# Estimation of the tail index of Pareto-type distributions using
regularisation 11footnotemark: 1

E. Ocran, R. Minkah, G. Kallah-Dagadu and K. Doku-Amponsah  Correspondence:
<EMAIL_ADDRESS>
The first named author (E. Ocran) would like to thank the University of Ghana
Building a New Generation of Academics in Africa (BANGA-Africa) Project with
funding from Carnegie Corporation of New York for providing support.
Department of Statistics $\&$ Actuarial Science,
School of Physical and Mathematical Sciences, University of Ghana, Legon,
Accra, Ghana

###### Abstract

In this paper, we introduce reduced-bias estimators for the estimation of the
tail index of a Pareto-type distribution. This is achieved through the use of
a regularised weighted least squares with an exponential regression model for
log-spacings of top order statistics. The asymptotic properties of the
proposed estimators are investigated analytically and found to be
asymptotically unbiased, consistent and normally distributed. Also, the finite
sample behaviour of the estimators are studied through a simulations theory.
The proposed estimators were found to yield low bias and MSE. In addition, the
proposed estimators are illustrated through the estimation of the tail index
of the underlying distribution of claims from the insurance industry.

###### keywords:

Statistics of extremes , tail index , Pareto-type distribution , large
deviations , Hill estimator , Strong law of large numbers , Limit theorem.

###### MSC:

[2010]: 62G32 , 60F10 , 60F25.

##  1 Introduction

Pareto-type distributions are often encountered in applications in the area of
Finance [1, 2, 3], re-insurance [4, 5, 6], risk management [7, 8, 9] and
telecommunication [10, 11]. This distribution type has tail function,

$1-F(x)=x^{-1/\gamma}\ell(x)\hskip 4.0pt\text{as}\hskip 4.0ptx\to\infty,$ (1)

or equivalently upper tail quantile function

$U(x)=x^{\gamma}\ell_{u}(x)\hskip 4.0pt\text{as}\hskip 4.0ptx\to\infty.$ (2)

The component $\ell_{F}$ and $\ell_{u}$ are slowly varying functions expressed
as

$\lim\limits_{t\to\infty}\dfrac{\ell(xt)}{\ell(t)}=1,\hskip 6.0ptt>1.$ (3)

The parameter $\gamma$, is strictly positive for Pareto-type distributions and
is also known as the tail index.

Suppose $X_{1},X_{2},...,X_{n}$ denote independent and identically distributed
(i.i.d) random variables drawn from a distribution belonging to the maximum
domain of attraction of the Pareto family of distributions, then for some
auxiliary sequences of constants $\left\\{a_{n}>0;n\geq 1\right\\}$ and
$\left\\{b_{n};n\geq 1\right\\}$ [12]

$\lim\limits_{n\to\infty}P\left(\dfrac{\max\left\\{X_{1},X_{2},...,X_{n}\right\\}-b_{n}}{a_{n}}\leq
x\right)=\exp\left\\{-\left(1+\gamma x\right)^{-1/\gamma}\right\\},\hskip
4.0pt1+\gamma x\geq 0,\hskip 4.0pt$ (4)

where $\gamma\geq 0.$ The estimation of $\gamma$ continues to receive
considerable attention in statistics of extremes as all inferences in extreme
value analysis depend on the tail index. In practice, we seek estimators with
less variance and bias as possible. A parametric or semi-parametric approach
can be employed to estimate the tail index,[13], [14], [15], [16]. However, in
this paper we employ the semi-parametric approach to develop reduced-bias
estimators since they result in bias reduction.

Under the semi-parametric framework, the tail index estimators are dependent
on the $k$ largest observations, with these assumption about $k$:

  1. $\bullet$

Assumption 1: $k(n)\to\infty$ as $n\to\infty$.

  2. $\bullet$

Assumption 2: $k=k(n)=O(n)$ as $n\to\infty$.

The most widely used semi-parametric tail index estimator is the Hill
estimator [17]. [17] approximates the top $k$ order statistics with a Pareto
distribution and estimates $\gamma>0$ using a maximum likelihood estimator
(MLE). The Hill estimator has the minimum asymptotic variance among the semi-
parametric estimators but it is very sensitive to the choice of $k$ [18]. This
drawback of the estimator makes its usage challenging in practice, especially
in the selection of the tail fraction, $k$. [17] defined the tail estimator as

$\hat{\gamma}_{k}^{H}=\dfrac{1}{k}\sum_{j=1}^{k}\log\left(\dfrac{X_{n-j+1,n}}{X_{n-k,n}}\right).$
(5)

The Hill estimator due to it popularity has received several generalisations:
See for example the works of [19], [20], [21], [22], [23], [24, 25] and [26].

Another estimator which is also a refinement of the Hill estimator is the
bias-corrected Hill estimator [27]. The authors proposed two approaches for
reducing the bias of the Hill estimator while maintaining the asymptotic
variance of the Hill estimator. Empirically, the bias-corrected Hill estimator
yields stable tail index estimates compared to the Hill estimator, i.e., the
bias-corrected Hill estimator is less sensitive to the choice of $k$ relative
to the Hill estimator.

In this study, we seek to propose alternative tail index estimators which
empirically yield much more stable tail index estimates and attain the minimum
asymptotic variance of the Hill estimator under some conditions. Tail index
estimators which yield stable estimates are highly sort after in practice
since they alleviate the problem of selecting an optimal $k$ to some extent.

