# Stochastic comparisons of record values based on their relative aging

Mohamed Kayid†,

Department of Statistics and Operations Research
College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi
Arabia

Corresponding author, Email address<EMAIL_ADDRESS>
###### Abstract

In this paper we examine some relative orderings of upper and lower records.
It is shown that if $m>n$, the $m$th upper record ages faster than the $n$th
upper record, where the data sets come from a sequence of independent and
identically distributed observations from a continuous distribution.
Sufficient conditions are also obtained to see whether the $m$th upper record
arisen from a continuous distribution ages faster in terms of the relative
hazard rate than the $n$th upper record arisen from another continuous
distribution. It is also shown that the reversed hazard rate of the $m$th
lower record decreases faster than the reversed hazard rate of the $n$th lower
record, when $m>n$. Preservation property of the relative reversed hazard rate
order at lower record values is investigated. Several examples are presented
to examine the results.

Key Words: Upper record values, Lower record values, Relative aging,
Stochastic order.

##  1 Introduction

Order statistics result from a random sample of finite size, independent of
the order of the observations in the sample. The extreme values of the sample
are referred to as minimum order statistic and maximum order statistic. Record
values are the extreme values of a sequence of random samples that converge to
the extremes of the distribution under consideration. Formally, successive
observations of a single random variable, called a sequence of random
variables, yield two statistics, namely the upper record value and the lower
record value. Extreme amounts of a random sequence of random variables are
available as record values and, as a result, record values are important
statistics that are useful to show the extremes of the distribution of
observations. The record values have been extensively studied in the
literature thus far. Because they are found in so many academic fields such as
climatology, athletics, medicine, transportation, industry and so on, records
are very popular. These records serve as time capsules. Through the history of
records, we can examine how science and technology have developed and judge
humanity based on historical achievements in a number of fields. Numerous
long-term records have inspired the development of various mathematical models
that reflect related record-keeping processes and predict future record
outcomes. Records have been thoroughly studied in the literature. Considerable
progress has been made in the field of stochastic orderings of record values.
We cite the following sources for a selection of results: Khaledi [26], Hu and
Zhuang [21], Raqab and Amin [39], Ahmadi and Arghami [3], Belzunce et al.
[10], Kayid et al. [25], Esna-Ashari et al. [16], Aswin et al. [7] and
references therein. Raqab and Asadi [40] have published more recent work on
the mean residual lives of data sets considering reliability, aging aspects
and other relevant properties. For more information on distribution theory and
its applicability to different types of data modeled by record values, see
Ahsanullah [2] and Arnold et al. [5].

Aging is a universal phenomenon that affects both mechanical systems and
living beings. The age of a living organism is the time $t$ at which this
organism is still alive and functioning. The aging process of an object is
usually due to the wear and tear of the object. It usually explains how a
system or living being gets better or worse as it gets older. In recent
decades, researchers have been intensively studying stochastic aging. Numerous
ideas on stochastic aging have been developed in the literature to
characterize various aging features of a system, such as the increasing
failure rate (IFR), the increasing failure rate average (IFRA), etc. There are
three categories of aging: no aging, negative aging and positive aging. Two
publications that provide a quick overview of different aging principles are
Barlow and Proschan [8] and Lai and Xie [30]. Comparative aging is a useful
idea that describes how one system ages relative to another and is similar to
these other aging terms.

The term “relative aging” refers to the aging of one system compared to
another. The commands for faster aging are those that compare the respective
ages of the two systems. The relative age of two systems can be described by
two additional sets of stochastic orders. The convex transformation order, the
quantitative mean inactive time order, the star-shaped order, the
superadditive order, the DMRL (decreasing mean residual life) order, the s-IFR
order, and other so-called transformation orders belong to the first set of
stochastic orders, which characterize whether one system ages faster than
another in terms of increasing failure rate, increasing failure rate on
average, new better than used, etc. Barlow and Proschan [8], Bartoszewicz [9],
Deshpande and Kochar [14], Kochar and Wiens [28], Arriaza et al [4], Nanda et
al [36] and the references therein provide a thorough treatment of these
orderings. The monotonicity of the ratios of certain reliability measures,
such as the mean residual life function, the inverted hazard rate function and
the hazard rate function, defines the second group of stochastic orders, often
referred to as faster aging orders. We recommend reading Kalashnikov and
Rachev [22], Sengupta and Deshpande [42], Di Crescenzo [15], Finkelstein [17],
Razaei et al. [41], Hazra and Nanda [19], Misra et al [34], Kayid et al [24]
and Misra and Francis [35] for information on the foundations and
applicability of these orders.

In reliability and survival analysis, the proportional hazard (PH) model, also
known as Cox’s PH model (see Cox [13]), is often used to analyze failure time
data. Later, other models were developed, including the proportional mean
remaining life model, the proportional odds model, the proportional inverse
hazard rate model, and others (see Marshall and Olkin [32], Lai and Xie [30],
and Finkelstein [18]). The phenomenon of crossing hazards and, further,
crossing mean residual lifetimes has been observed in numerous real-world
scenarios (see Mantel and Stablein [31], Pocock et al. [38] and Champlin et
al. [12]). Based on the idea of relative aging, Kalashnikov and Rachev [22]
developed a stochastic ordering (called faster aging ordering in the hazard
rate) to address this problem of crossing hazard rates. In fact, this strategy
could be seen as a solid replacement for the PH model. Sengupta and Deshpande
[42] give a thorough analysis of this order. They also presented two other
related types of stochastic ordering. Subsequently, Razaei et al. [41] offered
a similar stochastic ordering in the form of the reversed hazard rate
functions, while Finkelstein [17] proposed a stochastic ordering based on the
mean residual life functions characterizing the relative ages of two life
distributions. Hazra and Nanda [19] presented some generalized orderings in
this direction.

However, the problem of the relative aging ordering of record values has not
yet been considered in the literature. The aim of the present study is to
initiate such an investigation in order to develop preservation properties of
the relative hazard rate order and the relative reversed hazard rate order
under upper records and lower records, respectively.

The structure of the paper is as follows. In Section 2, we introduce some
preliminary concepts and definitions. In Section 3, we investigate the
preservation property of the relative hazard rate order under upper records.
In Section 4, preservation property of the relative reversed hazard rate order
under lower records is investigated. In Section 5, we conclude the paper with
further remarks and illustrations on current and future research.

##  2 Preliminaries

In this section, we bring some preliminaries that will be used throughout the
paper. The definitions of the ageing faster orders utilized in our paper are
provided below.

######  Definition 2.1.

Let $X$ and $Y$ be two absolutely continuous random variables with hazard rate
functions $h_{X}$ and $h_{Y}$, respectively, and reversed hazard rate
functions $r_{X}$ and $r_{Y}$, respectively. It is then said that $X$ ages
faster than $Y$ in

  * (i)

hazard rate, denoted by $X\preceq_{c}Y$, if

$\frac{h_{X}(t)}{h_{Y}(t)}\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{is
increasing in}\leavevmode\nobreak\ t\geq 0.$

  * (ii)

reversed hazard rate, denoted by $X\preceq_{b}Y$, if

$\frac{r_{X}(t)}{r_{Y}(t)}\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{is
decreasing in}\leavevmode\nobreak\ t\geq 0.$

Let us look at a technical system that experiences shocks like voltage peaks.
A set of independent and identically distributed (i.i.d.) rvs $\\{X_{i},i\geq
1\\}$ with a common continuous cdf $F_{X}$, pdf $f_{X}$ and survival function
$\bar{F}_{X}\equiv 1-F_{X}$ can then be used to simulate the shocks. The loads
on the system at different points in time are represented by the shocks. The
record statistics (values of the highest stresses reported so far) of this
sequence are of interest to us. As we consider the lifetime of devices in our
context, we thus suppose that the observations which produce record values are
non-negative. Hence, the record values are consequently non-negative. The
$i$-th order statistics from the first $n$ observations are labeled with the
symbol $X_{i:n}$.

Next, we define the upper record values $X_{U_{n}},$ and upper record timings
$\\{T_{n},n\geq 1\\}$, respectively, as follows:

$X_{U_{n}}=X_{T_{n}:T_{n}},\ n=0,1,\ldots,$

where

$T_{0}=1,\ T_{n}=\min\\{j:j>T_{n-1},X_{j}>X_{U_{n}}\\},\ n\geq 1.$

It is commonly known that the pdf and sf of $X_{U_{n}},$ represented by
$f_{X_{U_{n}}}(x)$ and $\bar{F}_{X_{U_{n}}}(x)$ respectively, are given by

$f_{X_{U_{n}}}(x)=\frac{[-\log(\bar{F}_{X}(x))]^{n-1}}{(n-1)!}f_{X}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ x\geq 0,$ (1)

and

$\displaystyle\bar{F}_{X_{U_{n}}}(x)$
$\displaystyle=\int_{-\log(\bar{F}_{X}(x))}^{+\infty}\frac{u^{n-1}}{(n-1)!}e^{-u}du$
$\displaystyle=\bar{F}_{X}(x)\,\sum_{k=0}^{n-1}\frac{[-\log(\bar{F}_{X}(x))]^{k}}{k!},\leavevmode\nobreak\
\leavevmode\nobreak\ x\geq 0,$ (2)

where the last identity is derived by using the expansion of incomplete gamma
function (see e.g. [6]). In contrast to the upper record values are the lower
record values. The $n$th lower record time $L(n),n=1,2,\ldots$ with $L(1)=1$
is stated as

$L(1)=1,\ L(n+1)=\min\\{j:j>L(n),X_{1:L(n)}>X_{1:j}\\},\ n=1,2,\ldots.$

and the $n$-th lower record is enumerated as
$X_{L_{n}}=X_{1:L(n)},n=1,2,\ldots.$ The pdf of $X_{L_{n}}$ can be acquired as

$f_{X_{L_{n}}}(x)=\frac{[-\ln(F_{X}(x))]^{n-1}}{(n-1)!}f_{X}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ x\geq 0.$ (3)

In addition, $X_{L_{n}}$ has cdf:

$\displaystyle F_{X_{L_{n}}}(x)$
$\displaystyle=\int_{-\log(F_{X}(x))}^{+\infty}\frac{u^{n-1}}{(n-1)!}e^{-u}du$
$\displaystyle=F_{X}(x)\,\sum_{k=0}^{n-1}\frac{[-\ln(F_{X}(x))]^{k}}{k!};\;x\geq
0.$ (4)

Three stochastic orders which consider magnitude of random variables rather
than their relative aging behaviours are defined below according to Shaked and
Shanthikumar [43].

