# Stochastic Comparisons of Random Extremes from non-identical Random
Variables

Amarjit Kundu  Department of Mathematics, Raiganj University, West Bengal,
India  Shovan Chowdhury111Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>Quantitative Methods and Operations Management Area, Indian
Institute of Management Kozhikode, India  Bidhan Modok  Department of
Mathematics, Raiganj University, West Bengal, India

###### Abstract

We propose some new results on the comparison of the minimum or maximum order
statistic from a random number of non-identical random variables. Under the
non-identical set-up, with certain conditions, we prove that random minimum
(maximum) of one system dominates the other in hazard rate (reversed hazard
rate) order. Further, we prove variation diminishing property (Karlin [8]) for
all possible restrictions to derive the new results.

Keywords and Phrases: Hazard rate order, Order statistics, $RR_{2}$ function,
Reversed hazard rate order, Stochastic order, $TP_{2}$ function.
AMS 2010 Subject Classifications: 60E15, 60K40

##  1 Introduction

The problem of stochastic comparison of extremes (maximum or minimum) has
always been an important topic in a variety of disciplines when sample size is
considered fixed or random. In a fixed sample size scenario, many researchers
have compared series (minimum order statistic) or parallel (maximum order
statistic) system when the components are identical or non-identical and are
ordered in some fashion (see [23],[13], [3], [5], [14]). Moreover, readers may
refer to [2] and the references therein for a comprehensive review til 2013.
However, in the field of finance and risk management, reliability optimization
and life testing experiment, actuarial science, biological Science,
engineering and manufacturing, hydrology and others, sample size may be
unobserved, or may depend on the occurrence of unsystematic events which makes
it random. Following examples justify the use of random sample size in
practice.
Lifetime experiments are always censored and hence sample size can hardly be
fixed. In actuarial science, the performance of a portfolio of insurance
contracts requires stochastic modeling of the largest claims while the number
of claims is random. In biological and agricultural experiments, some elements
of the sample under consideration may die for unknown reasons making sample
size random. In finance and risk management, one may be interested in the
minimum and maximum loss from a portfolio of loans or securities with random
number of assets or liabilities. The use of random sample size arises
naturally in reliability engineering to model lifetime of a parallel or series
system. One can think of a situation where failure (of a device for example)
occurs due to the presence of an unknown number (random) of initial defects of
the same kind (a number of semiconductors from a defective lot, for example)
where each defect can be detected only after causing failure, in which case it
is repaired perfectly. In reliability optimization, one system can dominate
the other under certain conditions when the components are chosen randomly
from two batches ( [6], [7], [16]). In hydrological applications ( [12])
concerning the distribution of annual maximum rainfall or flood, it may be
assumed that the number of storms or floods per year is not constant; in
traffic engineering applications ( [22]) velocity distribution of vehicles on
a particular spot on a road or a highway is obtained after recording
velocities of a random number of vehicles pass through the point; in
transportation theory ( [21]) modeling accident-free distance of a shipment of
explosives require random number of explosives which are defective and lead to
explosion and cause an accident; in biostatistical applications ( [11]
concerning the distribution of minimum time for clonogenic (carcinogenic) cell
to produce a detectable cancer mass, number of clonogens (carcinogenic cells)
left active after a cancer treatment is always random.
Results concerning stochastic comparison and ageing properties of random
minima and maxima can be found in [19], [21], [4], [15], [1], [16], [7]. For
an overview of stochastic comparisons with random sample size, see [17]. The
results in these papers are obtained for independent and identically
distributed ($iid$) random variables. So, the natural question arises whether
random extremes can be compared for independent and non-identically
distributed ($inid$) random variables. To the best of our knowledge, there has
been no attention paid to the case of non-identical set-up which motivates us
to compare random extremes of $inid$ random variables with suitable stochastic
ordering principle. To this endeavor, we have proven variation diminishing
property (Karlin [8]) for all possible restrictions which fills some
theoretical gap in the literature.
One may think of applying the key findings of the paper in the following real
life situations.
Let us consider two independent queueing systems with one server each. One may
wish to compare the service time of customers stochastically in order to
measure the operational efficiency of the servers. Time taken to serve the
$i^{th}$ customer ($i=1,2,...,n$) by the $j^{th}$ server ($j=1,2$) can be
assumed to be $inid$ random variable with cdf $F.$ It is known that the
minimum of the service time of $n$ customers served by server 1 is less than
that of the same by server 2 in hazard rate ordering. In practice, queue size
is random and hence it is important to answer what happens to the comparison
when the number of customers ($N$) served by each server is random and follows
any discrete distribution $p(n)$ with common parameter(s) with
$\mathcal{N^{+}}$ as support. According to Theorem 3.1, minimum of the service
time of $N$ customers served by server 1 is also less than that of the same by
server 2 in hazard rate ordering. The result confirms that the same ordering
relationship will hold for random number of customers under certain condition.
Suppose two portfolios of insurance contracts need to be compared
stochastically in terms of the maximum claim amounts (loss to the firm). The
actual number of claims is supposed to be always random and the portfolio
managers are interested to know which portfolio incurs maximum loss. If it is
known that the maximum loss from the first portfolio of $n$ claims is more
than the same from the second in reversed hazard rate ordering, according to
Theorem 3.3, the same result will hold under certain conditions for random
claim size $N$ defined with support $\mathcal{N^{+}}.$
The organization of the paper is as follows. Relevant definitions and existing
results are given in Section 2. In section 3, new lemmas are derived and the
results related to hazard rate order and reversed hazard rate order between
two random extremes of $inid$ random variables are established with examples.
Section 4 concludes the paper.
Throughout the paper, the word increasing (resp. decreasing) and nondecreasing
(resp. nonincreasing) are used interchangeably. The notations
$\mathcal{R}_{+}$ and $\mathcal{N}^{+}$ denote the set of positive real
numbers $\\{x:0<x<\infty\\}$ and natural numbers $\\{x:x=1,2,...,\infty\\}$.
The random variables considered in this paper are all nonnegative.

##  2 Preliminaries

For an absolutely continuous random variable $X$, we denote the cumulative
distribution function by $F_{X}(\cdot)$, the hazard rate function by
$r_{X}(\cdot)$ and the reversed hazard rate function
$\overline{r}_{X}(\cdot)$. The survival or reliability function of the random
variable $X$ is written as $\bar{F}_{X}(\cdot)=1-F_{X}(\cdot)$.
In order to compare different order statistics, stochastic orders are used for
fair and reasonable comparison. In literature different kinds of stochastic
orders are developed and studied. The following well known definitions can be
obtained in [20].

###### Definition 2.1

Let $X$ and $Y$ be two absolutely continuous random variables with respective
supports $(l_{X},u_{X})$ and $(l_{Y},u_{Y})$, where $u_{X}$ and $u_{Y}$ may be
positive infinity, and $l_{X}$ and $l_{Y}$ may be negative infinity. Then, $X$
is said to be smaller than $Y$ in

  1. 1.

hazard rate (hr) order, denoted as $X\leq_{hr}Y$, if

$\frac{\bar{F}_{Y}(t)}{\bar{F}_{X}(t)}\;\text{is increasing
in}\,t\in(-\infty,max(u_{X},u_{Y})),$

which can equivalently be written as $r_{X}(t)\geq r_{Y}(t)$ for all $t$;

  2. 2.

reversed hazard rate (rh) order, denoted as $X\leq_{rh}Y$, if

$\frac{F_{Y}(t)}{F_{X}(t)}\;\text{is increasing
in}\,t\in(min(l_{X},l_{Y}),\infty),$

which can equivalently be written as
$\overline{r}_{X}(t)\leq\overline{r}_{Y}(t)$ for all $t$;

  3. 3.

usual stochastic (st) order, denoted as $X\leq_{st}Y$, if
$\bar{F}_{X}(t)\leq\bar{F}_{Y}(t)$ for all
$t\in(-\infty,\infty).$ $\hfill\Box$

The next two theorems due to [20] are used to prove the key findings of the
paper.