##  2 Estimation Methods

We let $X_{1},X_{2},X_{3},...,X_{n}$ denote a sequence of i.i.d random
variables drawn from a population with distribution function $F$ and the
associated tail quantile function $U$. Let $X_{1,n}\leq X_{2,n}\leq...\leq
X_{n,n}$ be the order statistics associated with the sample. Using Eqn. (2),
the order statistics can be jointly expressed as,

$\log X_{n-j+1,n}\stackrel{{\scriptstyle d}}{{\sim}}\gamma\log
U^{-1}_{j,n}+\log\ell\left(U^{-1}_{j,n}\right),$ (6)

where $U^{-1}_{j,n}$, $j=1,2,3,...,n$ represent the order statistics of the
standard uniform distribution. Using Eqn. (6), we [28] demonstrated that

$\dfrac{\log X_{n-j+1,n}}{\log X_{n-k,n}}\stackrel{{\scriptstyle
d}}{{\sim}}\gamma\log\dfrac{U^{-1}_{k+1,n}}{U^{-1}_{j,n}}+\log\ell\left(\dfrac{U^{-1}_{j,n}}{U^{-1}_{k+1,n}}\right)$
(7)

$k\in\left\\{2,3,...,n-1\right\\}$ and we also obtained a more refined
expression of Eqn. (7) by imposing a second-order assumption on the rate of
convergence to Eqn. (3). This is stated in as an assumption as follows:

Assumption 3: There exists a real constant $\rho\leq 0$ and a rate function
$b$ satisfying $b(x)\to 0$ as $x\to\infty$ such that for all $u\geq 1$,

$\lim_{x\to\infty}\Big{[}\log\dfrac{\ell(xu)}{\ell(u)}\Big{]}=b(t)h_{\rho}(u),$
(8)

with $h_{\rho}(u)=\int_{1}^{u}y^{\rho-1}dy$ [28].

Under assumption 3, [28] showed that the weighted log-spacings of order
statistics,

$T_{j}=j\left\\{\log X_{n-j+1,n}-\log X_{n-j,n}\right\\},\hskip 4.0pt1\leq
j\leq k<n,$ (9)

are approximately exponentially distributed. They particularly obtained the
expression

$T_{j}=\left(\gamma+b_{n,k}\left(\dfrac{j}{k+1}\right)^{-\rho}\right)f_{j},\hskip
4.0pt1\leq j\leq k,$ (10)

where $b_{n,k}=b\left((n+1)/(k+1)\right)\to 0$ as $k,n\to\infty$, and
$f_{j}^{\prime}s$ are i.i.d exponentially distributed with a unit mean and
$\rho$ ($\rho<0$) is a second-order parameter. The authors employed MLE to the
estimate the parameters in Eqn. (10).

Using Eqn. (10) and assumptions 1 and 2, [29], demonstrated that $T_{j}$ can
be further approximated as a regression model,

$T_{j}=\gamma+b_{n,k}C_{j}+\epsilon_{j}\hskip 4.0pt\text{for}\hskip
4.0ptj\in\left\\{1,2,...,k\right\\},$ (11)

where $b_{n,k}=b\left(n/k\right)$ is the slope,
$C_{j}=\left(j/(k+1)\right)^{-\rho}$ is the covariate, $\gamma$ is the
intercept and $\epsilon_{j}$ are error terms with asymptotic mean, $0$, and
variance, $\gamma^{2}$.

[29] proposed the ordinary least squares estimator for the estimation of
$\gamma$ in Eqn. (11). Further, based on Eqn. (11), [13] have introduced the
ridge regression estimator for estimating $\gamma$. In this paper, we propose
the regularised weighted least squares estimators for estimating $\gamma$ in
Eqn. (11).

###  2.1 The proposed estimators

In order to estimate $\gamma$, the loss function of the regularised weighted
least squares for Eqn. (11) is defined as

$L_{k}\left(\gamma,b_{n,k};\lambda,W\right)=\sum_{j=1}^{k}W_{j}\left(T_{j}-\gamma-
b_{n,k}C_{j}\right)^{2}+\lambda kb_{n,k}^{2},\hskip 10.0pt\lambda\geq 0.$ (12)

Here, $W_{j}$ is the weight function defined as

$W_{j}=\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k+1}\right),j\in\\{1,2,...,k\\},$
(13)

where $\theta_{j}\stackrel{{\scriptstyle i.i.d}}{{\sim}}U(0,1)$. Thus,
$W_{j}\in(0,1)$ and decreases linearly with respect to $j$. The exponent
$\alpha(k)\geq 0$ is chosen such that

$\varDelta:=\lim\limits_{k\to\infty}\dfrac{\alpha(k)}{k}<\infty.$ (14)

In this study, we consider $0<\varDelta\leq 1$.

Thus, we would define $\alpha(k)$ such that $0<\alpha(k)\leq k$. Note that,
$W_{j}$ is random through $\theta_{j}$, and when the exponent is $0$, $W_{j}$
is deterministic. In particular, when $\alpha(k)=0$, we obtain the weight
function $g_{j}=1-j/(k+1)$ as introduced by [30]. Nevertheless, we can
approximate the weight $g_{j}$ as a limit of the current result by allowing
$\varDelta$ to approach $0$.