######  Definition 2.2.

Let us suppose that $X$ and $Y$ represent the lifetime of two devices. It is
said that $X$ is smaller than $Y$ in the

  * (i)

hazard rate order, denoted by $X\preceq_{hr}Y$, if

$h_{X}(t)\geq h_{Y}(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{for
all}\leavevmode\nobreak\ t\geq 0.$

  * (ii)

reversed hazard rate order, denoted by $X\preceq_{rh}Y$, if

$r_{X}(t)\leq r_{Y}(t)\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{for
all}\leavevmode\nobreak\ t>0.$

  * (iii)

usual stochastic order, denoted by $X\preceq_{st}Y$, if

$\bar{F}_{X}(t)\leq\bar{F}_{Y}(t)\leavevmode\nobreak\ \leavevmode\nobreak\
\textmd{for all}\leavevmode\nobreak\ t\geq 0.$

The following definition is regarding the aging property of a life unit.

######  Definition 2.3.

Let $X$ be a lifetime random variable with hazard rate $h_{X}$ and reversed
hazard rate $r_{X}$. It is said that $X$ has

  * (i)

increasing failure rate (denoted as $X\in IFR$), if $h_{X}(t)$ is increasing
in $t\geq 0$.

  * (ii)

decreasing reversed hazard rate (denoted as $X\in DRHR$) if $r_{X}(t)$ is
decreasing in $t>0$.

The following definition is due to Karlin [23] which be used frequently in the
sequel.

######  Definition 2.4.

Let $f(x,y)$ be a non-negative function. It is said that $f$ is Totally
positive of order 2 ($TP_{2}$) in $(x,y)\in\mathcal{X}\times\mathcal{Y}$ where
$\mathcal{X}$ and $\mathcal{Y}$ are two arbitrary subsets of
$\mathbb{R}=(-\infty,+\infty)$ whenever

$\begin{array}[]{|cc|}f(x_{1},y_{1})&f(x_{1},y_{2})\\\
f(x_{2},y_{1})&f(x_{2},y_{2})\\\ \end{array}\geq 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{for all}\leavevmode\nobreak\ x_{1}\leq
x_{2}\in\mathcal{X},\leavevmode\nobreak\ \textmd{and for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ y_{1}\leq y_{2}\in\mathcal{Y}.$
(5)

If the direction of the inequality given after determinant in (5) is reversed
then it is said that $f$ is Reverse regular of order 2 ($RR_{2}$) in
$(x,y)\in\mathcal{X}\times\mathcal{Y}$.

##  3 Relative aging of upper record values

Suppose that $X$ and $Y$ are two non-negative rvs with absolutely continuous
cdfs $F_{X}$ and $F_{Y}$, and the associated pdfs $f_{X}$ and $f_{Y}$,
respectively. We also denote by $\bar{F}_{X}$ and $\bar{F}_{Y}$ the sfs of $X$
and $Y$, respectively. Let us obtain the expression of hazard rate of
$X_{U_{m}}$ and $Y_{U_{n}}$, as the upper record values of two sequences of
rvs $\\{X_{i};i=1,2,\ldots\\}$ and $\\{Y_{i};i=1,2,\ldots\\}$, respectively.
From Eq. (1) and Eq. (2), the hazard rate of $X_{U_{m}}$ at point $t$ of time,
is derived as

$\displaystyle h_{X_{U_{m}}}(t)$
$\displaystyle=\frac{f_{X_{U_{m}}}(t)}{\bar{F}_{X_{U_{m}}}(t)}$
$\displaystyle=\displaystyle\frac{(-\ln(\bar{F}_{X}(t)))^{m-1}}{(m-1)!\sum_{i=0}^{m-1}(-\ln(\bar{F}_{X}(t)))^{i}/i!}h_{X}(t)$
$\displaystyle=\displaystyle\frac{1}{(m-1)!\sum_{i=0}^{m-1}(-\ln(\bar{F}_{X}(t)))^{i-m+1}/i!}h_{X}(t).$
(6)

In a similar manner, the hazard rate of $Y_{U_{n}}$ at time point $t$, is
derived as

$h_{Y_{U_{n}}}(t)=\displaystyle\frac{1}{(n-1)!\sum_{i=0}^{n-1}(-\ln(\bar{F}_{Y}(t)))^{i-n+1}/i!}h_{Y}(t).$
(7)

Next, we present a result on the preservation property of relative hazard rate
order under distribution of upper records.

######  Theorem 3.1.

Let $X\succeq_{st}Y$ and also let $m\geq n\in\mathbb{N}$. Then,
$X\preceq_{c}Y$ implies $X_{U_{m}}\preceq_{c}Y_{U_{n}}$.

###### Proof.

We first prove that $X_{U_{m}}\preceq_{c}X_{U_{n}}$ for every $n\in\mathbb{N}$
and for all $m=n,n+1,\ldots$. Then, we establish that
$X_{U_{n}}\preceq_{c}Y_{U_{n}}$. From transitivity property of the relative
hazard rate order, one concludes that $X_{U_{m}}\preceq_{c}Y_{U_{n}}$. For
$m\geq n\in\mathbb{N}$, let us denote

$K(u):=\frac{\sum_{i=0}^{n-1}u^{i-n+1}/i!}{\sum_{i=0}^{m-1}u^{i-m+1}/i!}.$

By using (3), one can then write

$\frac{h_{X_{U_{m}}}(t)}{h_{X_{U_{n}}}(t)}=\frac{(n-1)!}{(m-1)!}K(-\ln(\bar{F}_{X}(t))),\leavevmode\nobreak\
\leavevmode\nobreak\ t\geq 0.$

It suffices to show that $K(u)$ increases in $u\geq 0$. One can get

$\displaystyle K(u)$
$\displaystyle=\frac{\sum_{i=0}^{n-1}u^{i+m-n}/i!}{\sum_{i=0}^{m-1}u^{i}/i!}$
$\displaystyle=\frac{\sum_{i=m-n}^{m-1}u^{i}/(i-(m-n))!}{\sum_{i=0}^{m-1}u^{i}/i!}.$

Note that $K(u)$ in increasing in $u\geq 0$, if and only if,

$Q^{\ast}(j,u):=\sum_{i=0}^{m-1}Q(j,i)W(i,u)\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{is}\leavevmode\nobreak\
TP_{2}\leavevmode\nobreak\ \textmd{in}\leavevmode\nobreak\
(j,u)\in\\{1,2\\}\times[0,+\infty),$

where $Q(j,i):=I[i\geq 0]/i!$ for $j=1$ and $Q(j,i):=I[i\geq m-n]/(i-(m-n))!$,
for $j=2;$ and also $W(i,u):=u^{i}$. It is plain to see that $Q(j,i)$ is
$TP_{2}$ in $(j,i)\in\\{1,2\\}\times\\{0,1,\ldots,m-1\\}$ and $W(i,u)$ is also
$TP_{2}$ in $(i,u)\in\\{0,1,\ldots,m-1\\}\times[0,+\infty)$. Thus, by using
general composition theorem of Karlin [23], $Q^{\ast}(j,u)$ is $TP_{2}$ in
$(j,u)\in\\{1,2\\}\times[0,+\infty).$ Hence, we established that
$X_{U_{m}}\preceq_{c}X_{U_{n}}$ for every $m>n\in\mathbb{N}$.

Now, we prove that $X_{U_{n}}\preceq_{c}Y_{U_{n}}$, for all $n\in\mathbb{N}.$
From Eq. (3) and Eq. (7), one has the following

$\frac{h_{X_{U_{n}}}(t)}{h_{Y_{U_{n}}}(t)}=\frac{h_{X}(t)}{h_{Y}(t)}\cdot\frac{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}.$

Since $X\preceq_{c}Y$, thus $h_{X}(t)/h_{Y}(t)$ is increasing in $t\geq 0$.
Thus, it is sufficient to prove that

$\frac{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{is increasing in}\leavevmode\nobreak\
\leavevmode\nobreak\ t\geq 0.$

Now, we can write

$\displaystyle\frac{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}$
$\displaystyle=\sum_{i=0}^{n-1}\left(\frac{-\ln(\bar{F}_{Y}(t))}{-\ln(\bar{F}_{X}(t))}\right)^{i-n+1}\cdot\frac{(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}$
$\displaystyle=E[\Phi^{\ast}(I^{\ast}(t),t)],$

where
$\Phi^{\ast}(i,t):=\left(\frac{-\ln(\bar{F}_{Y}(t))}{-\ln(\bar{F}_{X}(t))}\right)^{i-n+1}$
and $I^{\ast}(t)$ is a discrete rv with pmf

$f^{\ast}(i|t)=\frac{(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(\bar{F}_{X}(t)))^{i-n+1}/i!},i=0,1,\ldots,n-1.$

Since $X\preceq_{c}Y,$ thus $\frac{h_{Y}(t)}{h_{X}(t)}$ is decreasing in
$t\geq 0.$ Hence, from Lemma 2.1 of Khaledi et al. [27],
$(-\ln(\bar{F}_{Y}(t)))/(-\ln(\bar{F}_{X}(t)))$ is decreasing in $t\geq 0$.
Therefore, since for every $i=0,1,\ldots,n-1$, $i-n+1<0$, thus
$\Phi^{\ast}(i,t)$ is increasing in $t\geq 0$. On the other hand, from
assumption we have $X\succeq_{st}Y$, and thus, for all $t\geq 0$, it holds
that $-\ln(\bar{F}_{Y}(t))\geq-\ln(\bar{F}_{X}(t)).$ As a result,
$\Phi^{\ast}(i,t)$ is an increasing function in $i=0,1,\ldots,n-1$, for every
$t\geq 0.$ Further, it can be shown readily that $f^{\ast}(i|t)$ is $TP_{2}$
in $(i,t)\in\\{1,2\\}\times[0,+\infty)$, which further implies that
$I^{\ast}(t_{1})\preceq_{st}I^{\ast}(t_{2}),$ for all $t_{1}\leq t_{2}$. Now,
by using Lemma 2.2(i) of Misra and van der Meulen [33],
$E[\Phi^{\ast}(I^{\ast}(t),t)]$ is increasing in $t\geq 0$. Now, the result is
proved. ∎

Stochastic comparison of convolutions of exponential random variables has been
an important subject of research in literature (see, e.g., Bon and Pãltãnea
[11] and Kochar and Ma [29]). However, relative ordering of convolutions of
exponential random variables has not been investigated in literature thus far.
In the next example, we make the relative hazard rate ordering of convolutions
of i.i.d. exponential random variables according to the relative hazard rate
order. Note that a non-negative rv $T$ is said to have Erlang distribution
with parameters $(n,\lambda)$ whenever it has pdf
$f_{T}(t)=\frac{\lambda(\lambda t)^{n-1}e^{-\lambda t}}{(n-1)!},$ for all
$t\geq 0$, where $n=1,2,\ldots$ and $\lambda>0$.