###### Theorem 2.1

$\left(\text{Shaked and
Santhikumar~{}\cite[cite]{[\@@bibref{}{shak}{}{}]},Theorem 1.B.28, Page
31}\right)\;$ If $X_{1},\ X_{2}\ldots,\ X_{m}$ are independent random
variables, then $X_{(k:m-1)}\geq_{hr}X_{(k:m)}$ for $k=1,2,\ldots,m-1.$

###### Theorem 2.2

$\left(\text{Shaked and
Santhikumar~{}\cite[cite]{[\@@bibref{}{shak}{}{}]},Theorem 1.B.57, Page
41}\right)\;$ Let $X_{1},\ X_{2}\ldots,\ X_{m}$ be independent random
variables. If $X_{m}\leq_{rh}X_{j}$ for all $j=1,2,\ldots,m-1$, then
$X_{(k-1:m-1)}\leq_{rh}X_{(k:m)}$ for $k=2,3,\ldots,m$.

The notion of Totally Positive of order $2$ ($TP_{2}$) and Reverse Regular of
order $2$ ($RR_{2}$) are of great importance in various fields of applied
probability and statistics (see Karlin [8], Karlin and Rinnot [9, 10]). It can
be recalled that a non-negative function $f:\Re^{2}\longmapsto\Re_{+}$ is said
to be $TP_{2}$ if

$f(u_{1},v_{1})f(u_{2},v_{2})-f(u_{2},v_{1})f(u_{1},v_{2})\geq 0,$ (2.1)

for all $u_{1}\leq u_{2}$ and $v_{1}\leq v_{2}$. This $f$ is said to be
$RR_{2}$ if the inequality in (2.1) is reversed. This can equivalently be
written as follows
$f$ is $TP_{2}$ ($RR_{2}$) if $f(u,v_{2})/f(u,v_{1})$ is increasing
(decreasing) in $u$ for all $v_{1}\leq v_{2}$.
The following Theorem on variation diminishing property is proved by Karlin
([8]) for some definite restrictions on $f(x,y)$ and $K(x,y)$.

###### Theorem 2.3

$\left(\text{Karlin~{}\cite[cite]{[\@@bibref{}{ka}{}{}]},Theorem 11.2, Page
324-25}\right)\;$ Let $K\left(x,y\right)>0$, defined on $X\times Y,$ be
$TP_{2}$, where $X$ and $Y$ are the subsets of the real line. Assume that a
function $f\left(x,y\right)$ is such that

  1. (i)

for each $x$, $f\left(x,y\right)$ changes sign at most once, and from negative
to positive values, as $y$ traverses $Y;$

  2. (ii)

for each $y$, $f\left(x,y\right)$ is increasing in $x$;

  3. (iii)

$g\left(x\right)=\int_{Y}f\left(x,y\right)K\left(x,y\right)d\mu(y)$ exists
absolutely and defines a continuous function on $x$.

Then $g\left(x\right)$ changes sign at most once, and from negative to
positive values.$\hfill\Box$

The Lemmas in the next section are based on Theorem 2.3 yielding results for
all possible restrictions on $f(x,y)$ and $K(x,y)$ where $y$ is defined on
$\mathcal{N^{+}}.$ The lemmas contribute to the literature in the field of
inequalities and are the building blocks for the key results of the paper.

##  3 Main Results

We begin this section with the following four lemmas which will be used to
prove the main results. Readers will find the proof of Lemma 3.1. The other
three lemmas can be proved in a similar line and hence are omitted.

###### Lemma 3.1

For any positive integer $n$ and positive real number $x$, let $K_{n}(x)>0$ be
a $RR_{2}$ function in $n\in\mathcal{N}^{+}$ and $x\in\mathcal{R}_{+}$. Assume
that any function $f_{n}(x)$ be such that

  * i)

for each $x\in\mathcal{R}_{+}$, $f_{n}(x)$ changes sign at most once and if
the change occurs, it is from negative to positive as $n$ traverses in
$\mathcal{N}^{+}$;

  * ii)

for each $n\in\mathcal{N}^{+}$, $f_{n}(x)$ decreases in $x$;

  * iii)

$w(x)=\sum_{n=1}^{\infty}{f_{n}(x)K_{n}(x)}$ converges absolutely and defines
a continuous function of $x$.

Then $w(x)$ changes sign at most once and if the change occurs, it is from
positive to negative.

Proof: For any $x_{0}\in\mathcal{R}_{+}$ let us assume that $w(x_{0})=0$. Then
there must exist one positive integer $n_{0}\in\mathcal{N^{+}}$ such that
$f_{n}(x_{0})\geq 0$ for all $n\geq n_{0}$ and $f_{n}(x_{0})\leq 0$ for all
$n\leq n_{0}.$ For any $x\in\mathcal{R}_{+}$ let us consider

$\displaystyle\frac{w(x)}{K_{n_{0}}(x)}$ $\displaystyle=$
$\displaystyle\frac{w(x)}{K_{n_{0}}(x)}-\frac{w(x_{0})}{K_{n_{0}}(x_{0})}\quad\left[\text{since}\;w(x_{0})=0\right]$
(3.1) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{\frac{K_{n}(x)f_{n}(x)}{K_{n_{0}}(x)}-\sum_{n=1}^{\infty}{\frac{K_{n}(x_{0})f_{n}(x_{0})}{K_{n_{0}}(x_{0})}}}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{\left[\frac{K_{n}(x)f_{n}(x)}{K_{n_{0}}(x)}-\frac{K_{n}(x)f_{n}(x_{0})}{K_{n_{0}}(x)}+\frac{K_{n}(x)f_{n}(x_{0})}{K_{n_{0}}(x)}-\frac{K_{n}(x_{0})f_{n}(x_{0})}{K_{n_{0}}(x_{0})}\right]}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{f_{n}(x_{0})\left[\frac{K_{n}(x)}{K_{n_{0}}(x)}-\frac{K_{n}(x_{0})}{K_{n_{0}}(x_{0})}\right]}+\sum_{n=1}^{\infty}{\left[f_{n}(x)-f_{n}(x_{0})\right]\frac{K_{n}(x)}{K_{n_{0}}(x)}}.$

Now two cases may arise.
Case-1: Let $x<x_{0}$ and $n>n_{0}$. Now as $K_{n}(x)$ is $RR_{2}$ in $n$ and
$x$, it can be written that $K_{n}(x_{0})K_{n_{0}}(x)\leq
K_{n_{0}}(x_{0})K_{n}(x),$ for all $x<x_{0}$ and $n>n_{0},$ which gives

$\frac{K_{n}(x)}{K_{n_{0}}(x)}-\frac{K_{n}(x_{0})}{K_{n_{0}}(x_{0})}\geq 0.$

Since $n>n_{0}$ implies $f_{n}(x_{0})>0$, the first term of (3.1) is positive.
Now, for each $n\in\mathcal{N}^{+},$ $f_{n}(x)$ is decreasing in $x$ and
$x<x_{0}$, yielding $f_{n}(x)-f_{n}(x_{0})\geq 0$. Hence, from the fact that,
for all $n\in\mathcal{N}^{+}$ and $x\in\mathcal{R}_{+}$ $K_{n}(x)>0$, it can
be concluded that the second term of (3.1) is also positive.
Case-2: Now, let $x<x_{0}$ and $n<n_{0}$. Since $K_{n}(x)$ is $RR_{2}$ in $n$
and $x$, then $K_{n}(x)K_{n_{0}}(x_{0})\leq K_{n_{0}}(x)K_{n}(x_{0})$, which
in turn gives

$\frac{K_{n}(x)}{K_{n_{0}}(x)}-\frac{K_{n}(x_{0})}{K_{n_{0}}(x_{0})}\leq 0.$

Since $n<n_{0}$ implies $f_{n}(x_{0})<0,$ the first term of (3.1) is positive.
Again, for each $n,$ $f_{n}(x)$ is decreasing in $x$, yielding
$f_{n}(x)-f_{n}(x_{0})\geq 0,$for all $x<x_{0}.$ Therefore, from the fact that
for all $n,x,$ $K_{n}(x)>0$, it can be concluded that the second term of (3.1)
is also positive and thus $\frac{w(x)}{K_{n_{0}}(x)}$ is positive.
Hence, for $x<x_{0},$ $w(x)>0$ in both the cases. Thus the Lemma is
proved.$\hfill\Box$

###### Lemma 3.2

For any positive integer $n$ and positive real number $x$, let $K_{n}(x)>0$ be
a $RR_{2}$ function in $n\in\mathcal{N}^{+}$ and $x\in\mathcal{R}_{+}$. Assume
that any function $f_{n}(x)$ be such that

  * i)

for each $x\in\mathcal{R}_{+}$, $f_{n}(x)$ changes sign at most once and if
the change occurs, it is from positive to negative as $n$ traverses in
$\mathcal{N}^{+}$;

  * ii)

for each $n\in\mathcal{N}^{+}$, $f_{n}(x)$ increases in $x$;

  * iii)

$w(x)=\sum_{n=1}^{\infty}{f_{n}(x)K_{n}(x)}$ converges absolutely and defines
a continuous function of $x$.