We minimize the loss function $L_{k}$ with respect to $\gamma$ and $b_{n,k}$
to obtain jointly estimates $\hat{\gamma}$ and $\hat{b}_{n,k}$

$\hat{b}_{n,k}(\lambda,\theta)=\dfrac{\sum_{j=1}^{k}\tilde{W}_{j}\left(C_{j}-\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)T_{j}}{2\kappa(\alpha(k))\lambda+\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)^{2}}$
(15)

and

$\hat{\gamma}_{RW}(\lambda,\theta)=\sum_{j=1}^{k}\tilde{W}_{j}T_{j}-\hat{b}_{n,k}(\lambda,\theta)\sum_{j=1}^{k}\tilde{W}_{j}C_{j}.$
(16)

while,

$\tilde{W}_{j}=\dfrac{W_{j}}{\sum_{j=1}^{k}W_{j}},1\leq j\leq k,$ (17)
$C_{j}=\left(\dfrac{j}{k+1}\right)^{-\rho},\hskip 12.0pt\text{for}\hskip
12.0ptj\in(1,2,...,k),\hskip 12.0pt\rho<0,$ (18)

and

$\kappa(\alpha(k))=\dfrac{\alpha(k)+1}{2\alpha(k)+1}.$ (19)

We substitute Eqn. (13) into Eqn. (17) to obtain an explicit expression for
Eqn. (17) as

$\tilde{W}_{j}\approxeq\dfrac{2\kappa(\alpha(k))}{k}\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k+1}\right).$
(20)

The parameter $\rho<0$ is estimated using the minimum variance approach
introduced in [13].

In addition, the parameter $\lambda$ in Eqn. (12) is the penalty that
regulates the bias coefficient $b_{n,k}$. The loss function, $L_{k}$,
minimises the weighted sum of squared residuals and also regulates the size of
the bias coefficient $b_{n,k}$. The penalty term shrinks the bias term,
$b_{n,k}$ to $0$ as the penalty parameter, $\lambda$, increases. Thus, the
larger the value of $\lambda$, the higher the contribution of the penalty term
to the loss function and the stronger the regularisation process. To obtain an
estimator for the penalty term, $\lambda$, we minimise the mean squared error
(MSE) of the proposed estimator, $\hat{\gamma}_{RW}(\lambda)$. See,example
[13].

Note that, since the weight function depends on the $\theta_{j}$’s, the
estimators in Eqn. (15) and Eqn. (16) also depend on the $\theta_{j}$’s
through the weight function. Therefore, we would find the MSE by conditioning
on the $\theta_{j}$’s. From Eqn. (15) and Eqn. (16), the MSE for
$\hat{\gamma}_{RW}(\lambda)$ is obtained as

where

$S_{1}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}C_{j},$ (21)

$S_{2}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)^{2},$
(22)

$\dot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right),$
(23)

$\ddot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right)^{2},$
(24)

and

$\phi(\alpha(k))=\dfrac{\left(\alpha(k)+1\right)\left(6\alpha^{2}(k)+4\alpha(k)+1\right)}{\left(2\alpha(k)+1\right)^{2}}.$
(25)

We now derive an expression for the penalty term, $\lambda$. Minimising
$MSE\left(\hat{\gamma}_{RW}(\lambda)|\theta_{j}=\theta\right)$ over $\lambda$,
the optimal value of $\lambda$ is obtained by solving the equation

$2b_{n,k}^{2}\kappa(\alpha(k))S_{1}(\theta)S_{2}(\theta)\lambda-2\kappa(\alpha(k))\gamma^{2}\dot{S}(\theta)\lambda-\gamma^{2}S_{2}(\theta)\dot{S}(\theta)-\gamma^{2}S_{1}(\theta)\ddot{S}(\theta)=0.$
(26)

we obtain,

$\lambda_{k}(\theta)=\dfrac{\gamma^{2}S_{2}(\theta)\dot{S}(\theta)+\gamma^{2}S_{1}(\theta)\ddot{S}(\theta)}{2b_{n,k}^{2}\kappa(\alpha(k))S_{1}(\theta)S_{2}(\theta)-2\kappa(\alpha(k))\gamma^{2}\dot{S}(\theta)}.$
(27)

In order to estimate $\hat{\lambda}_{k}(\theta)$, we assume the slowly varying
function $\ell$ in (8) is constant.Thus, we have

$b(x)=\gamma\beta x^{\rho}\\{1+O(1)\\}\hskip 6.0pt\text{as}\hskip
6.0ptk\to\infty$ (28)

for some $\beta\in\mathbb{R}$. We estimate $\beta$ via the estimator proposed
by [31] to obtain

$\hat{\lambda}_{k}(\theta)=\dfrac{S_{1}(\theta)\ddot{S}(\theta)+\dot{S}(\theta)S_{2}(\theta)}{2\kappa(\alpha(k))S_{1}(\theta)S_{2}(\theta)\hat{\beta}_{k}^{2}\left(\dfrac{n}{k}\right)^{2\hat{\rho}}-2\kappa(\alpha(k))\dot{S}(\theta)}.$
(29)

The penalty term, $\lambda$ is required to be non-negative; therefore we
define $\hat{\lambda}_{k}^{+}(\theta)=\max\\{\hat{\lambda}_{k}(\theta),0\\}$.
We the obtain a penalty term which does not depend on $\theta_{j}$’s,by
averaging $\hat{\lambda}_{k}(\theta)$ over the $\theta_{j}$’s, as follows:

$\hat{\lambda}_{k}=\operatorname{\mathbb{E}}_{\theta}\left\\{\hat{\lambda}_{k}(\theta)\mathds{1}_{\left\\{S_{1}(\theta)S_{2}(\theta)\hat{\beta}_{k}(n/k)^{2\hat{\rho}}>\dot{S}(\theta)\right\\}}\right\\}.$
(30)

####  2.1.1 Asymptotic Properties of the Proposed estimators

Unbiasedness, consistency and normality are desirable properties of a good
estimator. In this section, we investigate desirable properties of the
proposed estimator possesses.

