# Strong solutions and asymptotic behavior of bidomain equations with random
noise.

Oleksiy Kapustyan<EMAIL_ADDRESS>Department of Mathematics, Taras
Shevchenko National University of Kyiv, Ukraine  Oleksandr Misiats
<EMAIL_ADDRESS>Department of Mathematics, Virginia Commonwealth University,
Richmond, VA, 23284, USA  Oleksandr Stanzhytskyi<EMAIL_ADDRESS>Department
of Mathematics, Taras Shevchenko National University of Kyiv, Ukraine

(August 27, 2024)

###### Abstract

In this paper we study the conditions for the existence of strong solutions
(both local and global) for stochastic bidomain equations. To this end, we use
apriori energy estimates and Serrin-type theorems. We further address the
asymptotic behavior of the solutions, which includes the analysis of small
stochastic perturbations and large deviations. In a separate section we
specify the support of the invariant measure, whose existence was established
in [15].

##  1 Introduction.

In this paper we study the existence of strong solutions and the asymptotic
behavior of nonlocal reaction diffusion-type system of equations, called the
bidomain equation. The bidomain model was first proposed by Tung [27]. It is
now the generally accepted model of electrical behavior of cardiac tissue
[12]. The heart muscle consists of so-called excitable cells, which react to
electrical stimulus by depolarizing. In other words, the potential difference
across the cell membrane changes. This depolarization motivates us to consider
the tissue consisting of two separate domains: the intracellular and the
extracellular. The corresponding electric potentials are denoted with $u_{i}$
and $u_{e}$, and the difference between these potentials is the transmembrane
potential, denoted with $u:=u_{i}-u_{e}$. Since the current, which passes from
one domain to the other, must pass through the cell membrane, the assumptions
on the conductive properties of the cell membrane, combined with the
conservation of current principle, yield the system of equations. It consists
of a pair of partial differential equations of reaction-diffusion type,
coupled with an ordinary differential equation:

$\begin{cases}\frac{\partial u}{\partial t}+f(u,w)+A_{i}u_{i}=I_{i}\text{ in
}(0,+\infty)\times D,\\\ \frac{\partial u}{\partial
t}+f(u,w)-A_{e}u_{e}=-I_{e}\text{ in }(0,+\infty)\times D,\\\ \frac{\partial
w}{\partial t}+g(u,w)=0,\text{ in }(0,+\infty)\times D,\\\ \sigma_{i}\nabla
u_{i}\cdot n=\sigma_{e}\nabla u_{e}\cdot n=0,\text{ in
}(0,+\infty)\times\partial D,\\\
u(0):=u_{i}(0)-u_{e}(0)=u_{0},w(0)=w_{0}\text{ in }D.\end{cases}$ (1)

Here $D\subset\mathbb{R}^{3}$ is a smooth bounded set representing myocardium,
$A_{i,e}u:=-\nabla\cdot(\sigma_{i,e}\nabla u)$, $\sigma_{i,e}$ are uniformly
elliptic conductivity matrices, $f,g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$
are the functions representing transmembrane ionic currents, and $s_{i,e}$ are
the external current sources.

There is a number of models for the nonlinearities $f$ and $g$, which describe
the propagation of impulses. The most relevant are the FitzHugh - Nagumo model
[9]

$\begin{cases}f(u,w)=\eta[u(u-a)(u-1)+w],\\\ g(u,w)=bw-cu,\end{cases}$ (2)

the Aliev-Panfilov model [1]

$\begin{cases}f(u,w)=\eta[ku(u-a)(u-1)+wu],\\\ g(u,w)=ku(u-1-a)+w,\end{cases}$
(3)

and the Rogers-McCulloch model [25]

$\begin{cases}f(u,w)=\eta[bu(u-a)(u-1)+wu],\\\ g(u,w)=-(cu-dw).\end{cases}$
(4)

In these models the coefficients satisfy $0<a<1$ and
$\eta,b,c,d,k,\varepsilon>0$. Another important example of nonlinear
interaction is Allen-Cahn model

$\begin{cases}f(u)=\eta(u^{3}-u),\\\ g\equiv 0,\end{cases}$ (5)

in which the system (1) decouples. Following the procedure described, e.g. in
[3] and references therein, the system (1) may be written in the form

$\begin{cases}\frac{\partial u}{\partial t}+f(u,w)+{\mathbb{A}}u=I,\\\
\frac{\partial y}{\partial t}+g(u,w)=0,\\\ u(0)=u_{0},w(0)=w_{0},\end{cases}$
(6)

where ${\mathbb{A}}:=A_{i}(A_{i}+A_{e})^{-1}A_{e}$ is the bidomain operator
and $I:=I_{i}-A_{i}(A_{i}+A_{e})^{-1}(I_{i}+I_{e})$ is the modified source
term. The primary difficulty arising in the analysis of the system (6) is that
the operator ${\mathbb{A}}$ is nonlocal. In particulrar, it fails to satisfy
the maximum principle, and the comparison theorems do not apply. Nevertheless,
it was shown in [10] that this operator generates an analytic semigroup in
both $L^{p}$ and $L^{\infty}$. The paper [3] showed that the equation (6) is
locally well-posed in the strong sense and globally well-posed in the weak
sense for the nonlinearities $f$ and $g$ described above. The recent result
[14] established the existence of strong periodic solutions.

The equation (6) is an important model of cardiac defibrillation [16], [19].
In this model, $I$ is the extracellular stimulation current in the form
$I=\sum_{k=0}^{N}u_{k}(t)\chi_{\Omega_{k}}(x)$, where $\Omega_{k}(x)$ are the
characteristic functions of electrodes, and $u_{k}$ are the corresponding
pulses. During defibrillation various regions of heart tissue are in
different, usually random, phases of electrical activity (excited, refractory,
partially recovered etc), and the purpose of defibrillation is to give an
electric impulse that stimulates the entire heart and returns it to its normal
(e.g. stationary) state. Therefore, understanding long time behavior of
problems of type (6) is crucial in order to understand whether such recovery
can take place, and how quickly it happens.

In this article we consider the bidomain equations driven by white noise:

$\begin{cases}du=[-{\mathbb{A}}u-f(u,w)+I]dt+dW,\ t\geq 0,\\\
w_{t}=-g(u,w),\\\ u(0)=u_{0},w(0)=w_{0},\end{cases}$ (7)

where is an infinite-dimensional $Q$-Weiner process, defined below.

To the best of our knowledge, the results on stochastic bidomain equation of
type (7) are relatively recent, and include [13], [2] and [15]. In [2] the
authors established the existence of both martingale and weak solutions under
the monotonicity condition on nonlinearity, which holds, in particular, if we
pick FitzHugh-Nagumo model. In [15] we established the existence and
uniqueness of weak solutions without monotonicity condition. Its worth
mentioning that this result is applicable to Rogers-McCullogh and Aliev-
Panfilov models, which do not satisfy the monotonicity condition. The
existence of stationary solution and the corresponding invariant measures was
also obtained in [15]. Finally, in [13] the authors obtained the existence of
global strong solution for (7) with FitzHugh-Nagumo nonlinearity.

In the present paper we address the following questions:

  * •

In Section 3 we establish the existence of a local strong solution for a large
class of nonlinearities, containing not only FitzHugh-Nagumo and Allen-Cahn,
but also Rogers-McCulloch and Aliev-Panfilov nonlinearities;

  * •

Under slightly more restrictive assumptions on the noise, in Section 4 we
establish the existence of a global strong solution for the same class of
nonlinearities in dimension 2;

  * •

Finally, in Sections 5 and 6 we study the asymptotic behavior of the
solutions, such as small stochastic perturbations, large deviations principle,
and specify the support of the invariant measure.

##  2 Preliminaries.

Throughout the paper, $D\subset\mathbb{R}^{3}$ is a smooth bounded set,
$H=L^{2}(D)$,

$H_{0}:=\\{v\in H,\int_{D}vdx=0\\},$ (8)

and $V=H^{1}(D)=W^{1}_{2}(D)$. The pairing $(u,v)$ stands for the dot product
in $H$, and the pairing $\langle u,v\rangle$ stands for the dot product in
$V$.

We assume the uniform ellipticity assumption on the conductivities
$\sigma_{i}$ and $\sigma_{e}$, i.e. there are $\sigma_{2}>\sigma_{1}>0$ such
that for all $x\in D$ and $\xi\in\mathbb{R}^{3}$ we have

$\sigma_{1}|\xi|^{2}\leq\langle\sigma_{i,e}\xi,\xi\rangle\leq\sigma_{2}|\xi|^{2}$
(9)

It was shown in [14] and [3] that the bidomain operator ${\mathbb{A}}$ is well
defined on

$D({\mathbb{A}}):=\\{u\in H^{2}(D)\cap H_{0},\nabla u\cdot\nu=0\text{ on
}\partial D\\}.$ (10)

Furthermore, following [3], one can define the biliniear form $a(u,v)$ for all
$u,v\in V:=H^{1}(D)$, such that

$a(u,v)=({\mathbb{A}}u,v)$

if $u$ and $v$ are in $D({\mathbb{A}})$. This biliniar form implicitly defines
the bidomain operator in weak sense. Furthermore, under the assumption (9) it
is shown, e.g. in [3], that the bilinear form $a$ is symmetric, continuous and
coercive on $V$, i.e. there are constants $\alpha>0$ and $M>0$ such that

$\forall u\in V,\ \alpha\|u\|_{V}\leq a(u,u)+\alpha\|u\|_{H}^{2}$ (11)

and

$\forall u,v\in V,\ |a(u,v)|\leq M\|u\|_{V}\|v\|_{V}.$ (12)

Furthermore, there are
$0<\lambda_{0}\leq\lambda_{1}\leq...\leq\lambda_{i}\leq...$ in $\mathbb{R}$
and orthonormal (in $H$) basis of $\\{\psi_{i},i\geq 1\\}\in V$ s.t.

$\forall v\in V,a(\psi_{i},v)=\lambda_{i}(\psi_{i},v).$

It follows, e.g., from Theorem 13, [3], that $\psi_{i}\in D({\mathbb{A}})$ for
all $i\geq 1$, therefore ${\mathbb{A}}\psi_{i}=\lambda_{i}\psi_{i}$ in $H$. We
may now define the noise as

$W(t,x):=\sum_{i=1}^{\infty}\sqrt{\gamma_{i}}\psi_{i}(x)W_{i}(t),$

where $W_{i}$ are independent standard Wiener processes, and

$\gamma:=\sum_{i=1}^{\infty}\gamma_{i}<\infty.$ (13)

We next describe the class of the nonlinearities $f$ and $g$. Following [3],
we assume that the nonlinearities $f(u,w)$ and $g(u,w)$ are of the form

$f(u,w)=f_{1}(u)+f_{2}(u)w,\ g(u,w)=g_{1}(u)+g_{2}w.$ (14)

Here $g_{2}\in\mathbb{R}$, and $f_{1},f_{2}$ and $g_{1}$ are continuous real
functions, satisfying the following conditions:

  * •

[C1] there exist constants $c_{i}\geq 0\ (i=1...6)$ such that for any
$u\in\mathbb{R}$

$\displaystyle|f_{1}(u)|\leq c_{1}+c_{2}|u|^{3};$ (15)
$\displaystyle|f_{2}(u)|\leq c_{3}+c_{4}|u|;$ $\displaystyle|g_{1}(u)|\leq
c_{5}+c_{6}|u|^{2}.$

  * •

[C2] $f_{1}(u)$, $f_{2}(u)$ and $g_{1}(u)$ are locally Lipschitz in $u$;

  * •

[C3] there exist constants $a,b$ and $c\geq 0$ such that for any
$(u,w)\in\mathbb{R}^{2}$ we have

$uf(u,w)+wg(u,w)\geq au^{4}-b(u^{2}+w^{2})-c.$ (16)

Let $(\Omega,\mathcal{F},P)$ be a complete probability space, and
$\mathcal{F}_{t}$ be a right-continuous filtration such that $W(t,x)$ is
adapted to $\mathcal{F}_{t}$, and $W(t)-W(s)$ is independent of
$\mathcal{F}_{s}$ for all $s<t$.

######  Definition 1.

An $\mathcal{F}_{t}$-adapted random process $(u(t,\cdot),w(t,\cdot))\in
V\times H$ is called a weak solution of (7), if for a.e. $t>0$, for every
$v\in V$ and $y\in H$ we have

$(u(t),v)=(u(0),v)-\int_{0}^{t}[a(u(\tau),v)+(f(u(\tau),w(\tau)),v)-(I(\tau),v)]d\tau+(v,W(t))$
(17)

and

$(w(t),y)=(w(0),y)-\int_{0}^{t}(g(u(\tau),w(\tau)),y)\,d\tau.$ (18)

######  Definition 2.

An $\mathcal{F}_{t}$-adapted random process $(u(t,\cdot),w(t,\cdot))\in
D({\mathbb{A}})\times H$ is called a strong solution of (7), if for a.e. $t>0$

$u(t)=u(0)-\int_{0}^{t}[{\mathbb{A}}u(\tau)+f(u(\tau),w(\tau))-I(\tau)]d\tau+W(t),$
(19)

and

$w(t)=w(0)-\int_{0}^{t}g(u(\tau),w(\tau))\,d\tau.$ (20)

Let $S(t)$ be the semigroup generated by the operator $-{\mathbb{A}}$, i.e.

$S(t)u_{0}(x):=u(t,x),$

where $u(t,x)$ solves

$\begin{cases}u_{t}=-{\mathbb{A}}u,\\\ u(0,x)=u_{0}(x).\end{cases}$

It follows from [10] that $S(t)$ is a analytic semigroup in $L^{p}(D)$ for
$p>1$. Note that by definition of $\lambda_{k}$ and $\psi_{k}$ we have

$S(t)\psi_{k}=e^{-\lambda_{k}t}\psi_{k}\text{ for }k\geq 1.$ (21)

The semigroup $S(t)$ enables us to define the stochastic convolution

$W_{A}(t,x):=\int_{0}^{t}S(t-\tau)dW(\tau)=\sum_{i=1}^{\infty}\gamma_{i}\int_{0}^{t}S(t-\tau)\psi_{i}(x)dW_{i}(\tau).$

Due to Sobolev embedding, $V=H^{1}(D)\subset L^{p}(D)$ in $\mathbb{R}^{3}$ if
$2\leq p\leq 6$ [4].

We conclude this section with the following Lemma regarding the stochastic
convolution, which was established in [15]:

######  Lemma 1.

Assume

$\sum_{k=1}^{\infty}\gamma_{k}\lambda_{k}^{1/2}<\infty.$ (22)

Then

  * [i]

For $T\geq 0$ and almost all $\omega\in\Omega$ we have

$\sup_{t\in[0,T]}\|W_{A}\|_{L^{p}(D)}^{p}\leq C(T,\omega)<\infty.$ (23)

  * [ii]

For all $t\in[0,T]$ and almost all $\omega\in\Omega$ we have
$W_{A}(t,\cdot)\in D({\mathbb{A}})$.

