\rho}(A(w))
\stackrel{\eqref{a-derivatives}}{=}
\{(2w-1)\sigma_\rho-r(A(w))\} A'(w) f_W(w;\sigma_\rho).
\end{align*}

\begin{align*}
f'_W(w;\sigma_\rho)=0  \quad
\Longleftrightarrow\quad
(2w-1)\sigma_\rho=r(A(w)) \quad
\Longleftrightarrow\quad
\left[2 A^{-1}(A(w))-1\right]\sigma_\rho
\end{align*}

Therefore, by using (<ref>), a critical point $w$ of the ULS PDF (<ref>) is such that

\begin{align*}
\left[{2\over 1+\exp(-\sigma_\rho A(w))}-1\right]\sigma_\rho
\end{align*}

or, equivalently,

\begin{align}\label{eq-critical point}
\tanh\biggl({\sigma_\rho^2 \over 2}\, y\biggr)
{r(\sigma_\rho y)\over \sigma_\rho},
\quad \text{with} \ y={A(w)/ \sigma_\rho}.
\end{align}

§.§.§ Unit-log-normal

In this case, from Subsubsection <ref>, we have $Z_\rho\sim N(0,1)$, and then $r(z)=z$. So, from (<ref>), the equation for the critical points of the ULS PDF is given by

\begin{align*}
\tanh\biggl({\sigma_\rho^2 \over 2}\, y\biggr)
y, \quad \text{with} \ y={A(w)/ \sigma_\rho},
\end{align*}

if and only if

\begin{align}\label{tanh-equation}
{\sigma_\rho^2 \over 2}\, y={{\rm arctanh}(y)}.
\end{align}

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axis x line=middle,
axis y line=middle,

yticklabel style=inner ysep=0pt, anchor=east,
ymin=-2.0, ymax=2.0,
xmin=-1.25, xmax=1.25,

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Axis Style,
-1.0 ,0.0, 1.0
xlabel= $y$,

$-1$, $0$, $1$,

[name path=line 5] coordinates  (0,0) (0,0) ;

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[name path=line 1,thick,  smooth, black]  arctanh() node[above =24pt,pos=.63]${\rm arctanh}(y)$;
[name path=line 2, thick, smooth, black!30] 1.1*1.1*x/2 node[above=0,pos=.95]${\sigma_{\rho}^2\over 2}  y$;

name intersections=of=line 1 and line 2,total=]
in 1,...,
(intersection-) circle (2pt)

Point of intersection of the graphs of the functions  ${\sigma_\rho^2 y /2}$ and ${{\rm arctanh}(y)}$ when $\sigma_\rho\leqslant \sqrt{2}$.

/pgf/declare function=arctanh() = 0.5*(ln(1+)-ln(1-));

Axis Style,
-0.875, 0.0, 0.875
xlabel= $y$,

$y_-$, $0$,  $y_+$,

[name path=line 5] coordinates  (0,0) (0,0) ;

[name path=line 0] coordinates  (0,0) (0,0) ;

[name path=line 1, thick, smooth, black]  arctanh() node[above =24pt,pos=.63]${\rm arctanh}(y)$;
[name path=line 2, thick, smooth, black!30] 1.75*1.75*x/2 node[below =-0.2pt,pos=.95]${\sigma_{\rho}^2\over 2}  y$;

name intersections=of=line 1 and line 2,total=]
in 1,...,
(intersection-) circle (2pt)

Points of intersection of the graphs of the functions  ${\sigma_\rho^2 y /2}$ and ${{\rm arctanh}(y)}$ when $\sigma_\rho> \sqrt{2}$.

Observe that there always exists at least one solution for the above equation.  The number of solutions
of (<ref>) depends on whether ${\sigma_\rho}$ is larger or smaller than $\sqrt{2}$. If $\sigma_\rho\leqslant\sqrt{2}$, then (<ref>) has a unique solution, given by $y_0 = 0$ (Figure <ref>), and if $\sigma_\rho>\sqrt{2}$, then (<ref>) has two additional non-trivial solutions (Figure <ref>), $y_-=y_{-}(\sigma_\rho)<0$ and $y_+=y_{+}(\sigma_\rho)>0$ (which depend on $\sigma_\rho$), with $y_{+}=-y_{-}$; this means that
$f_W(w;\sigma_\rho)$ has one critical point at $w_0=1/2$,  or three critical points, $w_-=A^{-1}(\sigma_\rho y_-)$, $w_0=1/2$ and $w_+=A^{-1}(\sigma_\rho y_+)$.
As $\lim_{w\to 0^+} f_W(w;\sigma_\rho)=\lim_{w\to 1^-} f_W(w;\sigma_\rho)=0$,
we then obtain the following result.

Let $W\sim {\rm ULS}(\sigma_\rho,g_c)$, with $g_c(x)=\exp(-x/2)$.  Then, the following holds:

If $\sigma_\rho\leqslant\sqrt{2}$, then the ULS PDF $f_W(w;\sigma_\rho)$ is unimodal, with mode $w_0=1/2$.
On the other hand,
if $\sigma_\rho>\sqrt{2}$, then the ULS PDF $f_W(w;\sigma_\rho)$ is bimodal, with modes

\begin{align*}
w_-=A^{-1}(\sigma_\rho y_-)={1\over 1+\exp(-\sigma_\rho^2 y_{-})}
\quad \text{and} \quad
w_+=A^{-1}(\sigma_\rho y_+)={1\over 1+\exp(-\sigma_\rho^2 y_{+})},
\end{align*}

and minimum point  $w_0=1/2$, so that $0<w_-<w_0<w_+<1$.
Moreover, the graph of $f_W(w;\sigma_\rho)$ is symmetric around $w_0=1/2$  (Figure <ref> a).

§.§.§ Unit-log-Student-$t$

In this case, from Subsubsection <ref>, we have $Z_\rho\sim t_\nu$, and $r(z)=(\nu+1)z/(\nu+z^2)$. So, from  (<ref>), the equation for critical points of the ULS PDF is given by

\begin{align*}
\tanh\biggl({\sigma_\rho^2 \over 2}\, y\biggr)
{(\nu+1) y\over \nu+\sigma_\rho^2 y^2},
\quad \text{with} \ y={A(w)/ \sigma_\rho},
\end{align*}

or, equivalently,

\begin{align}\label{tanh-equation-1}
\tanh(z)
{2(\nu+1) z\over \nu\sigma_\rho^2+4z^2},
\quad \text{with} \ z=\sigma_\rho^2y/2.
\end{align}

Note that $w_0=1/2$ is a critical point of the ULS PDF because $z_0=0$ satisfies (<ref>).

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yticklabel style=inner ysep=0pt, anchor=east,
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Axis Style,
xlabel= $z$,

[name path=line 5] coordinates  (0,0) (0,0) ;

[name path=line 0] coordinates  (0,0) (0,0) ;

[name path=line 1,thick,  smooth, black]  tanh() node[above =-2.0pt,pos=.89]${\rm tanh}(z)$;
[name path=line 2, thick, smooth, black!30] 2*(5+1)*x/(5*1.8*1.8+4*x*x) node[right =15.pt,below=-24.0,pos=.25]${2(\nu+1) z\over \nu\sigma_\rho^2+4z^2}$;

name intersections=of=line 1 and line 2,total=]
in 1,...,
(intersection-) circle (2pt)

The graphs of the functions ${2(\nu+1) z/(\nu\sigma_\rho^2+4z^2)}$ and ${{\rm tanh}(z)}$ have a single point of intersection.