We shall summarise the asymptotic behaviour of the statistics used to build
the MSE of the proposed estimator in Lemma 1. These properties will be
required in the proof of the asymptotic consistency and sampling distribution
of the proposed estimator. Henceforth, anytime we use the term $a.s$ it is
with respect to the law of the i.i.d sequence
$\theta_{1},\theta_{2},\theta_{3},...,\theta_{k}$.

######  Lemma 1.

Assume that $\rho$ is estimated by a consistent estimator $\hat{\rho}$ and
(14) holds, then as $k\to\infty$ and $k/n\to 0$;

  1. i.

$S_{1}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}1/(1-\rho).$

  2. ii.

$S_{2}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)^{2}\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}\rho^{2}/(1-2\rho)(1-\rho)^{2}.$

  3. iii.

$\dot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right)\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}0$

  4. iv.

$\ddot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right)^{2}\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}0$.

######  Lemma 2.

Suppose $\rho<0$ and $\beta\in\mathbb{R}$ are estimated by their respective
consistent estimators $\hat{\rho}$ and $\hat{\beta}_{k}$, then as
$k,n\to\infty$ and $k/n\to 0$,

$\hat{\lambda}_{k}(\theta)\stackrel{{\scriptstyle a.s}}{{\longrightarrow}}0.$
(31)

It follows from Lemma 2 that the regularised weighted least estimator,
$\hat{\gamma}_{RW}(\lambda)$ is asymptotically unbiased. That is, as
$k\to\infty$, $bias\left(\hat{\gamma}_{RW}(\lambda)\right)\to 0$. The bias of
the proposed estimator is given as,

$bias\left(\hat{\gamma}_{RW}(\lambda)\right)=\operatorname{\mathbb{E}}_{\theta}\left\\{\dfrac{2b_{n,k}\kappa(\alpha(k))S_{1}(\theta)\lambda(\theta)}{2\kappa(\alpha(k))\lambda(\theta)+S_{2}(\theta)}\right\\}.$

where

$bias\left(\hat{\gamma}_{RW}(\lambda)|\theta=\hat{\theta}\right)=\dfrac{2b_{n,k}\kappa(\alpha(k))S_{1}(\hat{\theta})\lambda(\theta)}{2\kappa(\alpha(k))\lambda(\hat{\theta})+S_{2}(\hat{\theta})}.$

Since the term in the bracket converges to $0$ as $k\to\infty$ almost sure by
Lemma 2, the expectation of the term will converge to $0$ as $k\to\infty$.
Therefore,

$\displaystyle\lim\limits_{k\to\infty}bias\left(\hat{\gamma}_{RW}(\lambda)\right)$
$\displaystyle=\lim\limits_{k\to\infty}\int_{[0,1]^{k}}\left\\{\dfrac{2b_{n,k}\kappa(\alpha(k))S_{1}(\theta)\lambda(\theta)}{2\kappa(\alpha(k))\lambda(\theta)+S_{2}(\theta)}\right\\}d\theta$
$\displaystyle=\int_{[0,1]^{\infty}}\left\\{\lim\limits_{k\to\infty}\dfrac{2b_{n,k}\kappa(\alpha(k))S_{1}(\theta)\lambda(\theta)}{2\kappa(\alpha(k))\lambda(\theta)+S_{2}(\theta)}\right\\}d\theta$
$\displaystyle=0.$

which gives , $\lim_{k\to\infty}bias(\hat{\gamma}_{RW}(\lambda))=0$.

Similarly, we may use Lemma 1 and Lemma 2,to show that the MSE of
$\hat{\gamma}_{RW}(\lambda)\to 0$ as $k\to\infty$. Now, observe from Eqn.
(LABEL:E21) that

$MSE\left(\hat{\gamma}_{RW}(\lambda)\right)=\operatorname{\mathbb{E}}_{\theta}\left(MSE\left(\hat{\gamma}_{RW}(\lambda)|\hat{\theta}=\theta\right)\right).$

and therefore, we have

$\displaystyle\lim\limits_{k\to\infty}MSE\left(\hat{\gamma}_{RW}(\lambda)\right)$
$\displaystyle=\lim\limits_{k\to\infty}\int_{[0,1]^{k}}\left\\{(MSE\left(\hat{\gamma}_{RW}(\lambda)|\hat{\theta}=\theta\right)\right\\}d\theta$
$\displaystyle=\int_{[0,1]^{\infty}}\left\\{\lim\limits_{k\to\infty}(MSE\left(\hat{\gamma}_{RW}(\lambda)|\hat{\theta}=\theta\right)\right\\}d\theta$
$\displaystyle=0.$

This implies that the proposed estimator is asymptotically consistent under
some conditions.