Then $w(x)$ changes sign at most once and if the change occurs, it is from
negative to positive.

###### Lemma 3.3

For any positive integer $n$ and positive real number $x$, let $K_{n}(x)>0$ be
a $TP_{2}$ function in $n\in\mathcal{N}^{+}$ and $x\in\mathcal{R}_{+}$. Assume
that any function $f_{n}(x)$ be such that

  * i)

for each $x\in\mathcal{R}_{+}$, $f_{n}(x)$ changes sign at most once and if
the change occurs, it is from positive to negative as $n$ traverses in
$\mathcal{N}^{+}$;

  * ii)

for each $n\in\mathcal{N}^{+}$, $f_{n}(x)$ decreases in $x$;

  * iii)

$w(x)=\sum_{n=1}^{\infty}{f_{n}(x)K_{n}(x)}$ converges absolutely and defines
a continuous function of $x$.

Then $w(x)$ changes sign at most once and if the change occurs, it is from
positive to negative.

###### Lemma 3.4

For any positive integer $n$ and positive real number $x$, let $K_{n}(x)>0$ be
a $TP_{2}$ function in $n\in\mathcal{N}^{+}$ and $x\in\mathcal{R}_{+}$. Assume
that any function $f_{n}(x)$ be such that

  * i)

for each $x\in\mathcal{R}_{+}$, $f_{n}(x)$ changes sign at most once and if
the change occurs, it is from negative to positive as $n$ traverses in
$\mathcal{N}^{+}$;

  * ii)

for each $n\in\mathcal{N}^{+}$, $f_{n}(x)$ increases in $x$;

  * iii)

$w(x)=\sum_{n=1}^{\infty}{f_{n}(x)K_{n}(x)}$ converges absolutely and defines
a continuous function of $x$.

Then $w(x)$ changes sign at most once and if the change occurs, it is from
negative to positive.

Let $X_{i}$ ($Y_{i}$), $i=1,2,\ldots,n$, be $n$ independent and non-identical
random variables having distribution function $F_{i}\left(x\right)$
($G_{i}\left(x\right)$), for all $i=1,2,\ldots,n$. Also let
$F_{n:n}\left(x\right)$ and $G_{n:n}\left(x\right)$
($\bar{F}_{1:n}\left(x\right)$ and $\bar{G}_{1:n}\left(x\right)$) be the
distribution (survival) functions of $X_{n:n}$ and $Y_{n:n}$ ($X_{1:n}$ and
$Y_{1:n}$) respectively with $X_{1:n}$ and $X_{n:n}$ being defined as

$X_{1:n}=min\left\\{X_{1},X_{2},...,X_{n}\right\\}$

and

$X_{n:n}=max\left\\{X_{1},X_{2},...,X_{n}\right\\}.$

Let $N$ is a discrete random variable with probability mass function (pmf)
$p(n)$, $n\in\mathcal{N}^{+}$. Now, for the random variable $N$, if
$F_{N:N}\left(x\right)$ and $G_{N:N}\left(x\right)$
($\bar{F}_{1:N}\left(x\right)$ and $\bar{G}_{1:N}\left(x\right)$) denote the
distribution (survival) functions of $X_{N:N}$ and $Y_{N:N}$ ($X_{1:N}$ and
$Y_{1:N}$) respectively, then it can be written that

$F_{N:N}(x)=\sum_{n=1}^{\infty}F_{n:n}(x)p(n),\
G_{N:N}(x)=\sum_{n=1}^{\infty}G_{n:n}(x)p(n),$ (3.2)
$\bar{F}_{1:N}(x)=\sum_{n=1}^{\infty}\bar{F}_{1:n}(x)p(n)\ \text{and}\
\bar{G}_{1:N}(x)=\sum_{n=1}^{\infty}\bar{G}_{1:n}(x)p(n).$ (3.3)

The following theorems show that under certain restrictions, if there exists
hazard rate (reversed hazard rate) ordering between $X_{1:n}$ and $Y_{1:n}$
($X_{n:n}$ and $Y_{n:n}$), then the same ordering also exists between
$X_{1:N}$ and $Y_{1:N}$ ($X_{N:N}$ and $Y_{N:N}$).

###### Theorem 3.1

Suppose $X_{1:n}\sim\bar{F}_{1:n}(x)$ and $Y_{1:n}\sim\bar{G}_{1:n}(x).$ Also
let $N$ be a random variable having pmf $p(n).$ Now, for all $x\geq 0$, if
$\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}$ is increasing in
$n\in\mathcal{N}^{+}$, then $X_{1:n}\leq_{hr}Y_{1:n}$ implies
$X_{1:N}\leq_{hr}Y_{1:N}.$

Proof: Here, we need to prove that $\frac{\bar{F}_{1:N}(x)}{\bar{G}_{1:N}(x)}$
is decreasing in $x$ or equivalently
$\frac{\sum_{n=1}^{\infty}{\bar{F}_{1:n}(x)p(n)}}{\sum_{n=1}^{\infty}{\bar{G}_{1:n}(x)p(n)}}$
is decreasing in $x.$ Now for any $\lambda>0$, let

$\displaystyle w(x)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{\left[\bar{F}_{1:n}(x)-\lambda\bar{G}_{1:n}(x)\right]p(n)}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{\bar{G}_{1:n}(x)\left[\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}-\lambda\right]p(n)}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}{\bar{G}_{1:n}(x)f_{n}(x)p(n)}\
\text{(say),}$

where
$f_{n}(x)=\left[\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}-\lambda\right]$.
Now, $X_{1:n}\leq_{hr}Y_{1:n}$ implies that
$\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}$ is decreasing in $x$, and hence
$f_{n}(x)$ is also decreasing in $x.$
Since $\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}$ is increasing in $n$, it is
clear that $f_{n}(x)$ has at most one change of sign as $n$ traverses and if
it does occur, it is from negative to positive.
Also, Theorem 2.1 concludes that for any two positive integers $n_{1}\leq
n_{2}$, $X_{1:n_{1}}\geq_{hr}X_{1:n_{2}}$, which by definition of $hr$
ordering indicates that $\frac{\bar{G}_{1:n_{1}}(x)}{\bar{G}_{1:n_{2}}(x)}$ is
increasing in $x$, yielding that $\bar{G}_{1:n}(x)$ is $RR_{2}$ function in
$n$ and $x$.
Thus, by Lemma 3.1, $w(x)$ changes it’s sign at most once and if the changes
occurs, it is from positive to negative. So, it can be concluded that
$\frac{\sum_{n=1}^{\infty}{\bar{F}_{1:n}(x)}p(n)}{\sum_{n=1}^{\infty}\bar{G}_{1:n}(x)p(n)}$
is decreasing in $x$, proving the result.$\hfill\Box$
Next, we illustrate Theorem 3.1 with two examples. In the first example,
independent heterogenous location family distributed random variable is
assumed while $inid$ Weibull distributed random variable is chosen in the
second example.