Axis Style,
-2.77, 0.0,  2.77
xlabel= $z$,

$z_-$,  $0$,  $z_+$,

[name path=line 5] coordinates  (0,0) (0,0) ;

[name path=line 0] coordinates  (0,0) (0,0) ;

[name path=line 1, thick, smooth, black]  tanh() node[above =-2.0pt,pos=.89]${\rm tanh}(z)$;
[name path=line 2, thick, smooth, black!30] 2*(5+1)*x/(5*0.75*0.75+4*x*x) node[below =2pt,pos=.80]${2(\nu+1) z\over \nu\sigma_\rho^2+4z^2}$;

name intersections=of=line 1 and line 2,total=]
in 1,...,
(intersection-) circle (2pt)

The graphs of the functions ${2(\nu+1) z/(\nu\sigma_\rho^2+4z^2)}$ and ${{\rm tanh}(z)}$ have three points of intersection.

Because $\lim_{z\to\pm \infty} \tanh(z) = \pm 1$, by analyzing
the graphs of functions $z\mapsto\tanh(z)$ and $2(\nu+1) z/ (\nu\sigma_\rho^2+4z^2)$, we see that (<ref>) has one solution, given by $z_0=0$, or three  solutions, $z_-< z_0=0<z_+=-z_-$, that depend on the choice of parameters $\nu$ and $\sigma_\rho$ (Figures <ref> and <ref>). In other words, $f_W(w;\sigma_\rho)$ has one critical point at $w_0=1/2$, or three critical points, $
As $\lim_{w\to 0^+} f_W(w;\sigma_\rho)=\lim_{w\to 1^-} f_W(w;\sigma_\rho)=\infty$ and $w_0=1/2$ is a critical point (and point of symmetry, see Proposition <ref>) of $f_W(w;\sigma_\rho)$, we readily have the following result.

If $W\sim {\rm ULS}(\sigma_\rho,g_c)$, with $g_c(x)=(1+{x/\nu})^{-(\nu+2)/ 2}$, then the ULS PDF $f_W(w;\sigma_\rho)$ has a bathtub shape with minimum point
$w_0=1/2$, or
is decreasing-increasing-decreasing-increasing with
minimum points

\begin{align*}
w_-=A^{-1}(2z_-/\sigma_\rho)={1\over 1+\exp(-2z_-)}
\quad \text{and} \quad
w_+=A^{-1}(2z_+/\sigma_\rho)={1\over 1+\exp(-2z_+)},
\end{align*}

and maximum point  $w_0=1/2$.
Moreover, the graph of the  ULS PDF is symmetric with around $w_0=1/2$ (Figures <ref> c, d).

§.§.§ Unit-log-Laplace

From Subsubsection <ref>, $Z_\rho\sim {\rm Laplace}(0,1/\sqrt{2})$, and $r(z)=\sqrt{2} z/\vert z\vert$, $z\neq 0$.  The function $f_W(w,\sigma_{\rho})$ is not differentiable at $w_0=1/2$, and this point is a critical point with $f_W(w_0,\sigma_{\rho})=4/\sigma_\rho$.
So, Equation (<ref>) for critical points (excluding $w_0$) of the ULS PDF is given by

\begin{align}\label{eq-critical point-Laplace}
\tanh\biggl({\sigma_\rho^2 \over 2}\, y\biggr)
{\sqrt{2}\over \sigma_\rho}\, {y\over \vert y\vert}
\quad \text{with} \ y={A(w)/ \sigma_\rho},
\end{align}

which implies that
y_{\pm}=\pm {2} \,{\rm arctanh}({\sqrt{2}/ \sigma_{\rho}})/\sigma_{\rho}^2.
In other words, $f_W(w;\sigma_\rho)$ has one critical point at $w_0=1/2$, or three critical points,
$0<w_-=A^{-1}(\sigma_\rho y_-)<w_0=1/2<w_+=A^{-1}(\sigma_\rho y_+)<1$.

\begin{align*}
\lim_{w\to 0^+}f_W(w,\sigma_{\rho})=\lim_{w\to 1^-}f_W(w,\sigma_{\rho})=
\begin{cases}
0, & \sigma_{\rho}<\sqrt{2},
\\
1/\sqrt{2}, & \sigma_{\rho}=\sqrt{2},
\\
\infty, & \sigma_{\rho}>\sqrt{2},
\end{cases}
\end{align*}

and $w_0=1/2$ is a critical point of symmetry (Proposition <ref>) of $f_W(w;\sigma_\rho)$, we readily obtain the following result.

Let $W\sim {\rm ULS}(\sigma_\rho,g_c)$, with $g_c(x)=K_0(\sqrt{2x})$.  Then, the following holds:

If $\sigma_\rho\leqslant\sqrt{2}$ then the ULS PDF $f_W(w;\sigma_\rho)$ is unimodal, with mode $w_0=1/2$.
On the other hand,
if $\sigma_\rho>\sqrt{2}$ then the ULS PDF $f_W(w;\sigma_\rho)$ is decreasing-increasing-decreasing-increasing with
minimum points

\begin{align*}
&w_-=A^{-1}(\sigma_\rho y_-)={1\over 1+\exp({2}\, {\rm arctanh}({\sqrt{2}/ \sigma_{\rho}}))},
\\[0,2cm]
&w_+=A^{-1}(\sigma_\rho y_+)={1\over 1+\exp(-{2}\, {\rm arctanh}({\sqrt{2}/ \sigma_{\rho}}))},
\end{align*}

and maximum point  $w_0=1/2$, so that $0<w_-<w_0<w_+<1$.
Moreover,  $f_W(w;\sigma_\rho)$ is not differentiable and symmetric at $w_0=1/2$ (Figure <ref> b).

§.§ Quantile function

From (<ref>), the $p$-quantile function $Q_W(p)$ of $W\sim {\rm ULS}(\sigma_\rho,g_c)$, for $p\in(0,1)$, satisfies

\begin{align*}
\mathbb{P}(
\leqslant
%	\log\bigg[\Big({Q_W(p)\over 1-Q_W(p)}\Big)^{1/\sigma_\rho}\bigg]
\end{align*}

where $Z_\rho$ is as in (<ref>).

\begin{align}\label{eq:qf}
\quad \Longleftrightarrow \quad
Q_W(p)=A^{-1}(Q_{Z_\rho}(p))	\stackrel{\eqref{inverse-A}}{=}{1\over 1+\exp(-\sigma_\rho Q_{Z_\rho}(p))}.
\end{align}

As the quantiles of Gaussian, Student-$t$, and Laplace variables are well-known, by using the results of Subsubsections <ref>, <ref>, and <ref> on the distribution of $Z_\rho$, we have the quantiles as presented in Table <ref>.

Density generators $(g_c)$ and $p$-quantiles of some ULS distributions.

Distribution      $g_c$     $Q_W(p)$

height 1.0pt
$[1+\exp(-\sigma_\rho \Phi^{-1}(p))]^{-1}$

$(1+{x\over\nu})^{-(\nu+2)/ 2}$
$[1+\exp(-\sigma_\rho F_\nu^{-1}(p))]^{-1}$

$[1+(2p)^{-\sigma_\rho/\sqrt{2}}]^{-1} \mathds{1}_{(0,1/2]}(p)
[1+(2-2p)^{\sigma_\rho/\sqrt{2}}]^{-1} \mathds{1}_{[1/2, \infty)}(p)$

Here, $\Phi^{-1}_\nu$ and $F^{-1}_\nu$ denote the quantiles of the standard normal and Student-$t$ distributions, respectively.