##  3 Existence of local strong solutions.

In this section, we obtain the existence of a local strong solution for (7) in
the most general case, when the nonlinearities $f$ and $g$ satisfy the
conditions [C1]-[C3]. As follows from Lemma 1, $W_{A}(t,x)\in D({\mathbb{A}})$
provided (22) holds. Thus by Lemma 5.13 [6] we have
$W_{A}(t)=\int_{0}^{t}{\mathbb{A}}W_{A}(s)ds+W(t)$. We proceed with the change
of variables $U=u-W_{A}(t)$. This way, the pair $(U,y)$ solves

$\begin{cases}dU=(-{\mathbb{A}}U-f(U+W_{A}(t),y)+I(t))dt;\\\
dw=-g(U+W_{A}(t),w)dt.\end{cases}$ (24)

If $(U,w)$ is a strong solution of (24), then $U\in D({\mathbb{A}})$, which,
in turn, implies that $u=U+W_{A}\in D({\mathbb{A}})$, and hence $(u,w)$ is a
strong solution of (7). Since $S_{A}(t)$ is an analytic semigroup, the Theorem
5.14 [7] for $\alpha\in(0,1/2)$, the process $W_{A}(\cdot)$ has
$\alpha$-Holder continuous trajectories in $H=L^{2}(D)$. Following [3],
introduce

$Z:=H\times B,\ B=L^{\infty}(D)$

and $\mathcal{A}:D(\mathcal{A})\subset Z\to Z$,
$\mathcal{A}z=(-{\mathbb{A}}u,0)$ for $z=(u,w)\in Z$. Here
$D(\mathcal{A})=D({\mathbb{A}})\times B$, where

$D({\mathbb{A}})=\\{u\in
H:\sum_{i=1}^{\infty}\lambda_{i}^{2}(u,\Psi_{i})^{2}<\infty\\}.$

Next, define

$Z^{\alpha}=\\{u\in H,\sum_{i\geq
0}\lambda_{i}^{2\alpha}(u,\Psi_{i})^{2}<\infty\\}\times B$

and

$\mathcal{A}^{\alpha}(u,y):=\left(\sum_{i\geq
0}\lambda_{i}^{\alpha}(u,\Psi_{i})\Psi_{i},0\right).$

It follows from [12], that for $0\leq\alpha\leq\beta$ we have an embedding
$Z^{\beta}\subset Z^{\alpha}$, and if $\beta=1$, then $D(\mathcal{A})\subset
Z^{\beta}\subset Z^{\alpha}$. Hence

$W_{A}\in D((-{\mathbb{A}})^{\alpha}).$ (25)

If $\frac{d}{4}<\alpha<1$, then $Z^{\alpha}\subset B\times B$. Then by Lemma
16 [3] we have

$\mathcal{F}(z):=(u,w)\to(f(u,w),g(u,w))$

is a locally Lipschits map from $Z^{\alpha}$ to $Z$, provided $f$ and $g$ are
locally Lipschits functions.
The main result of this section is the following Theorem:

######  Theorem 1.

Assume $\\{\gamma_{k},k\geq 1\\}$ satisfy (22). Then for any initial
conditions $(u_{0},w_{0})\in Z^{\alpha}$ there exists $T=T(\omega)>0$ such
that there exists $(u(t,x,\omega),w(t,x,\omega)),$ which is a strong solution
of (7) on the interval $[0,T(\omega))$ in the sense of Definition 2.

###### Proof.

Using the local Lipschitz condition, we get

$\|f(U+W_{A}(t),w)-f(U_{1}+W_{A}(t_{1}),w_{1})\|_{H}\\\ \leq
L(\|U-U_{1}\|_{Z^{\alpha}}+\|w-w_{1}\|_{Z^{\alpha}}+L_{1}\|W_{A}(t)-W_{A}(t_{1})\|_{D(-{\mathbb{A}})}),$
(26)

where $L_{1}$ is the constant in the embedding $D(\mathcal{A})\subset Z$. But

$\|W_{A}(t)-W_{A}(t_{1})\|_{D(-{\mathbb{A}})}=\|W_{A}(t)-W_{A}(t_{1})\|_{H}+\|{\mathbb{A}}(W_{A}(t)-W_{A}(t_{1}))\|_{H}.$

Thus it follows from Theorem 5.14 [6] that $W_{A}(t)$ is $\alpha$ \- Holder,
with $\alpha\in(0,1/2)$, i.e.

$\|W_{A}(t)-W_{A}(t_{1})\|_{H}\leq C(\omega)|t-t_{1}|^{\alpha}$

with probability 1. Furthermore, since

${\mathbb{A}}W_{A}=\sum_{k=1}^{\infty}\gamma_{k}\int_{0}^{t}(-\lambda_{k})e^{-\lambda_{k}(t-s)}\psi_{k}(x)dW_{k}(s),$

we have

$\displaystyle\mathbb{E}\|{\mathbb{A}}(W_{A}(t)-W_{A}(t_{1}))\|_{H}^{2}=$
$\displaystyle\mathbb{E}\left\|\sum_{k=1}^{\infty}\gamma_{k}(-\lambda_{k})\left(\int_{0}^{t}(-\lambda_{k})e^{-\lambda_{k}(t-s)}\psi_{k}(x)dW_{k}(s)-\int_{0}^{t_{1}}(-\lambda_{k})e^{-\lambda_{k}(t_{1}-s)}\psi_{k}(x)dW_{k}(s)\right)\right\|_{H}^{2}\leq$
$\displaystyle\mathbb{E}\left\|\sum_{k=1}^{\infty}\int_{0}^{t_{1}}(-\lambda_{k})\gamma_{k}(e^{-\lambda_{k}(t-s)}-e^{-\lambda_{k}(t_{1}-s)})\psi_{k}(x)dW_{k}(s)+\int_{t_{1}}^{t}\gamma_{k}(-\lambda_{k})\psi_{k}e^{-\lambda_{k}(t-s)}dW_{k}(s)\right\|_{H}^{2}\leq$
$\displaystyle\leq
2\left(\mathbb{E}\sum_{k=1}^{\infty}\gamma_{k}\lambda_{k}\int_{0}^{t_{1}}(e^{-\lambda_{k}(t-s)}-e^{\lambda_{k}(t_{1}-s)})dW_{k}(s)\right)^{2}$
$\displaystyle+2\mathbb{E}\left(\sum_{k=1}^{\infty}\gamma_{k}\lambda_{k}\int_{t_{1}}^{t}e^{-\lambda_{k}(t-s)}dW_{k}\right)^{2}:=2J_{1}+2J_{2}.$

Using the properties of the stochastic integrals ([6], pp.132-133), we have
the following estimate:

$J_{1}\leq\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}\int_{0}^{t_{1}}\left(e^{-\lambda_{k}(t-s)}-e^{-\lambda_{k}(t_{1}-s)}\right)^{2}ds=\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}\int_{0}^{t_{1}}\left(\int_{t_{1}-s}^{t-s}-\lambda_{k}e^{-\lambda_{k}\tau}d\tau\right)^{2}$
$=\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}\int_{0}^{t_{1}}\left(\int_{t_{1}-s}^{t-s}-\lambda_{k}\tau
e^{-\lambda_{k}\tau}\frac{1}{\tau}d\tau\right)^{2}ds\leq
C_{0}\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}\int_{0}^{t_{1}}\left(\int_{t_{1}-s}^{t-s}\frac{d\tau}{\tau}\right)^{2}ds.$
(27)

Now, for fixed $\gamma\in(0,1/2)$, we estimate the last term in (27) as

$\int_{t_{1}-s}^{t-s}\frac{d\tau}{\tau}=\int_{t_{1}-s}^{t-s}\frac{\tau^{\gamma-1}d\tau}{\tau^{\gamma}}\leq\int_{t_{1}-s}^{t-s}\frac{\tau^{\gamma-1}d\tau}{(t_{1}-s)^{\gamma}}=\frac{(t-s)^{\gamma}-(t_{1}-s)^{\gamma}}{\gamma(t_{1}-s)^{\gamma}}\leq
C\frac{(t-t_{1})^{\gamma}}{(t_{1}-s)^{\gamma}}.$ (28)

Combining (27) and (28), we get

$J_{1}\leq C\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}(t-t_{1})^{2}.$

Next, estimating $J_{2}$ as

$J_{2}\leq\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}\int_{t_{1}}^{t}e^{-2\lambda_{k}(t-s)}ds\leq
C(t-t_{1}),$

altogether we have

$\mathbb{E}\|{\mathbb{A}}W_{A}(t)-{\mathbb{A}}W_{A}(t_{1})\|^{2}_{H}\leq
C(t-t_{1})^{2\gamma}.$

By Proposition 3.15 [6],

$\|{\mathbb{A}}W_{A}(t)-{\mathbb{A}}W_{A}(t_{1})\|\leq
C(\omega)(t-t_{1})^{\alpha},\,\alpha\in\left(0,\frac{1}{4\gamma}\right).$

Hence, by Theorem 20 [3], for any $(u_{0},w_{0})\in Z^{\alpha}$ there is
$T(\omega)$ such that the system (7) has a strong solution. ∎

##  4 Existence of global strong solution

We will next use the regularity theory for critical spaces (Serrin-type
results) for parabolic semilinear equations. We will need the following
notation:

  * •

For $\beta\in[0,1]$ the interpolation space
$X_{\beta}=(X_{0},X_{1})_{\beta}:=D({\mathbb{A}}^{\beta})\times H$, where
$X_{0}=H\times H$ and $X_{1}=D({\mathbb{A}})\times H$.

  * •

Time-weighed spaces: if $Y$ is a Banach space, $p>1$, $a\geq 0$ and
$1\geq\mu>1/p$, then

$u\in L^{p,\mu}((0,a),Y)\text{ if }t^{1-\mu}u\in L^{p}((0,a),Y)$

and

$u\in W^{1}_{p,\mu}((0,a),Y)\text{ if }t^{1-\mu}u\in W^{1,p}((0,a),Y).$

  * •

For $p\geq 2$, define the space $X_{\mu}:=W^{2/p}_{2}(D)\times H$, and
$E_{\mu}(a):=L^{\infty}((0,a),H)\cap L^{2}((0,a),V)\times L^{p}((0,a),H).$ The
Sobolev inequality (see, e.g. [4]) yields $E_{\mu}(a)\subset
L^{p}([0,a],X_{\mu})$ if $D\subset\mathbb{R}^{2}$.

Let $A$ be a sectorial operator $A:X_{1}\to X_{0}$, and $F:X_{\beta}\to
X_{0}$, where $\beta\in(0,1)$ is chosen such that $X_{\beta}\subset V\times
H.$ Consider the following evolution equation:

$\begin{cases}v_{t}+Av=F(v+\varphi(t,x)),\\\ v(0)=v_{0}\in
X_{\beta}.\end{cases}$ (29)

Assume

  * [i]

For every $a\geq 0$, $\varphi\in L^{2p,\mu}([0,a),X_{\beta})$.

  * [ii]

There exists $C>0$ such that

$\|F(\cdot)\|_{X_{0}}\leq C(1+\|\cdot\|_{X_{\beta}}^{3}).$ (30)

Then the following Theorem holds:

######  Theorem 2.

Assume the conditions [i] and [ii] hold, and the solution of (29) is defined
on the maximal interval of existence $[0,T)$. Then $u\in L^{p}((0,a),X_{\mu})$
for each $0\leq a<T$, and if $T<\infty$, then $u\notin L^{p}([0,T),X_{\mu})$.

###### Proof.

The proof follows the idea from [24], Corollary 2.3. For

$\mu-\frac{1}{p}>2\beta-1,$ (31)

let

$\alpha:=\frac{\beta-(\mu-1/p)}{1-(\mu-1/p)}.$

Assume $T<\infty$. Using the condition [ii], for any $a<T$ we have

$\|F(v+\varphi)\|_{L^{p,\mu}((0,a),X_{\beta})}\leq
C_{1}(1+\|v+\varphi\|^{3}_{L^{2p,\tau}((0,a),X_{\beta})})\leq
4C_{1}\|v\|^{3}_{L^{2p,\tau}((0,a),X_{\beta})}+C_{2},$ (32)

where

$C_{2}:=C_{1}+4C_{1}\|\varphi\|^{3}_{L^{2p,\tau}((0,T),X_{\beta})}<\infty.$

Next, by interpolation inequality we have

$\|v\|^{3}_{L^{2p,\tau}((0,a),X_{\beta})}\leq
C_{3}\|v\|_{L^{p}((0,a),X_{\mu})}^{3(1-\alpha)}\|v\|_{E_{\mu}(0,a)}^{3\alpha}.$
(33)

Finally, let $M$ be the constant of maximal regularity for the interval
$[0,T)$. Combining (32) and (33), we have

$\|v\|_{E_{\mu}(0,a)}\leq
M(\|v(0)\|_{X_{\mu}}+C_{2}+C_{4}\|v\|_{L^{p}((0,a),X_{\mu})}^{3(1-\alpha)}\|v\|_{E_{\mu}(0,a)}^{3\alpha}).$

For the subcritical choice of parameters $\mu$, $\beta$ and $p$, given by (31)
such that $3\alpha\leq 1$, we have $\|v\|_{E_{\mu}(0,a)}$ is bounded uniformly
in $a$ for all $a<T$. This means that the solution can be continued beyond
$T$, which is a contradiction with the fact that $[0,T]$ is the interval of
maximal existence.

∎

The main result of this section is the following Theorem:

######  Theorem 3.

Assume

$\sum_{k=1}^{\infty}\gamma_{k}^{2}\lambda_{k}^{2}<\infty,$

and the equation (7) is considered in dimension 2. Then for any initial
conditions $(u_{0},w_{0})\in X_{\beta}$ there exists a strong solution
$(u(t,x,\omega),w(t,x,\omega))$ of (7) in the sense of Definition 2, which is
defined for all $t\geq 0$.