######  Example 3.1.

Let us suppose that $X_{1},X_{2},\ldots$ and $Y_{1},Y_{2},\ldots$ are two
i.i.d. sequences of exponential random variables with means $1/\lambda_{1}$
and $1/\lambda_{2}$, respectively, such that $\lambda_{1}<\lambda_{2}$. From
Eq. (1), $X_{U_{m}}$ and $Y_{U_{n}}$ have pdfs

$f_{X_{U_{m}}}(x)=\frac{\lambda_{1}(\lambda_{1}x)^{m-1}e^{-\lambda_{1}x}}{(m-1)!}\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\
f_{Y_{U_{n}}}(y)=\frac{\lambda_{2}(\lambda_{2}y)^{n-1}e^{-\lambda_{2}y}}{(n-1)!},$

which are the pdfs of two Erlang distributions with parameters
$(m,\lambda_{1})$ and $(n,\lambda_{2}),$ respectively. On the other hand, it
is a well-known fact that partial sums of i.i.d. exponential random variables
$X_{1},\ldots,X_{m}$ and $Y_{1},\ldots,Y_{n}$ follow Erlang distribution with
parameters $(m,\lambda_{1})$ and $(n,\lambda_{2})$, respectively. Therefore,
$X_{U_{m}}$ and $Y_{U_{n}}$ are equal in distribution with
$\sum_{i=1}^{m}X_{i}$ and $\sum_{i=1}^{n}Y_{i}$, respectively. From Eq. (3)
and Eq. (7), one gets

$h_{\sum_{i=1}^{m}X_{i}}(t)=\frac{\lambda_{1}}{(m-1)!\sum_{i=0}^{m-1}(\lambda_{1}t)^{i-m+1}/i!},\leavevmode\nobreak\
\leavevmode\nobreak\
h_{\sum_{i=1}^{n}Y_{i}}(t)=\frac{\lambda_{2}}{(n-1)!\sum_{i=0}^{n-1}(\lambda_{2}t)^{i-n+1}/i!}.$

Since $\lambda_{1}<\lambda_{2}$, thus $X_{1}\succeq_{st}Y_{1}$. Thus, for
every $m\geq n\in\mathbb{N}$, by using Theorem 3.1, one concludes that
$\sum_{i=1}^{m}X_{i}\preceq_{c}\sum_{i=1}^{n}Y_{i}.$ In Figure 1, we have
plotted the hazard rate ratio of convolutions of exponential random variables,
which is an increasing function.

Figure 1: The plot of the function
$\frac{h_{\sum_{i=1}^{m}X_{i}}(t)}{h_{\sum_{i=1}^{n}Y_{i}}(t)}$ in Example 3.1
for $m=3,n=2,\lambda_{1}=3$ and $\lambda_{2}=5$.

In Theorem 3.1 there is an order condition that $X\succeq_{st}Y$, However,
this is a strong condition that may not be satisfied. In the residual part of
this section, we want to relax the condition $X\succeq_{st}Y$ to obtain the
preservation property of the relative hazard rate order under upper records.
The following lemma is useful to prove such result.

######  Lemma 3.1.

Let $\xi_{n}(u)=(n-1)!\sum_{i=0}^{n-1}\frac{(-\ln(u))^{i-n+1}}{i!},$ for
$u\in(0,1)$. Then, $\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}$ is a non-negative
increasing function of $u\in(0,1)$ for every $n=1,2,\ldots$.

###### Proof.

Let us suppose that $I(u)$ is a discrete random variable with the following
probability mass function

$f(i\mid
u):=\frac{(-\ln(u))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(u))^{i-n+1}/i!},\leavevmode\nobreak\
i=0,1,\ldots,n-1.$

Then, one has the following

$\displaystyle\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}$
$\displaystyle=\frac{\sum_{i=0}^{n-1}(n-i-1)(-\ln(u))^{i-n}/i!}{\sum_{i=0}^{n-1}(-\ln(u))^{i-n+1}/i!}$
$\displaystyle=\sum_{i=0}^{n-1}\left(\frac{n-i-1}{-\ln(u)}\right)\cdot\frac{(-\ln(u))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(u))^{i-n+1}/i!}$
$\displaystyle=\sum_{i=0}^{n-1}\Psi(i,u)\cdot f(i\mid u)$
$\displaystyle=E[\Psi(I(u),u)],$

where $\Psi(i,u):=\frac{n-i-1}{-\ln(u)}$ is a decreasing function of
$i=1,2,\ldots,n-1$ and it is an increasing function of $u\in(0,1)$. Note that
$f(i\mid u)$ is an $RR_{2}$ function in $(i,u)\in\\{1,2\\}\times(0,1)$. Hence,
$I(u_{1})\geq_{st}I(u_{2}),$ for every $u_{1}\leq u_{2}\in(0,1)$, i.e., $I(u)$
is stochastically decreasing in $u\in(0,1)$. Thus, by using Lemma 2.2(i) of
Misra and van der Meulen [33], $E[\Psi(I(u),u)]$ is an increasing function of
$u$. This completes the proof of lemma. ∎

The limiting behaviour of hazard rates ratio is an important problem in
reliability analysis (see, e.g., Navarro and Sarabia [37]). Let us define

$c_{0}=\lim_{t\rightarrow 0^{+}}\frac{h_{X}(t)}{h_{Y}(t)},\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\
c_{1}=\lim_{t\rightarrow+\infty}\frac{h_{X}(t)}{h_{Y}(t)}.$

In what follows, we assume that $c_{0}$ and $c_{1}$ are positive and finite.
The following theorem is the main result of this section.

######  Theorem 3.2.

Let $\psi_{n}(u)=\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}$ where
$\xi_{n}(u)=(n-1)!\sum_{i=0}^{n-1}\frac{(-\ln(u))^{i-n+1}}{i!},$ for
$u\in(0,1)$. If $\sup_{0<u<1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}\leq
c_{0}$ then, $X\preceq_{c}Y$ implies $X_{U_{m}}\preceq_{c}Y_{U_{n}}.$

###### Proof.

From Eq. (3) and Eq. (7), for all $t\geq 0,$ one has the following

$h_{X_{U_{m}}}(t)=\frac{h_{X}(t)}{\xi_{m}(\bar{F}_{X}(t))},\leavevmode\nobreak\
\leavevmode\nobreak\
h_{Y_{U_{n}}}(t)=\frac{h_{Y}(t)}{\xi_{n}(\bar{F}_{Y}(t))}.$

Thus, it is evident that

$\frac{h_{X_{U_{m}}}(t)}{h_{Y_{U_{n}}}(t)}=\frac{h_{X}(t)}{h_{Y}(t)}\cdot\frac{\xi_{n}(\bar{F}_{Y}(t))}{\xi_{m}(\bar{F}_{X}(t))}.$

Since from assumption $X\preceq_{c}Y,$ thus it is enough to show that
$\frac{\xi_{n}(\bar{F}_{Y}(t))}{\xi_{m}(\bar{F}_{X}(t))}$ is increasing in
$t\geq 0$, or equivalently demonstrate that

$\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(\bar{F}_{Y}(t))}{\xi_{m}(\bar{F}_{X}(t))}\right)\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{for every
}\leavevmode\nobreak\ t\geq 0.$ (8)

We have

$\displaystyle\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(\bar{F}_{Y}(t))}{\xi_{m}(\bar{F}_{X}(t))}\right)$
$\displaystyle=-\frac{f_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}+\frac{f_{X}(t)\xi^{\prime}_{m}(\bar{F}_{X}(t))}{\xi_{m}(\bar{F}_{X}(t))}$
$\displaystyle=-\frac{f_{Y}(t)}{\bar{F}_{Y}(t)}\frac{\bar{F}_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}+\frac{f_{X}(t)}{\bar{F}_{X}(t)}\frac{\bar{F}_{X}(t)\xi^{\prime}_{m}(\bar{F}_{X}(t))}{\xi_{m}(\bar{F}_{X}(t))}$
$\displaystyle=h_{X}(t)\cdot\frac{\bar{F}_{X}(t)\xi^{\prime}_{m}(\bar{F}_{X}(t))}{\xi_{m}(\bar{F}_{X}(t))}-h_{Y}(t)\cdot\frac{\bar{F}_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}.$

Note that since $\frac{h_{X}(t)}{h_{Y}(t)}$ is increasing in $t\geq 0$, thus
$\frac{h_{X}(t)}{h_{Y}(t)}\geq c_{0}$, for all $t\geq 0,$ and
$\frac{h_{X}(t)}{h_{Y}(t)}\leq c_{1}$, for all $t\geq 0,$ which further imply
that $h_{X}(t)\geq c_{0}h_{Y}(t),$ for all $t\geq 0$ and that
$\bar{F}_{X}(t)\geq\bar{F}^{c_{1}}_{Y}(t),$ for all $t\geq 0,$ respectively.
The second implication is due to the recursive formulas
$\bar{F}_{X}(t)=e^{-\int_{0}^{t}h_{X}(x)dx}$ and
$\bar{F}_{Y}(t)=e^{-\int_{0}^{t}h_{Y}(x)dx}$ so that if $h_{X}(t)\leq
c_{1}h_{Y}(t),$ for all $t\geq 0$, then one concludes
$\bar{F}_{X}(t)\geq\bar{F}^{c_{1}}_{Y}(t),$ for all $t\geq 0.$ Now, one has
the following

$\displaystyle\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(\bar{F}_{Y}(t))}{\xi_{m}(\bar{F}_{X}(t))}\right)$
$\displaystyle=h_{X}(t)\cdot\frac{\bar{F}_{X}(t)\xi^{\prime}_{m}(\bar{F}_{X}(t))}{\xi_{m}(\bar{F}_{X}(t))}-h_{Y}(t)\cdot\frac{\bar{F}_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}$
$\displaystyle\geq
h_{Y}(t)\cdot\left(c_{0}\frac{\bar{F}_{X}(t)\xi^{\prime}_{m}(\bar{F}_{X}(t))}{\xi_{m}(\bar{F}_{X}(t))}-\frac{\bar{F}_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}\right)$
$\displaystyle\geq
h_{Y}(t)\cdot\left(c_{0}\frac{\bar{F}^{c_{1}}_{Y}(t)\xi^{\prime}_{m}(\bar{F}^{c_{1}}_{Y}(t))}{\xi_{m}(\bar{F}^{c_{1}}_{Y}(t))}-\frac{\bar{F}_{Y}(t)\xi^{\prime}_{n}(\bar{F}_{Y}(t))}{\xi_{n}(\bar{F}_{Y}(t))}\right)$
$\displaystyle=h_{Y}(t)\cdot\left(c_{0}\psi_{m}(\bar{F}^{c_{1}}_{Y}(t))-\psi_{n}(\bar{F}_{Y}(t))\right)$

where the last inequality holds true because
$\bar{F}_{X}(t)\geq\bar{F}^{c_{1}}_{Y}(t),$ for all $t\geq 0$ and, further,
since from Lemma 3.1, $\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}$ is increasing
in $u\in(0,1).$ Now, one can see that (8) is satisfied if
$c_{0}\psi_{m}(\bar{F}^{c_{1}}_{Y}(t))\geq\psi_{n}(\bar{F}_{Y}(t)),$ for all
$t\geq 0$. Since $\psi_{m}$ is a non-negative function, and moreover, since
$\sup_{t\geq
0}\frac{\psi_{n}(\bar{F}_{Y}(t))}{\psi_{m}(\bar{F}^{c_{1}}_{Y}(t))}=\sup_{0<u<1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}$
thus the recent inequality holds if, and only if,