Example 1: Let, $X\sim\bar{F}(x)$ is a random variable having decreasing
failure rate function $r(x)$. Also let, for $n\in\mathcal{N}^{+}$,
$X_{i}\sim\bar{F}\left(x-\lambda_{i}\right)$ and
$Y_{i}\sim\bar{F}\left(x-\mu_{i}\right),~{}i=1,2,...,n$ are two sequences of
$inid$ random variables where $\lambda_{i}\geq\mu_{i}$. Then, the survival and
failure rate functions of $X_{1:n}$ and $Y_{1:n}$ are obtained as

$\bar{F}_{1:n}(x)=\prod_{i=1}^{n}\bar{F}\left(x-\lambda_{i}\right),\
\bar{G}_{1:n}(x)=\prod_{i=1}^{n}\bar{F}\left(x-\mu_{i}\right),$ (3.4)

and

$r_{1:n}(x)=\sum_{i=1}^{n}r\left(x-\lambda_{i}\right)\ \text{and}\
s_{1:n}(x)=\sum_{i=1}^{n}r\left(x-\mu_{i}\right)$ (3.5)

respectively.
Since, $\lambda_{i}\geq\mu_{i}$ and $r(x)$ is decreasing in $x$, it can be
written that $r\left(x-\lambda_{i}\right)\geq r\left(x-\lambda_{i}\right)$ for
$i=1,2,\ldots,n.$ Hence, from (3.5), it is clear that

$r_{1:n}(x)=\sum_{i=1}^{n}r\left(x-\lambda_{i}\right)\geq\sum_{i=1}^{n}r\left(x-\mu_{i}\right)=s_{1:n}(x),$

proving that $X_{1:n}\leq_{hr}Y_{1:n}$.
Now, for all $n\in\mathcal{N}^{+},$ $\lambda_{i}\geq\mu_{i}$ gives
$\bar{F}\left(x-\lambda_{i}\right)\geq\bar{F}\left(x-\mu_{i}\right).$ From
(3.4), it canm be immediately concluded that

$\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}=\prod_{i=1}^{n}\frac{\bar{F}\left(x-\lambda_{i}\right)}{\bar{F}\left(x-\mu_{i}\right)}$

is increasing in $n$.
Furthermore, from (3.4) it can be written that for all $n_{1}\leq n_{2}$,

$\displaystyle\frac{\bar{G}_{1:n_{1}}(x)}{\bar{G}_{1:n_{2}}(x)}=\frac{\prod_{i=1}^{n_{1}}\bar{F}\left(x-\mu_{i}\right)}{\prod_{i=1}^{n_{2}}\bar{F}\left(x-\mu_{i}\right)}=\frac{1}{\prod_{i=n_{1}+1}^{n_{2}}\bar{F}\left(x-\mu_{i}\right)},$

which is clearly increasing in $x$. Thus all the conditions of Theorem 3.1 are
satisfied. Hence, it can be concluded that $X_{1:N}\leq_{hr}Y_{1:N}$.

Example 2: Let us assume that $X$ follows Weibull distribution with survival
function

$\bar{F}(x)=e^{-2x^{0.5}}.$

Clearly the failure rate function of $X$, $r(x)=\frac{1}{\sqrt{x}}$ is
decreasing in $x.$
For $i=1,2\ldots,5$, suppose $X_{i}\sim\bar{F}\left(x-\lambda_{i}\right)$ and
$Y_{i}\sim\bar{F}\left(x-\mu_{i}\right)$ where
$\left(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}\right)=(0.1,0.2,0.3,0.4,0.5)$
and
$\left(\mu_{1},\mu_{2},\mu_{3},\mu_{4},\mu_{5}\right)=(0.05,0.15,0.25,0.35,0.45)$,
which implies that $\lambda_{i}\geq\mu_{i}$. So, for $n=1,2\ldots,5$, it can
be written that
$\bar{F}_{1:n}(x)=\prod_{i=1}^{n}\bar{F}(x-\lambda_{i})$ and
$\bar{G}_{1:n}(x)=\prod_{i=1}^{n}\bar{F}(x-\mu_{i})$. Now, if it is assumed
that

$X_{1},X_{2},X_{3}~{}(Y_{1},Y_{2},Y_{3})~{}\text{are selected with
probability}P(N=3)=p(3)=1/5,$
$X_{1},X_{2},X_{3},X_{4}~{}(Y_{1},Y_{2},Y_{3},Y_{4})~{}\text{are selected with
probability}P(N=4)=p(4)=2/5~{}\text{and}$
$X_{1},X_{2},X_{3},X_{4},X_{5}~{}(Y_{1},Y_{2},Y_{3},Y_{4},Y_{5})~{}\text{are
selected with probability}P(N=5)=p(5)=2/5,$

then from Figure 1, it can be verified that

$\frac{\bar{F}_{1:N}(x)}{\bar{G}_{1:N}(x)}=\frac{{\bar{F}_{1:3}(x)}p(3)+{\bar{F}_{1:4}(x)}p(4)+{\bar{F}_{1:5}(x)}p(5)}{{\bar{G}_{1:3}(x)}p(3)+{\bar{G}_{1:4}(x)}p(4)+{\bar{G}_{1:5}(x)}p(5)}$

is decreasing in $x$, proving the result. Here, substitution $x=-\ln y$,
$0\leq y\leq 1$ is used to capture the real line.

Figure 1: Graph of $\frac{\bar{F}_{1:N}(-\ln y)}{\bar{G}_{1:N}(-\ln y)}$

###### Remark 3.1

It can be checked that, in Theorem 3.1, if there exists stochastic ($st$)
ordering between $X_{1:n}$ and $Y_{1:n}$ in place of $hr$ ordering, then
without any condition as of the previous theorem, there will also exists $st$
ordering between $X_{1:N}$ and $Y_{1:N}$. The proof of it is very trivial.
Same can be observed for Theorem 3.2-3.4 also.

The next few theorems can be proved in the same fashion as above and hence the
details are omitted.

###### Theorem 3.2

Suppose $X_{1:n}\sim\bar{F}_{1:n}(x)$ and $Y_{1:n}\sim\bar{G}_{1:n}(x).$ Also
let $N$ be a random variable having pmf $p(n).$ Now, for all $x\geq 0$, if
$\frac{\bar{F}_{1:n}(x)}{\bar{G}_{1:n}(x)}$ is decreasing in
$n\in\mathcal{N}^{+}$, then $X_{1:n}\geq_{hr}Y_{1:n}$ implies
$X_{1:N}\geq_{hr}Y_{1:N}.$

Proof: The theorem can be proved using Lemma 3.2, Theorem 2.1 and following
the proof of Theorem 3.1.

###### Theorem 3.3

Suppose $X_{n:n}\sim\bar{F}_{n:n}(x)$ and $Y_{n:n}\sim\bar{G}_{n:n}(x).$ Also
let $N$ be a random variable having pmf $p(n).$ Now, for all $x\geq 0$, if
$\overline{r}_{Y_{n:n}}(x)$, the reversed hazard rate function of $Y_{n:n}$ is
increasing in $n\in\mathcal{N}^{+}$ and $\frac{F_{n:n}(x)}{G_{n:n}(x)}$ is
decreasing in $n\in\mathcal{N}^{+}$, then $X_{n:n}\leq_{rh}Y_{n:n}$ implies
$X_{N:N}\leq_{rh}Y_{N:N}.$

Proof: If $\overline{r}_{Y_{n:n}}(x)$ is increasing in $n$, then for any two
positive integers $n_{1}\leq n_{2}$ and for all $x\geq 0$ it can be written
that $\overline{r}_{Y_{n_{1}:n_{1}}}(x)\leq\overline{r}_{Y_{n_{2}:n_{2}}}(x)$,
which can equivalently conclude that
$\frac{G_{n_{1}:n_{1}}(x)}{G_{n_{2}:n_{2}}(x)}$ is decreasing in $x\geq 0$.
Thus, the theorem can be proved using Lemma 3.3 and following the line of the
proof of Theorem 3.1.