§.§ Moments

For $W\sim {\rm ULS}(\sigma_\rho,g_c)$, since $W=T_1/(T_1+T_2)$ with $(T_1,T_2)\sim {\rm BLS}(\boldsymbol{\theta},g_c)$, it is clear that $0\leqslant\mathbb{E}(W^r)\leqslant 1$, $r>0$.
Consequently, all positive moments of $W$ always exist.

In general, the moments of $W$ admit the following representation:

\begin{align*}
\mathbb{E}(W^r)
\mathbb{E}(
[A^{-1}( Z_\rho)]^r
\stackrel{\eqref{inverse-A}}{=}
\mathbb{E}\left[
{1\over (1+\exp(-\sigma_\rho Z_\rho))^r}
\,
\right], \quad r\in\mathbb{R},
\end{align*}

where $Z_\rho$ is as given in (<ref>).

Upon using binomial expansion, the negative integer moments of $W$ can be written as

\begin{align*}
\mathbb{E}(W^{-n})
\mathbb{E}\left[
\left(
{1+\exp(-\sigma_\rho Z_\rho)}
\right)^n
\,
\right]
\sum_{k=0}^{n}\binom{n}{k} M_{Z_\rho}(-k\sigma_\rho), \quad n\in\mathbb{N},
\end{align*}

where $M_{Z_\rho}(t)$ is the moment generating function (MGF) of $Z_\rho$.

As the MGFs of Gaussian and Laplace variables are avaliable in some simple explicit forms, by using the results of Subsubsections <ref> and <ref> on the distribution of $Z_\rho$, we have the  negative moments corresponding to unit-log-normal and unit-log-Laplace (Table <ref>).
On the other hand, since $Z_\rho\sim t_\nu$ as $g(x)=(1+{x/\nu})^{-(\nu+2)/ 2}$ (Subsubsection <ref>), it is clear that the  negative moments

corresponding to unit-log-Student-$t$ do not exist.

Density generators $(g_c)$ and negative moments for some ULS distributions.

Distribution      $g_c$     $\mathbb{E}(W^{-n})$     Restriction

height 1.0pt
$\sum_{k=0}^{n}\binom{n}{k} \exp({1\over 2}\, k^2\sigma_\rho^2)$     -

$(1+{x\over\nu})^{-(\nu+2)/ 2}$
$\nexists$     -

$\sum_{k=0}^{n}\binom{n}{k} (1-{1\over 2} \, k^2\sigma_\rho^2)^{-1}$      $n<\sqrt{2}/\sigma_\rho$, $\sigma_\rho\leqslant\sqrt{2}/2$

§.§ Shannon entropy

The entropy of a random variable can be interpreted as the average
level of uncertainty inherent in the possible outcomes of the variable.
Following [18], the entropy $H$ of a random variable $W\sim {\rm ULS}(\sigma_\rho,g_c)$, which takes
values in the interval $(0,1)$ and is distributed according to
$f_W(\cdot,\sigma_{\rho})$ in (<ref>), is given by

\begin{align*}
H(W)=-\mathbb{E}[\log f_W(W,\sigma_{\rho})].
\end{align*}

Upon using (<ref>) and the representation in (<ref>), we observe that

\begin{align*}
\sigma[(1+\rho)\mathbb{E}(Z_1)+\sqrt{1-\rho^2}\mathbb{E}(Z_2)]
\end{align*}

where $Z_1$ and $Z_2$ are as given in Subsection <ref> and $Z_\rho$ is as in (<ref>). Because $Z_1$ and $Z_2$ are identically distributed <cit.>, the above identity can be expressed as

\begin{align*}
\sigma(1+\rho+\sqrt{1-\rho^2})\mathbb{E}(Z_1)
\end{align*}

It is known that the entropies of Gaussian, Student-$t$, and Laplace variables all exist. Then, by using the results of Subsubsections <ref>, <ref>, and <ref> on the distribution of $Z_\rho$, we obtain the entropies for the three models as presented in <ref>.

Density generators $(g_c)$ and entropies for some ULS distributions.

Distribution      $g_c$     $H(W)$

height 1.0pt
{1\over 2}\, [\log(2\pi)+1]$

$(1+{x\over\nu})^{-(\nu+2)/ 2}$
{\nu+1\over 2}\, \left[\Psi\left({\nu+1\over 2}\right)-\Psi\left({\nu\over 2}\right)\right]
\log\left[\sqrt{\nu}\, {\rm B}\left({\nu\over 2}, {1\over 2}\right)\right]$

\log(\sqrt{2}{\rm e})$

Here,  $\Psi(x)=\Gamma'(x)/\Gamma(x)$ is the digamma function and ${\rm B}(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$ is the complete beta function, with $\Gamma(t)=\int_0^\infty x^{t-1} \exp(-x) \,{\rm d}x$, $t>0$, being the gamma function.

§.§ Maximum likelihood estimation

Let $\{W_{i}; i=1,\ldots,n\}$ be a univariate random sample of size $n$ from the ${\rm ULS}(\sigma_\rho,g_c)$ distribution with PDF as in (<ref>), and let $w_{i}$ be the sample observations of $W_{i}$. Then, the log-likelihood function for $\sigma_\rho$ is given by

\begin{align*}%\label{general pdf formula}
\ell(\sigma_\rho)
-\sum_{i=1}^{n} \log[w_i(1-w_i)]
\sum_{i=1}^{n}
\log
\end{align*}

where the random variable $Z_\rho$ is as defined in (<ref>) and $A$ is as in (<ref>).

In the case when a supremum $\widehat{\sigma}_\rho$ exists, it must be such that

\begin{align}\label{likelihood equation}
{\partial \ell(\sigma_\rho)\over\partial\sigma_\rho}
\bigg\vert_{\sigma_\rho=\widehat{\sigma}_\rho}
\quad \text{and} \quad
{\partial^2 \ell(\sigma_\rho)\over\partial\sigma_\rho^2}
\bigg\vert_{\sigma_\rho=\widehat{\sigma}_\rho}
\end{align}

\begin{align}\label{2-derivative}
{\partial \ell(\sigma_\rho)\over\partial\sigma_\rho}
-{n\over \sigma_\rho}
\frac{1}{\sigma_\rho^2}
\sum_{i=1}^{n}
\log\Big({w_i\over 1-w_i}\Big)
G(A(w_i)), \nonumber
\\[0,2cm]
{\partial^2 \ell(\sigma_\rho)\over\partial\sigma_\rho^2}
%{n\over \sigma_\rho^2}
%\log\Big({w_i\over 1-w_i}\Big)
%\log^2\Big({w_i\over 1-w_i}\Big)
%G'(A(w_i)) \nonumber
-{1\over \sigma_\rho}\, \left[2\, {\partial\ell(\sigma_\rho) \over\partial\sigma_\rho}+{n\over\sigma_\rho }\right]
\frac{1}{\sigma_\rho^4}
\sum_{i=1}^{n}
\log^2\Big({w_i\over 1-w_i}\Big)
\end{align}

We shall now adopt the notation

\begin{align}\label{def-x-rho}
\end{align}

Observe now that the likelihood equation $	{\partial \ell(\sigma_\rho)/\partial\sigma_\rho}
\big\vert_{\sigma_\rho=\widehat{\sigma}_\rho}=0$ in (<ref>) can be written as

\begin{align}\label{eq-sigma-rho}
\sum_{i=1}^{n}
\log\Big({w_i\over 1-w_i}\Big)
\bigg\vert_{\sigma_\rho=\widehat{\sigma}_\rho}
\widehat{\sigma}_\rho.
\end{align}

Any nontrivial root $\widehat{\sigma}_\rho$ of the above equation is the ML estimate of ${\sigma}_\rho$ in the loose sense. When the parameter value provides the absolute maximum of the log-likelihood function, it is called an ML estimate in the strict sense.