######  Theorem 3.

Suppose Eqn.(3), Eqn.(8) and Eqn.(14) are satisfied. Assume also that $\rho$
is estimated by a consistent estimator $\hat{\rho}$, then if assumptions
$A_{1}$ and $A_{2}$ holds, and $\sqrt{k}b_{n,k}\rightarrow_{p}0$,

$\sqrt{k}\left(\hat{\gamma}_{RW}(\lambda)-\gamma\right)\rightarrow_{d}N(0,\gamma^{2}).$

######  Corollary 4.

Suppose (3) and (8) holds and also
$\lim\limits_{k\to\infty}\alpha(k)/k\to\varDelta\in(0,\infty)$. Assume $\rho$
is estimated by a consistent estimator $\hat{\rho}$, then as $k,n\to\infty$
and $\sqrt{k}b_{n,k}\longrightarrow_{p}0$,

$\sqrt{k}\left(\hat{\gamma}_{RW}(\tau)-\gamma\right)/\gamma\to_{d}N(0,1)$

###### Proof.

The proof of Corollary 4 follow from the proof of Theorem 3. ∎

Theorem 3 discusses the asymptotic normality of $\hat{\gamma}_{RW}(\lambda)$
defined in Eqn.(16). To prove Theorem 3 we require the following properties in
addition.

We write

$\mathfrak{M}_{\lambda}(\theta):=\dfrac{\tilde{W}_{j}(C_{j}-\sum_{j=1}^{k}\tilde{W}_{j}C_{j})}{2\kappa(\alpha(k))\lambda+\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)^{2}}.$
(32)

######  Lemma 5.

Let
$C_{j}=\left(\dfrac{j}{k+1}\right)^{-\rho},\tilde{W}_{j}=\dfrac{W_{j}}{\sum_{j=1}^{k}W_{j}},j\in\\{1,2,...,k\\}$
and $\rho<0.$ Then as $k\to\infty$,

$\mathfrak{M}_{\lambda}(\theta)\to O(1/k^{1+\omega}),$

where $0<\omega\leq 0.1$.

Lemma 5 is required in the proof of Lemma 6.

######  Lemma 6.

Let $T_{1},T_{2},T_{3},...$ be independent random variables from an
exponential distribution with mean $\mu_{i}<\infty$, for all $i$. Then for any
$\epsilon>0$

$\lim\limits_{k\rightarrow\infty}\operatorname{\mathbb{P}}\left(\sqrt{k}b_{n,k}\geq\epsilon\right)=0,$
(33)

where $b_{n,k}$ is defined by Eqn.(15).

######  Remark 7.

Lemma 6 shows the statistics $\sqrt{k}b_{n,k}\to_{p}0$ as $k\to\infty$.

The next Lemma is about the satisfaction of the Lyapunov’s version of the
central limit theorem. The Lyapunov’s variant of the central limit theorem
assumes the existence of a finite moment of an order higher than two.

######  Lemma 8.

Suppose that $Y_{1},Y_{2},...$ are independent random variables such that
$\operatorname{\mathbb{E}}(Y_{k})=\hat{\mu}_{k}$ and
$Var(Y_{k})=\sigma_{k}^{2}<\infty$, then there exists $r>0$ such that

$\lim\limits_{k\rightarrow\infty}\dfrac{1}{s_{n}^{3}}\sum_{j=1}^{k}\operatorname{\mathbb{E}}\left(|Y_{j}-\hat{\mu}_{j}|^{r}\right)=0,$
(34)

where $Y_{k}=\tilde{W}_{k}(\theta_{k})T_{k}.$

######  Remark 9.

Setting the penalty term to 0 reduces the regularised weighted least squares
estimator to a weighted least squares estimator. The difference between this
weighted least squares estimator and the one introduced by [30] is that, this
weighted least squares estimator has smaller asymptotic variance and this is
due to the introduction of randomness into the weight function. The resulting
weighted least squares estimator is also asymptotically unbiased, consistent,
and normally distributed with mean $0$ and variance $\gamma^{2}$.

##  3 Simulation Study

In the previous section, we proposed the regularised weighted least squares
estimators under the semi-parametric setting to estimate the tail index of the
underlying distribution of a given data from the Pareto-type of distributions.
In this section, we perform a simulation study to compare the performance of
our proposed estimators to other existing semi-parametric tail index
estimators. Particularly, the regularised weighted least squares, RWLS, the
reduced-bias weighted least squares with modified weight function, WLS, the
ridge regression, RR [13], the least squares, LS [29], the Hill estimator,
HILL [17] and the bias-corrected Hill, BCHILL [27] in the case of Pareto-type
distributions are compared.

###  3.1 Simulation design

We consider the Fréchet and Burr XII from the Pareto-type distributions as
shown in Table 1. For each distribution $F$, we generate 1000 repetitions of
samples of size $n=50,500$ and $2000$. For the Fréchet distributions, we
consider $\alpha=10,2$ and $1.0$; and for the Burr XII we consider the
mixtures

  1. i.

$\xi=\sqrt{10}\hskip 6.0pt\&\hskip 6.0pt\tau=\sqrt{10}$,

  2. ii.

$\xi=\sqrt{2}\hskip 6.0pt\&\hskip 6.0pt\tau=\sqrt{2}$ and

  3. iii.

$\xi=2\hskip 6.0pt\&\hskip 6.0pt\tau=1/2$

to obtain the tail index values $\gamma=0.1,0.5$ and $1.0$, respectively. We
consider the finite sample behaviour of the proposed estimators, RWLS and WLS,
and also compared these estimators with RR, LS, BCHILL and HILL. The MSE and
bias are plotted as a function of the number of top-order statistics, $k$, to
investigate the estimators’ sample path behaviour.