###### Proof.

Let $t\in[0,T)$ be the maximal interval of existence of
$(u(t,x,\omega),w(t,x,\omega))$. We argue by contradiction, and assume
$T<\infty$. Using generalized Ito’s formula [18], we have the following energy
bounds:

$\displaystyle\begin{cases}\|u(t)\|^{2}=\|u_{0}\|^{2}+&2\int_{0}^{t}\langle-{\mathbb{A}}u,u\rangle
d\tau-2\int_{0}^{t}(f,u)d\tau+2\int_{0}^{t}(I,u)d\tau\\\ &+\gamma
t+2\int_{0}^{t}(u(\tau),dW(\tau)),\\\
\|w(t)\|^{2}=\|w_{0}\|^{2}-&2\int_{0}^{t}(g,w)d\tau.\end{cases}$

It follows from (11) that

$\langle-{\mathbb{A}}u,u\rangle\leq-\alpha\|u\|_{V}^{2}+\alpha\|u\|_{H}^{2}.$
(34)

Combining (34) and the condition (16), we get

$\|u(t)\|^{2}+\|w(t)\|^{2}\leq\|u_{0}\|^{2}_{H}+\|w_{0}\|_{H}^{2}+\\\
+2\int_{0}^{t}\left[-\alpha\|u(s)\|_{V}^{2}+\alpha\|u(s)\|_{H}^{2}-2a\int_{D}u^{4}(x,s)dx+b(\|u(s)\|_{H}^{2}+\|w(s)\|_{H}^{2})+c|D|\right]ds\\\
+2\int_{t_{0}}^{t}(I,u)d\tau+\gamma t+2\int_{0}^{t}(u(\tau),dW(\tau)).$ (35)

Using Lemma 7.2 [6], we have

$\left(\mathbb{E}\sup_{s\in[0,t]}\left|\int_{0}^{s}(u(\sigma),dW(\sigma))\right|\right)^{2}\leq\mathbb{E}\left(\sup_{s\in[0,t]}\left|\int_{0}^{s}(u(\sigma),dW(\sigma))\right|\right)^{2}=\mathbb{E}\sup_{s\in[0,t]}\left|\int_{0}^{s}(u(\sigma),dW(\sigma))\right|^{2}\leq$
$\leq C\int_{0}^{t}\mathbb{E}\|u(s)\|^{2}_{H}ds\leq
C\int_{0}^{t}\mathbb{E}\sup_{\sigma\in[0,s]}(\|u(\sigma)\|^{2}_{H}+\|w(\sigma)\|^{2}_{H})ds.$

Define

$\psi(s):=\mathbb{E}\sup_{\sigma\in[0,s]}(\|u(\sigma)\|_{H}^{2}+\|w(\sigma)\|_{H}^{2}),$

we have

$\mathbb{E}\sup_{s\in[0,t]}\left|\int_{0}^{s}u(\sigma)dW(\sigma)\right|\leq
C\sqrt{\int_{0}^{t}\varphi(s)ds}\leq c_{1}+c_{2}\int_{0}^{t}\varphi(s)ds.$

It follows from (35) that for $t\in[0,T]$

$\psi(t)+2\alpha\mathbb{E}\int_{0}^{t}\|u\|_{V}^{2}ds+2a\mathbb{E}\int_{0}^{t}\int_{D}u^{4}dxds\leq
c_{1}+c_{2}\int_{0}^{t}\psi(s)ds.$

Hence, by Gronwalls inequality,

$\psi(t)\leq c_{1}e^{c_{2}t},t\in[0,T],$

as well

$\mathbb{E}\int_{0}^{t}\|u\|_{V}^{2}ds+\mathbb{E}\int_{0}^{t}\int_{D}u^{4}dxds<\infty.$
(36)

Consequently, for almost all $\omega\in\Omega$,

$u\in L^{\infty}((0,T),H)\cap L^{2}((0,T),V)$ (37)

and

$w\in L^{\infty}((0,T),H).$

Introduce $z:=u-W$. Then $z$ satisfies

$\begin{cases}dz=[-{\mathbb{A}}z-f(z+W,w)+I]dt,\ t\geq 0,\\\
dw=-g(z+W,w)dt.\end{cases}$ (38)

Since

$W(t,x)=\sum_{k=1}^{\infty}\gamma_{k}\Psi_{k}(x)W_{k}(t),$

we have

${\mathbb{A}}W(t,x)=\sum_{k=1}^{\infty}\gamma_{k}{\mathbb{A}}\Psi_{k}(x)W_{k}(t),$

hence

$W\in
D({\mathbb{A}})\iff\sum_{k=1}^{\infty}(\gamma_{k}\lambda_{k})^{2}<\infty.$

Thus $W$ belongs to $V$, and for a.e. $\omega\in\Omega$ we have

$\|W\|_{L^{2}((0,T),V)}<\infty.$

Furthermore, by Theorem 2.5 [5], we have

$\mathbb{E}\sup_{0\leq t\leq T}\|W(t)\|^{2}_{H}\leq
4\mathbb{E}\|W(T)\|_{H}^{2}<\infty,$

thus for a.e. $\omega\in\Omega$ we have

$W\in L^{\infty}((0,T),H)\cap L^{2}((0,T),V).$ (39)

Combining (37) and (39) we get that

$z=u-W\in L^{\infty}((0,T),H)\cap L^{2}((0,T),V).$

We claim that

$w\in L^{p}([0,T],H).$ (40)

To this end, we represent the solution of

$w_{t}=-g_{2}w-g_{1}(u)$

as

$w=\frac{1}{e^{g_{2}t}}\left(-\int_{0}^{t}e^{g_{2}s}g_{1}(u(s))ds+w_{0}\right).$

Hence,

$\displaystyle\int_{0}^{t}\|w(s)\|_{H}^{p}ds\leq
C_{1}+C_{2}\int_{0}^{t}\left\|\int_{0}^{s}e^{g_{2}\tau}g_{1}(u(\tau))d\tau\right\|_{H}^{p}ds\leq$
(41) $\displaystyle\leq
C_{1}+C_{2}\int_{0}^{t}\left(\int_{0}^{t}\|g_{1}(u(\tau))\|_{H}d\tau\right)^{p}ds=C_{1}+C_{2}t\left(\int_{0}^{t}\|g_{1}(u(\tau))\|_{H}d\tau\right)^{p}=$
$\displaystyle=C_{1}+C_{2}t\left(\int_{0}^{t}\left(\int_{D}g_{1}^{2}(u(\tau))dx\right)^{1/2}d\tau\right)^{p}\leq
C_{1}+C_{2}t\left(\int_{0}^{t}\left(\int_{D}g_{1}^{2}(u(\tau))dx\right)^{1/2}d\tau\right)^{p}\leq$
$\displaystyle\leq
C_{1}+C_{2}t\left(\int_{0}^{t}\left(\int_{D}u^{4}(\tau))dx\right)^{1/2}d\tau\right)^{p}<\infty$

a.s., where we used (36). We are now in position to use the embedding for
$d=2$:

$L^{\infty}((0,T),H)\cap L^{2}((0,T),V)\subset L^{p}((0,T),W^{2/p}_{2}(D)).$

This way for $v=(u,w)$ we have

$v\in L^{p}((0,T),W_{2}^{2/p}(D)\times L^{q}(D))=L^{p}((0,T),X_{\mu})$

It remains to verify that $\varphi(t,x):=[-W(t,x),0]$ satisfies the condition
[i]. Note that by Theorem 2.3, ch 6 [5], we have

$\mathbb{E}\sup_{0\leq\tau\leq
T}\|W(\tau,x)\|_{X_{\beta}}\leq\left(\frac{2p}{2p-1}\right)^{2p}\|W(T,x)\|_{X_{\beta}}<\infty,$

since $W(t,x)\in D({\mathbb{A}})$. Therefore for a.e. $\omega$

$W(t,x)\in L^{\infty}((0,T),X_{\beta})\subset L^{2p,\mu}((0,T),X_{\beta}),$

thus [i] holds. Hence, by Theorem 2, $T$ must be equal to $+\infty$, leading
to contradiction. Therefore, we have global existence of the solution. ∎

##  5 Support of the invariant measure

The long time behavior of solutions of stochastic evolution equations in some
Hilbert space $H$ is a question of a separate interest. It is often addressed
in terms of establishing existence and, in certain cases, uniqueness of
invariant measures, which, in turn, is the crucial item in establishing the
ergodic behavior of the underlying physical systems. For the reader’s
convenience, we briefly remind the concept of invariant measure. Suppose
$v_{0}\in H$, and $B$ is a Borel subset of $H$. For $t\geq 0$ define the
probability transition semigroup

$P_{t}(v_{0},B):=\mathbb{P}(v(t,v_{0})\in B).$

Furthemore, for any Borel measurable bounded function $\varphi\in M_{b}(H)$,
define the Markov semigroup

$P_{t}\varphi(v_{0})=\mathbb{E}\varphi(v(t,v_{0})=\int_{H}\varphi(v)P_{t}(v_{0},dv).$

######  Definition 3.

Let the set $Pr(H)$ be the set of probability measures on $H$. An element
$\mu\in Pr(H)$ is called an invariant measure for the Markov semigroup $P_{t}$
if

$\int_{H}\varphi(v_{0})d\mu(v_{0})=\int_{H}P_{t}\varphi(v_{0})d\mu(v_{0}).$

The works [6, 7] provide conditions for the existence and uniqueness of
invariant measures in for general evolution equation of reaction-diffusion
type. Their approach is based on the results of Krylov and Bogoliubov [17] on
the tightness of a family of measures. The key ingredients of this approach
include the Feller property and stochastic continuity of the Markov semigroup
$P_{t}$, and the existence of least one solution which is globally bounded in
certain probability sense [20, 22, 21]. The existence of invariant measures
for bidomain equation (7) using the aforementioned procedure was established
in [15]. In particular, we showed that the equation (7) has a stationary
solution, which, in turn, implies the existence of invariant measure, defined
on the functional space $(u,w)\in H\times H$. The sufficient conditions for
the uniqueness of this invariant measure were also derived in [15].

A different approach to establishing the existence of invariant measures and
asymptotic behaviour of stochastic evolution equations is based on the
coupling method (see e.g. [23], [8] and references therein). The study of the
asymptotic behavior based on the exponential dichotomy of the differential
operator was carried on in [26].

The main goal of this section is to show, that the invariant measure,
established in [15], Theorem 5.1, is supported on the functional class $(u,w)$
with $u$ having regularity of at least $V$. Recall that for $v=(u,w)$ we
denote

$\|v\|_{\tilde{V}}^{2}:=\|u\|_{V}^{2}+\|w\|^{2}_{H}$

and

$\|v\|_{\tilde{H}}^{2}:=\|u\|_{H}^{2}+\|w\|^{2}_{H}.$

The main result of this section is the following theorem:

######  Theorem 4.

Assume the conditions of Theorem 5.1 [15] holds. Let $\mu\in Pr({\tilde{H}})$
be the invariant measure for (7). Then

$\int_{\tilde{H}}\|v\|_{\tilde{V}}^{2}d\mu(v)<\infty.$

In other words, any invariant measure is supported in $\tilde{V}=V\times H$.

###### Proof.

The idea of the proof was inspired by the work [11], in which the authors
derived a similar result for the invariant measure emerging in stochastic
primitive equations. As it was shown in [15] (see proof of Theorem 5.1), if
$v=(u,w)$ is a solution of (7), then

$\int_{0}^{T}\|v\|_{\tilde{V}}^{2}dt=\int_{0}^{T}(\|u\|_{V}^{2}+\|w\|^{2}_{H})dt<K_{1}\|v_{0}\|_{\tilde{H}}^{2}+K_{2}T$

for some $K_{1}$ and $K_{2}$ independent of $T$. Denote
$f_{R}(v):=\|v\|_{\tilde{V}}^{2}\land R$. By definition of invariant measure

$\int_{\tilde{H}}f(v_{0})d\mu(v_{0})=\int_{\tilde{H}}\int_{\tilde{H}}\frac{1}{T}\int_{0}^{T}P_{t}(v_{0},dv)f(v)dtd\mu(v_{0})$
(42)

for any $T>0$. Now, for any $\rho\geq 1$ and any $v_{0}\in
B_{\tilde{H}}(\rho)$, the ball of radius $\rho$ about the origin in
${\tilde{H}}$, we have

$\left|\frac{1}{T}\int_{0}^{T}\int_{\tilde{H}}P_{t}(v_{0},dv)f_{R}(v)dtd\mu(v)dt\right|=\left|\frac{1}{T}\int_{0}^{T}\mathbb{E}f_{R}(v(t,v_{0}))dt\right|\leq
K_{2}+\frac{K_{1}\|v_{0}\|_{\tilde{H}}^{2}}{T}\leq
K_{3}(1+\frac{\rho^{2}}{T}).$ (43)

Combining (42) and (43) we have

$\int_{\tilde{H}}f_{R}(v_{0})d\mu(v_{0})\leq\int_{B_{\tilde{H}}(\rho)}\frac{1}{T}\left|\int_{0}^{T}\int_{\tilde{H}}P_{t}(v_{0},dv)f_{R}(v)d\mu(v)dt\right|d\mu(v_{0})+\\\
\int_{{\tilde{H}}\setminus
B_{\tilde{H}}(\rho)}\frac{1}{T}\left|\int_{0}^{T}\int_{\tilde{H}}P_{t}(v_{0},dv)f_{R}(v)d\mu(v)dt\right|d\mu(v_{0})\leq
K_{3}(1+\frac{\rho^{2}}{T})+R\mu({\tilde{H}}\setminus B_{\tilde{H}}(\rho)).$
(44)

First, for sufficiently large $\rho$, depending on $R$, we have

$R\mu({\tilde{H}}\setminus B_{\tilde{H}}(\rho))\leq 1.$

Then for such $\rho$ we choose $T$ large enough so that

$\frac{\rho^{2}}{T}\leq 1.$

Altogether, (44) yields an estimate, which is independent of $R$:

$\int_{\tilde{H}}f_{R}(v_{0})d\mu(v_{0})\leq 2K_{3}+1.$ (45)

Applying the Monotone Convergence Theorem to (45) we get

$\int_{\tilde{H}}\|v_{0}\|^{2}_{\tilde{V}}d\mu(v_{0})<\infty,$

which completes the proof of the theorem. ∎

##  6 Small Perturbations and Large Deviations

###  6.1 Small perturbations

In certain physical problems it is natural to assume that the random
perturbations are small compared to the deterministic component of the
dynamics (see [28] and references therein). In this section, we consider such
perturbations for bidomain equation, which leads to the following equation for
any $\varepsilon>0$:

$\begin{cases}du_{\varepsilon}=[-{\mathbb{A}}u_{\varepsilon}-f(u_{\varepsilon},w_{\varepsilon})+I]dt+\varepsilon
dW,\ t\geq 0,\\\ dw_{\varepsilon}=-g(u_{\varepsilon},w_{\varepsilon})dt,\\\
u_{\varepsilon}(0)=u_{0},w_{\varepsilon}(0)=w_{0}.\end{cases}$ (46)

Along with (46), we consider the limiting equation

$\begin{cases}du=[-{\mathbb{A}}u-f(u,w)+I]dt,\ t\geq 0,\\\ dw=-g(u,w)dt,\\\
u(0)=u_{0},w(0)=w_{0}.\end{cases}$ (47)

Let $y:=(u,w)^{T}\in\mathbb{R}^{2}.$ Define $\mathcal{F}$ by

$\mathcal{F}(y)=\mathcal{F}(u,w):=\left(\begin{array}[]{c}-f(u,w)+I\\\
-g(u,w)\\\ \end{array}\right).$

We assume that there exists constants $c_{1}\in\mathbb{R}$ and
$c_{2}\in\mathbb{R}$ such that for all
$(u_{1},w_{1}),(u_{2},w_{2})\in\mathbb{R}^{2}$ the function $\mathcal{F}$
satisfies the monotonicity condition