$\sup_{0<u<1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}\leq c_{0}.$

∎

######  Remark 3.1.

It is to be mentioned that the result of Theorem 3.2 in situations where $X$
and $Y$ are equal in distribution, could also be achieved by using Theorem
3.1, since the assumption $X\succeq_{st}Y$ is satisfied when $X$ and $Y$ have
a common distribution. It is remarkable in the context of Theorem 3.2, that if
$X$ and $Y$ are equal in distribution, then $c_{0}=c_{1}=1$. Note that
$\psi_{n}(u)$ is increasing in $n\geq 1$, if and only if for every $m\geq
n\in\mathbb{N}$,

$\psi_{m}(u):=\frac{u\xi^{\prime}_{m}(u)}{\xi_{m}(u)}\geq\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}=\psi_{n}(u),\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{for all}\leavevmode\nobreak\ u\in(0,1).$

The above inequality is also equivalent to

$\frac{\xi^{\prime}_{m}(u)}{\xi_{m}(u)}\geq\frac{\xi^{\prime}_{n}(u)}{\xi_{n}(u)},\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{for all}\leavevmode\nobreak\ u\in(0,1),$

which means that $\frac{\xi_{m}(u)}{\xi_{n}(u)}$ is increasing in $u\in(0,1).$
We can write

$\displaystyle\frac{\xi_{m}(u)}{\xi_{n}(u)}$
$\displaystyle=\frac{(m-1)!\sum_{i=0}^{m-1}(-\ln(u))^{i-m+1}/i!}{(n-1)!\sum_{i=0}^{n-1}(-\ln(u))^{i-n+1}/i!}$
$\displaystyle=\frac{(m-1)!}{(n-1)!}\cdot\frac{\sum_{i=0}^{m-1}(-\ln(u))^{i}/i!}{\sum_{i=0}^{n-1}(-\ln(u))^{i-n+m}/i!}$
$\displaystyle=\frac{(m-1)!}{(n-1)!}\cdot\frac{\sum_{i=0}^{m-1}(-\ln(u))^{i}/i!}{\sum_{i=m-n}^{m-1}(-\ln(u))^{i}/(i-(m-n))!},$

in which
$\frac{\sum_{i=0}^{m-1}(-\ln(u))^{i}/i!}{\sum_{i=m-n}^{m-1}(-\ln(u))^{i}/(i-(m-n))!}$
is increasing in $u\in(0,1)$, if and only if,

$q^{\star}(j,u):=\sum_{i=0}^{m-1}q(j,i)w(i,u)\leavevmode\nobreak\
\textmd{is}\leavevmode\nobreak\ TP_{2}\leavevmode\nobreak\
\textmd{in}\leavevmode\nobreak\ (j,u)\in\\{1,2\\}\times(0,1)$

where $q(j,i):=I[i\geq 0]/i!$ for $j=2$ and $q(j,i):=I[i\geq m-n]/(i-(m-n))!$
for $j=1$, and $w(i,u):=(-\ln(u))^{i}$. Since $q(j,i)$ is $RR_{2}$ in
$(j,i)\in\\{1,2\\}\times\\{0,1,\ldots,m-1\\}$ and $w(i,u)$ is also $RR_{2}$ in
$(i,u)\in\\{1,2\\}\times(0,1),$ thus the required result follows from general
composition theorem of Karlin [23]. Now, since $c_{0}=c_{1}=1$ as discussed
before, thus for every $m=n,n+1,\ldots$ where $n\in\mathbb{N}$ one has

$\displaystyle\sup_{0\leq u\leq 1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}$
$\displaystyle=\sup_{0\leq u\leq 1}\frac{\psi_{n}(u)}{\psi_{m}(u)}$
$\displaystyle\leq 1=c_{0},$

where the last inequality holds because $\psi_{n}(u)\leq\psi_{m}(u),$ for
every $u\in[0,1].$ Therefore, from Theorem 3.2, we deduced that if $m\geq
n\in\mathbb{N},$ then $X_{U_{m}}\preceq_{c}X_{U_{n}}$.

The following example presents an example where Theorem 3.2 is applicable.
However, we can not deduce the result using Theorem 3.1 because
$X\nsucceq_{st}Y$.

######  Example 3.2.

Let us consider $X$ and $Y$ as two random lifetimes with Lomax distributions
having sfs $\bar{F}_{X}(t)=\frac{1}{(1+t)^{3}}$ and
$\bar{F}_{Y}(t)=\frac{9}{(3+4t)^{2}}.$ The hr functions of $X$ and $Y$ are
then calculated as $h_{X}(t)=\frac{3}{1+t}$ and $h_{Y}(t)=\frac{8}{3+4t}$,
which yields $\frac{h_{X}(t)}{h_{Y}(t)}=\frac{9+12t}{8+8t}.$ The function
$\frac{h_{X}(t)}{h_{Y}(t)}$ is increasing in $t\geq 0$. Thus, according to
definition, $X\preceq_{c}Y$. It is seen that $c_{0}=\lim_{t\rightarrow
0^{+}}\frac{h_{X}(t)}{h_{Y}(t)}=\frac{9}{8}$ and
$c_{1}=\lim_{t\rightarrow+\infty}\frac{h_{X}(t)}{h_{Y}(t)}=\frac{3}{2}$. It is
trivial that, for all $t\geq 0$, $h_{X}(t)\geq h_{Y}(t)$ which means
$X\preceq_{hr}Y$, and consequently, $X\preceq_{st}Y$. Therefore, the condition
of Theorem 3.1 does not hold. We want to compare the second upper record from
a sequence of i.i.d. random lifetimes adopted from $F_{X}$ to the third upper
record from a sequence of i.i.d. random lifetimes following $F_{Y}$. Hence,
$n=2$ and $m=3$. We show using the setting of Theorem 3.2 that
$X_{U_{m}}\preceq_{c}Y_{U_{n}}.$ For $u\in[0,1]$, we observe that

$\xi_{n}(u)=\frac{1}{-\ln(u)}+1,\leavevmode\nobreak\
\textmd{and}\leavevmode\nobreak\
\xi_{m}(u)=1+\frac{2}{-\ln(u)}+\frac{2}{(-\ln(u))^{2}},$

from which one further obtains

$\xi^{\prime}_{n}(u)=\frac{1}{u(-\ln(u))^{2}},\leavevmode\nobreak\
\textmd{and}\leavevmode\nobreak\
\xi^{\prime}_{m}(u)=\frac{2}{u(-\ln(u))^{2}}+\frac{4}{u(-\ln(u))^{3}}.$

After some routine calculation, we derive

$\psi_{n}(u)=\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}=\frac{1}{(-\ln(u))\cdot(-\ln(u)+1)}$

and, similarly,

$\psi_{m}(u)=\frac{u\xi^{\prime}_{m}(u)}{\xi_{m}(u)}=\frac{-2\ln(u)+4}{(-\ln(u))\cdot((-\ln(u))^{2}-2\ln(u)+2)}.$

Therefore,

$\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}=\frac{\psi_{n}(u)}{\psi_{m}(u^{\frac{3}{2}})}=\frac{27(-\ln(u))^{3}+36(-\ln(u))^{2}+24(-\ln(u))}{(-\ln(u))\cdot(1-\ln(u))\cdot(32-24\ln(u))}.$

It can be checked using website _https://www.dcode.fr/maximum-function_ that
$\sup_{0\leq u\leq 1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}=\frac{9}{8}$. One
can also see Figure 2. Hence, the condition of Theorem 3.2 is satisfied and
hence the required result follows.

In Example 3.2, one can further obtain

$h_{X_{U_{m}}}(t)=\frac{h_{X}(t)}{\xi_{m}(\bar{F}_{X}(t))}=\frac{27\ln^{2}(1+t)}{(1+t)\cdot(2+6\ln(1+t)+9\ln^{2}(1+t))},$

and

$h_{Y_{U_{n}}}(t)=\frac{h_{Y}(t)}{\xi_{n}(\bar{F}_{Y}(t))}=\frac{8(-\ln(9)+2\ln(3+4t))}{(3+4t)\cdot(1-\ln(9)+2\ln(3+4t))}.$

In Figure 3 we plot the graph of $\frac{h_{X_{U_{m}}}(t)}{h_{Y_{U_{n}}}(t)}$
to exhibit that it is an increasing function in $t$.

Figure 2: The graph of the function $\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}$
in Example 3.2.  Figure 3: The graph of
$\frac{h_{X_{U_{m}}}(t)}{h_{Y_{U_{n}}}(t)}$ in Example 3.2.

##  4 Relative comparison of lower record values

In this section, we consider the lower records $X_{L_{m}}$ and $Y_{L_{n}}$
from two sequences of rvs $\\{X_{i}:i=1,2,\ldots\\}$ and
$\\{Y_{i}:i=1,2,\ldots\\}$, each containing i.i.d. observations. The rhr of
$X_{L_{m}}$ is obtained as

$\displaystyle r_{X_{L_{m}}}(t)$
$\displaystyle=\frac{f_{X_{L_{m}}}(t)}{F_{X_{L_{m}}}(t)}$
$\displaystyle=\displaystyle\frac{(-\ln(F_{X}(t)))^{m-1}}{(m-1)!\sum_{i=0}^{m-1}(-\ln(F_{X}(t)))^{i}/i!}r_{X}(t)$
$\displaystyle=\displaystyle\frac{1}{(m-1)!\sum_{i=0}^{m-1}(-\ln(F_{X}(t)))^{i-m+1}/i!}r_{X}(t).$
(9)

Analogously, the rhr of $Y_{L_{n}}$ at time point $t$, is acquired as

$r_{Y_{L_{n}}}(t)=\displaystyle\frac{1}{(n-1)!\sum_{i=0}^{n-1}(-\ln(F_{Y}(t)))^{i-n+1}/i!}r_{Y}(t).$
(10)

Next, we present a result on the preservation property of relative reversed
hazard rate order under distribution of lower records.