§.§.§ Unit-log-normal

From Subsubsection <ref>,  $Z_\rho\sim N(0,1)$. Then,  $G(x)$ in (<ref>) and its derivative are given by $G(x)=-x$ and $G'(x)=-1$.
Thus, by using (<ref>), the ML estimate of ${\sigma}_\rho^2$ in the loose sense   is given by

\begin{align*}
\widehat{\sigma}_\rho^2
\sum_{i=1}^{n}
\log^2\Big({w_i\over 1-w_i}\Big)}.
\end{align*}

From  (<ref>), it is then easy to verify that ${\partial^2 \ell(\sigma_\rho)/\partial\sigma_\rho^2}
\big\vert_{\sigma_\rho=\widehat{\sigma}_\rho}<0$, and so $\widehat{\sigma}_\rho^2$ is an ML estimate in the strict sense.

Let $\widehat{\mathcal{O}}_\rho^2$ be the corresponding ML estimator of ${\sigma}_\rho^2$. By using Proposition <ref>,  we have $A(W)=Z_\rho\sim N(0,1)$, for $W\sim {\rm ULS}(\sigma_\rho,g_c)$. Then (for $k=1,2,\ldots$),

\begin{align}\label{Unbiasedness}
\mathbb{E}\big[(\widehat{\mathcal{O}}_\rho^2)^k\big]
\mathbb{E}\left[\log^{2k}\Big({W\over 1-W}\Big)\right]
\mathbb{E}\left[A^{2k}(W)\right]
\mathbb{E}\left(Z_\rho^{2k}\right)
{\sigma}_\rho^{2k}\, {(2k)!\over 2^k k!}.
\end{align}

Let $W_1, \ldots, W_n$ be independent and identically distributed random variables with the unit-log-normal distribution in (<ref>), where $\sigma_\rho$ is unknown. Then, the ML estimator $\widehat{\mathcal{O}}_\rho^2$ of $\sigma_\rho^2$  is

* unbiased;

* consistent;

* asymptotically normal; that is, $\sqrt{n}(\widehat{\mathcal{O}}_\rho^2- {\sigma}_\rho^2)$ converges in distribution to a normal distribution with
mean zero and variance $2\sigma_\rho^2$;

* efficient.

By taking $n=1$ in (<ref>), the unbiasedness of $\widehat{\mathcal{O}}_\rho^2$ follows.

By the Strong Law of Large Numbers and
the Central Limit Theorem, it follows that $\widehat{\mathcal{O}}_\rho^2$ is consistent and asymptotically normal.

Finally, since the variance of $\widehat{\mathcal{O}}_\rho^2$ coincides with the Cramér-Rao lower bound $[n I(\sigma_\rho)]^{-1}$, where
I(\sigma_\rho)=-\mathbb{E}[{\partial^2 \log f_W(w;\sigma_\rho) /\partial(\sigma_\rho^2)^2}]
{1/(2 \sigma_\rho^2)}
is the Fisher information in one observation from $f_W(w;\sigma_\rho)$,
the efficiency of $\widehat{\mathcal{O}}_\rho^2$ follows readily.

§.§.§ Unit-log-Student-$t$

From Subsubsection <ref>,  $Z_\rho\sim t_\nu$. Then,  $G(x)$ in (<ref>) and its derivative are given by $G(x)=-(\nu+1)x/(\nu+x^2)$ and $G'(x)=-(\nu+1)(\nu-x^2)/(\nu+x^2)^2$.
Thus, by using (<ref>), the ML estimate of ${\sigma}_\rho^2$ in the loose sense satisfies

\begin{align}\label{id-ML-est}
\frac{\nu+1}{n}
\sum_{i=1}^{n}
\dfrac{\displaystyle \log^2\Big({w_i\over 1-w_i}\Big)}{\nu \widehat{\sigma}_\rho^2+\log^2\Big({\displaystyle w_i\over \displaystyle 1-w_i}\Big)}
\end{align}

Any ML estimate of ${\sigma}_\rho^2$ in the loose sense
is an ML estimate in the strict sense.

Suppose $\widehat{\sigma}_\rho^2$ is an ML estimate of ${\sigma}_\rho^2$ in the loose sense.
From (<ref>), it follows that $\widehat{\sigma}_\rho^2$ is an ML estimate in the strict sense whenever

\begin{align*}
&{\nu+1\over n}
\sum_{i=1}^{n}
\log^2\Big({ w_i\over 1-w_i}\Big)\,
\dfrac{
\nu\widehat{\sigma}_\rho^2-\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)
\left[\nu\widehat{\sigma}_\rho^2+\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)\right]^2
\end{align*}

By using the partial fraction expansion

\begin{align*}
{x-y\over (x+y)^2}={2x\over (x+y)^2}-{1\over x+y},
\end{align*}

the above inequality is written as

\begin{align*}
{	2\nu(\nu+1)\widehat{\sigma}_\rho^2\over n}
\sum_{i=1}^{n}
\dfrac{
\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)
\left[\nu\widehat{\sigma}_\rho^2+\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)\right]^2
{\nu+1\over n}
\sum_{i=1}^{n}
\dfrac{
\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)
\nu\widehat{\sigma}_\rho^2+\log^2\Big({\displaystyle w_i\over\displaystyle 1-w_i}\Big)
\end{align*}

As $\widehat{\sigma}_\rho^2$ is an ML estimate in the loose sense, it satisfies  (<ref>), and consequently, the above inequality is true. This completes the proof of the result.

§.§.§ Unit-log-Laplace

In this case, note that $f_W(w;\sigma_\rho)=\exp[\theta T(w)-\psi(\theta)] h(w)$, $W\sim {\rm ULS}(\sigma_\rho,g_c)$, with  $\theta=1/\sigma_\rho$,

\begin{align*}
T(w)=-{\sqrt{2}} \biggl\vert\log\Big({w\over 1-w}\Big)\biggr\vert,
\quad
\psi(\theta)=-\log(\theta)
\quad \text{and} \quad
h(w)={1\over \sqrt{2}w(1-w)}.
\end{align*}

Then, $f_W$ belongs to the one-parameter exponential family with $\Theta=\{\theta:\theta>0\}$ parameter space.

* For $\theta\in\Theta^{\rm o}$,  the interior of $\Theta$, all moments of $T(W)$ exist, and $\psi(\theta)$ is infinitely differentiable at any such $\theta$. Furthermore,

\begin{align*}
\mathbb{E}\left(\sqrt{2}\biggl\vert\log\Big({W\over 1-W}\Big)\biggr\vert\right)=\sigma_{\rho}
\quad \text{and} \quad
{\rm Var}\left(\sqrt{2}\biggl\vert\log\Big({W\over 1-W}\Big)\biggr\vert\right)
\sigma_{\rho}^2;
\end{align*}

* Given an i.i.d. sample of size $n$ from $f_W(w;\sigma_\rho)$,

\begin{align*}
-{\sqrt{2}} \sum_{i=1}^{n} \biggl\vert\log\Big({W\over 1-W}\Big)\biggr\vert
\end{align*}

is minimally sufficient;

* The Fisher information function exists, is finite at all $\theta\in\Theta^{\rm o}$ and equals
$I (\theta)=[I(\sigma_\rho)]^{-1} = \sigma_{\rho}^{2}$.

The proof follows by direct application of Proposition 16.1 of  [6], and is therefore omitted for the sake of conciseness.