###### Theorem 3.4

Suppose $X_{n:n}\sim\bar{F}_{n:n}(x)$ and $Y_{n:n}\sim\bar{G}_{n:n}(x).$ Also
let $N$ be a random variable having pmf $p(n).$ Now, for all $x\geq 0$, if
$\overline{r}_{Y_{n:n}}(x)$, the reversed hazard rate function of $Y_{n:n}$ is
increasing in $n\in\mathcal{N}^{+}$ and $\frac{F_{n:n}(x)}{G_{n:n}(x)}$ is
decreasing in $n\in\mathcal{N}^{+}$, then $X_{n:n}\geq_{rh}Y_{n:n}$ implies
$X_{N:N}\geq_{rh}Y_{N:N}.$

Proof: The theorem can be proved using Lemma 3.4 and following the proof of
Theorem 3.3.

###### Remark 3.2

It is to be noted that Theorem 3.3 and 3.4 are proved with an additional
condition that “for all $x\geq 0$, the reversed hazard rate function
($\overline{r}_{Y_{n:n}}(x)$) of $Y_{n:n}$ is increasing in $n$”, while no
such condition is required to prove Theorems 3.1 and 3.2. This can be
rationalized by Theorem 1.B.28 of Shaked and Shanthikumar ([20]) (see Theorem
2.1 above) which states that for all $x\geq 0$, the hazard rate function
($r_{Y_{1:n}}(x)$) of $Y_{1:n}$ is always increasing in $n$, which is not
always true for reversed hazard rate function. In fact, Theorem 1.B.57 of
Shaked and Shanthikumar ([20]) (see Theorem 2.2 above) uses more restrictive
sufficient condition $X_{m}\leq_{rh}X_{j}$ for all $j=1,2,\ldots,m-1$ to prove
similar result. Hence, one additional condition is used in Theorems 3.3 and
3.4 to avoid such restrictive sufficient conditions.

##  4 Conclusion

The paper aims to provide some new results on the comparison of extremes with
random number of independent and non-identically distributed random variables.
The variation diminishing property (Karlin [8]) is extended to all possible
restrictions and the associated lemmas are proved which are used to derive the
key results. The findings of this paper have not been studied in the
literature to the best of our knowledge. Stochastic comparison of systems
(series/parallel) where the components are non-identically distributed and are
randomly chosen (randomized components) from two different batches can be an
interesting problem for future research.

Disclosure statement
No potential conflict of interest was reported by the author(s).

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††thanks: E.Y and P.G contributed equally to this work.

Present address: ]Center for Integrated Nanostructure Physics (CINAP),
Institute for Basic Science (IBS), Sungkyunkwan University, Suwon 16419,
Republic of Korea.

# Ferromagnetism and Metal-Insulator transition in F-doped LaMnO3

Ekta Yadav  Department of Physics, Indian Institute of Technology, Kanpur
208016, India.  Pramod Ghising∗ [  Department of Physics, Indian Institute of
Technology, Kanpur 208016, India.  K. P. Rajeev  Department of Physics, Indian
Institute of Technology, Kanpur 208016, India.  Z. Hossain<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
Institute of Low Temperature and Structure Research, Polish Academy of
Sciences, Okólna 2, 50-422 Wrocław, Poland.

###### Abstract

We present our studies on polycrystalline samples of fluorine doped LaMnO3
(LaMnO3-yFy). LaMnO2.5F0.5 exhibits remarkable magnetic and electrical
properties. It shows ferromagnetic and metallic behavior with a high Curie
temperature of $\approx 239$ K and a high magnetoresistance of $-$64$\%$. This
drastic change in magnetic properties in comparison to pure LaMnO3 is ascribed
to the presence of mixed-valence Mn ions driven by the F-doping at the
O-sites, which enables double exchange (DE) in LMOF. Furthermore, the
resistivity data exhibits two resistivity peaks at 239 K and 213 K,
respectively. Our results point towards the possibility of multiple double
exchange hopping paths of two distinct resistances existing simultaneously in
the sample below 213 K.

**footnotetext: These authors contributed equally to this work

##  I Introduction

Perovskite manganites have been known to exhibit exotic phenomena such as
colossal magnetoresistance (CMR), ferromagnetism, half-metallic behavior [1,
2, 3] etc. Perovskites have the chemical formula ABO3. In perovskite
manganites the smaller B ions (transition metals) sit at the center of an
oxygen octahedra BO6 as shown in Fig. 1. The exotic properties of manganites
are derived from the parent compound LaMnO3 (LMO), which is an insulating
A-type antiferromagnet with a Néel temperature of 140 K [4, 5, 6]. LMO being
an antiferromagnetic insulator, is “interesting but useless”, to quote Louis
Néel from his Nobel prize speech. A common strategy to extract “useful”
properties out of LMO is doping the A-sites with divalent atoms. A large
number of these exciting phenomena results from doping the La-site of the
parent compound LaMnO3. Prominent examples include compounds such as
La1-xSrxMnO3 (LSMO), La1-xPrxCa0.375MnO3 (LPCMO), La1-xCaxMnO3 (LCMO) etc.,
which exhibit rich phase diagrams at various dopant concentrations [7, 8, 9,
10, 11]. It is observed that for a certain range of the value of the doping
concentration $x$, we can induce ferromagnetism and metallic properties in
LMO. In the case of LSMO, the range of $x$ is $0.2<x<0.5$ [12, 13].

Doping at the A-site or the La-site with divalent ions results in a mixed
valency of the Mn ions i.e. Mn3+/Mn4+ [14, 15]. The Mn3+/Mn4+ ions occupy the
B-site of the LMO lattice. Mixed valency of Mn is essential for the
ferromagnetic and metallic behavior, which result from the double exchange
(DE) hopping of electrons along the Mn${}^{3+}-$O$-$Mn4+ chains from one
B-site to another B-site in the manganite lattice [16]. The DE hopping depends
on the Mn${}^{3+}-$O$-$Mn4+ bond angle, $\theta$ [17]. Double exchange hopping
is maximum for $\theta_{\circ}=180^{\circ}$, and decreases as $\theta$
deviates from $\theta_{\circ}$. Consequently, ferromagnetic and metallic
properties of manganites are very sensitive to distortions of the MnO6
octahedra.

An alternate route to tune the properties of manganites is doping at the anion
sites. Although La site doping of LMO has been studied extensively, there are
very few reports on O-site doping of LMO. One possible way is to dope the
O-site with fluorine [18, 19]. Here, we present our studies on F doping of LMO
sample. We observe, that on F doping, the LMO sample exhibits ferromagnetic
and metallic behavior with high values of magnetoresistance (MR). The
ferromagnetic properties are at par with La-site doped manganites.

Figure 1: Structure of a perovskite oxide with the chemical formula ABO3. The
B-ion sits at the center of an oxygen octahedra BO6.

##  II Experimental Details

Figure 2: Powder X-ray diffraction result of LaMnO3 and $y=$ 0.5 F-doped
LaMnO3 sample.

LaMnO3-yFy (LMOF) samples (with $y=0.25,0.50,0.75$) were synthesized by the
solid state reaction method. Powders of La2O3, Mn2O3 and LaF3 were taken in
stoichiometric ratios and were ground and mixed thoroughly using a mortar and
pestle for 45 minutes. The ground powder was sintered in air for 24 hours at a
fixed temperature of 600∘C, 700∘C, 800∘C, 900∘C, 1000∘C and 1100∘C. Before
each sintering process, the powder was thoroughly ground in a mortar and
pestle. Finally, the mixed powder was pressed into a pellet and annealed in
air at 900∘C.

X-ray diffraction (XRD) and Energy-dispersive X-ray spectroscopy (EDX) was
used for structural characterization and elemental analysis of the samples,
respectively. XRD was performed using a Panalytical Expert Pro X-ray
diffractometer with a Cu$-$Kα1 radiation. JXA-8230 Electron Probe
Microanalyzer (JEOL Ltd.) was used for determining the chemical composition.
Iodometric titration was also used for determination of oxygen content. The
electrical and magnetotransport measurements were carried out in a Quantum
design physical property measurement system (PPMS). Standard four-probe
technique was used for the electrical resistivity measurements. X-ray
photoemission spectroscopy (XPS) measurement (using Al$-$Kα X-ray source,
$h\nu=1486.6$ eV, PHI 5000 Versa Probe II, FEI Inc.) was carried out to
determine the valence states of Mn and La in LMOF.