######  Theorem 4.1.

Let $X\preceq_{st}Y$ and $m\geq n\in\mathbb{N}$. Then, $X\preceq_{b}Y$ yields
$X_{L_{m}}\preceq_{b}Y_{L_{n}}$.

###### Proof.

Firstly, we demonstrate that $X_{L_{m}}\preceq_{b}X_{L_{n}}$ holds true for
$n\in\mathbb{N}$ and for every $m\geq n$ such that $m\in\mathbb{N}$. Then, it
will be proved that $X_{L_{n}}\preceq_{b}Y_{L_{n}}$. Due to the transitivity
property of the relative reversed hazard rate order, it is deduced that
$X_{L_{m}}\preceq_{b}Y_{L_{n}}$. Keeping the notations of Theorem 3.1 in mind,
and in view of (4), we can write

$\frac{r_{X_{L_{m}}}(t)}{r_{X_{L_{n}}}(t)}=\frac{(n-1)!}{(m-1)!}K(-\ln(F_{X}(t))),\leavevmode\nobreak\
\leavevmode\nobreak\ t\geq 0.$

In the proof of Theorem 3.1, it was shown that $K(u)$ in increasing in $u\geq
0$. Hence, since $-\ln(F_{X}(t))$ is decreasing in $t>0$, thus
$\frac{r_{X_{L_{m}}}(t)}{r_{X_{L_{n}}}(t)}$ is decreasing in $t>0$. Therefore,
$X_{L_{m}}\preceq_{b}X_{L_{n}}$ for every $m=n,n+1,\ldots$. To finalize the
proof, we need to prove $X_{L_{n}}\preceq_{b}Y_{L_{n}}$, for every
$n\in\mathbb{N}.$ In the spirit of Eq. (4) and Eq. (10), we get

$\frac{r_{X_{L_{n}}}(t)}{r_{Y_{L_{n}}}(t)}=\frac{r_{X}(t)}{r_{Y}(t)}\cdot\frac{\sum_{i=0}^{n-1}(-\ln(F_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(F_{X}(t)))^{i-n+1}/i!}.$

Following assumption, we have $X\preceq_{b}Y$. Hence $r_{X}(t)/r_{Y}(t)$ is
decreasing in $t>0$. It thus suffices to establish that

$\frac{\sum_{i=0}^{n-1}(-\ln(F_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(F_{X}(t)))^{i-n+1}/i!}\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{is decreasing in}\leavevmode\nobreak\
\leavevmode\nobreak\ t>0.$

Let us write

$\displaystyle\frac{\sum_{i=0}^{n-1}(-\ln(F_{Y}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(F_{X}(t)))^{i-n+1}/i!}$
$\displaystyle=\sum_{i=0}^{n-1}\left(\frac{-\ln(F_{Y}(t))}{-\ln(F_{X}(t))}\right)^{i-n+1}\cdot\frac{(-\ln(F_{X}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(F_{X}(t)))^{i-n+1}/i!}$
$\displaystyle=E[\Phi_{\ast}(I_{\ast}(t),t)],$

where
$\Phi_{\ast}(i,t):=\left(\frac{-\ln(F_{Y}(t))}{-\ln(F_{X}(t))}\right)^{i-n+1}$
and $I_{\ast}(t)$ is an rv with pmf

$f_{\ast}(i|t)=\frac{(-\ln(F_{X}(t)))^{i-n+1}/i!}{\sum_{i=0}^{n-1}(-\ln(F_{X}(t)))^{i-n+1}/i!},i=0,1,\ldots,n-1.$

Since $X\preceq_{b}Y,$ thus from Hazra and Misra [20],
$\frac{-\ln(F_{Y}(t))}{-\ln(F_{X}(t))}$ is increasing in $t>0$. Thus,
$\Phi_{\ast}(i,t)$ is decreasing in $t>0$. From assumption, it holds that
$X\preceq_{st}Y$, and as a result, for all $t>0$,
$-\ln(F_{Y}(t))\geq-\ln(F_{X}(t)).$ Therefore, $\Phi_{\ast}(i,t)$ is
increasing in $i=0,1,\ldots,n-1$, for all$t>0.$ It can be seen readily that
$f_{\ast}(i|t)$ is $RR_{2}$ in $(i,t)\in\\{1,2\\}\times(0,+\infty)$, which in
turn yields $I_{\ast}(t_{1})\succeq_{st}I_{\ast}(t_{2}),$ for all $t_{1}\leq
t_{2}$. On applying Lemma 2.2(ii) of Misra and van der Meulen [33],
$E[\Phi_{\ast}(I_{\ast}(t),t)]$ is decreasing in $t>0$. Now, the result is
proved. ∎

The following example presents a situation where Theorem 4.1 is applicable.

######  Example 4.1.

We suppose that $X$ and $Y$ have inverse Weibull distribution with cdfs
$F_{X}(t)=\exp(-\frac{4}{t^{2}})$ and $F_{Y}(t)=\exp(-\frac{5}{t^{2}}),$ where
$t>0.$ We can get $r_{X}(t)=\frac{8}{t^{3}}$ and $r_{Y}(t)=\frac{10}{t^{3}}.$
Since $\frac{r_{X}(t)}{r_{Y}(t)}=\frac{4}{5},$ thus $X\preceq_{b}Y$. It is
also trivial to see that $X\preceq_{st}Y$. Let us choose $n=2$ and $m=3$. The
result of Theorem 4.1 implies that $X_{L_{m}}\preceq_{b}Y_{L_{n}}.$ One has
the following

$r_{X_{L_{m}}}(t)=\frac{r_{X}(t)}{\xi_{m}(F_{X}(t))}=\frac{64}{t^{3}(t^{4}+4t^{2}+8)}\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\
r_{Y_{L_{n}}}(t)=\frac{r_{Y}(t)}{\xi_{n}(F_{Y}(t))}=\frac{50}{t^{3}(t^{2}+5)}.$

In Figure 4 we plot the graph of $\frac{r_{X_{L_{m}}}(t)}{r_{Y_{L_{n}}}(t)}$
and it is shown that this function decreases in $t>0$ which validates the
result of Theorem 4.1.

Figure 4: The graph of $\frac{r_{X_{L_{m}}}(t)}{r_{Y_{L_{n}}}(t)}$ in Example
4.1.

The extremal behaviour of reversed hazard rates ratio is considered here. Let
us define

$c^{\ast}_{0}=\lim_{t\rightarrow
0^{+}}\frac{r_{X}(t)}{r_{Y}(t)},\leavevmode\nobreak\ \leavevmode\nobreak\
\textmd{and}\leavevmode\nobreak\ \leavevmode\nobreak\
c^{\ast}_{1}=\lim_{t\rightarrow+\infty}\frac{r_{X}(t)}{r_{Y}(t)}.$

In the sequel, we assume that $c^{\ast}_{0}$ and $c^{\ast}_{1}$ are positive
and finite. In the following theorem the condition of $X\preceq_{st}Y$ in
Theorem 4.1 is relaxed.

######  Theorem 4.2.

If $\sup_{0<u<1}\frac{\psi_{n}(u^{\frac{1}{c_{0}^{\ast}}})}{\psi_{m}(u)}\leq
c^{\ast}_{1}$ then, $X\preceq_{b}Y$ implies $X_{L_{m}}\preceq_{b}Y_{L_{n}}.$

###### Proof.

In view of Eq. (4) and Eq. (10), for all $t>0,$ we get

$r_{X_{L_{m}}}(t)=\frac{r_{X}(t)}{\xi_{m}(F_{X}(t))},\leavevmode\nobreak\
\leavevmode\nobreak\ r_{Y_{L_{n}}}(t)=\frac{r_{Y}(t)}{\xi_{n}(F_{Y}(t))}.$

Therefore,

$\frac{r_{X_{L_{m}}}(t)}{r_{Y_{L_{n}}}(t)}=\frac{r_{X}(t)}{r_{Y}(t)}\cdot\frac{\xi_{n}(F_{Y}(t))}{\xi_{m}(F_{X}(t))}.$

By assumption we have $X\preceq_{b}Y$ which means $\frac{r_{X}(t)}{r_{Y}(t)}$
is decreasing. Thus, we only need to show that
$\frac{\xi_{n}(F_{Y}(t))}{\xi_{m}(F_{X}(t))}$ is decreasing in $t\geq 0$, or
equivalently, we can prove that

$\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(F_{Y}(t))}{\xi_{m}(F_{X}(t))}\right)\leq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \textmd{for
all}\leavevmode\nobreak\ t>0.$ (11)

One has

$\displaystyle\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(F_{Y}(t))}{\xi_{m}(F_{X}(t))}\right)$
$\displaystyle=\frac{f_{Y}(t)\xi^{\prime}_{n}(F_{Y}(t))}{\xi_{n}(F_{Y}(t))}-\frac{f_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}$
$\displaystyle=\frac{f_{Y}(t)}{F_{Y}(t)}\frac{F_{Y}(t)\xi^{\prime}_{n}(F_{Y}(t))}{\xi_{n}(F_{Y}(t))}-\frac{f_{X}(t)}{F_{X}(t)}\frac{F_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}$
$\displaystyle=r_{Y}(t)\cdot\frac{F_{Y}(t)\xi^{\prime}_{n}(F_{Y}(t))}{\xi_{n}(F_{Y}(t))}-r_{X}(t)\cdot\frac{F_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}.$

Since $\frac{r_{X}(t)}{r_{Y}(t)}$ is decreasing in $t>0$, thus
$\frac{r_{X}(t)}{r_{Y}(t)}\leq c^{\ast}_{0}$, for all $t>0,$ and
$\frac{r_{X}(t)}{r_{Y}(t)}\geq c^{\ast}_{1}$, for all $t>0.$ Hence,
$r_{X}(t)\leq c^{\ast}_{0}r_{Y}(t),$ for all $t>0$ and also $F_{Y}(t)\leq
F^{\frac{1}{c_{0}^{\ast}}}_{X}(t),$ for all $t>0$ where the second inequality
follows from the identities $F_{X}(t)=e^{-\int_{t}^{+\infty}r_{X}(x)dx}$ and
$F_{Y}(t)=e^{-\int_{t}^{+\infty}r_{Y}(x)dx}$. This is because if $r_{X}(t)\leq
c^{\ast}_{0}r_{Y}(t),$ for all $t>0$, then $F_{Y}(t)\leq
F^{\frac{1}{c_{0}^{\ast}}}_{X}(t),$ for all $t>0.$ Therefore,