Let $\theta = \theta_0 \in \Theta^{\rm o}$ be the true value. Then, for all large $n$, with probability $1$, a unique ML estimator of
$\sigma_{\rho}$ exists, and is consistent and asymptotically normal.

As $\psi''(\theta)=\sigma_{\rho}^2>0$,
by Theorem 16.1 of [6], a unique ML estimator of
$\theta=1/\sigma_{\rho}$ exists, and is consistent and asymptotically normal. Hence, by the invariance property of the ML estimator, by the continuous application theorem for convergence in probability and by delta method, the proof of the result gets completed.

Furthermore, in the unit-log-Laplace case, the likelihood function for $\sigma_\rho$ is given by

\begin{align*}
\prod_{i=1}^{n}
{1\over w_i(1-w_i)\sigma_\rho}\,
\end{align*}

where $A$ is as given in (<ref>) and $f_\ell$ is the PDF of Laplace distribution with location parameter $0$ and scale parameter $1/\sqrt{2}$. Because $f_\ell(x)$ is not differentiable at $x=0$ we cannot use the log-likelihood function differentiation method as in the last two cases. In what follows, we maximize $L(\sigma_\rho)$.

Indeed, an algebraic manipulation yields

\begin{align*}
{1\over 2^{n/2}}
\left[\prod_{i=1}^{n} {1\over w_i(1-w_i)}\right]
\dfrac{\displaystyle\exp\left(-{\sqrt{2}\over\sigma_\rho}\, 	\sum_{i=1}^{n}
\left\vert\log\Big({w_i\over 1-w_i}\Big)\right\vert \right)}{\sigma_\rho^n}.
\end{align*}

As the function $\sigma_\rho\longmapsto\exp(-t/\sigma_\rho)/\sigma_\rho^n$, $t>0$, reaches a maximum at $\sigma_\rho=t/n$, the right-hand expression in the above equation is at most

\begin{align*}
{\exp(-n)\over 2^{n}}
\left[\prod_{i=1}^{n} {1\over w_i(1-w_i)}\right]
\Bigg/
\left[
{1\over n} \sum_{i=1}^{n} \biggl\vert\log\Big({w_i\over 1-w_i}\Big)\biggr\vert
\right]^{n}.
\end{align*}

So, the ML estimate of ${\sigma}_\rho^2$ is

\begin{align*}
\widehat{\sigma}_\rho^2
{\sqrt{2}\over n} \sum_{i=1}^{n} \biggl\vert\log\Big({w_i\over 1-w_i}\Big)\biggr\vert.
\end{align*}

This corroborates the uniqueness of the  ML estimator evidenced in Proposition <ref>. Moreover, from Item 1 of Proposition <ref>, the  ML estimator of ${\sigma}_\rho^2$ is unbiased and consistent.

§ SIMULATION RESULTS

In this section, we present the results of Monte Carlo simulation studies for unit-log-normal model (the results of the unit-log-Student-$t$ and unit-log-Laplace are somewhat similar and so are omitted here due to space limitations), considering different scenarios of parameters and sample sizes. The first part of the study presents the evaluation of the ML estimates, while the second part evaluates the empirical distribution of the generalized Cox-Snell (GCS) and randomized quantile (RQ) residuals, which are given, respectively, by

\begin{equation*}
\hat{r}_i^{\text{GCS}} = -\log[1-\widehat{F}_W(w_i;\widehat{\sigma}_{\rho})]\quad \text{and} \quad
\hat{r}_i^{\text{RQ}} = \Phi^{-1}(\widehat{F}_W(w_i;\widehat{\sigma}_{\rho})), \quad  i = 1,\dots,n,
\end{equation*}

where $\widehat{F}_W$ is the fitted unit-log-normal or unit-log-Student-$t$ CDF, and $\widehat{\sigma}_{\rho}$ is the ML estimate of $\sigma_{\rho}$.  When the models are correctly specified, the GCS is asymptotically standard exponentially distributed, while the RQ is asymptotically standard normally distributed. The distributions of these residuals are important as they will be used to assess the goodness-of-fit of the considered model.

Both studies consider simulated data generated from each of the unit-log-normal and unit-log-Student-$t$ models according to the following stochastic relation:

\begin{equation*}
\stackrel{\eqref{inverse-A}}{=}
{1\over 1+\exp(-\sigma_\rho F_W^{-1}(U))},
\end{equation*}

where ${F}_W$ is the unit-log-normal or unit-log-Student-$t$ CDF and $U\sim U[0,1]$. The
Monte Carlo simulation experiments were performed using the  environment; see

§.§ Maximum likelihood estimation

The simulation scenario used the following settings: sample size $n \in {50,100,150,250,600}$ and true parameter $\sigma_{\rho} \in \{0.15,0.25,0.50,1.0,1.5\}$, with $500$ Monte Carlo replications for each sample size. To study the performance of the ML estimators, we  computed the bias and root mean square error (RMSE), defined by

\begin{eqnarray*}
\widehat{\textrm{Bias}}(\widehat{\theta}) = \frac{1}{M} \sum_{i = 1}^{M} \widehat{\theta}^{(i)} - \theta, \quad
\widehat{\mathrm{RMSE}}(\widehat{\theta}) = {\sqrt{\frac{1}{M} \sum_{i = 1}^{M} (\widehat{\theta}^{(i)} - \theta)^2}}, \\
\end{eqnarray*}

where $\theta$ and $\widehat{\theta}^{(i)}$ are the true parameter value and its respective $i$-th ML estimate, and $M$ is the number of Monte Carlo replications.

The ML estimation results are shown in Figure <ref> wherein the empirical bias and RMSE are both presented. It can be seen that the simulations produced the expected outcomes, namely, that the bias and RMSE both approach zero as $n$ increases.

Monte Carlo simulation results for the unit-log-normal model.

§.§ Empirical distribution of residuals

In this subsection, we evaluate the performance of GCS and RQ residuals. We present the empirical means of the following descriptive statistics: mean, median, standard deviation (Sd), coefficient of skewness and coefficient of kurtosis, whose values are expected to be 1, 0.69, 1, 2 and 6, respectively, for the GCS residuals, and 0, 0, 1, 0 and 0, respectively, for the RQ residuals. From Figures <ref> and <ref>, we note that the considered residuals conform well with their reference distributions, and we can therefore use the RQ and GCS residuals to verify well the fit of the proposed models.

Monte Carlo simulation results of the RQ residuals for the unit-log-normal model.

Monte Carlo simulation results of the GCS residuals for the unit-log-normal model.

§ ILLUSTRATION WITH INTERNET ACCESSS DATA

In this section, a real-world data set, corresponding to the share of the population using internet, is analyzed. Figure <ref> shows the share of the population that is accessing internet for 201 countries of the world in 2015. Here, the internet can be accessed through a computer, mobile phone, games machine, digital TV, etc.; see <https://ourworldindata.org/internet>. From Figure <ref>, we observe high rates of people online in richer countries and much lower rates in the developing world. For example, three-quarters (74.55%) of people in the US were online, while in India only 14.9% used the internet.

Share of the population using the internet, 2015.

In terms of stochastic representation, the share of the population using internet, represented by the random variable $W$, can be expressed as

\begin{equation}\label{stoch}
W=\frac{T_1}{T_1 + T_2}
\end{equation}

where $T_1 \in \mathbb{R}^+$ and $T_2 \in \mathbb{R}^+$ are two random variables following log-symmetric distributions representing the number of people with internet access and the number of people without internet access, respectively; in other words, the sum $T_1 + T_2$ represents the total population while the ratio in (<ref>) has support in the unit interval $(0,1)$ with PDF in (<ref>).