$(\mathcal{F}(u_{1},w_{1})-\mathcal{F}(u_{2},w_{2}))\cdot((u_{1},w_{1})-(u_{2},w_{2}))\leq-
c_{1}(u_{1}-u_{2})^{2}-c_{2}(w_{1}-w_{2})^{2}.$ (48)

######  Theorem 5.

Assume $f$ and $g$ satisfy the conditions [C1]-[C3], and the monotonicity
condition (48). Then there is $C=C(T)$ such that

$\mathbb{E}\sup_{t\in[0,T]}(\|u_{\varepsilon}(t)-u(t)\|^{2}+\|w_{\varepsilon}(t)-w(t)\|^{2})\leq\varepsilon
C(T).$

###### Proof.

Introduce $v_{\varepsilon}:=u_{\varepsilon}-u$ and
$z_{\varepsilon}:=w_{\varepsilon}-w.$ Then the pair
$(v_{\varepsilon},z_{\varepsilon})$ solves

$\begin{cases}dv_{\varepsilon}=[-{\mathbb{A}}u_{\varepsilon}-(f(u_{\varepsilon},w_{\varepsilon})-f(u,w))]dt+\varepsilon
dW,\ t\geq 0,\\\
dz_{\varepsilon}=-(g(u_{\varepsilon},w_{\varepsilon})-g(u,w))dt,\\\
v_{\varepsilon}(0)=0,z_{\varepsilon}(0)=0.\end{cases}$ (49)

Using Ito’s formula (see, e.g.[18]), we have

$\|v_{\varepsilon}(t)\|^{2}=2\int_{0}^{t}\langle-{\mathbb{A}}v_{\varepsilon},v_{\varepsilon}\rangle
ds-2\int_{0}^{t}(f(u_{\varepsilon},w_{\varepsilon})-f(u,w),u_{\varepsilon}-u)ds+\varepsilon^{2}\gamma
t+2\varepsilon\int_{0}^{t}(v_{\varepsilon},dW_{s})$

as well as

$\|z_{\varepsilon}\|^{2}=-2\int_{0}^{t}(g(u_{\varepsilon},w_{\varepsilon})-g(u,w),w_{\varepsilon}-w)ds.$

Altogether, using the monotonicity condition (48) and the estimate (34),

$\|v_{\varepsilon}(t)\|^{2}+\|z_{\varepsilon}(t)\|^{2}\leq
2\int_{0}^{t}(-\alpha\|v_{\varepsilon}\|_{V}^{2}+\alpha\|v_{\varepsilon}\|^{2})ds+C_{1}\int_{0}^{t}(\|v_{\varepsilon}\|^{2}+\|z_{\varepsilon}\|^{2})ds\\\
+\varepsilon^{2}\gamma t+2\varepsilon\int_{0}^{t}(v_{\varepsilon},dW_{s}).$

Hence

$\mathbb{E}\sup_{t\in[0,T]}(\|v_{\varepsilon}(t)\|^{2}+\|z_{\varepsilon}(t)\|^{2})\leq\sigma\int_{0}^{T}(\|v_{\varepsilon}\|^{2}+\|z_{\varepsilon}\|^{2})ds+\varepsilon^{2}\gamma
T+2\varepsilon\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(v_{\varepsilon},dW_{s})\right|,$

for some $\sigma>0$. Here

$\int_{0}^{t}(v_{\varepsilon},dW_{s})=\int_{0}^{t}(v_{\varepsilon},\sum_{j}\gamma_{i}\Psi_{i}(x)dW_{i}(s))$

is a martingale, which can further be estimated as follows:

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(v_{\varepsilon},dW_{s})\right|\leq\sqrt{\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(v_{\varepsilon},dW_{s})\right|^{2}}$
(50)
$\displaystyle=\sqrt{\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}\left(v_{\varepsilon},\sum_{i}\gamma_{i}\Psi_{i}dW_{i}(s)\right)\right|^{2}}=\sqrt{\mathbb{E}\sup_{t\in[0,T]}\left|\sum_{i}\gamma_{i}^{2}\int_{0}^{t}(v_{\varepsilon},\Psi_{i})dW_{i}(s)\right|^{2}}$
$\displaystyle\leq
C\sqrt{\sum_{i}\gamma_{i}^{2}\int_{0}^{T}\mathbb{E}(v_{\varepsilon},\Psi_{i})^{2}ds}\leq
C\sqrt{\int_{0}^{T}\mathbb{E}\|v_{\varepsilon}(s)\|^{2}ds}\leq
C+C\int_{0}^{T}\mathbb{E}\|v_{\varepsilon}(s)\|^{2}ds,$

where we used the elementary inequality $\sqrt{a}\leq a+1.$ This way, for some
$\sigma_{1}>0$, we have

$\mathbb{E}\sup_{t\in[0,T]}(\|v_{\varepsilon}(t)\|^{2}+\|z_{\varepsilon}(t)\|^{2})\leq\sigma_{1}\int_{0}^{T}\sup_{s\in[0,t]}(\|v_{\varepsilon}(s)\|^{2}+\|z_{\varepsilon}(s)\|^{2})dt+\varepsilon^{2}\gamma
T+2\varepsilon C.$

It follows from Gronwall inequality that

$\mathbb{E}\sup_{t\in[0,T]}(\|u_{\varepsilon}(t)-u(t)\|^{2}+\|w_{\varepsilon}(t)-w(t)\|^{2})\leq
C(T)\varepsilon.$

∎

###  6.2 Large deviations

The influence of small random perturbations on large time intervals is a
question of special interest. Generally speaking, in this case small
perturbations have a significant influence on the macroscopic behavior of the
underlying physical system. In order to study this influence, we have to
estimate the probabilities of unlikely events. In other words, we need to
study the asymptotic behavior of large deviations for random processes.

Let $C_{p}$ be the constant in Poincare’s inequality, i.e.,

$\|u\|_{H}^{2}\leq C_{p}\|\nabla u\|_{H}^{2},\quad u\in V\cap H_{0}.$ (51)

with $H_{0}$ defined in (8). The main result of this subsection is the
following Theorem:

######  Theorem 6.

Let the conditions [C1], [C2, [C3] and the monotonicity condition (48) holds.
Assume, in addition, that the constants $c_{1}$ and $c_{2}$, appearing in
(48), satisfy $c_{2}\geq 0$ and

$c_{1}\geq-\frac{\alpha}{C_{p}}$ (52)

with $\alpha$ and $C_{p}$ given by (34) and (51) respectively. Then

$P\\{\sup_{0\leq t\leq
T}\\{\|u_{\varepsilon}(t)-u(t)\|^{2}+\|w_{\varepsilon}(t)-w(t)\|^{2}\\}\geq
r^{2}\\}\leq 3\exp\left[-\frac{r^{2}}{4\gamma\varepsilon^{2}T}\right],$ (53)

with $\gamma$ given in (13).

###### Proof.

We first note that, using (34), (48), (51) and (52):

$\displaystyle\langle-{\mathbb{A}}v_{\varepsilon},v_{\varepsilon}\rangle-(f(u_{\varepsilon},w_{\varepsilon})-f(u,w),u_{\varepsilon}-u)-(g(u_{\varepsilon},w_{\varepsilon})-g(u,w),w_{\varepsilon}-w)$
(54)
$\displaystyle\leq-\alpha\|v_{\varepsilon}\|_{V}^{2}+\alpha\|v_{\varepsilon}\|_{H}^{2}-c_{1}\|v_{\varepsilon}\|_{H}^{2}-c_{2}\|z_{\varepsilon}\|_{H}^{2}=-\alpha\|\nabla
v_{\varepsilon}\|^{2}_{H}-c_{1}\|v_{\varepsilon}\|_{H}^{2}-c_{2}\|z_{\varepsilon}\|_{H}^{2}\leq$
$\displaystyle-\frac{\alpha}{C_{p}}\|v_{\varepsilon}\|_{H}^{2}-c_{1}\|v_{\varepsilon}\|_{H}^{2}-c_{2}\|z_{\varepsilon}\|_{H}^{2}\leq
0.$

Following [5], introduce the functional

$\Phi_{\lambda}(v,z)=(1+\lambda(\|v\|^{2}+\|z^{2}\|^{2})^{1/2},\ v,z\in H.$

Using Ito’s formula, we have

$\displaystyle\Phi_{\lambda}(v_{\varepsilon},z_{\varepsilon})=1+\lambda\int_{0}^{t}\Phi_{\lambda}^{-1}(v_{\varepsilon},z_{\varepsilon})[\langle-{\mathbb{A}}v_{\varepsilon},v_{\varepsilon}\rangle$
(55)
$\displaystyle-(f(u_{\varepsilon},w_{\varepsilon})-f(u,w),u_{\varepsilon}-u)-(g(u_{\varepsilon},w_{\varepsilon})-g(u,w),w_{\varepsilon}-w)]ds+$
$\displaystyle+\lambda\varepsilon\int_{0}^{t}\Phi_{\lambda}^{-1}(v_{\varepsilon},y_{\varepsilon})(v_{\varepsilon},dW_{s})+\frac{\varepsilon^{2}}{2}\int_{0}^{t}\left\\{\lambda\Phi_{\lambda}^{-1}(v_{\varepsilon},z_{\varepsilon})\gamma-\lambda^{2}\Phi_{\lambda}^{-3}(v_{\varepsilon},z_{\varepsilon})(Qv_{\varepsilon},v_{\varepsilon})\right\\}ds.$

Here $Q$ is a trace class covariance operator, such that
$Qe_{k}=\lambda_{k}e_{k}$, $Q$ is given in the definition of $W_{t}$. It
follows from (55) and (54) that

$\Phi_{\lambda}(v_{\varepsilon},z_{\varepsilon})\leq 1+\eta_{t}^{\lambda}+\\\
\frac{\varepsilon^{2}}{2}\int_{0}^{t}\left[\lambda\Phi^{-1}_{\lambda}(v_{\varepsilon},z_{\varepsilon})\gamma+\lambda^{2}(Qv_{\varepsilon},v_{\varepsilon})(\Phi^{-2}_{\lambda}(v_{\varepsilon},z_{\varepsilon})-\Phi^{-3}_{\lambda}(v_{\varepsilon},z_{\varepsilon}))\right]ds,$
(56)

where $\gamma=Tr(Q)$ and

$\eta_{t}^{\lambda}=\lambda\varepsilon\int_{0}^{t}\Phi_{\lambda}^{-1}(v_{\varepsilon},z_{\varepsilon})(v_{\varepsilon},dW_{s})-\frac{1}{2}\lambda^{2}\varepsilon^{2}\int_{0}^{t}\Phi_{\lambda}^{-2}(v_{\varepsilon},z_{\varepsilon})(Qv_{\varepsilon},v_{\varepsilon})ds.$
(57)

Note that $\forall\lambda>0$ $\forall\alpha>0$, with
$0<\Phi_{\lambda}^{-\alpha}\leq 1$, we have

$\Phi_{\lambda}^{-2}(v,z)[1-\Phi_{\lambda}^{-1}(v,z)]\leq 1$

and

$(Qv,v)\leq\gamma\|v\|^{2}.$

Therefore,

$\Phi_{\lambda}^{-2}(v,z)(Qv,v)\leq\frac{\gamma}{\lambda}\frac{\|v\|^{2}}{\frac{1}{\lambda}+\|v\|^{2}+\|z\|^{2}},$

which for $t\leq T$ implies

$\Phi_{\lambda}(v_{\varepsilon},z_{\varepsilon})\leq 1+\varepsilon^{2}\lambda
T\gamma+\eta_{t}^{\lambda}.$ (58)

Therefore,

$\displaystyle P\\{\sup_{0\leq t\leq
T}\\{\|u_{\varepsilon}-u\|^{2}+\|w_{\varepsilon}-w\|^{2}\\}\geq
r^{2}\\}=P\\{\sup_{t\in[0,T]}\Phi_{\lambda}(v_{\varepsilon},z_{\varepsilon})\geq(1+\lambda
r^{2})^{\frac{1}{2}}\\}\leq$ (59) $\displaystyle P\\{1+\varepsilon^{2}\lambda
T\gamma+\sup_{t\in[0,T]}\eta_{t}^{\lambda}\geq(1+\lambda
r^{2})^{\frac{1}{2}}\\}=P\\{e^{\sup_{t\in[0,T]}\eta_{t}^{\lambda}}\geq
e^{(1+\lambda r^{2})^{\frac{1}{2}}-\varepsilon^{2}\lambda T\gamma-1}\\}.$

It follows from Ito’s formula $e^{\eta_{t}^{\lambda}}$ is a martingale, which,
in addition, satisfies

$\mathbb{E}e^{\eta_{t}^{\lambda}}=\mathbb{E}e^{\eta_{0}^{\lambda}}=1.$

Now, using Chebyshev inequality, Doob’s martingale inequality, and the fact
that $\sup_{t}e^{a(t)}=e^{\sup_{t}a(t)}$, we get

$P\\{\sup_{0\leq t\leq
T}\\{\|u_{\varepsilon}(t)-u(t)\|^{2}+\|w_{\varepsilon}(t)-w(t)\|^{2}\\}\geq
r^{2}\\}\leq e^{1+(1+\varepsilon^{2}\lambda T\gamma-\lambda
r^{2})^{\frac{1}{2}}}.$ (60)

It remains to optimize the latter expression in $\lambda$. Setting

$\lambda:=\left(\frac{r}{2\gamma\varepsilon^{2}T}\right)^{2}-\frac{1}{r^{2}},$

the exponent can be estimated as

$1+(1+\varepsilon^{2}\lambda T\gamma-\lambda r^{2})^{\frac{1}{2}}\leq
1-\frac{r^{2}}{4\varepsilon^{2}\gamma T},$

yielding the desired result

$P\\{\sup_{0\leq t\leq
T}\\{\|u_{\varepsilon}(t)-u(t)\|^{2}+\|w_{\varepsilon}(t)-w(t)\|^{2}\\}\geq
r^{2}\\}\leq 3e^{-\frac{r^{2}}{4\varepsilon^{2}\gamma T}}.$

∎

We conclude this subsection with some examples of typical nonlinearities,
satisfying the conditions of Theorem 6.

######  Example 1.

Allen-Cahn nonlinearity

$f(u):=\eta(u^{3}-u).$

The conditions (16) and (48) read as

$\displaystyle uf(u)$ $\displaystyle\geq au^{4}+bu^{2}+K,\quad\mbox{ and}$
(61) $\displaystyle(f(u_{1})-f(u_{2}))(u_{1}-u_{2})$ $\displaystyle\geq
c_{1}(u_{1}-u_{2})^{2}.$ (62)

Clearly, condition (61) is satisfied. In order to verify (62), note that

$(f(u_{1})-f(u_{2}))(u_{1}-u_{2})\geq\eta(u_{1}-u_{2})^{2}(u_{1}^{2}+u_{1}u_{2}+u_{2}^{2}-1)\geq-\eta(u_{1}-u_{2})^{2}.$

Thus the coefficient condition (52) in (48) holds, provided
$\eta\leq\frac{\alpha}{C_{p}}$.