$\displaystyle\frac{\partial}{\partial
t}\ln\left(\frac{\xi_{n}(F_{Y}(t))}{\xi_{m}(F_{X}(t))}\right)$
$\displaystyle=r_{Y}(t)\cdot\frac{F_{Y}(t)\xi^{\prime}_{n}(F_{Y}(t))}{\xi_{n}(F_{Y}(t))}-r_{X}(t).\frac{F_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}$
$\displaystyle\leq
r_{X}(t)\cdot\left(\frac{1}{c_{1}^{\ast}}\frac{F_{Y}(t)\xi^{\prime}_{n}(F_{Y}(t))}{\xi_{n}(F_{Y}(t))}-\frac{F_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}\right)$
$\displaystyle\leq
r_{X}(t)\cdot\left(\frac{1}{c_{1}^{\ast}}\frac{F^{\frac{1}{c_{0}^{\ast}}}_{X}(t)\xi^{\prime}_{n}(F^{\frac{1}{c_{0}^{\ast}}}_{X}(t))}{\xi_{n}(F^{\frac{1}{c_{0}^{\ast}}}_{X}(t))}-\frac{F_{X}(t)\xi^{\prime}_{m}(F_{X}(t))}{\xi_{m}(F_{X}(t))}\right)$
$\displaystyle=r_{X}(t)\cdot\left(\frac{1}{c_{1}^{\ast}}\psi_{n}(F^{\frac{1}{c_{0}^{\ast}}}_{X}(t))-\psi_{m}(F_{X}(t))\right)$

where the first inequality holds because
$r_{Y}(t)\leq\frac{1}{c^{\ast}_{1}}r_{X}(t),$ for all $t>0$ and the last
inequality is due to $F_{Y}(t)\leq F^{\frac{1}{c_{0}^{\ast}}}_{X}(t),$ for all
$t>0,$ and since by Lemma 3.1, $\frac{u\xi^{\prime}_{n}(u)}{\xi_{n}(u)}$ is
increasing in $u\in(0,1).$ It is now realized that (11) is fulfilled if
$c^{\ast}_{1}\psi_{m}(F_{X}(t))\geq\psi_{n}(F^{\frac{1}{c}}_{X}(t)),$ for all
$t>0$. We know that $\psi_{m}$ is a non-negative function, and further, it is
trivial that $\sup_{t\geq
0}\frac{\psi_{n}(F^{\frac{1}{c_{0}^{\ast}}}_{X}(t))}{\psi_{m}(F_{X}(t))}=\sup_{0<u<1}\frac{\psi_{n}(u)}{\psi_{m}(u^{c_{1}})}$
thus the recent inequality holds if, and only if,

$\sup_{0<u<1}\frac{\psi_{n}(u^{\frac{1}{c_{0}^{\ast}}})}{\psi_{m}(u)}\leq
c^{\ast}_{1}.$

∎

It is remarkable here that when $X$ and $Y$ are equal in distribution then,
$X\preceq_{st}Y$ and further $X\preceq_{b}Y$. In the spirit of Remark 3.1,
since $\psi_{n}(u)$ is increasing in $u$ for every $u\in[0,1],$ and because
$c_{0}^{\ast}=c_{1}^{\ast}=1$, thus for every $m=n,n+1,\ldots,$ one observes
that

$\sup_{0<u<1}\frac{\psi_{n}(u^{\frac{1}{c_{0}^{\ast}}})}{\psi_{m}(u)}=\sup_{0<u<1}\frac{\psi_{n}(u)}{\psi_{m}(u)}\leq
1.$

Hence, using Theorem 4.1, one obtains $X_{L_{m}}\preceq_{b}X_{L_{n}}.$ This
result could also be concluded by applying Theorem 3.2.

The next example illustrates a situation where Theorem 4.1 is not applicable
while the result of Theorem 4.2 works.

######  Example 4.2.

Let $X$ and $Y$ be two random lifetimes with Inverse Weibull distributions
having cdfs $F_{X}(t)=\exp(-\frac{4}{t^{4}})$ and
$F_{Y}(t)=\exp(-\frac{2}{t^{4}}).$ The rhr functions of $X$ and $Y$ are
obtained as $r_{X}(t)=\frac{16}{t^{5}}$ and $r_{Y}(t)=\frac{8}{t^{5}}$, which
further implies that $\frac{r_{X}(t)}{r_{Y}(t)}=2.$ The function
$\frac{r_{X}(t)}{r_{Y}(t)}$ is, therefore, non-increasing in $t>0$. By
definition, $X\preceq_{b}Y$. We can see that
$c^{\ast}_{0}=c_{1}^{\ast}=\lim_{t\rightarrow
0^{+}}\frac{r_{X}(t)}{r_{Y}(t)}=2$. For all $>0$, $r_{X}(t)\geq r_{Y}(t)$
i.e., $X\succeq_{rh}Y$ and moreover $X\succeq_{st}Y$. This means that the
condition of Theorem 4.1 is not satisfied. Here, we take $n=2$ and $m=3$. By
using Theorem 4.2 we shall conclude that $X_{L_{m}}\preceq_{b}Y_{L_{n}}.$ For
$u\in[0,1]$, in the sprit of the proof of Theorem 3.2 in Example 3.2, one gets

$\frac{\psi_{n}(u^{\frac{1}{c_{0}^{\ast}}})}{\psi_{m}(u)}=\frac{\psi_{n}(u^{\frac{1}{2}})}{\psi_{m}(u)}=\frac{(-\ln(u))^{2}-2\ln(u)+2}{(4-2\ln(u)).(\frac{1}{2}-\frac{1}{4}\ln(u))}$

We can check using website _https://www.dcode.fr/maximum-function_ that
$\sup_{0\leq u\leq 1}\frac{\psi_{n}(\sqrt{u})}{\psi_{m}(u)}=2$ as is shown in
Figure 5. The condition of Theorem 4.2 holds and the result follows.

In Example 4.2, we can obtain the rhr functions of $X_{L_{m}}$ and $Y_{L_{n}}$
as follows:

$r_{X_{L_{m}}}(t)=\frac{r_{X}(t)}{\xi_{m}(F_{X}(t))}=\frac{256}{t^{4}(16+8t^{4}+2t^{8})},$

and

$r_{Y_{L_{n}}}(t)=\frac{r_{Y}(t)}{\xi_{n}(F_{Y}(t))}=\frac{16(t^{4}+2}{16+8t^{4}+2t^{8}}.$

In Figure 6 we have plotted the graph of
$\frac{r_{X_{L_{m}}}(t)}{r_{Y_{L_{n}}}(t)}$ to exhibit that it decreases in
$t$.

Figure 5: The graph of the function
$\frac{\psi_{n}(u^{\frac{1}{c_{0}^{\ast}}})}{\psi_{m}(u)}$ in Example 4.2.
Figure 6: The graph of $\frac{r_{X_{L_{m}}}(t)}{r_{Y_{L_{n}}}(t)}$ in Example
4.2.

##  5 Conclusion

In this paper, we first presented conditions under which the relative hazard
rate order between $X$ and $Y$ with the respective distributions $F$ and $G$
is translated into the relative hazard rate order of the upper record values
$X_{U_{m}}$ and $Y_{U_{n}}$ resulting from sequences of random lifetimes
following the base distributions $F$ and $G$. The first result states that if
$m\geq n$ and $X\succeq_{st}Y$, then $X\preceq_{c}Y$ implies
$X_{U_{m}}\preceq_{c}Y_{U_{n}}$, which means that if $X$ ages faster than $Y$
in terms of the hazard rate function, then $X_{U_{m}}$ ages correspondingly
faster than $Y_{U_{n}}$. In situations where $X_{U_{m}}$ and $Y_{U_{n}}$ have
an IFR distribution, $X_{U_{m}}\preceq_{c}Y_{U_{n}}$ means that the hazard
rate function of $X_{U_{m}}$ increases faster than the hazard rate function of
$Y_{U_{n}}$. It is also concluded that if $Y_{U_{n}}\in IFR$ then
$X_{U_{m}}\in IFR$. It has been shown that $X\succeq_{st}Y$ is not a necessary
condition to obtain the preservation property of order $\preceq_{c}$ under
upper record values. We have secondly given conditions under which the
relative reversed hazard rate order between $X$ and $Y$ is translated into the
relative reversed hazard rate order of lower record values $X_{L_{m}}$ and
$Y_{L_{n}}$. We have shown that if $m\geq n$ and $X\preceq_{st}Y$, then
$X\preceq_{b}Y$ results in $X_{L_{m}}\preceq_{b}Y_{L_{n}}$. If then
$X_{L_{m}}$ and $Y_{L_{n}}$ have a DRHR distribution, then
$X_{L_{m}}\preceq_{b}Y_{L_{n}}$ is equivalent to saying that the reversed
hazard rate function of $X_{L_{m}}$ decreases faster than the reversed hazard
rate function of $Y_{L_{n}}$. In this context, if $Y_{L_{n}}\in DRHR$ then
$X_{L_{m}}\in DRHR$. We have also proved that $X\preceq_{st}Y$ is not a
necessary condition to obtain the preservation property of order $\preceq_{b}$
under lower record values.

The results obtained in this study can be used to describe further findings on
record values. The hazard rate function and reversed hazard rate function of a
lifetime random variable $Z$ are proportional to the probabilities $P(X\leq
t+\delta|X>t)$ and $P(X>t-\delta|X\leq t)$ respectively, where $\delta$ is a
very small positive number. Usually it is important to predict future upper or
lower records. Therefore, it is a natural question whether a record higher
than $t$ will be reached directly after $t$. Let us assume $X$ and $Y$ have
the cdfs $F$ and $G$ respectively. For example,

$X_{U_{m}}\preceq_{c}Y_{U_{n}}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \frac{P(X_{U_{m}}\leq
t_{1}+\delta|X_{U_{m}}>t_{1})}{P(Y_{U_{n}}\leq
t_{1}+\delta|Y_{U_{n}}>t_{1})}\leq\frac{P(X_{U_{m}}\leq
t_{2}+\delta|X_{U_{m}}>t_{2})}{P(Y_{U_{n}}\leq
t_{2}+\delta|Y_{U_{n}}>t_{2})},\leavevmode\nobreak\ \leavevmode\nobreak\
\textmd{for all}\leavevmode\nobreak\ t_{1}\leq t_{2}\in\mathbb{R}^{+},$

indicates that if the $m$th upper record from the sequence of i.i.d. lifetimes
$X_{1},X_{2},\ldots$ from $F$ (i.e., $X_{U_{m}}$) has not retained until time
$t_{2}$ and also if the $n$th upper record from the sequence of i.i.d.
lifetimes $Y_{1},Y_{2},\ldots$ from $G$ (i.e., $Y_{U_{n}}$) has not retained
until time $t_{2}$, then it is more likely in the time $t_{2}$ in comparison
with the time $t_{1}$ that $X_{U_{m}}$ is equal to a value in a small
neighborhood after $t_{2}$ than that $Y_{U_{n}}$ to be equal to such value in
a small neighborhood after $t_{2}$. Moreover, there is another question
whether a lower record less than $t$ has retained right before $t$. We observe
that

$X_{L_{m}}\preceq_{b}Y_{L_{n}}\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \frac{P(X_{L_{m}}>t_{1}-\delta|X_{L_{m}}\leq
t_{1})}{P(Y_{L_{n}}>t_{1}-\delta|Y_{L_{n}}\leq
t_{1})}\geq\frac{P(X_{L_{m}}>t_{2}-\delta|X_{L_{m}}\leq
t_{2})}{P(Y_{L_{n}}>t_{2}-\delta|Y_{L_{n}}\leq t_{2})},\leavevmode\nobreak\
\leavevmode\nobreak\ \textmd{for all}\leavevmode\nobreak\ t_{1}\leq
t_{2}\in\mathbb{R}^{+},$

which shows that if the $m$th lower record from the sequence of i.i.d.
lifetimes $X_{1},X_{2},\ldots$ from $F$ (i.e., $X_{L_{m}}$) has retained until
time $t_{1}$ and further if the $n$th lower record from the sequence of i.i.d.
lifetimes $Y_{1},Y_{2},\ldots$ from $G$ (i.e., $Y_{L_{n}}$) has also retained
before time $t_{2}$, then it is more likely in the time $t_{1}$ compared with
the time $t_{2}$ that $X_{L_{m}}$ is equal to a value in a small neighborhood
before $t_{1}$ than that $Y_{L_{n}}$ to be equal to such value in a small
neighborhood before $t_{1}$.