Table <ref> provides descriptive statistics for the share of the population using the internet, including the mean, median, standard deviation (SD), coefficient of variation (CV), skewness (CS), (excess) kurtosis (CK), and minimum and maximum values. From this table, we note that the mean is almost equal to the median. In addition, the CV (dispersion around the mean) is less than 100%, indicating a low dispersion of data around the mean. Finally, we observe that the data show skewness near zero and low degree of kurtosis.

Summary statistics for the internet data.

Mean       Median            SD          CV            CS        CK        minimum     maximum     size

0.480        0.497             0.289      60.276%       0.008     -1.316     0.011     0.983      201

Table <ref> presents the ML estimates and SEs for the unit-log-normal, unit-log-Student-$t$ and unit-log-Laplace model parameters. This table also presents the log-likelihood value, and Akaike (AIC) and Bayesian information (BIC) criteria. For comparative purpose, the results of the beta model [9] are provided as well. The results in Table <ref> reveal that the proposed unit-log-normal and unit-log-Student-$t$ models provide better adjustments than the beta model based on the values of log-likelihood, AIC and BIC. Furthermore, the unit-log-normal model provides the best fit to these data.

ML estimates (with SE in parentheses) and model selection measures for fit to the internet data.

Unit-log-normal      Unit-log-Student-$t$      Unit-log-Laplace          Beta

$\sigma_{\rho}\, (\text{or}\ \mu)$      1.6645       1.5390      1.9794        -0.7386

(0.0830)     (0.0860)     (0.2341)      (0.0397)

$\nu \, (\text{or}\ \phi)$                           10                      2.1860

Log-lik.                               3.2534       1.2758       -21.9354     1.1450

AIC                                    -4.5068      -0.5516      45.8709      1.7098

BIC                                    -1.2035      2.7517       49.1742      8.3164

Figure <ref> shows the QQ plots with simulated envelope of the GCS and RQ residuals for the unit-log-normal, unit-log-Student-$t$, unit-log-Laplace and beta models considered in Table <ref>. We observe that the unit-log-normal model provides better fit than the unit-log-Student-$t$, unit-log-Laplace and beta models. Figure <ref> displays the histogram and superimposed fitted PDFs, and fitted CDFs (empirical CDF in gray).

QQ plot and its envelope for the GCS (top) and RQ (bottom) residuals in the indicated model for the internet data.

Histogram of the internet data with fitted PDFs (left) and fitted CDFs (right).

§ CONCLUDING REMARKS

In the present paper, we have introduced a family of flexible models over the interval $(0, 1)$. By suitably defining the density generator, we can transform any distribution over the real line into a bounded distribution over the interval $(0, 1)$. In this way, we have transformed the normal, Student-$t$ and Laplace distributions into  unit-log-normal, unit-log-Student-$t$ and unit-log-Laplace distributions, respectively. These three cases have been studied at great length.  However, many other unit-log-symmetric distributions, such as the unit-log Kotz type, unit-log Pearson Type VII, unit-log hyperbolic, unit-log slash, unit-log-logistic and unit-log-power exponential distributions can all be studied similarly. These may be useful alternatives to “conventional” models like beta generalizations, Kumaraswamy and the Topp-Leone distributions.

This study was financed in part by the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES)
(Finance Code 001). Roberto Vila gratefully acknowledges financial support from FAP-DF, Brazil.

Disclosure statement
There are no conflicts of interest to disclose.

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Davison, A. C.,
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Fang, K. T., Kotz, S., Ng, K. W.,
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Chapman and Hall, London, 1990.

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Ferrari, S., Cribari-Neto, F.,
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The Laplace Distribution and Generalizations,
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# Higher-order ferromagnetic resonances in periodic arrays of synthetic-
antiferromagnet nanodiscs

V. Yu. Borynskyi  Institute of Magnetism of NAS of Ukraine and MES of Ukraine,
36-b Akad. Vernadsky blvd., Kyiv 03142, Ukraine  D. M. Polishchuk  Institute
of Magnetism of NAS of Ukraine and MES of Ukraine, 36-b Akad. Vernadsky blvd.,
Kyiv 03142, Ukraine  Nanostructure Physics, Royal Institute of Technology,
Stockholm 10691, Sweden  A. K. Melnyk  Institute for Sorption and Problems of
Endoecology of NAS of Ukraine, 13 Naumov str., Kyiv 03164, Ukraine  A. F.
Kravets  Institute of Magnetism of NAS of Ukraine and MES of Ukraine, 36-b
Akad. Vernadsky blvd., Kyiv 03142, Ukraine  Nanostructure Physics, Royal
Institute of Technology, Stockholm 10691, Sweden  A. I. Tovstolytkin
Institute of Magnetism of NAS of Ukraine and MES of Ukraine, 36-b Akad.
Vernadsky blvd., Kyiv 03142, Ukraine  V. Korenivski  Nanostructure Physics,
Royal Institute of Technology, Stockholm 10691, Sweden

###### Abstract

We investigate spin dynamics in nanodisc arrays of synthetic-antiferromagnets
(SAF) made of Py/NiCu/Py trilayers, where the NiCu spacer undergoes a Curie
transition at about 200 K. The observed ferromagnetic resonance spectra have
three distinct resonance modes at room temperature, which are fully recreated
in our micromagnetic simulations showing also how the intra-SAF asymmetry can
be used to create and control the higher-order resonances in the structure.
Below the Curie temperature of the spacer, the system effectively transitions
into a single-layer nanodisc array with only two resonance modes. Our results
show how multi-layering of nano-arrays can add tunable GHz functionality
relevant for such rapidly developing fields as magnetic meta-materials,
magnonic crystals, arrays of spin-torque oscillators and neuromorphic
junctions.

††preprint: AIP/123-QED

The last decade has seen large research efforts focused on collective spin
dynamics of magnonic crystals and logic circuits aimed for various high-speed
applications Chumak  _et al._ (2015); Liu  _et al._ (2013); Lenk  _et al._
(2011); Haldar and Adeyeye (2016). Controllable size of the individual
elements used in such metamaterials, down to the 10 nm range, enables rich
higher-order FMR dynamics such as standing spin waves and edge-mode
resonances. The typical geometry of a periodic array offers high flexibility
in engineering novel properties. At the same time, the increased complexity
requires systematic investigations of various effects and parameters involved
Saha  _et al._ (2012); Liu  _et al._ (2008); Schneider  _et al._ (2007); Verba
_et al._ (2020).

Relatively simple periodic arrays of single-layer ferromagnetic nanodiscs
exhibit anisotropic spin wave modes (commonly, with four-fold in-plane
anisotropy) affected by the inter-element dipolar interactions Mathieu  _et
al._ (1997); Kakazei  _et al._ (2006); Carlotti (2019). The properties become
more intriguing for arrays of multilayered elements such as SAF’s, which are
of interest in antiferromagnetic spintronics Duine  _et al._ (2018) as well as
magnonics Etesamirad  _et al._ (2021). Their collective spin dynamics is
dictated by the inter-element interactions, the intra-particle dipolar
coupling, and the interplay of the two Pauselli  _et al._ (2017); Kamimaki
_et al._ (2020); Chatterjee  _et al._ (2018).