Figure 3: EDX average area points scan of various LMO samples: (a) undoped,
(b) 0.25 F-doped, (c) 0.5 F-doped, (d) 0.75 F-doped.

##  III RESULT AND DISCUSSION

The results of the XRD measurement carried out on the LMOF sample with $y=0.5$
is shown in Fig. 2(a) (unless stated explicitly, LMOF denotes F doping
concentration of $y=0.5$ in the rest of the text). For comparison the XRD
result of an LMO control sample is also shown in Fig. 2 (a) (red line). The
XRD data reveals coinciding LMO and LMOF peaks indicating that LMOF also
maintains the same perovskite structure as LMO. The absence of extra peaks in
the XRD data confirms the absence of any secondary phases in the LMOF sample.
The lattice parameters of LMO sample was determined to be $a=$ 0.5483 nm, $b=$
0.7803 nm, $c=$ 0.5543 nm which are comparable to the values reported by
others [20]. The lattice parameters of LMOF are $a=$ 0.5477 nm, $b=$ 0.7802
nm, $c=$ 0.5542 nm. The grain size of the different LMOF samples determined
using Scherrer’s formula are 29 nm, 39 nm and 32 nm for different F
concentrations with $y=$ 0.25, $y=$ 0.50 and $y=$ 0.75, respectively. EDX
measurements (at room temperature) on different regions of the LMOF sample
show La deficiency. EDX data for undoped and F-doped LMO samples are shown in
Fig. 3 along with atomic ratio of all elements. For undoped LMO, EDX [Fig.
3(a)] reveals La : Mn : O = 1.02 $\pm$ 0.01 : 1.00 $\pm$ 0.01 : 3.50 $\pm$
0.01 with excess oxygen and with atomic concentration ($\pm$ standard error)
La = 18.48 $\pm$ 0.46, Mn = 18.04 $\pm$ 0.78, O = 63.44 $\pm$ 1.99. The atomic
ratio averaged over many point scans for different F-doped samples are
depicted in the Fig. 3(b)-(d). For $y=$ 0.5 F-doped LMO it is La : Mn : O : F
= 0.93 $\pm$ 0.03 : 1 $\pm$ 0.04 : 3.41 $\pm$ 0.05 : 0.17 $\pm$ 0.02 (with at.
concentration La = 17.00 $\pm$ 0.60, Mn = 18.10 $\pm$ 0.65, F = 3.21 $\pm$
0.35, O = 61.54 $\pm$ 0.97). For undoped LMO, at$\%$ of the cations are
La=18.48 and Mn=18.04. Similarly, for F-doped LMO it is La=17.00 and Mn=18.10.
It is clearly seen that the Mn content is more or less the same in both the
doped and undoped LMO, whereas La content decreases in the F-doped sample.
Thus, it is reasonable to normalize with respect to Mn rather than La. For all
F-doped samples, fluorine concentration is seen to increase monotonically
(0.11 $\pm$ 0.02, 0.17 $\pm$ 0.02, 0.21 $\pm$ 0.03) but is less than the
intended value of y=0.25, 0.5 and 0.75, respectively. EDX data shows
considerably higher value of oxygen concentration for both undoped and F-doped
LMO.

The low atomic number of oxygen makes it difficult for EDX measurements to
make reliable estimate of its concentration in oxides. Therefore, we also
performed iodometric titration[21, 22] to further determine the oxygen
concentration in both doped and undoped LMO samples. Titration involves slowly
adding a chemical substance to a reaction mixture till the chemical change is
complete. For undoped LMO, iodometric titration yields a concentration of 3.21
with an excess oxygen of 0.21. The excess oxygen obtained with this method is
much less than the EDX value of 0.5 for undoped LMO. Similarly, for $y=$ 0.5
F-doped LMO, it shows a concentration of 3.15, with excess oxygen of 0.15.
This is again less than the excess oxygen value obtained from EDX for $y=$ 0.5
F-doped LMO (which is 0.41). Combining the EDX and iodometric titration we
conclude that the undoped and doped samples have some excess oxygen and there
is a systematic increase of F-content for different F-doped samples. The
excess oxygen is due to sintering and annealing of the samples in air.

Figure 4: (a) Results of FC and ZFC-$M(T)$ measurements carried out on 0.5-F
doped LMO. Two magnetic transitions in the $M(T)$ data are apparent at
$T_{C1}=238$ K and $T_{C2}=217$ K. (b) $M(H)$ measurement of the 0.5-F doped
LMO sample at 120 K ($<T_{C1}$, $T_{C2}$) exhibits a hysteresis loop, which
confirms the ferromagnetic nature of 0.5-F doped LMO. The inset shows the
blown up view of the central region of the hysteresis loop. The coercivity is
64 Oe. Figure 5: (a) Electrical transport measurement on the LMOF sample
($y=0.50$) exhibits metallic behavior with two distinct resistivity peaks at
$T_{p1}=239$ K and $T_{p2}=213$ K. On the other hand, pure LMO sample exhibits
insulating behavior as is observed from its resistivity data shown in the
upper inset. The bottom inset shows the Hall measurement on the LMOF sample at
300 K. The positive slope in the Hall measurement indicates hole-type carriers
in LMOF. (b) Resistivity measurement on samples with different F
concentrations ($y=$0.25 and $y=$0.75) also exhibit metallic behavior with
similar double peaks. Figure 6: (a) Magnetic field dependent resistivity of
the LMOF sample. At $T_{p1}$ and $T_{p2}$ a large magnetoresistance of
$-$64$\%$ and $-$50$\%$ is observed. MR (shown on the right axis) is
independent of temperature below $T_{p2}$. (b) Comparison of the resistivity
peaks with the magnetic transition temperature of LMOF. It is observed that
the two magnetic transition temperatures in $M(T)$ correspond to the two
resistivity peaks in $\rho(T)$.

A very interesting feature of our LMOF sample is the observation of a
ferromagnetic transition close to room temperature. Figure 4(a) displays the
results of the field cooled (FC) and zero field cooled (ZFC) magnetization
measurement as a function of temperature. For the FC measurement the sample
was cooled in an applied field of 2000 Oe and the data acquisition was carried
out during the heating cycle. The Curie temperature $T_{C1}$ of the
ferromagnetic transition is 238 K. Another ferromagnetic transition is also
observed in the $M(T)$ data at $T_{C2}=$ 217 K, indicating the presence of two
distinct ferromagnetic exchanges in the LMOF sample. $M(H)$ measurement done
at 120 K [shown in Fig. 4(b)] exhibits a hysteresis loop typical of
ferromagnets, further confirming the ferromagnetic nature of LMOF below the
Curie temperature.

To further probe the nature of ferromagnetism we carried out electrical
transport measurements on the LMOF sample, the results of which are shown in
Fig. 5(a). The sample exhibits a metallic behavior with a metal-insulator
transition (MIT) at $T_{p1}=239$ K and another slightly broad resistivity peak
at $T_{p2}=213$ K. It is to be noted that the undoped LMO control sample
exhibits an insulating behavior without any MIT as can be seen in the upper
inset of Fig. 5(a). Hall measurement carried out at 300 K on the LMOF sample
reveals hole-type carriers, inferred from the positive slope of the
$\rho_{xy}-H$ curve [shown in lower inset of Fig. 5(a)]. The carrier
concentration was determined to be 0.18 $\times$ 1022 cm-3. For comparison, we
also carried out resistivity measurements on two additional LaMnO3-yFy samples
with $y=$ 0.25 and $y=$ 0.75 . Their resistivity data also exhibit metallic
behavior with similar double peaks as shown in Fig. 5(b). The field dependent
resistivity data of LMOF is shown in Fig. 6(a). In the presence of a 7 T
magnetic field both the resistivity peaks at $T_{p1}$ and $T_{p2}$ are
suppressed and are also shifted to higher temperatures by an amount of 40 K
and 22 K, respectively, suggestive of a similar magnetic origin of the two
peaks. Moreover, a large magnetoresistance
[MR$=\frac{{\rho(H)-\rho(0)}}{{\rho(0)}}$] of about $-$64$\%$ and $-$50$\%$ is
also observed at $T_{p1}$ and $T_{p2}$, respectively. The MR below $T_{p2}$
show no significant change with temperature. Interestingly, the onset of the
ferromagnetic transition $T_{C1}$ and the MIT temperature $T_{p1}$ coincide
[shown in Fig. 6(b)]. This result strongly suggests that the ferromagnetic and
the metallic properties of LMOF are related. One such example of concurrent
ferromagnetic and metallic behavior in A-site doped LMO (such as LSMO, LCMO)
results from double exchange mechanism [13, 23], where the charge carriers are
holes. Such DE mechanism in LMOF is only possible in the presence of mixed
valence of Mn. Also, the resistivity peak $T_{p2}$ coincides with the
ferromagnetic transition in the $M(T)$ data at $T_{C2}=217$ K, which further
indicates that $T_{p2}$ is related to ferromagnetic properties of the sample.