In future research, one may consider simplifying the condition of Theorem 3.2
with respect to $n,m,c_{0}$ and $c_{1}$ and also simplifying the condition of
Theorem 4.2 on the basis of $n,m,c_{0}^{\ast}$ and $c_{1}^{\ast}$. The
existence of the supremum in Theorem 3.2 was guaranteed in Remark 3.1 for the
case that $X$ and $Y$ are equally distributed, where $c_{0}=c_{1}=1$ and also
after the proof of Theorem 4.2 it was guaranteed that the supremum exists in
the cases where $X$ and $Y$ have a common distribution leading to
$c^{\ast}_{0}=c^{\ast}_{1}=1$. However, as explained in Example 3.2 and
Example 4.2, there are also more complicated situations in which the suprema
exists. Further implications of the relative ordering of dataset values, e.g.
finding bounds for the survival functions of the upper datasets and the
distribution function of the lower datasets, can also be considered in the
future.

Acknowledgments

This work was supported by Researchers Supporting Project number
(RSP2024R392), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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# Blockchain-Powered Supply Chain Management for Kidney Organ Preservation
††thanks: * Authors Contributed Equally

Kapil Panda* University of North Texas Denton, United States of America
<EMAIL_ADDRESS>Anirudh Mazumder* University of North Texas Denton,
United States of America<EMAIL_ADDRESS>
###### Abstract

Due to the shortage of available kidney organs for transplants, handling every
donor kidney with utmost care is crucial to preserve the organ’s health,
especially during the organ supply chain where kidneys are prone to
deterioration during transportation. Therefore, this research proposes a
blockchain platform to aid in managing kidneys in the supply chain. This
framework establishes a secure system that meticulously tracks the organ’s
location and handling, safeguarding the health from donor to recipient.
Additionally, a machine-learning algorithm is embedded to monitor organ health
in real-time against various metrics for prompt detection of possible kidney
damage.

###### Index Terms:

cybersecurity, blockchain, supply chain, kidneys, machine learning, artificial
intelligence, organ preservation

##  I Introduction

Kidney transplantation has emerged as a life-saving medical intervention for
individuals suffering from end-stage organ failure [2]. Due to the lack of
kidneys available for organ donation, however, every kidney must be handled
with the utmost care to ensure that the organ’s health is preserved. The organ
supply chain is one such instance where organ health has been prone to
degradation due to various issues in transportation. Thus, ensuring the timely
and secure preservation of kidneys throughout their journey from donor to
recipient is an intricate logistical challenge that demands utmost precision
and transparency to avoid any detriments to the organ [4].

Blockchain technology has emerged as a transformative solution, offering the
potential to revolutionize supply chain management across various industries
[3]. At its core, blockchain represents a decentralized digital ledger that
securely records data in a series of interconnected blocks, ultimately
fostering trust and integrity in exchanging information. In recent years, the
medical industry has started to look towards this technology to enhance data
security, interoperability, and transparency in healthcare systems, while also
improving patient data management and secure sharing of electronic health
records among healthcare providers [5].

Therefore, this research aims to leverage blockchain technology to aid in
preserving and managing kidneys in the supply chain. By harnessing the
inherent features of blockchain, including decentralization and immutability,
a secure and transparent system is sought to be established that meticulously
tracks the organ’s condition, location, and handling at each stage of the
supply chain, safeguarding their health and viability from donor to recipient.
Furthermore, a machine learning algorithm is embedded that constantly checks
the health of the kidneys in real-time against various metrics to alert
stakeholders in case of any possible damage to the kidney[18].

##  II Methodology

###  II-A Materials

A dataset for predicting the state of a kidney was utilized as it allows the
platform to create a red flag stopping the transport of the kidney when it has
degraded past a threshold in which it is safe for the patient. This dataset
can be found at [1].

###  II-B Blockchain Platform

####  II-B1 Base Class

The base class serves as the foundation of the blockchain platform and
encompasses a blockchain class equipped with various essential methods which
create blocks, manage transactions, and ensure that the chain is valid.

####  II-B2 Block Creation

A few metrics come in when a new block is created within the platform. The
first is the time in which the block was created. The timestamp is crucial for
understanding a chain of events, as it gives an indication of the transaction
between the donor and the recipient, which is crucial for ordering the group
of blocks where the transactions can validate the chain [6].

Utilizing block creation, a transaction addition method was created, which
allows the user to add their data to the platform securely. The transaction
contains data about the conditions of the kidney from the donor, medical
information about the recipient and donor, and the location of the kidney.
Utilizing this information and the block creation method, the transaction can
be added to the platform instantiating the real-time abilities of the platform
[7].

The platform utilizes a node creation algorithm that enables new nodes to join
the decentralized network. Existing nodes can discover and connect to the new
node once it comes online. The new node then shares its IP address so other
nodes can connect to it. Next, the new node follows the network’s consensus
protocol to ensure it will operate consistently with other nodes. After
meeting consensus requirements, the node undergoes an identity verification
process to prevent unauthorized access to sensitive data. Upon successful
registration, the new node receives current blockchain data from peers to stay
synced. This node-creation process promotes decentralization by allowing new
nodes to join, making the network more resilient and scalable. The algorithm
also provides fault tolerance critical to the network’s reliability and
ability to handle increased transaction volume as it expands [8].

####  II-B3 Chain Validation

To ensure that the block is secure, a hashing method is added, which makes the
hash act as an identifier of the block, thus facilitating a link to the
previous block in the chain. This makes the data immutable by making it
incredibly complex to alter without breaking the cryptographic link to the
other blocks in the platform, while also establishing a consensus among the
different nodes in a decentralized network. The hash ensures that the nodes
can verify each other by recomputing it and comparing it to the previous hash
in the network. The consensus mechanism ensures that all the nodes agree on
the validity of the chain and the transactions it makes [9].

A proof of work method ensures the decentralized nodes can achieve a
consensus. This was done through data mining which has to find a random number
combined with the data of a new block. Using these two, the data miner has to
generate a hash that meets a specific condition. Then, the minder uses the
hash found and pushes a new block to the network, where other nodes verify the
proof of the network by recomputing the hash and seeing if the new block meets
a condition. The proof of work method is essential for securing the
blockchain, preventing tampering with previous blocks. The method also ensures
that within a decentralized network, all the nodes can work on the consensus
process, and no singular node controls the process. Lastly, this method allows
the nodes to achieve consensus about the chronological order of the
transactions and the validity of new blocks [10].

Then, chain validation attempts to check if anything has been tampered with,
in the blockchain. The method verifies the validity of each block in the chain
to ensure that the blocks are linked together in the correct order. The method
is vital to ensuring that nothing fraudulent gets into the platform, improving
the accuracy of the blockchain framework. The method first checks that the
hash is consistent throughout the chain. The check ensures that every block
has the same hash as the previous one unless it is the first one. Then, the
algorithm writer puts the hash in. Additionally, the algorithm checks the
validity of each block individually by ensuring they all meet a specific set
of criteria, including a valid work of proof. Then within each block,
transactions must be verified by ensuring they have valid signatures and
adhere to any protocol. To ensure that the chain is valid, it checks to ensure
they all meet consensus rules across all blocks. The chain validation process
is essential to ensure the chain is valid and accurate. Additionally, it is
critical to find invalid blocks which may occur due to issues such as
malicious attacks, so this method ensures that they are avoided and kept out
of the network to ensure that the platform is consistent and accurate [11].

Implementing an effective conflict resolution mechanism was critical as
inconsistencies could occur due to multiple factors, such as network delays,
accidental forks, or malicious activities. Thus, a way to reconcile these
discrepancies is needed to maintain a trustworthy distributed ledger. A
consensus mechanism like Proof of Work or Proof of Stake into the algorithm.
This allows the nodes in the network to agree on the valid canonical chain -
the most extended version of the blockchain. By leveraging consensus to
resolve conflicts, the nodes can converge on a consistent state of the ledger.
This prevents forks and ensures the integrity of the blockchain is preserved.
Through robust conflict resolution capabilities, the algorithm can enhance
security, immutability, and reliability. The algorithm can then function as a
robust decentralized system resistant to manipulation or corruption. Building
effective conflict resolution into the blockchain algorithm design is
essential for enabling key benefits like consistency across distributed
ledgers. This innovation makes the algorithm a viable backbone for
decentralized applications requiring strong consistency guarantees [12].

###  II-C Data Representation

####  II-C1 Blockchain Data

The blockchain consists of a chronological sequence of blocks
cryptographically linked together in a chain-like structure. Each block
contains several critical pieces of data. First, there is an index number that
indicates the unique position of the block within the chain. Second, a
timestamp records when the block was created and added. Next, a proof value
represents the computational work required to validate and append the block to
the chain while also upholding network security [13]. Additionally, each block
contains the previous block’s hash in the chain, chaining the blocks together
in order. Finally, the block includes a list of transaction data detailing the
kidney transfers from donors to patients. This transaction information
provides the critical utility of the blockchain for tracking the ownership and
provenance of the kidneys. The specific structure of blockchain blocks,
chaining them together with hashes and timestamps, is essential for the
decentralized ledger’s integrity, security, and suitability.