Arrays of multilayered nanomagnets can exhibit distinct spin dynamic modes not
found in single-layer systems. The intra-particle dipolar interactions lead to
complex demagnetizing field distributions, altering the collective spin
dynamics in such systems, resulting in mode-frequency shifts Talapatra and
Adeyeye (2021) or even splitting of the resonant modes Carlotti  _et al._
(2015). For example, a systematic micromagnetic study performed by Pauselli et
al. Pauselli  _et al._ (2017) for a sub-40-nm magnetic tunnel junction read
head explained additional higher-order modes as originating from a strong
localization of spin excitations to the junction’s edges, where an excitation
in one of the junction’s magnetic layers drives a localized resonance in the
neighboring, dipole-coupled magnetic layer. Multi-layering is thus a capable
tool for varying spin wave modes and therefore spectral modification Haldar
and Adeyeye (2020); Begari and Haldar (2018), additional to element grouping
or sequencing in arrays of single-layer nanoparticles Carlotti  _et al._
(2015); Gartside  _et al._ (2021). Furthermore, particle shape imperfections
unavoidable on the experiment, such as edge deviations Nembach  _et al._
(2021) or a sidewall angle Maranville  _et al._ (2007), can have a significant
effect on the spin dynamics of the system. Strongly localized edge-mode
excitations are highly sensitive to this kind of distortions, which can lead
to lifting of their degeneracy for both single-layer Chia  _et al._ (2012) and
multilayered Zhang  _et al._ (2019) systems. While arrays of singe-layer
nanoparticles are well studied Hu  _et al._ (2015); Ding  _et al._ (2012);
Shaw  _et al._ (2009), a systematic investigation of arrays of multilayered
nano-elements is still pending.

In this work, we investigate higher-order FMR excitations in periodic arrays
of multilayered SAF nanodiscs, designed with an in-situ thermal control of the
interlayer coupling capable of making the nanodiscs effectively single-
layered. This in-situ control of the SAF versus single-layer character of the
array elements allows us to identify the SAF-unique properties from a direct
comparison of the FMR spectra measured above and below the Curie point of the
spacer. Using detailed micromagnetic simulations, we explain the occurrence,
positions, and anisotropy of the measured resonances and how they are affected
by strong localizations of the spin excitations and a complex interplay
between the inter- and intra-particle interactions, pronounced in SAF-based
nanostructures.

We focus our investigation on arrays of SAF nanodiscs Py(7 nm)/NiCu(10
nm)/Py(7 nm) with periodicity 250 nm and nominal diameter $d~{}=~{}150$ nm.
Here, Py and NiCu stand for Ni80Fe20 (Permalloy) and Ni60Cu40, respectively.
The spacer is paramagnetic at room temperature (RT) and mediates no direct
exchange between the outer Py layers, which interact through magnetostatic
coupling only, thereby forming a SAF element. At low temperature, the spacer
is ferromagnetic, which allows in-situ switching between the SAF and the
single-layer states in the system Kravets  _et al._ (2012, 2014). Multilayers
Py/NiCu/Py were grown by magnetron sputtering (Orion, AJA Int.) onto thermally
oxidized Si (001) wafers at room temperature. The multilayers were then used
to fabricate arrays of SAF nanodiscs using electron-beam lithography (Voyager,
Raith Inc.). The arrays were etched using Ar-plasma via a double-layer hard
mask, TaN(60 nm)/Al(20 nm). Here, thin Al was used as a lift-off layer to form
a mask (with the nominal diameter $d_{0}~{}=~{}150$ nm) for selective reactive
plasma etching with SF6 of the thick TaN layer underneath. Additional array
geometries were fabricated and tested for calibration and verification
purposes, in the process of optimizing the SAF arrays, with most interesting
properties presented herein.

Figure 1: (a) Top-view SEM image of select region in planar array of 150-nm
SAF nanodiscs (Py/NiCu/Py). Inset indicates particle periodicity and in-plane
orientation of external magnetic field. (b) 45$\degree$ SEM view of same
sample. (c) Typical M-H loops, measured using MOKE at 300 and 170 K; inset
illustrates vertical cross-section of single SAF nanodisc. TaN is used as
etching mask and for capping.

SEM images of the fabricated arrays in Fig. 1(a–b) show good uniformity in the
element size (the spread is less than 5 %) and good edge morphology. Angled
SEM in Fig. 1(b) shows a side-wall angle of 20–30$\degree$, formed in the
process of plasma etching, making the effective diameter of the top Py disc
about 15 % smaller than that of the bottom disc. The top dome-like cap,
visible as the central concentric circles in the top-view and contrasted-top
in the side-view SEM images, Fig. 1(a–b), is the remaining TaN mask that
protects the fabricated particles from oxidization.

Cavity-FMR measurements were carried out using an X-band ELEXSYS E500
spectrometer (Bruker Inc.) equipped with an automatic goniometer for precise
angle control. The measurements were performed at a constant operating
frequency of 9.36 GHz. The magnetic volume of the arrays of area $1.4\times
1.4$ mm2 was large enough for a high signal-to-noise ratio. Additionally,
magnetostatic properties of the arrays were measured using a custom-made
variable-temperature magneto-optical Kerr-effect (MOKE) setup.

Figure 1(c) shows two characteristic MOKE loops measured in-plane
($\varphi_{H}=0\degree$). Zero remnant magnetization at $H=0$ and relatively
high saturation field $H_{s}\approx 300$ Oe for RT loop indicate a pronounced
antiferromagnetic-type interlayer coupling of the top and bottom magnetic
moments of the individual SAF nanodiscs. The double-step shape and small
hysteresis can arise from the presence of magnetic anisotropy Polishchuk  _et
al._ (2018) as well as a non-ideal circular shape of the nanodiscs Koop  _et
al._ (2017); Holmgren  _et al._ (2017).

Figure 2: (a) FMR spectra measured at RT for 150-nm SAF array at 0, 45, and
90$\degree$-oriented external field. Lm, Le1 and Le2 mark resonance lines
attributed to main and two edge-modes, respectively. (b) Resonance fields of
individual modes versus in-plane angle. (c) FMR spectra calculated for
individual single-layer discs with $d_{0}=150$ nm and $d\approx 0.85d_{0}$,
corresponding to bottom and top layers of SAF nanodiscs. Insets show
excitation regions for two eigen-modes. (d) Calculated FMR spectrum for
individual SAF particle with $d_{0}$ and $d\approx 0.85d_{0}$. Insets display
excitation regions in both layers of SAF nanodisc. All spectra were calculated
with in-plane field at $\varphi_{H}=0\degree$ (black arrow in left inset to
c). (e) In-plane angular dependence of resonance fields simulated for SAF disc
array with $d_{0}=150$ nm, $d\approx 0.85d_{0}$, and array periodicity 200 nm.

Figure 2(a) shows the FMR spectra measured at RT for the array of 150-nm SAF
nanodiscs with different in-plane orientations of the external magnetic field,
$\varphi_{H}=0,45$, and 90$\degree$. Each curve shows three well-defined
resonance lines marked as Lm, Le1, and Le2. We associate the Lm line with the
quasi-uniform (hereafter main) FMR mode since its position coincides with that
of the resonance line for a continuous Py film. With respect to the position
of Lm, lines Le1 and Le2 are observed at higher resonance fields. A similar
higher-field mode, along with the main mode, was reported previously Carlotti
(2019) for arrays of single-layer nanodiscs and was attributed to the edge-
mode resonance. For our arrays of SAF nanodiscs, we observe two higher-field
modes, both of which we associate with higher-order edge-mode resonances in
the SAF nanostructure – the interpretation supported by our micromagnetic
simulations, detailed below. We note that any FMR modes related to potential
domains or vortices forming in the SAF particles are excluded from
consideration since the resonance fields of Lm, Le1, and Le2 ($H_{r}>1$ kOe)
are much higher than the SAF saturation field [$H_{s}\approx 300$ Oe; Fig.
1(c)].