######  Example 2.

FitzHugh-Nagumo nonlinearity

$\begin{cases}f(u,w)=\eta[u(u-a)(u-1)+w],\\\ g(u,w)=kw-du,\end{cases}$

with $0<a<1$ and $k,d>0$. Note that

$u\eta[u^{3}-(a+1)u^{2}+au]+\eta uw+kw^{2}-duw\geq\frac{\eta}{2}u^{4}+a\eta
u^{2}-|\eta-d|(\frac{1}{2}w^{2}+\frac{1}{2}u^{2})+kw^{2}.$

Hence (16) holds for $\frac{|\eta-d|}{2}\leq k$. In order to verify the
condition (52) in (48), note that

$\displaystyle-\eta(-u_{1}^{3}+u_{2}^{3}+(1+a)(u_{1}^{2}-u_{2}^{2})-a(u_{1}-u_{2}))(u_{1}-u_{2})+$
$\displaystyle+(\eta-d)(w_{1}-w_{2})(u_{1}-u_{2})+k(w_{1}-w_{2})^{2}\geq-\eta\big{(}(1+a)^{2}/3-a\big{)}(u_{1}-u_{2})^{2}$
$\displaystyle-|\eta-d|\Big{(}\frac{(u_{1}-u_{2})^{2}}{2}+\frac{(w_{1}-w_{2})^{2}}{2}\Big{)}+k(w_{1}-w_{2})^{2}$
$\displaystyle=-\Big{(}(1+a)^{2}/3-a)\eta+\frac{|\eta-d|}{2}\Big{)}(u_{1}-u_{2})^{2}+\Big{(}k-\frac{|\eta-d|}{2}\Big{)}(w_{1}-w_{2})^{2},$

where we used the elementary inequality

$-(u_{1}^{2}+u_{1}u_{2}+u_{2}^{2})+(1+a)(u_{1}+u_{2})-a\leq\frac{1}{3}(1+a)^{2}-a\in\left[\frac{1}{4},\frac{1}{3}\right),0<a<1.$

Consequently, (52) holds for $\eta$ and $d$ satisfying
$\left(\frac{(1+a)^{2}}{3}-a\right)\eta+\frac{|\eta-d|}{2}\leq\frac{\alpha}{C_{p}}$
and $\frac{|\eta-d|}{2}\leq k$.

###  6.3 Convergence of stationary solutions

$\begin{cases}du_{\varepsilon}=[-{\mathbb{A}}u_{\varepsilon}-f(u_{\varepsilon},w_{\varepsilon})+I(x)]dt+\varepsilon
dW,\ t\geq 0,\\\
dw_{\varepsilon}=-g(u_{\varepsilon},w_{\varepsilon})dt.\end{cases}$ (63)

Assume the conditions of Theorem 5.2 [15] hold. Then for any $\varepsilon\geq
0$ the equation (63) has a unique stationary solution
$z_{\varepsilon}^{*}:=(u_{\varepsilon}^{*},w_{\varepsilon}^{*})$.

######  Remark 1.

Note that $z_{\varepsilon}$ may have different initial conditions for every
$\varepsilon$.

######  Theorem 7.

Under the conditions of Theorem 5.2 [15], we have

$\sup_{t\in[0,T]}\mathbb{E}\|z^{*}_{\varepsilon}(t)-z^{*}_{0}(t)\|\to
0,\varepsilon\to 0.$

Following the lines of Theorem 6.3.2 of [7], we extend the Wiener process $W$
for negative $t$ by

$\overline{W}(t)=\begin{cases}W(t),\text{ for }t\geq 0,\\\ V(-t),\text{ if
}t\leq 0,\end{cases}$ (64)

and set $\overline{\mathcal{F}(t)}=\sigma\big{(}\overline{W}(s),s\leq
t\big{)}$ for $t\in\mathbb{R}$, where $V(t),t\geq 0$ is another Wiener
process, independent of $W(t)$. For simplicity of our notation, we still
denote the process $\overline{W(t)}$ by $W(t)$. For $n\geq 1$ denote
$z_{\varepsilon}(t,-n,z_{0})$ to be the solution of (63) with the initial
condition $z_{\varepsilon}(-n)=z_{0}=(u_{0},w_{0})\in H\times H$. Using the
energy estimate from Theorem 5.2 [15], we have

$\frac{1}{2}\frac{d}{dt}\mathbb{E}\|z_{\varepsilon}(t)\|^{2}\leq\frac{1}{2}\varepsilon^{2}\gamma-\mu\mathbb{E}\|z_{\varepsilon}(t)\|^{2}+K_{2},$
(65)

which, by Gronwall’s inequality, implies the uniform in $\varepsilon\in[0,1]$
bound

$\mathbb{E}\|z_{\varepsilon}\|^{2}\leq K_{3}(1+\|z(0)\|^{2}_{H}).$

Furthermore, for $\varepsilon_{1}\geq 0$ and $\varepsilon_{2}\geq 0$,

$\begin{cases}d(u_{\varepsilon_{1}}-u_{\varepsilon_{2}})=[-{\mathbb{A}}(u_{\varepsilon_{1}}-u_{\varepsilon_{2}})-[f(u_{\varepsilon_{1}},w_{\varepsilon_{1}})-f(u_{\varepsilon_{2}},w_{\varepsilon_{2}})]dt+(\varepsilon_{1}-\varepsilon_{2})dW,\
t\geq 0,\\\
d(w_{\varepsilon_{1}}-w_{\varepsilon_{2}})=-(g(u_{\varepsilon_{1}},w_{\varepsilon_{1}})-g(u_{\varepsilon_{2}},w_{\varepsilon_{2}}))dt.\end{cases}$
(66)

Using the same reasoning as in (65), we have

$\frac{d}{dt}\left(\mathbb{E}\|z_{\varepsilon_{1}}-z_{\varepsilon_{2}}\|^{2}\right)\leq-\mu\mathbb{E}\|z_{\varepsilon_{1}}-z_{\varepsilon_{2}}\|^{2}+(\varepsilon_{1}-\varepsilon_{2})^{2}\gamma,$

yielding

$\mathbb{E}(\|u_{\varepsilon_{1}}(0,-n,u_{0})-u_{\varepsilon_{2}}(0,-n,u_{0})\|^{2}+\|w_{\varepsilon_{1}}(0,-n,u_{0})-w_{\varepsilon_{2}}(0,-n,u_{0})\|^{2})\\\
\leq(\varepsilon_{1}-\varepsilon_{2})^{2}\gamma.$ (67)

Furthermore, as it was shown in the proof of Theorem 5.2 [15], we have

$\mathbb{E}\|u_{\varepsilon_{1}}(0,-n,u_{0})-u_{\varepsilon_{2}}(0,-n,u_{0})\|^{2}\to\mathbb{E}\|u_{\varepsilon_{1}}^{*}(0)-u_{\varepsilon_{2}}^{*}(0)\|^{2},\
n\to\infty$

and

$\mathbb{E}\|w_{\varepsilon_{1}}(0,-n,u_{0})-w_{\varepsilon_{2}}(0,-n,u_{0})\|^{2}\to\mathbb{E}\|w_{\varepsilon_{1}}^{*}(0)-w_{\varepsilon_{2}}^{*}(0)\|^{2},\
n\to\infty.$

Passing to the limit as $n\to\infty$ in (67), we obtain

$\mathbb{E}(\|u_{\varepsilon_{1}}^{*}(0)-u_{\varepsilon_{2}}^{*}(0)\|^{2}+\|w_{\varepsilon_{1}}^{*}(0)-w_{\varepsilon_{2}}^{*}(0)\|^{2})\leq(\varepsilon_{1}-\varepsilon_{2})^{2}\gamma.$

The bound above implies that the sequence
$\\{(u_{\varepsilon}^{*}(0),w_{\varepsilon}^{*}(0)\\}$ is Cauchy, and
therefore

$\\{(u_{\varepsilon}^{*}(0),w_{\varepsilon}^{*}(0)\\}\to\\{(u_{0}^{*}(0),w_{0}^{*}(0)\\},\
\ \varepsilon\to 0.$

We are now in position to establish

$\mathbb{E}\sup_{t\in[0,T]}\|z_{\varepsilon}^{*}(t)-z_{0}^{*}(t)\|\to
0,\varepsilon\to 0.$

To this end, setting $v_{\varepsilon}=u_{\varepsilon}^{*}-u_{0}^{*}$ and
$y_{\varepsilon}=w_{\varepsilon}^{*}-w_{0}^{*}$. Then

$\displaystyle\|v_{\varepsilon}(t)\|^{2}+\|y_{\varepsilon}(t)\|^{2}=\|u_{\varepsilon}^{*}(0)-u_{0}^{*}(0)\|^{2}+\|w_{\varepsilon}^{*}(0)-w_{0}^{*}(0)\|^{2}+$
$\displaystyle
2\int_{0}^{t}[\langle-{\mathbb{A}}v_{\varepsilon},v_{\varepsilon}\rangle-(f(u_{\varepsilon}^{*},w_{\varepsilon}^{*})-f(u_{0}^{*},w_{0}^{*}),u_{\varepsilon}^{*}-u_{0}^{*})-$
$\displaystyle-(g(u_{\varepsilon}^{*},w_{\varepsilon}^{*})-g(u_{0}^{*},w_{0}^{*}),w_{\varepsilon}^{*}-w_{0}^{*})]ds+\varepsilon^{2}\gamma
t+2\varepsilon\int_{0}^{t}(v_{\varepsilon}^{*},dW_{s}).$

We may now proceed with estimating the stochastic analogously to (50), and
then use the Gronwall’s inequality the same way as in (65) to conclude that

$\mathbb{E}\sup_{t\in[0,T]}(\|u_{\varepsilon}^{*}(t)-u_{0}^{*}(t)\|^{2}+\|w_{\varepsilon}^{*}(t)-w_{0}^{*}(t)\|^{2})\\\
\leq\mathbb{E}(\|u_{\varepsilon}^{*}(0)-u_{0}^{*}(0)\|^{2}+\|w_{\varepsilon}^{*}(0)-w_{0}^{*}(0)\|^{2})+C_{1}(T)\varepsilon\to
0,\ \varepsilon\to 0,$

which completes the proof.

## Acknowledgement

The research of Oleksandr Misiats was supported by Simons Collaboration Grant
for Mathematicians No. 854856. The work of Oleksiy Kapustyan and Oleksandr
Stanzhytskyi was partially supported by the National Research Foundation of
Ukraine No. F81/41743 and Ukrainian Government Scientific Research Grant No.
210BF38-01.

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# The Exponentiated Hypoexponential Disribution.

Anass Nassabein
Department of Mathematics, Lebanese International University
Therrar Kadri
Department of Science, Northwestrern Polytecgnic
Seifideen Kadry
Department of Applied Computing and Technology, Noroff University College
Khaled Smaili
Department of Mathematics, Lebanese University  The authors gratefully
acknowledge please remember to list all relevant funding sources in the
unblinded version

###### Abstract

In this paper we study the Exponentiated Hypoexponential Distribution with
different parameters. The distribution added a parameter to the $n$ parameters
of the Hypoexponenial distribution. We first derive a closed expression of the
probability density function and the cumulative distribution function of
Maximum Exponentiated Exponential distribution. These functions are used to
determine an exact expression of the probability density function, the
cumulative function, reliability function, and hazard function of our
Exponentiated Hypoexponential Distribution. We discuss estimation of the
parameters by maximum likelihood estimators. The distribution has been fitted
to a real life data set and the fit has been found to be a serious competitor
to the others.

The Exponentiated Hypoexponential Disribution

Keywords: Maximum Exponentiated Exponential Distribution, Exponentiated
Hypoexponential Distribution, Maximum Likelihood Estimation.

##  1 Introduction

The Exponential random variable like its alternatives, Erlang and Gamma, plays
a significant role in both queueing and reliability problems [1] and [17]. The
time between independent events with constant average or time to failure are
typical examples of the Exponential random variable. Some examples in many
fields of science are: how long a cell-phone call lasts [5], how long it takes
a computer network to transmit a message from one node to another, and the
distance between mutations on a DNA stand…, see, [12].

In addition to that, the function of random variable possesses an important
role in the event modeling in many applied problems of engineering, physics,
economics, biology, genetics, medicine, see, [4] and [12]. One of the
important functions of random variable is the sum of independent random
variables especially the sum of independent Exponential distributions which is
known to be the Hypoexponential distribution. The Hypoexponential distribution
that also plays a significant role in modeling many events in most domains of
science, Trivedi [20], Markov process, Kadri et al. [11].

There are many ways to add one or more parameters to a parent distribution
function. Adding the parameters makes the resulting distribution more flexible
and richer for modeling data. Marshall and Olkin [14] added a positive
parameter to a general reliability function (RF). In their consideration of a
countable mixture of Pascal $(r,p)$ mixing proportion and positive integer
powers of RF, they obtained a reliability function with two extra parameters
in AL-Hussaini and Ghitany [2]. AL-Hussaini and Gharib [3] obtained a new
family of distributions as a countable mixture with Poisson by adding a
parameter.

The idea of Exponentiated distribution was introduced by Gupta et al. [6], who
discussed a new family of distributions called Exponentiated Exponential
distribution. The family has two parameters similar to that of Weibull or
Gamma’s family. Gupta and Kundu studied some properties of the distribution
see, [7]. They concluded that this distribution can be an alternative
possibility to Gamma or Weibull family, because their observation was clear
that several properties of the new family are similar to those of the Weibull
or Gamma family. They also examined the estimation and inference aspects of
the distribution see, [8], and [9]. The distribution has been further studied
by Nadarajah and Kotz [15]. A class of goodness-of-fit tests for the
distribution with estimated parameter has been proposed by Hassan [10]. Pal et
al. [16] studied the Exponentiated Weibull family as an extension of the
Weibull and Exponentiated Exponential families.

In this paper, we introduce the Exponentiated Hypoexponential distribution. We
start by characterizing the related functions that well describe our new
distribution. We show it by relating it to the Maximum Exponentiated
Exponential distribution. Next, we give an estimate of the new parameter k, by
MLE method. Eventually we run a simulation of data showing the
EHypoexponential distribution fits better than the original distribution. The
problem is solved completely by considering the different parameters of
Hypoexponential random variable.