The core level XPS spectra of the LMOF sample is shown in Fig. 7. Mn in LMOF
exhibits a mixed valence state as is apparent from its XPS spectra displayed
in Fig. 7(a). Analysis of the XPS spectra for Mn reveals a valence state of
Mn2+, Mn3+ and Mn4+. The Mn2+, Mn3+ and Mn4+ peak positions occur at 641.0,
642.1, 643.6 and 653.0, 653.6, 654.6 for the 2$p_{3/2}$ and 2$p_{1/2}$ spin-
orbit split peaks, respectively, which are similar to earlier reported values
[24, 25]. The XPS spectra of La [shown in Fig. 7(b)] determines its valence
state to be La3+ [26]. The deconvoluted La 3$d_{5/2}$ and 3$d_{3/2}$ show
multiple peaks due to La binding with oxide and hydroxyl species and also
final state effects. The peaks A, C, E and G correspond to La binding with
oxides while the peaks B, D and F correspond to La binding with hydroxyl
species [26]. F 1$s$ and O 1$s$ XPS spectra are shown in Fig. 7(c) and (d),
respectively. The additional peak at 531.5 eV in the O 1$s$ XPS spectra is
ascribed to the formation of hydroxyl species [27].

Thus, F doping induces a mixed valence state of the Mn ions. Using resonant
inelastic X-Ray Spectroscopy Orgiani et al., have established that Mn2+ ions
occupy the La site in LMO [28], which probably explains the La deficiency in
our LMOF sample. The origin of the Mn2+ is due to the sample’s tendency to
maintain charge neutrality upon F doping. Mn2+ ions occupying the La-sites is
analogous to A-site doping of manganites by divalent atoms (such as in LSMO,
LCMO etc.). It leads to the generation of Mn4+ ions at the B (or the Mn) sites
of LMOF. With the assumption that the La deficient sites (we deduce La
deficiency of 0.07 per unit cell from EDX measurement) are occupied by Mn2+,
we obtain a doping level of 0.07 Mn2+ ions per unit cell at the A-site. In
analogy with A-site doped manganites (with divalent atoms such as Sr, Ca
etc.), we obtain a carrier concentration of 0.07 holes per unit cell.
Furthermore, Hall measurement at 300 K confirms hole-type carriers with a
concentration of 0.18 $\times$ 1022 cm-3. Using the volume of the unit cell
obtained from XRD data, we determine the carrier concentration to be 0.11
holes per unit cell, which is comparable with the EDX result. This result
further supports Mn2+ occupancy of the A-sites.

The presence of Mn3+/Mn4+ in LMOF does provide a fertile ground for the
existence of double exchange, which would explain its metallic and
ferromagnetic properties below $T_{C1}$. On the other hand, F- doping at the
O2- sites will break the Mn${}^{3+}-$O$-$Mn4+ DE chain, which should degrade
the metallic and ferromagnetic properties. The Mn ions occupying the A-sites
have a considerably longer Mn$-$O bond lengths (2.7 Å) as compared to when the
Mn ions occupy the B-sites (1.9 Å) [28]. Additionally, the occupation of the
La sites by the smaller Mn2+ ions would result in lattice distortions which
would also lead to a decrease in DE hopping. Against this conflicting
backdrop, the observed persisting ferromagnetism and metallic behavior of LMOF
is very much intriguing.

Figure 7: XPS spectrum of LMOF. The XPS data were fitted using a combination
of Gaussian-Lorentzian peak shapes. (a) Mn 2$p$ spectrum with its two spin
orbit doublet. The presence of Mn2+, Mn3+ and Mn4+ peaks are observed in the
XPS spectra. (b) La 3$d$ spectrum showing the two spin-orbit peaks, 3d5/2 and
3d3/2. Additional peaks due to La bonding with hydroxyl groups and final state
effects are also observed. (c) F 1s spectrum (d) O 1s XPS spectrum with an
additional peak at 531.5 eV due to the formation of hydroxyl species.

A fascinating feature of the LMOF sample is the presence of two resistivity
peaks in $\rho(T)$ data at $T_{p1}$ and $T_{p2}$. Double peak in the
resistivity of manganites have been reported earlier. Such double peaks were
ascribed to the presence of excess oxygen in the manganites samples [29],
grain size [30] and grain boundary effects [31, 32]. The results of EDX and
iodometric titration show oxygen excess in both undoped and F-doped LMO
samples. However, double peaks in $\rho(T)$ is observed only in F-doped LMO.
This confirms that the double peak feature originates exclusively from
F-doping. The peak at $T_{p2}$ that originate due to small polycrystal grain
size exhibit low MR which is typically less than 20$\%$ as reported in
literature [30]. However, the MR at $T_{p2}$ (at 7 T) is high in the LMOF
sample, with a value of 50$\%$. The grain boundary effects manifest themselves
as a gradual increase in MR as the temperature is lowered, with MR increasing
to a value which is comparable to the MR value at $T_{p1}$ at the lowest
temperatures [33]. No such MR behavior is observed in the LMOF sample [please
see Fig. 6(a)]. MR is more or less independent of temperature below $T_{p2}$.
The last two observations discount the possibility of grain size and grain
boundary effects as the origin of double peak in the resistivity data. Another
important feature to note is the presence of an additional magnetic transition
in the $M(T)$ data at $T_{C2}$, which roughly corresponds to the second
resistivity peak at $T_{p2}$ [see Fig. 6(b)]. Moreover, a large MR of
$-$50$\%$ is observed at $T_{p2}$. These results indicate that the second
resistivity peak is related to ferromagnetic properties of the sample.

Figure 8: Schematic of the two DE hopping paths in LMOF. Top panel represents
DE hopping in the temperature range $T_{p2}<T<T_{p1}$. Figure on the left
shows a lattice with La3+ at the A site, whereas on the right we have a
lattice with Mn2+ at the A-site. Hopping takes place by conventional DE along
the low resistance path B$-$B′ for both the lattice. The bottom panel depicts
the hopping mechanism in the temperature range $T<T_{p2}$. At $T_{p2}$, local
orbital ordering of Mn3+ $e_{g}$ (along z direction) takes place in the
lattice with Mn2+ at the A-site. As a result orbital overlap takes place along
the $z$-direction. Consequently, DE hopping takes place along the high
resistance path B$-$A$-$B′. In this lattice, conventional DE hopping along
B$-$B′ is forbidden due to lack of orbital overlap along $x$-direction. On the
contrary, for lattice with La3+ at the A-site, there is no orbital ordering
and hence conventional DE hopping along B$-$B′ survives.