The blockchain’s data structure is vital for upholding the system’s integrity,
security, and transparency. By organizing data into blocks containing unique
attributes, each block can be identified and cryptographically linked to the
preceding block. This creates an immutable sequence of transactions. Hashing
each block to produce a unique fingerprint ensures tampering can be easily
detected, as any modification will alter the hash and break the chain. This
makes manipulating historical data on the blockchain practically infeasible.
Further, the data structure and proof-of-work consensus mechanism promote
trust and agreement among decentralized participants. This enables the
distributed network to function reliably without a central authority. Overall,
the ingenious data representation of blockchains is the foundation for their
ability to serve as open, secure, and tamper-proof ledgers, facilitating
mutually distrusting entities to reach a consensus on the authoritative state
of the system [14].

####  II-C2 Transaction Data

Each transaction on the blockchain contains extensive data regarding the
donor, recipient, and transportation of the kidney. Specifically, donor
information includes name, age, blood type, and other pertinent medical
details. Recipient data captures similar attributes for the patient getting
the transplant. Additionally, the real-time location of the kidney is recorded
while in transit to trace provenance. The blockchain also stores real-time
condition monitoring of the kidney using a machine-learning algorithm
developed to predict kidney viability. This rich transactional data provides
end-to-end tracking and oversight of the kidney journey from donor to
recipient. By immutably capturing comprehensive details at each stage, the
blockchain enables full transparency and audibility to instill trust in the
transplantation system. The granular data representation unlocks detailed
analytics on supply chain optimization; transport impacts on organ health, and
other insights to improve outcomes.

The transaction data structure is critical to keeping this information as it
keeps it secure [15]. Firstly, the data is easily accessible as people
utilizing the platform could retrieve information about the transactions
easily using a key to access the data. Additionally, it makes the data stay
consistent throughout the blockchain as it ensures that the transactions are
consistent with the standards throughout the blockchain. Lastly, the
algorithm’s ability to use real-time analysis is critical to understanding
where the kidney is within its transport and ensuring that the kidney is still
usable by the time it is given to the patient who requires the transplant.

####  II-C3 Machine Learning Data

The machine learning algorithm utilizes data about the kidney to dictate
whether or not the kidney is still in a condition where it would help the
recipient of the transplant. The data is represented within the transaction
data, as the transaction must contain the data as it has to be updated within
real-time, and it is vital for the blockchain to keep the data secure. Within
the blockchain, the data is utilized to update the predicted health condition
of the transaction data.

###  II-D Machine Learning

####  II-D1 Data Preprocessing

Categorical data in the dataset is encoded into numerical values using one-hot
or label encoding techniques. The resulting dataset is structured as a pandas
DataFrame, where each row represents an individual organ recipient, and the
columns correspond to various health measurements. The target variable,
indicating the health outcome (i.e., whether the kidney has kidney disease),
is represented as a series containing binary values (0 or 1).

####  II-D2 Machine Learning Model

The RandomForestClassifier is chosen due to its capability to handle non-
linearity, feature interactions, and complex decision boundaries, making it
well-suited for the intricate task of predicting organ health outcomes. During
the training phase, the dataset is split into training and test sets, with the
former used to train the model and the latter to evaluate its performance on
unseen data. The RandomForestClassifier is initialized with several decision
trees (n_estimators) and other hyperparameters [16].

####  II-D3 Performance Analysis

Once trained, the model can make predictions on new data. For each organ
recipient in the test set, the model predicts the health outcome based on
their health measurements. The model’s performance is then evaluated by
comparing the predicted outcomes with the actual health outcomes in the test
set. The study aims to improve decision-making processes in organ
transplantation, leading to better patient outcomes and more effective
allocation of organs.

Figure 1: Flow chart depicting blockchain algorithm’s framework

##  III Results

###  III-A Platform Implementation

A functional blockchain-based system was developed to enable the tracking of
organ transportation and assessment of real-time organ viability predictions.
The decentralized network consists of connected blocks, each containing key
data fields like a timestamp, proof of work hash, previous block hash, donor
and recipient details, transportation location data, and up-to-date predicted
organ health scores generated by an integrated machine learning model.

New organ transportation transactions are added to the network by creating a
new block that captures comprehensive details on the organ donor and recipient
along with current location-tracking information. Additionally, real-time
measurements of the transported organ’s biomarkers and clinical indicators are
fed as inputs into a pre-trained Random Forest classifier to generate updated
predictions on the kidney’s viability. The Random Forest model outputs a
probability score between 0 and 1, with higher values indicating a viable
organ suitable for transplantation. For example, in an initial test
transaction, a donor kidney with average red blood cell count, platelet count,
and other biomarkers was classified by the model as viable with 100%
probability.

Proof-of-work consensus is employed to validate each new block before
appending it to the chain. Cryptographic hashing of each block also ensures
the immutability of historical records, preventing tampering or errors. The
system was rigorously tested by adding multiple new sample transactions,
mining the transactions into new blocks, and visually inspecting the resulting
blockchain ledger. The decentralized architecture enforced trust,
transparency, and data integrity without needing a central authority.

Overall, a working prototype blockchain platform was successfully developed,
demonstrating the potential of blockchain technology to transform the
management of time-sensitive medical products like donor organs. Integrating
predictive analytics and machine learning into the tamper-proof ledger could
enable real-time tracking of organ health and transportation chain of custody.
This could dramatically improve clinical decision-making for complex medical
procedures like transplantation surgeries requiring organs to be viability
assessed up until the point of the implant.

###  III-B Machine Learning Accuracy

The machine learning model developed in this study achieved an impressive 100%
accuracy in diagnosing chronic kidney disease using kidney health metric data
that was recorded on the blockchain platform. The test set utilized to
evaluate the model’s performance consisted of 60 total kidney health profiles,
with 26 profiles from healthy individuals and 14 from individuals with chronic
kidney disease. As evident in the confusion matrix provided in Figure 2, the
model correctly classified all 26 healthy kidney profiles as negative for
chronic kidney disease and also correctly classified all 14 diseased kidney
profiles as testing positive for chronic kidney disease, with absolutely no
false positives or false negatives observed across the 40-sample test set.
This finding highlights the exceptional diagnostic capability of the machine
learning model.

Figure 2: Confusion matrix showing the predicted vs the actual values

The kidney health metric data that was securely recorded on the blockchain and
subsequently used to develop, train, and test the machine learning model
contained relevant biomarkers and indicators of kidney function and health
status. These metrics included critical biomarkers such as glomerular
filtration rate, albumin levels, creatinine levels, and other kidney function
parameters. By discerning subtle patterns and predictive cutoffs from these
kidney health metrics, the machine learning model could reliably differentiate
normal healthy kidney function from chronic kidney disease states.

In summary, the machine learning model achieved flawless 100% diagnostic
accuracy on the initial test set of 40 kidney health profiles. This remarkable
finding underscores the potential of integrating blockchain-based
decentralized health data recording with machine learning analysis to enable
accurate and automated screening for chronic kidney disease. Further
validation on larger datasets is warranted to confirm the model’s clinical
viability as a diagnostic tool moving forward.

##  IV Discussion

###  IV-A Conclusion

This study demonstrates the significant potential of blockchain technology to
transform and modernize the management of donor’s kidneys within the organ
supply chain. The blockchain-based platform developed in this research
provides comprehensive end-to-end tracking of kidney provenance throughout
transportation, enables real-time condition monitoring, and promotes
transparency across all stakeholders. By securely recording extensive data on
organ transportation events, handling, and predicted viability on a tamper-
proof decentralized ledger, this blockchain system allows for accountability,
security, coordination, and data-driven optimization across the organ supply
chain. The immutable nature of the blockchain ledger fosters trust and process
integrity for sensitive medical products like donor organs.

Additionally, the machine learning algorithm integrated into the blockchain
achieves an impressive 100% accuracy in diagnosing kidney disease using input
kidney health metrics. This finding highlights the tremendous value of
predictive analytics in enabling proactive, data-driven decision-making in
time-sensitive medical logistics. The machine learning model’s real-time organ
health monitoring capabilities could significantly improve clinical outcomes
through prompt intervention when any deterioration in organ viability is
detected. This proactive approach to dynamically preserving and maximizing
organ health throughout the supply chain could reduce waste and improve
efficiency in organ transplantation.

While further rigorous testing and validation on larger datasets is warranted,
this research provides a compelling proof-of-concept for modernizing and
optimizing organ supply chain management through integrating blockchain and
artificial intelligence technologies. By fostering trust, transparency, and
data-driven predictive insights, these technologies can help address
considerable challenges in equitable organ allocation, delivery, and
leveraging of available organs. The blockchain platform and machine learning
framework developed in this study lay the groundwork for scalable
implementation of these technologies in healthcare to coordinate complex
medical procedures like organ transplants better and ultimately save lives
through improved clinical outcomes.

###  IV-B Future Works

This study presents several important directions for further developing the
blockchain and machine learning framework. First and foremost, substantially
expanding the performance evaluation and validation of the blockchain platform
and integrated machine learning model on much larger and more diverse organ
transportation datasets is critical. Thoroughly testing the system across
various organ shipment scenarios with numerous sample transactions recorded on
the blockchain will be essential for comprehensive real-world validation
across edge cases. Carefully assessing generalizability will indicate where
additional training data may be beneficial.

Second, integrating Internet-of-Things (IoT) sensors for real-time tracking of
environmental conditions such as temperature, humidity, vibration, and other
metrics during organ transportation could provide valuable multivariate sensor
data to feed into the machine learning algorithm’s organ viability
predictions. Incorporating streaming real-time sensor metrics alongside
clinical biomarkers may enhance the early detection of organ changes.
Extensive experimentation will be essential to determine optimal strategies
for fusing IoT and clinical data at scale. The impact on model accuracy,
training requirements, and prediction latency requires careful evaluation.

Additionally, investigating alternative state-of-the-art machine learning
techniques such as neural networks, graph-based models, and ensemble methods
would be worthwhile to compare against the baseline Random Forest classifier’s
diagnostic performance. Determining if modern deep learning architectures
could boost detection accuracy over the Random Forest merits in-depth
exploration. Rigorously benchmarking a diverse set of models to identify the
architecture that yields the highest precision in predicting emerging organ
problems will be critical.

Furthermore, expanding the feature space considered in the model to
incorporate relevant biological traits and health data of the organ recipient
could better account for individual-specific variability. Tailoring organ
viability predictions based on a recipient’s clinical status and biomarkers
may improve clinical outcomes. Thoughtful feature engineering to identify key
recipient health attributes to integrate into the model is another important
direction for performance enhancement. Evaluating predictive value versus
model complexity will inform feature selection.

##  V Acknowledgment

We would like to thank the University of North Texas for providing us with the
resources and support to conduct this research. The invaluable guidance and
encouragement from our professors and mentors have been instrumental in
shaping the direction and scope of this study. We would also like to
acknowledge the National Kidney Foundation for their support and inspiration
in conducting this project. We would also like to thank our families for
supporting us throughout our research.

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