In the case of the weight function, the $\theta_{j}$’s will be replaced with
their point estimate, in this case, the mean of a standard uniform
distribution. In the case of $\alpha(k)$, we select $\alpha(k)$ such that
$\varDelta\to 0$, as $k\to\infty$. This choice of $\alpha(k)$ is made because
in practice we have observed that it yields much more stable estimates
compared to when $\alpha(k)$ is selected such that $\varDelta\to c>0$ in
application.

Table 1: Heavy-tailed distributions from the Pareto-type distribution Distribution | $1-F(x)$ | $\ell_{F}(x)$ | $\gamma$
---|---|---|---
Burr type XII | $\left(1+x^{\tau}\right)^{-\xi}$ | $\left(1+x^{-\tau}\right)^{-\xi}$ | $1/\tau\xi$
Fréchet | $1-F(x)=1-exp(-x^{-\alpha})$ | $1-\frac{x^{-\alpha}}{2}+O(x^{-\alpha})$ | $1/\alpha$

###  3.2 Discussion of simulation results

In this section, we discuss the behaviour of RWLS and WLS relative to RR, LS,
HILL and BCHILL. The Mean Square Error (MSE) and bias are the performance
measures in the simulation studies. The simulation results for the Burr
distribution with different tail indexes are shown in Figures 1 \- 3. Also,
Figures 4 \- 6 present the simulation results for the Fréchet distribution
with varying tail indexes.

From these figures, the plots of WLS and RWLS follow the same sample path for
$k\leq 0.4n$, i.e., their performance are relatively the same on that
interval. WLS and LS are very close to each other, though generally, WLS
slightly outperforms LS in terms of MSE and bias. Thus, generally the WLS can
be considered the most appropriate estimator of the tail index among the
regression-based estimators (i.e., RR, LS, WLS and RWLS) since it mostly has
smallest bias and MSE across all samples.

Additionally, the MSE plots of the proposed estimators are low and near
constant over the central part of $k$, except in the case of Burr XII with
$\gamma=1.0$. With the exception of the HILL estimator (which globally has the
highest MSE), the MSE curves of the estimators are mostly close to each other
in the central $k$ region, especially in the case of the Fréchet distribution.
This implies that the proposed estimators are competitive with the existing
estimators. However, the proposed estimators, WLS and RWLS, generally attain
the lowest bias for small samples, i.e., $n=50$. Furthermore, for medium to
large values of $k$, the sample paths of RWLS in the MSE and bias plots are
between HILL and RR. Even though the BCHILL estimator mostly has the smallest
MSE and bias, the proposed estimators (RWLS and WLS) outperform it for large
values of $k$.

Hence, from the simulation results, WLS and RWLS are appropriate for the
estimation of the tail index of the Pareto-type distributions in terms of MSE
and bias.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Results for Burr type XII distribution with $\gamma=0.1$: Bias(top
row) and MSE(bottom row). First column: $n=50$; second column: $n=500$; and
third column: $n=2000$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Results for Burr type XII distribution with $\gamma=0.5$: Bias(top
row) and MSE(bottom row). First column: $n=50$; second column: $n=500$; and
third column: $n=2000$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Results for Burr type XII distribution with $\gamma=1.0$: Bias(top
row) and MSE(bottom row). First column: $n=50$; second column: $n=500$; and
third column: $n=2000$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Results for Fréchet distribution with $\gamma=0.1$: Bias(top row)
and MSE(bottom row). First column: $n=50$; second column: $n=500$; and third
column: $n=2000$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Results for Fréchet distribution with $\gamma=0.5$: Bias(top row)
and MSE(bottom row). First column: $n=50$; second column: $n=500$; and third
column: $n=2000$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Results for Fréchet distribution with $\gamma=1.0$: Bias(top row)
and MSE(bottom row). First column: $n=50$; second column: $n=500$; and third
column: $n=2000$.

##  4 Applications

In this section, we consider the estimation of the tail index of the
underlying distribution of two datasets from the insurance industry. First,
the SOA Group Medical Insurance dataset consists of over 170,000 claims
recorded from 1991 to 1992. In this study, we consider the 1991 dataset, which
comprised 75,789 claims and have been studied widely in the extreme value
context (see e.g [18],[4]). Considering the large size of this dataset, we
focus on the extreme tail of the data and hence consider the top 10% data
points, (i. e, $0<k\leq 0.1n$). The SOA dataset is available at
https://lstat.kuleuven.be/Wiley/Data/soa.txt.

Second, the automobile insurance data from Ghana consists of 452 claims from
July 7, 2020, to May 11, 2021 and can be found at
https://github.com/kikiocran/TailEstimators We will refer to this dataset as
the GH claims in this study. To the best of our knowledge this dataset has
never been used in the extreme value theory literature.

The scatter plots of the SOA, and the GH claims are shown in Figure 7. We
observe that two claims and one claim in the SOA and GH claims, respectively,
appear to be far detached from the bulk of the data. These observations can
also be seen to deviate from linearity and far removed from the bulk of the
points respectively in the Pareto and exponential Q-Q plots (Figure 8) of the
two datasets. Such large observations are suspected outliers and may
significantly influence the tail index estimates (see e.g [4]). The convex
curvature of the exponential Q-Q plots and the near linearity of the Pareto
Q-Q plots of the datasets indicate that the datasets suggest that they belong
to the Pareto-type distributions.