The persistent ferromagnetism and metallic properties of LMOF may have its
origin in the presence of Mn2+ ions at the La site. Such occupancy of the
A-site by divalent Mn2+ would explain the hole-type carriers and the mixed-
valent Mn3+/Mn4+ (in the LMOF sample) observed in the Hall and XPS
measurements, respectively. The observed ferromagnetism may also originate
from excess oxygen in LMOF. The excess oxygen can increase Mn4+ concentration
thereby facilitating double exchange. However, we do not observe
ferromagnetism and metallic behavior in the undoped LMO sample with a similar
oxygen excess. Thus, ferromagnetism in LMOF originates exclusively from
F-doping. In LMO containing Mn2+, it has been shown that an additional double
exchange hopping path via an intermediate Mn2+ ion
(Mn${}^{3+}-$O$-$Mn${}^{2+}-$O$-$Mn4+) exists [28], other than the
conventional Mn3+/Mn4+ (Mn${}^{3+}-$O$-$Mn4+) hopping path. This DE hopping
path involves electron hopping along B$-$A$-$B′ sites (B′ is a neighboring B
site) in contrast to conventional DE, where the hopping sites are B$-$B′ in
the manganite lattice (shown in Fig. 8). We suspect that the second peak in
the resistivity data at $T_{p2}$ is on account of double exchange hopping
along this additional path. The FM ordering at $T_{C2}$, further lends some
credibility to our assumption. We could model the LMOF sample as consisting of
some Mn-ions at B sites (Mn${}^{3+}/$ Mn4+) and some at A sites (Mn2+), which
exhibit ferromagnetic ordering at different temperatures $T_{C1}$ and
$T_{C2}$, respectively. The ferromagnetic ordering of the A and B sites
originate from DE hopping of electrons via the B$-$B′ and B$-$A$-$B′ hopping
paths. Furthermore, the two DE paths exhibit different resistance. The higher
resistance path include DE hopping via the Mn2+ ions located at the A-sites.
Due to the decreased angle ($<$180∘) between the Mn ions and the longer
hopping path, it will have a higher resistance. Whereas the lower resistance
path involves DE hopping via the conventional Mn${}^{3+}-$O$-$Mn4+ chains
which occupy the B-sites with angle between the neighboring Mn ions close to
180∘. In the temperature range of $T_{p2}<T<T_{p1}$ the metallic behavior is
due to DE hopping via the low resistance path. At $T_{p2}$ the Mn2+ ions at
the A-sites start to exhibit ferromagnetic order, which implies that a DE
hopping via the high resistance path is also activated. One possible trigger
for the activation of DE hopping along B$-$A$-$B′ is local orbital ordering of
Mn3+ $e_{g}$ orbitals. Local ordering of Mn3+ $e_{g}$ orbitals have been
observed in thin films of LSMO and LMO [34, 35]. Here due to the presence of
Mn2+ at the A-sites, $e_{g}$ orbitals of Mn3+ (at the B-sites) order along the
$z$ direction (i.e. electrons occupy the $d_{z^{2}-r^{2}}$ orbital) [34, 35].
Consequently, for those lattices where Mn2+ ions occupy the A-site, hopping
takes place via the $d_{z^{2}-r^{2}}$ orbital. It is also interesting to note
that this orbital ordering prevailed despite the presence of a tensile strain
from the substrate on the LSMO film which would have otherwise preferred an
in-plane (i.e. $d_{x^{2}-y^{2}}$) orbital ordering [35]. Thus, such local
orbital ordering of Mn3+ is attributed to the presence of Mn2+ at the A-sites.
Such a local orbital ordering of Mn3+ $e_{g}$ orbitals is the only possible
scenario that favor hopping along B$-$A$-$B′ as shown in Fig. 8 (since DE
hopping takes place only in the presence of orbital overlap between the
participating Mn ions and the intermediate O ion). Mn3+ being Jahn-Teller (JT)
active, its $e_{g}$ orbitals ($d_{x^{2}-y^{2}}$ and $d_{z^{2}-r^{2}}$) are non
degenerate, whereas Mn2+ and Mn4+ are JT inactive and hence their $e_{g}$
orbitals are degenerate. In the temperature range $T_{p2}<T<T_{p1}$, electrons
occupy the lower energy $d_{x^{2}-y^{2}}$ in-plane orbital. As a result the
orbital overlap is in the in-plane direction leading to conventional DE (as
shown in top panel of Fig. 8). At $T_{p2}$ the Mn3+ $e_{g}$ orbitals in the
lattice containing Mn2+ ions at the A-sites order locally along the out-of
plane direction, i.e. the energy of the $d_{z^{2}-r^{2}}$ orbital is lowered.
Consequently, orbital overlap with the O ions take place along the out-of
plane direction as shown in the bottom panel of Fig. 8. This results in a DE
hopping along the high resistance B$-$A$-$B′ path. Thus, below $T_{p2}$ the
electron hopping is along the low resistance B$-$B′ and high resistance
B$-$A$-$B′ paths in series. Such low resistance and high resistance paths in
series leads to a re-emergent MIT in manganites [36]. The second resistivity
upturn at $T_{p2}$ in the LMOF sample can be explained in terms of addition of
the resistance of the two paths. Initially at $T_{p2}$, the resistivity of the
sample increases due to the inclusion of the high resistance B$-$A$-$B′ path.
Below $T_{p2}$, more number of hopping paths via Mn2+ are activated. As a
result, the resistance starts to decrease again after an initial increase at
$T_{p2}$ as temperature is further lowered.

##  IV Conclusion

To conclude, we prepared flourine doped LaMnO3 polycrystalline samples using
solid state reaction method. The LMOF samples exhibit ferromagnetic and
metallic properties which are similar to La-site doped manganites. The Curie
temperature is 239 K and a high MR of $-$64$\%$ is observed. The ferromagnetic
and metallic behavior is ascribed to DE mechanism. Furthermore, the
resistivity data exhibits two resistivity peaks around the same temperature at
which the $M(T)$ data exhibit two magnetic transitions. Our XPS studies show
the presence of Mn2+, Mn3+ and Mn4+ ions in the LMOF sample. The two peaks in
the resistivity data are attributed to the presence of multiple DE hopping
paths i.e. one along Mn${}^{3+}-$O$-$Mn4+ and another along
Mn${}^{3+}-$O$-$Mn${}^{2+}-$O$-$Mn4+. Our study shows that anion site doping
of perovskite manganite can be a robust technique to explore new and exciting
physics in perovskite manganites.

##  V Acknowledgement

We gratefully acknowledge financial support from IIT Kanpur, India and
Department of Science and Technology, India. ZH would like to acknowledge the
Polish National Agency for Academic Exchange (NAWA) for Ulam Fellowship. We
also thank Namrata Singh for helping us with iodometric titration.

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the features of the original datasets into binary for our experiments.
Concretely, the continuous features are discretized into quantiles, and each
quantile is represented by a binary feature. Also, a categorical feature with
$C$ categories is converted to $C$ binary features. More specifically, for the
“adults” dataset, we transform the 13 features of it into 122 binary features
and choose the feature of “gender” as the protected feature. For the “law
school” dataset, we transform the 11 features of it into 379 binary features
and choose the feature of “gender” as the protected feature. For the “bank
marketing” dataset, we transform 16 features of it into 3879 binary features
and choose “marital” as the protected feature.

For experiments, we tune the stepsize of EG from $\\{0.01,0.05,0.1,0.5\\}$ and
run it with 4000, 20000 and 40000 iterations for the “adults”, “law school”
and “Bank market” respectively as warm up to obtain ${\bf{z}}_{0}$ as initial
point. Then we evaluate all algorithms (including the baseline algorithm EG)
by starting with ${\bf{z}}_{0}$ and achieve the result shown in Figure 2.
Since each algorithm has the identical behavior in the warm up stage, we only
present the curves of iterations vs. $\left\|{\bf{g}}({\bf{z}})\right\|_{2}$
and CPU time vs. $\left\|{\bf{g}}({\bf{z}})\right\|_{2}$ after warm up stage
in Figure 2.

In addition, we set ${\bf G}_{0}=2{\bf I}$ for the proposed quasi-Newton
methods. For RaBFGSv1-G (Algorithm 5), we set $M=0$ for “adults”, $M=1$ for
“law school” and “bank market”. For RaBFGSv2-G (Algorithm 6), we set $M=1$ for
“adults” and “law school”. For RaSR1-G (Algorithm 7), we set $M=1$ for all
three datasets.