##  2 Preliminaries

The Exponentiated Exponential (EE) distribution, and known as the generalized
Exponential distribution, is a particular case of Gompertz-Verhulst
distribution function when the parameter $\rho=1$. Thus, $X\sim
EE\left(\lambda_{i},\alpha_{i}\right)$ is a two- parameter Exponentiated
Exponential random variable when it has the following comulative distribution
function (CDF);

$F(x;\alpha;\lambda)=\left(1-e^{-\lambda x}\right)^{\alpha};\text{ }x>0$ (1)

and therefore its probability density function (PDF) is of the form

$f(x;\alpha;\lambda)=\alpha\lambda\left(1-e^{-\lambda
x}\right)^{\alpha-1}e^{-\lambda x};x>0$ (2)

###  2.1 The Hypoexponential Distribution

The Hypoexponential random variable is the sum of independent Exponential
random variable. This distribution is examined accoding to the stages of the
Exponential random variable. Whenever the stages are identical the
distribution obtained is Erlang distribution. The other case of the
Hypoexponential distribution is when the stages are different or the
parameters are exponential distribution are different. This case was examined
completlet by Samili et al. in [19]. They denoted this distribution as the
Hypoexpoential distribution with different parameters and given as follows:

Let $X_{1\text{ }},X_{2\text{ }},...,X_{n\text{ }}$be independent Exponential
random variables with different, respective parameters $\alpha_{i},$
$i=1,2,...,n$. Thus $S_{n}=\sum\limits_{i=1}^{n}X_{i\text{ }}$is the
Hypoexponential random variable with parameters and written as $S_{n}\sim
hypoexp(\overrightarrow{\alpha})$, where
$\overrightarrow{\alpha}_{i}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$.
Samili et al. in [19] showed that the density function of $S_{n}$ is given as

$f_{S_{n}}(t)=\sum\limits_{i=1}^{n}A_{i}f_{X_{i\text{ }}}(t)$

where

$A_{i}=\prod\limits_{j=1,\,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)$ (3)

is denoted as the coefficients of Hypoexponential. The author as a consequence
detemined the other related function to $S_{n}$ as the CDF as

$F_{S_{n}}(t)=\sum\limits_{i=1}^{n}A_{i}F_{X_{i\text{ }}}(t)$ (4)

###  2.2 Exponentiated Distributions

Let X be a random variable with probability density function
$\left(PDF\right)$ $f\left(x\right)$ and the cumulative distributions function
$\left(CDF\right)$ $F\left(x\right)$, $x\in R^{1}$. Consider a random variable
$Z$ with the $CDF$ given by

$G_{\alpha}\left(z\right)=\left[F\left(z\right)\right]^{\alpha},\text{ }z\in
R^{1},\alpha>0$ (5)

Then Z is said to have an exponentiated distribution.

The $pdf$ of $Z$ is given by

$g_{\alpha}\left(z\right)=\alpha\left[F\left(z\right)\right]^{\alpha-1}f\left(z\right)$
(6)

The survival function $R_{\alpha}\left(z\right)$ and the failure rate
$h_{\alpha}\left(z\right)$ of the distribution of Z are defined by

$R_{\alpha}\left(z\right)-1-G_{\alpha}\left(z\right)$

$h_{\alpha}\left(z\right)=\frac{g_{\alpha}\left(z\right)}{1-G_{\alpha}\left(z\right)}$

##  3 Maxmumim Exponentiated Exponential Distribution

In this section we find a simplified expressions of PDF and CDF of the Maximum
Exponentiated Exponential distribution. The expressions obtained are used
later to derive the function of our new distribution the EHypoexponential
distribution.

In the following theorem we find a simplified expression of PDF and CDF of the
Maximum Exponentiated Exponential distribution when the exponents of the EE
distribution is a positive real number.

######  Theorem 1.

Let $X_{i}\sim EE(\lambda_{i},\alpha_{i})\ $where
$\alpha_{i}\in\mathbb{R}_{+}^{\ast},$ $\lambda_{i}>0,$ and
$\lambda_{i}\neq\lambda_{j}$ for all $i\neq j,i=1,2,...,n$ and let
$N=\max\left\\{X_{1},X_{2},...,X_{n}\right\\}$ denoted as $N\sim
MEE(\overrightarrow{\lambda},\overrightarrow{\alpha}),$
$\overrightarrow{\lambda}=\left(\lambda_{1},\lambda_{2},...,\lambda_{n}\right),$
$\overrightarrow{\alpha}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right).$
Then the CDF of $N$ is

$F_{N}(t)=\prod_{j=1}^{n}\left(\sum_{i=0}^{\infty}\binom{\alpha_{j}}{i}(-1)^{(\alpha_{j}-i)}e^{-\lambda_{j}(\alpha_{j}-i)t}\right)$

and the PDF

$f_{N}(t)=\left(\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{\alpha_{j}}\right)\left(\sum_{j=1}^{n}\frac{\lambda_{j}\alpha_{j}e^{-\lambda_{j}t}}{1-e^{-\lambda_{j}t}}\right)$

###### Proof.

Let $X_{i}$ be an Exponentiated Exponential distribution with rate parameter
$\lambda_{i}$ and exponent parameter $\alpha_{i}$, $i=1,2,...,n$, where
$\lambda_{i}\neq\lambda_{j}$ for all $i\neq j$. Then the CDF of the Maximum
Exponentiated Exponential distribution is given by

$\displaystyle F_{N}(t)$ $\displaystyle=$ $\displaystyle
F_{X_{1}}(t)F_{X_{2}}(t)...F_{X_{n}}(t)$ $\displaystyle=$
$\displaystyle(1-e^{-\lambda_{1}t})^{\alpha_{1}}(1-e^{-\lambda_{2}t})^{\alpha_{2}}...(1-e^{-\lambda_{n}t})^{\alpha_{n}}$
$\displaystyle=$
$\displaystyle\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{\alpha_{j}}$

∎

######  Corollary 1.

Let $X_{i}\sim EE(\lambda_{i},g_{i})$ where $g_{i}\in\mathbb{N}^{\ast}$,
$\lambda_{i}>0$, and $\lambda_{i}\neq\lambda_{j}$ for all $i\neq
j,i=1,2,...,n$. Let $N=\max\\{X_{1},X_{2},...,X_{n}\\}$ denoted as $N\sim
MEE(n,\overrightarrow{\lambda},\overrightarrow{g})$,
$\overrightarrow{\lambda}=(\lambda_{1},\lambda_{2},...,\lambda_{n})$,
$\overrightarrow{g}=(g_{1},g_{2},...,g_{n})$. Then the CDF of $N$ is

$F_{N}(t)=\prod_{j=1}^{n}\left(\sum_{i=0}^{\infty}\binom{\alpha_{{}_{j}}}{i}(-1)^{(\alpha_{{}_{j}}-i)}e^{-\lambda_{j}(\alpha_{{}_{j}}-i)t}\right)$

where
$\binom{\alpha_{{}_{j}}}{i}=\frac{(\alpha)_{j}}{i!}=\frac{\alpha(\alpha-1)..\left(\alpha-j+1\right)}{i!}$
is the generalized binomial coefficient. The PDF of $N$ is given by

$f_{N}(t)=\left(\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{\alpha_{j}}\right)\left(\sum_{j=1}^{n}\frac{\lambda_{j}\alpha_{j}e^{-\lambda_{j}t}}{1-e^{-\lambda_{j}t}}\right)$

Next, we consider the same problem by find a simplified expression of PDF and
CDF of the Maximum Exponentiated Exponential distribution but the exponents of
the EE distribution is a positive integer.

######  Corollary 2.

Let $X_{i}\sim EE(\lambda_{i},g_{i})$ where $g_{i}\in\mathbb{N}^{\ast}$,
$\lambda_{i}>0$, and $\lambda_{i}\neq\lambda_{j}$ for all $i\neq
j,i=1,2,...,n$. Let $N=\max\\{X_{1},X_{2},...,X_{n}\\}$ denoted as $N\sim
MEE(n,\overrightarrow{\lambda},\overrightarrow{g})$,
$\overrightarrow{\lambda}=(\lambda_{1},\lambda_{2},...,\lambda_{n})$,
$\overrightarrow{g}=(g_{1},g_{2},...,g_{n})$. Then the CDF of $N$ is

$F_{N}(t)=\prod_{j=1}^{n}\left(\sum_{i=0}^{g_{{}_{j}}}\binom{g_{{}_{j}}}{i}(-1)^{(g_{{}_{j}}-i)}e^{-\lambda_{j}(g_{{}_{j}}-i)t}\right)$
(7)

and the PDF is

$f_{N}(t)=\left(\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{g_{j}}\right)\left(\sum_{j=1}^{n}\frac{\lambda_{j}g_{j}e^{-\lambda_{j}t}}{1-e^{-\lambda_{j}t}}\right)$
(8)

###### Proof.

The steps of the proof are similar to the proof of Theorem 1 but we use
instead of the generalized binomial expansion, the usual binomial expansion
where

$(1-e^{-\lambda_{j}t})^{g_{j}}=\sum_{i=0}^{g_{{}_{j}}}\binom{g_{{}_{j}}}{i}\left(-1\right)^{(g_{{}_{j}}-i)}e^{-\lambda_{j}(g_{{}_{j}}-i)t}.$

∎

In the following we give the reliability and hazard function of the Maximum
Exponentiated Exponential distribution in the case when the exponents of the
EE distribution is positive real number and positive integer.

We have $R_{N}(t)=1-F_{N}\left(t\right)$, thus when the exponents of the EE
distribution is positive real number, from Theorem 1 we get

$R_{N}(t)=1-\prod_{j=1}^{n}\left(\sum_{i=0}^{\alpha_{j}}\binom{\alpha_{j}}{i}(-1)^{(\alpha_{j}-i)}e^{-\lambda_{j}(\alpha_{j}-i)t}\right)$

Also when the exponents of the EE distribution is positive integer, from
Corollary 2, we get

$R_{N}(t)=1-\prod_{j=1}^{n}\left(\sum_{i=0}^{g_{j}}\binom{g_{j}}{i}(-1)^{(g_{j}-i)}e^{-\lambda_{j}(g_{j}-i)t}\right)$
(9)

and we have $h_{N}(t)=\frac{f_{N}\left(t\right)}{R_{N}\left(t\right)}$, thus
when the exponents of the EE distribution are positive real number, from
Theorem 1 we get

$h_{N}(t)=\frac{\left(\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{\alpha_{j}}\right)\left(\sum_{j=1}^{n}\frac{\lambda_{j}\alpha_{j}e^{-\lambda_{j}t}}{1-e^{-\lambda_{j}t}}\right)}{1-\prod_{j=1}^{n}\left(\sum_{i=0}^{\infty}\binom{\alpha_{j}}{i}(-1)^{(\alpha_{j}-i)}e^{-\lambda_{j}(\alpha_{j}-i)t}\right)}$

Also when the exponents of the EE distribution is positive integer, from
Corollary 2, we get

$h_{N}(t)=\frac{\left(\prod_{j=1}^{n}(1-e^{-\lambda_{j}t})^{g_{j}}\right)\left(\sum_{j=1}^{n}\frac{\lambda_{j}g_{j}e^{-\lambda_{j}t}}{1-e^{-\lambda_{j}t}}\right)}{1-\prod_{j=1}^{n}\left(\sum_{i=0}^{g_{j}}\binom{g_{j}}{i}(-1)^{(g_{j}-i)}e^{-\lambda_{j}(g_{j}-i)t}\right)}$
(10)

##  4 Exponentiated Hypoexponential Distribution

Let $S_{n}\sim hypoexp(\overrightarrow{\alpha})\ $where
$n\in\mathbb{N}^{\ast}$ and $\alpha_{i}>0,$ $i=1,2,...,n$ and let $Y$ be the
exponentiated distribution of $S_{n}$, with exponent $k$. We name $Y$ as the
Exponentiated Hypoexponential distribution or EHypoexponential with exponent
$k$ and denoted as $Y\sim
EHypo\exp\left(\overrightarrow{\alpha},k\right),\overrightarrow{\alpha}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$
and $k\in\mathbb{N}^{\ast}$. Thus Exponentiated Hypoexponential distribution
has $n+1$ parameters $\left(\alpha_{1},\alpha_{2},...,\alpha_{n},k\right)$. We
aim to examine this new distribution, and find simple and closed expressions
of some statistical functions. We start by relating the CDF of the
Exponentiated Hypoexponential distribution to CDF of the maximum Exponentiated
Exponential distribution, that we discussed and determined in the prevous
sections. As mentioned previously this relation will facilitate our work in
determing other expressions.

######  Theorem 2.

Let $S_{n}\sim\text{hypoexp}(\overrightarrow{\alpha})$ with coefficients
$A_{i}$ defined in Equation 3, and let
$Y\sim\text{EHypoexp}(\overrightarrow{\alpha},k)$, where
$\overrightarrow{\alpha}=(\alpha_{1},\alpha_{2},...,\alpha_{n})$ and
$k\in\mathbb{N}^{\ast}$. Then the CDF of $Y$ is

$F_{Y}(t)=\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}F_{N_{i}}(t)$ (11)

where $E_{k}=\\{\overrightarrow{g}_{i}=(g_{1,i},g_{2,i},\ldots,g_{n,i})\ |\
0\leq g_{j,i}\leq k,\ \sum_{j=1}^{n}g_{j,i}=k\\}$. and

$B_{i}=\binom{k}{g_{1,i},\ldots,g_{n,i}}\prod_{j=1}^{n}A_{j}^{g_{j,i}}$

and denoted by the coefficient of the Exponentiated Hypoexponential
distribution, and $F_{N_{i}}(t)$ is the CDF of $N_{i}\sim
MEE(\overrightarrow{\alpha},\overrightarrow{g}_{i},k)$.

###### Proof.

Let $S_{n}\sim\text{hypoexp}(\overrightarrow{\alpha})$ and $\alpha_{i}>0$,
$i=1,2,\ldots,n$. From Eq. 4, the CDF of $S_{n}$ is given by
$F_{S_{n}}(t)=\sum_{i=1}^{n}A_{i}F_{X_{i}}(t)$, where
$X_{i}\sim\text{Exp}(\alpha_{i})$ and $A_{i}=\prod_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)$. By
exponentiating the distribution $S_{n}$ mentioned in Theorem 5, we get
$F_{Y}(t)=\left(\sum_{i=1}^{n}A_{i}F_{X_{i}}(t)\right)^{k}$. Using the
multinomial expansion, we have:

$\displaystyle F_{Y}(t)$ $\displaystyle=$
$\displaystyle\sum_{\overrightarrow{g}_{i}\in
E_{k}}\binom{k}{g_{1,i},g_{2,i},\ldots,g_{n,i}}\left(A_{1}F_{X_{1}}(t)\right)^{g_{1,i}}\left(A_{2}F_{X_{2}}(t)\right)^{g_{2,i}}\ldots\left(A_{n}F_{X_{n}}(t)\right)^{g_{n,i}}$
$\displaystyle=$ $\displaystyle\sum_{\overrightarrow{g}_{i}\in
E_{k}}\binom{k}{g_{1,i},g_{2,i},\ldots,g_{n,i}}(A_{1}^{g_{1,i}}A_{2}^{g_{2,i}}\ldots
A_{n}^{g_{n,i}})(F_{X_{1}}^{g_{1,i}}(t)F_{X_{2}}^{g_{2,i}}(t)\ldots
F_{X_{n}}^{g_{n,i}}(t))$

where $E_{k}=\\{\overrightarrow{g}_{i}=(g_{1,i},g_{2,i},\ldots,g_{n,i})\ |\
0\leq g_{j,i}\leq k,\ \sum_{j=1}^{n}g_{j,i}=k\\}$. But from Corollary 2,
$F_{X_{1}}^{g_{1,i}}F_{X_{2}}^{g_{2,i}}\ldots F_{X_{n}}^{g_{n,i}}$ is the CDF
of $N_{i}$, where
$N_{i}\sim\text{MEE}(\overrightarrow{\alpha},\overrightarrow{g_{i}})$. Then we
can write the CDF of $Y$ as

$F_{Y}(t)=\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}F_{N_{i}}(t)$

where
$B_{i}=\binom{k}{g_{1,i},\ldots,g_{n,i}}(A_{1}^{g_{1,i}}A_{2}^{g_{2,i}}\ldots
A_{n}^{g_{n,i}})$. ∎

In the following theorem, we determine a simplified expression of PDF of the
Exponentiated Hypoexponential distribution of different parameters.