(a)

(b)

Figure 7: Scatter plots: SOA claims data (left panel); GH claims data (right
panel).

(a)

(b)

(c)

(d)

Figure 8: Exponential and Pareto Q-Q plots: SOA claims (left panel); GH claims
(right panel).

Figure 9 shows the sample paths of the tail index estimators for the
underlying distributions of the two datasets. The plot of HILL diverges as $k$
increases, i.e., it is very sensitive to the changes in $k$. Hence, it is not
an appropriate estimator for estimating the tail index. The other estimators
exhibit some form of stability, however, the sample paths of the proposed
estimators (i.e., RWLS and WLS) are smooth, that is, these estimators are less
sensitive to changes in $k$. All the tail index estimators considered are very
unstable for small values of $k$ due to the small number of exceedances. A
specific tail index estimate can be obtained from the plots of WLS and RWLS
for both datasets.

(a)

(b)

Figure 9: Tail index estimates as a function of $k$: (a) SOA data; (b) GH
claims.

##  5 Conclusion

In this paper, we proposed tail index estimators for the Pareto-type of
distributions using the regression model. In addition to the ordinary least
squares and the ridge regression estimators, we proposed the regularised
weighted least squares and the weighted least squares estimators as
alternative reduced-bias estimators. The tail index estimates by the proposed
estimators are generally stable and smooth across a broader path of $k$. The
characteristics of the proposed estimators are as follows;

  1. $\bullet$

they are asymptotically consistent, unbiased and normally distributed with
mean $0$ and variance $\gamma^{2}$.

  2. $\bullet$

the MSE curves are low and flat over the central part of $k$.

  3. $\bullet$

the plots of their tail index estimates are more stable, smooth and horizontal
than the Hill, ordinary least squares and the bias-corrected Hill estimators.

In conclusion, comparatively, the proposed estimators are competitive to the
existing estimators and can be considered as appropriate estimators of the
tail index with MSE, bias and in real-life application.

##  6 Proofs

###### Proof of Lemma 1.

  1. i.

From Eqn. (20) and Eqn. (21), we have

$\displaystyle S_{1}(\theta)$ $\displaystyle=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}$
$\displaystyle=2\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)\int_{0}^{1}\left(1-\operatorname{\mathbb{E}}(\theta^{\alpha(k)})u\right)u^{-\rho}\,du+O(1)$
$\displaystyle=\dfrac{2\left(2\frac{\alpha(k)}{k}-\rho\frac{\alpha(k)}{k}+\frac{1}{k}\right)}{\left(2\frac{\alpha(k)}{k}+\frac{1}{k}\right)(1-\rho)(2-\rho)}+O(1).$

It follows that,

$\displaystyle\lim\limits_{k\to\infty}\Big{[}\dfrac{2\left(2\frac{\alpha(k)}{k}-\rho\frac{\alpha(k)}{k}+\frac{1}{k}\right)}{\left(2\frac{\alpha(k)}{k}+\frac{1}{k}\right)(1-\rho)(2-\rho)}+O(1)\Big{]}$
$\displaystyle=\dfrac{2\left(2\varDelta-\rho\varDelta\right)}{\left(2\varDelta\right)(1-\rho)(2-\rho)}$
$\displaystyle=\dfrac{1}{\left(1-\rho\right)},$

where $\varDelta=\lim\limits_{k\to\infty}\alpha(k)/k$ and hence, as
$k\to\infty$, we have $S_{1}(\theta)\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}1/(1-\rho).$

  2. ii.

From Eqn. (22),

$S_{2}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right)^{2}=\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}-\left(\dfrac{1}{1-\rho}\right)^{2}.$

Using Eqn. (20), the first term can be written as

$\displaystyle\sum_{j=1}^{k}\tilde{W}_{j}C_{j}^{2}$
$\displaystyle=2\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)\int_{0}^{1}\left(1-\operatorname{\mathbb{E}}(\theta^{\alpha(k)})u\right)u^{-2\rho}\,du+O(1)$
$\displaystyle=\dfrac{\left(2\frac{\alpha(k)}{k}-2\rho\frac{\alpha(k)}{k}+\frac{1}{k}\right)}{\left(2\frac{\alpha(k)}{k}+\frac{1}{k}\right)(1-\rho)(1-2\rho)}+O(1).$

It follows that,

$\displaystyle\lim\limits_{k\to\infty}\Big{[}\dfrac{\left(2\frac{\alpha(k)}{k}-2\rho\frac{\alpha(k)}{k}+\frac{1}{k}\right)}{\left(2\frac{\alpha(k)}{k}+\frac{1}{k}\right)(1-\rho)(1-2\rho)}+O(1)\Big{]}$
$\displaystyle=\dfrac{\left(2\varDelta-2\rho\varDelta\right)}{2\varDelta(1-\rho)(1-2\rho)}=\dfrac{1}{\left(1-2\rho\right)},$

where $\varDelta=\lim\limits_{k\to\infty}\alpha(k)/k$. Therefore,

$S_{2}(\theta)\longrightarrow\dfrac{1}{1-2\rho}-\left(\dfrac{1}{1-\rho}\right)^{2}=\dfrac{\rho^{2}}{\left(1-2\rho\right)\left(1-\rho\right)^{2}}$

as $k\to\infty$. That is, $S_{2}\left(\theta\right)\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}\rho^{2}/(1-2\rho)(1-\rho)^{2}$ as $k\to\infty$.