The resonance field, $H_{r}$, of each resonance line show a pronounced angular
dependence, as seen in Fig. 2(b). All lines show bi-axial anisotropy with,
however, opposite hard and easy axes for Lm and Le1,e2. While Lm exhibits easy
axes (minimum $H_{r}$) at 45$\degree$ with respect to the main $x$ and $y$
axes of the array [Fig. 1(a)], the edge-modes, Le1 and Le2, have easy axes
along $x$ and $y$. Such anisotropic behavior was explained in Refs. Mathieu
_et al._ , 1997; Kakazei  _et al._ , 2015 by the inter-element dipole coupling
leading to a non-uniform effective field distribution within the individual
discs, even at rather high external fields.

In order to explain the origin and behavior of the three FMR modes observed,
we performed detailed micromagnetic simulations McMichael and Stiles (2005),
using MuMax3 Vansteenkiste  _et al._ (2014); Exl  _et al._ (2014). The layout
of each element as well as the base diameter ($d_{0}$) used were those
observed experimentally [Fig. 1(c)]. Figure 2(c) displays the calculated
spectra for two isolated single-layer Py discs (7 nm thick) with diameters
$d_{0}=150$ nm (blue line) and $d=0.85d_{0}$ (red dashed line). This
difference in size corresponds to the SEM-measured difference in the diameters
of the bottom and top layers of our SAF nanodiscs; cf. Fig. 1(b). Both
simulated spectra show two resonance lines corresponding to the main and edge-
resonance modes, illustrated by the insets to Fig. 2(c). Of note is that
despite line Lm is associated with the uniform mode of a continuous Py film,
the related excitation within a nanodisc is not fully spatially uniform, which
is why we call it main rather than uniform FMR mode. Importantly, neither
spectrum shows any trace of a third line observed on the experiment, nor a
related additional line can be obtained when the two spectra are superposed.

The simulated spectra are substantially different when the two discs of
$d_{0}=150$ nm and $d=0.85d_{0}$ form a SAF element, as shown in Fig. 2(d).
Three resonance lines are visible, with the positions and intensities
resembling the experimental data. The insets to Fig. 2(d) clarify why we
associate the high-field lines with edge-mode resonances. Whereas the Le1 mode
is excited mostly in the bottom layer, the Le2 mode is more intensive in the
top layer. Importantly, when the top and bottom layers are of equal diameter,
only two modes, Lm and Le1, are excited; the third mode appears only when the
top and bottom diameters become noticeably different (by 5–10 %).

Figure 2(e) shows the simulated angle-dependence of the resonance fields of
the Lm, Le1, and Le2 modes obtained for an array of SAF nanodiscs. The
simulations are in excellent agreement with the experimental resonance-field
versus angle data of Fig. 2(b), which further validates our interpretation of
the observed modes and their properties.

Figure 3(a) compares the SAF FMR spectra measured as the temperature is
lowered from room to below the Curie point of the NiCu spacer. Our SAF design
is rather unique as it effectively transforms the nano-elements from trilayers
to single-layers when the spacer transitions from para- to ferromagnetic state
at about 220 K Kravets  _et al._ (2012, 2014, 2015, 2016). The associated
transition in the FMR spectrum is very illuminating – the three-mode resonance
is reduced to effectively two modes, which is a known characteristic of
single-layer nano-arrays. Figure 3(b) shows the fits to the extracted high-
order resonances (after subtracting the main peak): the pronounced double-
resonance at room temperature with dominating Le1 (red) and essentially a
single broad peak at 170 K where the Le1 peak is <1% of its RT value. The
transition from two high-order peaks to one takes place precisely at the Curie
temperature of the spacer, deduced independently from our MOKE magnetization
data shown in Fig. 3(c). We thus confirm, in a direct experiment, that the
origin of the double spin-wave resonance is unique to SAF nano-arrays.
Specifically, the high-order (non-uniform) double resonance is a result of
hybridization of predominantly acoustic and optical (in-phase and out-phase)
oscillations in the two ferromagnetic layers comprising the SAF. These modes
collapse into one on direct exchange coupling the outer layers below the
$T_{C}$ of the spacer and, generally, can be tuned by changing the SAF
geometry and the spacer properties (via magnetic dilution and/or temperature).

Figure 3: (a) FMR spectra of SAF arrays measured at select temperatures. (b)
Fits of high-order resonances (Le1,Le2) at above and below spacer’s Curie
point. (c) Normalized remnant magnetization versus temperature obtained from
MOKE loops.

The above results demonstrate that the magnetization dynamics of periodic
arrays of SAF nanomagnets have key distinctions from the known behavior of
single-layered nano-arrays. Our variable field-angle and temperature FMR and
MOKE studies, supported by morphological characterization and micromagnetic
simulations, enable us to identify the origins of the distinct SAF properties
observed. The three FMR modes and their pronounced in-plane anisotropy are two
such distinct properties that, however, are governed by different factors. Our
micromagnetic simulations indicate that the splitting of the higher-order edge
mode into two is due to the dipolar coupling within the individual, asymmetric
SAF particles. Here, the inter-particle interaction within the array is a
secondary effect, which on the other hand dominates the anisotropic properties
of the array. We find that the high-order spin excitations are hybridized
acoustic (predominantly in-phase for Le1) and optical (predominantly out-of-
phase for Le2) oscillations in the two layers comprising the SAF.

The occurrence of similar edge modes in single-layer nanodiscs is usually
explained by a non-uniform internal field distribution, with its minima
located at the edges orthogonal to the static magnetization of the discs
Pauselli  _et al._ (2017). Following similar logic for our asymmetric SAFs, we
associate the two edge resonances with non-uniform field distributions that
are different in each of the two nanodiscs owing to their different internal
and external (inter-layer) demagnetization field profiles. The demagnetization
is stronger in the smaller-diameter top layer, which results in a higher
external field needed to excite the edge mode in this layer. In a symmetric
SAF, with symmetric demagnetization fields, only one edge mode is excited.

The four-fold anisotropy of the three FMR modes is due to the bi-axial
symmetry of our periodic arrays forming a square lattice. For our circular
150-nm SAF nanodisc arrays, this anisotropy is clearly caused by the inter-
particle dipolar interactions. The observed 45$\degree$-angle between the easy
axes of the main mode and the edge modes can be explained by the difference in
the spacing of the respective excitation nodes for single-layer nanodot arrays
Kakazei  _et al._ (2015). We are able to recreate this type of anisotropy in
our micromagnetic simulations of SAF nano-arrays (to be discussed elsewhere).

In summary, we have investigated experimentally and micromagnetically the spin
dynamics of arrays of three-layer SAF nanoparticles where the intra-SAF
coupling can be in-situ controlled by varying temperature. The results show
how the intra- and inter-particle dipolar interactions combine to produce
rather unique GHz properties of the system, such as additional spin-wave modes
that can be tuned by the SAF geometry and/or temperature, which are relevant
for the emerging field of nanostructured magnetic metamaterials and magnonic
devices.

###### Acknowledgements.

We thank Dr. Roman Verba for fruitful discussions. Support from the Swedish
Research Council (VR 2018-03526), the Olle Engkvist Foundation
(2020-207-0460), the Volkswagen Foundation (90418), and the National Academy
of Sciences of Ukraine (Projects 0119U100469, and 0120U100457) are gratefully
acknowledged.

## Data Availability

The data supporting the findings of this study are available from the
corresponding author upon request.

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