######  Theorem 3.

Let $Y\sim
EHypo\exp\left(n,\overrightarrow{\alpha},k\right),\overrightarrow{\alpha}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right)$
for $1\leq i\leq n$. Then the PDF of $Y$ is

$f_{Y}(t)=\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}f_{N_{i}}(t)$ (12)

######  Proposition 1.

Let $B_{i}$ be the coefficient of the Exponentiated Hypoexponential
distribution, defined as
$B_{i}=\binom{k}{g_{1,i},g_{2,i},\ldots,g_{n,i}}(A_{1}^{g_{1,i}}A_{2}^{g_{2,i}}\ldots
A_{n}^{g_{n,i}})$, where $A_{i}=\prod_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)$. Let
$E_{k}=\left\\{\overrightarrow{g}_{i}=(g_{1,i},g_{2,i},\ldots,g_{n,i})\middle|0\leq
g_{j,i}\leq k,\ \sum_{j=1}^{n}g_{j,i}=k\right\\}$, and let
$N_{i}\sim\text{MEE}(\overrightarrow{\alpha},\overrightarrow{g_{i}})$. Then,
$\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}=1$.

###### Proof.

From Theorem 2, we have $F_{Y}\left(t\right)\ =\sum_{E_{k}}B_{i}F_{N_{i}}(t)$
and when we differentiate both sides we obtain knowing

$f_{Y}(t)=F_{Y}^{\prime}\left(t\right)\
=\frac{d\left(\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}F_{N_{i}}(t)\right)}{dt}=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}\frac{dF_{N_{i}}(t)}{dt}=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}f_{N_{i}}(t).$

∎

######  Proposition 2.

Let $B_{i}$ be the coefficient of the Exponentiated Hypoexponential
distribution. Then $\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}=1$, where
$E_{k}=\\{\overrightarrow{g}_{i}=(g_{1,i},g_{2,i},...,g_{n,i})\ |\ 0\leq
g_{j,i}\leq k,\ \sum_{j=1}^{n}g_{j,i}=k\\}$.

###### Proof.

Let $F_{Y}(t)$ and $F_{N_{i}}(t)$ be the CDFs of
$\text{EHypoexp}(n,\overrightarrow{\alpha},k)$ and
$\text{MEE}(\overrightarrow{\alpha}\overrightarrow{g_{i}},k)$, respectively.
The CDF for any random variable $X$, $P(X<t)=1$ as $t\rightarrow+\infty$.
Thus, $\lim_{t\rightarrow+\infty}F_{Y}(t)=1$ and
$\lim_{t\rightarrow+\infty}F_{N_{i}}(t)=1$. But from Eq. (11), we have:

$\displaystyle F_{Y}(t)$ $\displaystyle=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}F_{N_{i}}(t)$ $\displaystyle\lim_{t\rightarrow+\infty}F_{Y}(t)$
$\displaystyle=\lim_{t\rightarrow+\infty}\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}F_{N_{i}}(t)$ $\displaystyle=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}\lim_{t\rightarrow+\infty}F_{N_{i}}(t).$

Hence, $\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}=1$. ∎

######  Theorem 4.

Let $S_{n}\sim Hypoexp(\overrightarrow{\alpha})\ $where
$n\in\mathbb{N}^{\ast}$ and $\alpha_{i}>0,$ $i=1,2,...,n$ and let $Y$ be the
Exponentiated Hypoexponential distribution with exponent parameter
$k\in\mathbb{N}^{\ast}$ denoted as $Y\sim
EHypo\exp\left(n,\overrightarrow{\alpha},k\right),\overrightarrow{\alpha}=\left(\alpha_{1},\alpha_{2},...,\alpha_{n}\right).$
Then the reliability function of $Y$ is

$R_{Y}\left(t\right)=\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}R_{N_{i}}(t)$
(13)

and hazard function of $Y$ is

$h_{Y}\left(t\right)=\frac{\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}f_{N_{i}}(t)}{\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}R_{N_{i}}(t)}$ (14)

###### Proof.

We have $R_{Y}(t)=1-F_{Y}\left(t\right)$, then and by Theorem 2, we get
$R_{Y}(t)=1-\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}F_{N_{i}}(t)$. Now,
since $F_{N_{i}}(t)=1-R_{N_{i}}\left(t\right)$ and
$\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}=1$ from Proposition 2, we obtain
that $R_{Y}\left(t\right)=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}R_{N_{i}}(t).$

On the other hand, the hazard function of $Y$ is given as
$h_{Y}(t)=\frac{f_{Y}(t)}{R_{Y}(t)}$. Also from Theorem 12 we have the
expression of $f_{Y}\left(t\right)$ and substituting the expression of
$R_{Y}(t)$ in Eq. (13) we the obtain our result. ∎

###  4.1 Applications

######  Example 4.1.

Suppose that $X_{1},$ $X_{2}\ $and $X_{3}$ are the task times of $n=3$ Markov
chain that are Exponentially distributed as $X_{j}\sim Exp(\alpha_{j}),$
$j=1,2,3$ with $\alpha_{1}=5,$ $\alpha_{2}=4,$ $\alpha_{3}=3$.
$S_{3}=X_{1}+X_{2}+X_{3}$ is the proposed total task time of this system which
is a Hypoexponential distribution. Now, let $Y$ be the Exponentiated
Hypoexponential distribution with exponent parameter $k=3$. From Theorem 12

$f_{Y}(t)=\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}f_{N_{i}}(t).$

First $E_{3}=\\{\overrightarrow{g}_{i}=(g_{1,i},g_{2,i},g_{3,i})/0\leq
g_{j,i}\leq 3,\sum_{j=1}^{3}g_{j,i}=3\\}=\\{\overrightarrow{g}_{1}=(3,0,0),$
$\overrightarrow{g}_{2}=(0,3,0),$ $\overrightarrow{g}_{3}=(0,0,3),$
$\overrightarrow{g}_{4}=(2,1,0),$ $\overrightarrow{g}_{5}=(2,0,1),$
$\overrightarrow{g}_{6}=(1,2,0),$ $\overrightarrow{g}_{7}=(0,2,1),$
$\overrightarrow{g}_{8}=(1,0,2),$ $\overrightarrow{g}_{9}=(0,1,2),$
$\overrightarrow{g}_{10}=(1,1,1)\\}$.

Having $A_{i}=\prod\limits_{j=1,j\neq
i}^{n}(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}),$ then $A_{1}=6,$
$A_{2}=-15,$ $A_{3}=10,$

and
$B_{i}=\binom{3}{g_{1,i},g_{2,i},g_{3,i}}(A_{1}^{g_{1,i}}A_{2}^{g_{2,i}}A_{3}^{g_{3,i}}),$

and the PDF of $N_{i}\sim MEE(\overrightarrow{\alpha},\overrightarrow{g}_{i})$
derived from 1 is

$f_{N_{i}}(t)=\prod\limits_{j=1}^{3}(1-e^{-\alpha_{j}t})^{g_{i,j}}\times\sum_{j=1}^{3}\frac{g_{i,j}\alpha_{j}e^{-\alpha_{j}t}}{1-e^{-g\alpha_{j}t}},$

and then the PDF of $Y$ will have the form

$\displaystyle f_{Y}(t)$ $\displaystyle=$
$\displaystyle\sum_{\overrightarrow{g}_{i}\in
E_{k}}\binom{3}{g_{1,i},g_{2,i},g_{3,i}}(6)^{g_{1,i}}(-15)^{g_{2,i}}(10)^{g_{3,i}}$
$\displaystyle\times\prod\limits_{j=1}^{3}(1-e^{-\alpha_{j}t})^{g_{i,j}}\sum_{j=1}^{3}\frac{g_{i,j}\alpha_{j}e^{-\alpha_{j}t}}{1-e^{-g\alpha_{j}t}}$
$\displaystyle=$ $\displaystyle
90e^{-15t}(-1+e^{t})^{8}(6+3e^{t}+e^{2t})^{2}.$

Its figure is shown in Fig. 1.

Figure 1: The PDF of the EHyp.

Next, we use Theorem 11 to find the CDF of $Y$ as

$\displaystyle F_{Y}(t)$ $\displaystyle=$ $\displaystyle\sum_{\vec{g}_{i}\in
E_{k}}B_{i}F_{N_{i}}(t)$ $\displaystyle=$ $\displaystyle\sum_{\vec{g}_{i}\in
E_{k}}\binom{3}{g_{1,i},g_{2,i},g_{3,i}}(6)^{g_{1,i}}(-15)^{g_{2,i}}(10)^{g_{3,i}}\times\prod_{j=1}^{3}(1-e^{-\lambda_{j}t})^{\alpha_{j}}$
$\displaystyle=$ $\displaystyle e^{-15t}(-1+e^{t})^{9}(6+3e^{t}+e^{2t})^{3}$

Note that, if we integrate $f_{Y}(t)$, we get the same result. The figure of
$F_{Y}(t)$ is represented in Fig 2.

Figure 2: The CDF of the EHyp.

Now, from Theorem 13, we can find reliability function for $Y$ as

$\displaystyle R_{Y}(t)=\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}R_{N_{i}}(t).$

However, the expression of reliability function for Maximum Exponentiated
Exponential Distribution is given in Eq. (9) as
$R_{N}(t)=1-\prod_{j=1}^{n}\sum_{i=0}^{g_{j}}\binom{g_{j}}{i}(-1)^{i}e^{-\lambda_{i}(n-g_{j})t_{j}}.$

$\displaystyle R_{Y}\left(t\right)$ $\displaystyle=$
$\displaystyle\sum_{\overrightarrow{g}_{i}\in E_{k}}B_{i}R_{N_{i}}(t)$
$\displaystyle=$ $\displaystyle
1-e^{(-15t)}(-1+e^{t})^{9}(6+3e^{t}+e^{(2t)})^{3}$

The figure of $R_{Y}(t)$ is represented in Fig. 3.

Figure 3: The Reliability of the EHyp.

To understand the concept of $R_{Y}(2)=0.0588886,$ which means that the
probability that the system is still functioning (survive) after 2 years is
$0.0588886$ which is a low probability. Thus, we can consider that this system
will stop after 2 years or more.

Next, to find the hazard function of $Y,$ use Theorem 14. The hazard’s
expression is

$\displaystyle h_{Y}(t)$ $\displaystyle=$
$\displaystyle\frac{f_{Y}(t)}{R_{Y}(t)}$ $\displaystyle=$
$\displaystyle\frac{\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}f_{N_{i}}(t)}{\sum_{\overrightarrow{g}_{i}\in
E_{k}}B_{i}R_{N_{i}}(t)}$ $\displaystyle=$
$\displaystyle\frac{3(1-6e^{(-5t)}+15e^{(-4t)}-10e^{(-3t)})^{2}(30e^{(-5t)}-60e^{(-4t)}+30e^{(-3t)})}{1-e^{(-15t)}(-1+e^{t})^{9}(6+3e^{t}+e^{(2t)})^{3}}$

Figure 4: The Hazard of the EHyp.

In Figs. 3 and 4, we show the curves of reliability and hazard functions of
$Y.$ Note that, Fig. 4 shows that the hazard function of $Y$ is not constant
unlike the hazard of the Exponential distribution. The hazard function of $Y$
is a monotonic increasing function of $t$, then the system is described to be
”wear” out with time, see, [12]. In other words, the system is damaged as a
result of normal wearing out or aging with time. Fig. 4 for $h_{Y}(t)$ reveals
that the system takes approximately 10 years to wear out.

##  5 Maximum Likelihood Estimation Of Parameters In Exponentiated
Hypoexponential Distribution

Let $X_{1},X_{2},...,X_{N}$ be a random sample from the
$EHypo\left(\overrightarrow{\alpha},k\right)$ distribution with unknown
parameters $\alpha_{1},\alpha_{2},...,\alpha_{n},k$. We consider the maximum
likelihood estimation providing the maximum likelihood estimators (MLEs)
$\widehat{\alpha}_{1},\widehat{\alpha}_{2},...,\widehat{\alpha}_{n},\widehat{k}$
for the parameters $\alpha_{1},\alpha_{2},...,\alpha_{n},k$.

Let us recall that the MLEs have some statistical desirable properties (under
regularity conditions) as the sufficiency, invariance, consistency, efficiency
and asymptotic normality. Using the observed information matrix, asymptotic
confidence interval for
$\widehat{\alpha}_{1},\widehat{\alpha}_{2},...,\widehat{\alpha}_{n},$ and
$\widehat{k}$ can be constructed.

Let $Y\sim EHyp\left(\overrightarrow{\alpha},k\right)$, $\alpha_{i}>0,$
$i=1,2,...,n,$ and $N$ is a size of a random sample. Then the maximum
likelihood estimators of $\alpha_{1},\alpha_{2},...,\alpha_{n},$ and $k$ is
denoted respectively by
$\widehat{\alpha}_{1},\widehat{\alpha}_{2},...,\widehat{\alpha}_{n}$ and
$\widehat{k}$ and has the form:

$\widehat{k}=\frac{-N}{\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)}$

and $\widehat{a}_{p}$ verify the following implicit equation:

$\begin{split}&\sum_{j=1}^{N}\frac{\sum_{i=1}^{n}\left(\left(\sum_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}f_{X}(x_{i})+A_{i}\left((1-\alpha_{q}x_{j})e^{-\alpha_{q}x_{j}}\right)\right)}{\sum_{i=1}^{n}A_{i}f_{X}(x_{i})}\\\
&+(k-1)\sum_{j=1}^{N}\frac{\sum_{i=1}^{n}\left(\left(\sum_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}F_{X}(x_{i})+A_{i}(x_{j}e^{-\alpha_{p}x_{j}})\right)}{\sum_{i=1}^{n}A_{i}F_{X}(x_{i})}\equiv
0\end{split}$

In finding the estimators, the first thing we will do is write the probability
density function as a function of $k$.

$\displaystyle f_{Y}\left(x_{i},\alpha_{i},k\right)$ $\displaystyle=$
$\displaystyle
k\left(\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{i})\right)\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{i})\right)^{k-1}$
$\displaystyle=$ $\displaystyle
k\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}})\right)\left(\alpha_{i}e^{-\alpha_{i}x_{i}}\right)\right)$
$\displaystyle\times\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}})\right)\left(1-e^{-\alpha_{i}x_{i}}\right)\right)^{k-1}$

for $k>0$. Now, that makes the likelihood function:

$\displaystyle L\left(k,\overrightarrow{\alpha}\right)$ $\displaystyle=$
$\displaystyle\prod\limits_{j=1}^{N}k\left(\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)^{k-1}$
$\displaystyle=$ $\displaystyle
k^{N}\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)^{k-1}$
$\displaystyle=$ $\displaystyle
k^{N}\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(\alpha_{i}e^{-\alpha_{i}x_{i}}\right)\right)$
$\displaystyle\times\left(\prod\limits_{j=1}^{N}\left(\prod\limits_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(1-e^{-\alpha_{i}x_{i}}\right)\right)^{k-1}$

Where $i=1,2,...,n$ and $j=1,2,...,N.~{}$and therefore the log of the
likelihood function:

$\displaystyle\log\left(L\left(k,\overrightarrow{\alpha}\right)\right)$
$\displaystyle=$
$\displaystyle\log\left(k^{N}\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)^{k-1}\right)$
$\displaystyle=$ $\displaystyle
N\log\left(k\right)+\log\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)$
$\displaystyle+\log\left(\left(\prod\limits_{j=1}^{N}\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)^{k-1}\right)$
$\displaystyle=$ $\displaystyle
N\log\left(k\right)+\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(\alpha_{i}e^{-\alpha_{i}x_{i}}\right)\right)$
$\displaystyle+(k-1)\left(\sum\limits_{j=1}^{N}\log\left(\left(\prod\limits_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(1-e^{-\alpha_{i}x_{i}}\right)\right)\right)$

Now, upon taking the partial derivative of
$\log\left(L\left(k,\alpha_{i}\right)\right)$ with respect to the parameters
$k$ and then setting it to $0$.

$\frac{d}{dk}\log\left(L\left(k,\alpha_{i}\right)\right)=0$
$\frac{d}{dk}\left(n\log\left(k\right)+\sum\limits_{j=1}^{n}\log\left(\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)+(k-1)\left(\sum\limits_{j=1}^{n}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)\right)\right)\equiv
0$
$\begin{array}[]{c}\frac{d}{dk}(N\log\left(k\right))+\frac{d}{dk}\left(\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}f_{X_{i}}(x_{j})\right)\right)\\\
+\frac{d}{dk}\left((k-1)\left(\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)\right)\right)\equiv
0\end{array}$
$\frac{N}{k}+\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)\equiv
0$

Now, multiplying through by $k$, and distributing the summation, we get

$N+k\left(\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)\right)\equiv
0$

Now, solving for $k$, and putting its estimate, we have shown that the MLE of
$k$ is

$\widehat{k}=\frac{-N}{\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)}$

Again, by taking the partial derivative of the log likelihood with respect to
$\alpha_{q}$, and setting it $0$, leaving us with:

$\frac{\partial}{\partial\alpha_{q}}\log\left(L\left(k,\alpha_{i}\right)\right)=0$

$\frac{\partial}{\partial\alpha_{q}}\left(\begin{array}[]{c}N\log\left(k\right)+\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\left(\alpha_{i}e^{-\alpha_{i}x_{j}}\right)\right)\\\
+(k-1)\left(\sum\limits_{j=1}^{N}\log\left(\prod\limits_{j=1}^{N}\left(\prod\limits_{j=1,j\neq
i}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\left(1-e^{-\alpha_{i}x_{j}}\right)\right)\right)\end{array}\right)\equiv
0$
$\displaystyle\frac{\partial}{\partial\alpha_{p}}(N\log\left(k\right))+\frac{\partial}{\partial\alpha_{p}}\left(\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\left(\alpha_{i}e^{-\alpha_{i}x_{j}}\right)\right)\right)$
$\displaystyle+\frac{\partial}{\partial\alpha_{p}}\left((k-1)\left(\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}})\right)\left(1-e^{-\alpha_{i}x_{j}}\right)\right)\right)\right)\equiv
0$
$\begin{array}[]{c}\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
q}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{q}}\right)\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)\left(\alpha_{i}e^{-\alpha_{i}x_{j}}\right)+\left(\prod\limits_{j=1,j\neq
q}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{q}}\right)\left(\left(1-\alpha_{q}x_{j}\right)e^{-\alpha_{q}x_{j}}\right)}{\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\left(\alpha_{i}e^{-\alpha_{i}x_{i}}\right)}+(k-1)\\\
\times\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\left(\left(\prod\limits_{j=1,j\neq
q}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{q}}\right)\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)\right)\left(1-e^{-\alpha_{i}x_{j}}\right)+\left(\prod\limits_{j=1,j\neq
q}^{n}\frac{\alpha_{j}}{\alpha_{j}-\alpha_{q}}\right)\left(x_{j}e^{-\alpha_{p}x_{j}}\right)\right)}{\sum\limits_{i=1}^{n}\left(\prod\limits_{j=1,j\neq
i}^{n}(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}})\right)\left(1-e^{-\alpha_{i}x_{j}}\right)}\equiv
0\end{array}$
$\begin{array}[]{c}\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}f_{X}\left(x_{i}\right)+A_{i}\left(\left(1-\alpha_{q}x_{j}\right)e^{-\alpha_{q}x_{j}}\right)\right)}{\sum\limits_{i=1}^{n}A_{i}f_{X}\left(x_{i}\right)}\\\
+(k-1)\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}F_{X}\left(x_{i}\right)+A_{i}\left(x_{j}e^{-\alpha_{p}x_{j}}\right)\right)}{\sum\limits_{i=1}^{n}A_{i}F_{X}\left(x_{i}\right)}\equiv
0\end{array}$

Some numerical iterative methods, such as Newton-Raphson method can be used
since this equation is not solvable analytically. The solution can be
approximate numerically by using software such as MAPLE, R, and MATHEMATICA.
Here we worked with MATHEMATICA, see, Wolfram (1999).

###  5.1 ILLUSTRATIVE REAL DATA EXAMPLES

In this subsection, we analyze real data sets to show that the
$EHypo\exp\left(\overrightarrow{\alpha},k\right)$ distribution can be a better
model than other existing distributions.

Let $Y\sim EHypo\exp\left(\overrightarrow{\alpha},k\right)$, where
$\overrightarrow{\alpha}$ $=\left(\alpha_{1},\alpha_{2}\right)$ is the
EHypoexponential with exponent parameter $k$. The data set contains $n=128$
measures on the remission times in months of bladder cancer patients. It is
extracted from Lee and Wang [13]:

$0.08,2.09,3.48,4.87,6.94,8.66,13.11,23.63,0.20,2.23,3.52,4.98,6.97,9.02,13.29,$
---
$0.40,2.26,3.57,5.06,7.09,9.22,13.80,25.74,0.50,2.46,3.64,5.09,7.26,9.47,14.24,$
$25.82,0.51,2.54,0.08,2.09,3.48,4.87,6.94,8.66,13.11,23.63,0.20,2.23,3.52,4.98,$
$6.97,9.02,13.29,0.40,2.26,3.57,5.06,7.09,9.22,13.80,25.74,0.50,2.46,3.64,5.09,$
$7.26,9.47,14.24,25.82,0.51,2.54,3.70,5.17,7.28,9.74,14.76,26.31,0.81,2.62,3.82,$
$5.32,7.32,10.06,14.77,32.15,2.64,3.88,5.32,7.39,10.34,14.83,34.26,0.90,2.69,$
$4.18,5.34,7.59,10.66,15.96,36.66,1.05,2.69,4.23,5.41,7.62,10.75,16.62,43.01,$
$1.19,2.75,4.26,5.41,7.63,17.12,46.12,1.26,2.83,4.33,7.66,11.25,17.14,79.05,$
$1.35,2.87,5.62,7.87,11.64,17.36,1.40,3.02,4.34,5.71,7.93,11.79,18.10,1.46,4.40,$
$5.85,8.26,11.98,19.13,1.76,3.25,4.50,6.25,8.37,12.02,2.02,3.31,4.51,6.54,8.53,$
$12.03,20.28,2.02,3.36,6.76,12.07,21.73,2.07,3.36,6.93,8.65,12.63,22.69,5.49.$

The probability density function of $Y$ is:

$\displaystyle f_{Y}(x_{i},\alpha_{i},k)$ $\displaystyle=$ $\displaystyle
k\left(\sum_{i=1}^{n}\left(\prod_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(\alpha_{i}e^{-\alpha_{i}x_{i}}\right)\right)$
$\displaystyle\times\left(\sum_{i=1}^{n}\left(\prod_{j=1,j\neq
i}^{n}\left(\frac{\alpha_{j}}{\alpha_{j}-\alpha_{i}}\right)\right)\left(1-e^{-\alpha_{i}x_{i}}\right)\right)^{k-1}$

for $-\infty<x<\infty,$ with parameters $k~{}$and $\alpha_{i}.$ Thus, the
likelihood function is:

$\displaystyle L(x_{i},\alpha_{i},k)$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{N}k\left(\frac{e^{-x_{j}\alpha_{2}}\alpha_{1}\alpha_{2}}{\alpha_{1}-\alpha_{2}}-\frac{e^{-x_{j}\alpha_{1}}\alpha_{1}\alpha_{2}}{\alpha_{2}-\alpha_{1}}\right)$
$\displaystyle\times\left(1-\frac{e^{-x_{j}\alpha_{2}}\alpha_{1}}{\alpha_{1}-\alpha_{2}}-\frac{e^{-x_{j}\alpha_{1}}\alpha_{2}}{\alpha_{2}-\alpha_{1}}\right)^{k-1}$
$\displaystyle=$ $\displaystyle
k^{N}\left(\prod_{j=1}^{N}\left(\frac{e^{-x_{j}\alpha_{2}}\alpha_{1}\alpha_{2}}{\alpha_{1}-\alpha_{2}}-\frac{e^{-x_{j}\alpha_{1}}\alpha_{1}\alpha_{2}}{\alpha_{2}-\alpha_{1}}\right)\right)$
$\displaystyle\times\left(\prod_{j=1}^{N}\left(1-\frac{e^{-x_{j}\alpha_{2}}\alpha_{1}}{\alpha_{1}-\alpha_{2}}-\frac{e^{-x_{j}\alpha_{1}}\alpha_{2}}{\alpha_{2}-\alpha_{1}}\right)^{k-1}\right)$

Where $i=1,2,...,n$ and $j=1,2,...,N.$

It can be shown, upon maximizing the likelihood function with respect to $k$
,$\alpha_{1}$ and $\alpha_{2}$, that the maximum likelihood estimator of $k$
is:

$\widehat{k}=\frac{-N}{\sum\limits_{j=1}^{N}\log\left(\sum\limits_{i=1}^{n}A_{i}F_{X_{i}}(x_{j})\right)}$

and $\alpha_{1}$ and $\alpha_{2}$ verify the following implicit Eq.

$\begin{array}[]{c}\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}f_{X}\left(x_{i}\right)+A_{i}\left(\left(1-\alpha_{q}x_{j}\right)e^{-\alpha_{q}x_{j}}\right)\right)}{\sum\limits_{i=1}^{n}A_{i}f_{X}\left(x_{i}\right)}\\\
+(k-1)\sum\limits_{j=1}^{N}\frac{\sum\limits_{i=1}^{n}\left(\left(\sum\limits_{v=1,v\neq
q}^{n}\frac{1}{\alpha_{v}-\alpha_{q}}\right)A_{q}F_{X}\left(x_{i}\right)+A_{i}\left(x_{j}e^{-\alpha_{p}x_{j}}\right)\right)}{\sum\limits_{i=1}^{n}A_{i}F_{X}\left(x_{i}\right)}\equiv
0\end{array}$

Based on the given sample, and using MATHEMATICA the maximum likelihood
estimate of $k$ ,$\alpha_{1}$ and $\alpha_{2}$ are:

$\displaystyle\widehat{k}$ $\displaystyle=$ $\displaystyle 0.51541$
$\displaystyle\widehat{\alpha}_{1}$ $\displaystyle=$ $\displaystyle 0.105418$
$\displaystyle\widehat{\alpha}_{2}$ $\displaystyle=$ $\displaystyle 0.718769$

For the data set, we compare the fitted distributions using the criteria:
-2log(L), Akaike information criterion (AIC), Akaike information criterion
corrected (AICC), Bayesian information criterion (BIC), Anderson-Darling
($A^{\ast}$), and Cramer-von Mises ($W^{\ast}$). Let us be accurate that
$\log(L)$ is the log-likelihood taken with the estimated values,
AIC=-2log(L)+2c, AICC = AIC + $\frac{2c(c+1)}{v-c-1}$, and BIC = -2log(L) +
clog(v), where c denotes the number of estimated parameters and v denotes the
sample size. The best-fitted distribution corresponds to lower -2log(L), AIC,
AICC, BIC, $A^{\ast}$, and $W^{\ast}$. We see in Tables 1 and 2 that the
$EHyp\exp(\left\\{a_{1},a_{2}\right\\},k)$ distribution has the smallest
-2log(L), AIC, AICC, BIC, $A^{\ast}$, and $W^{\ast}$ for the two data sets
(except for the data set where the Hypexp distribution has the smallest BIC),
indicating that it is a serious competitor to the other considered
distributions. Also, graphically, the behaviors of our model, Figure 2, show
the fitted densities superimposed on the histogram of the given data sets.

Model | Estimate of parameters | $-2log(L)$ | $AIC$ | $AICC$ | $BIC$
---|---|---|---|---|---
$Hyp\exp$ |  | $\widehat{\alpha}_{1}=0.114536$
---
$\widehat{\alpha}_{2}=1.57547$
$826.09$ | $1082.09$ | $822.058$ | $914.813$
$EHyp\exp$ |  | $\widehat{\alpha}_{1}=0.105418$
---
$\widehat{\alpha}_{2}=0.718769$
$\widehat{k}=0.51541$
$823.33$ | $1079.33$ | $817.234$ | $963.952$
Model | $A^{\ast}$ | $W^{\ast}$
---|---|---
$Hyp$ | $0.541467$ | $0.0889889$
$EHyp$ | $0.455972$ | $0.0767364$
Table 1: Results for Model Parameters Figure 5: The Fitted Densities
Superimposed on the Histogram of Data Sets.

##  6 Conclusion And Open Problems

CHARACTERIZATION & ESTIMATION

In this article we investigated the problem of characterizing the PDF, some
statistical functions and maximum likelihood estimation of parameters of the
EHypoexponential random variables that have applications in many scientific
areas.

We derived expressions for PDF, CDF, and some statistical parameters of
Maximum Exponentiated Exponential random variable with different parameters
that are later used to obtain our results. Moreover, we determined closed
expressions for PDF, CDF, some statistical parameters and examined the Maximum
Likelihood Estimation of parameters of EHypoexponential Distribution.

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