  3. iii.

The expression
$\dot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right)$
can also be written as

$\displaystyle\dot{S}(\theta)$
$\displaystyle=\dfrac{4}{k}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)^{2}\left\\{\dfrac{1}{k}\sum_{j=1}^{k}\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k+1}\right)^{2}\left(S_{1}(\theta)-\left(\dfrac{j}{k+1}\right)^{-\rho}\right)\right\\}$
$\displaystyle=\dfrac{4}{k}\left(\dfrac{\frac{\alpha(k)}{k}+\frac{1}{k}}{2\frac{\alpha(k)}{k}+\frac{1}{k}}\right)^{2}\int_{0}^{1}\left(1-\operatorname{\mathbb{E}}(\theta^{\alpha(k)})u\right)^{2}\left(\dfrac{1}{1-\rho}-u^{-\rho}\right)\,du+O(1).$

Therefore, as $k\to\infty$, $\dot{S}(\theta)\stackrel{{\scriptstyle
a.s}}{{\longrightarrow}}0$.

  4. iv.

$\ddot{S}(\theta)=\sum_{j=1}^{k}\tilde{W}_{j}^{2}\left(S_{1}(\theta)-C_{j}\right)^{2}$
can also be expressed as

$\displaystyle\ddot{S}(\theta)$
$\displaystyle=\dfrac{4}{k}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)^{2}\left\\{\dfrac{1}{k}\sum_{j=1}^{k}\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k+1}\right)^{2}\left(S_{1}(\theta)-\left(\dfrac{j}{k+1}\right)^{-\rho}\right)^{2}\right\\}$
$\displaystyle=\dfrac{4}{k}\left(\dfrac{\frac{\alpha(k)}{k}+\frac{1}{k}}{2\frac{\alpha(k)}{k}+\frac{1}{k}}\right)^{2}\int_{0}^{1}\left(1-\operatorname{\mathbb{E}}(\theta^{\alpha(k)})u\right)^{2}\left(\dfrac{1}{1-\rho}-u^{-\rho}\right)^{2}\,du+O(1).$

It also follows that, as $k\to\infty$,
$\ddot{S}(\theta)\stackrel{{\scriptstyle a.s}}{{\longrightarrow}}0$.

∎

###### Proof of Lemma 2.

The proof of Lemma 2 easily follows by using Lemma 1. ∎

###### Proof of Lemma 5.

We observe that

$\dfrac{\\-1}{|k^{1+\omega}+5\rho|}\leq\mathfrak{M}_{\lambda}(\theta)\leq\dfrac{C_{j}^{2}/S_{2}(\theta)}{|k^{1+\omega}+5\rho|}.$

Therefore, we have $\mathfrak{M}_{\lambda}(\theta)\to O(1/k^{1+\omega}),\hskip
4.0pt0<\omega\leq 0.1$, as $k\to\infty$, which completes the proof of Lemma 5.
∎

###### Proof of Lemma 6.

The proof requires the use of large deviation principles (LDP). From Eqn.(15)
and Eqn.(32),

$b_{n,k}=\sum_{j=1}^{k}\mathfrak{M}_{\lambda}(\theta)T_{j}.$

$T_{j}$ is exponentially distributed with mean $\mu_{j}=\gamma
e^{D\left(\frac{j}{n+1}\right)^{-\rho}}$, therefore

$M_{T_{k}}(t)=\dfrac{1}{1-\mu_{k}t}.$ (35)

Using Lemma 6 and Eqn.(35), the moment generating function of $b_{n,k}$ is

$\displaystyle M_{b_{n,k}}(t)$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left\\{\prod_{j=1}^{k}\operatorname{\mathbb{E}}\left(e^{T_{j}t}|\theta_{j}=\theta\right)\right\\}=\operatorname{\mathbb{E}}_{\theta}\left\\{\prod_{j=1}^{k}M_{T_{j}}\left(\mathfrak{M}_{\lambda}(\theta)t\right)\right\\}$

It follows that

$M_{b_{n,k}}(kt)=\operatorname{\mathbb{E}}_{\theta}\left\\{\prod_{j=1}^{k}\left(\dfrac{1}{1-k\mu_{j}\mathfrak{M}_{\lambda}(\theta)t}\right)\right\\}.$

Note, from Lemma 5, we have $k\mu_{j}\mathfrak{M}_{\lambda}(\theta)\to
O(1/k^{1+\omega})$ as $k\to\infty$ and hence, there exists $\delta>0$ such
that $\displaystyle-\delta\leq
k\mu_{j}\mathfrak{M}_{\lambda}(\theta)\leq\delta.$ This implies that

$\displaystyle\operatorname{\mathbb{E}}_{\theta}\left[\left(1-\delta+\delta^{2}-\delta^{3}+...\right)^{k}\right]\leq
M_{b_{n,k}}(kt)\leq\operatorname{\mathbb{E}}_{\theta}\left[\left(1+\delta+\delta^{2}+\delta^{3}+...\right)^{k}\right]$
$\displaystyle k\log\left(1-\delta+\delta^{2}-\delta^{3}+...\right)\leq\log
M_{b_{n,k}}(kt)\leq k\log\left(1+\delta+\delta^{2}+\delta^{3}+...\right)$
$\displaystyle\log\left(1-\delta+\delta^{2}-\delta^{3}+...\right)\leq\lim\limits_{k\to\infty}\dfrac{1}{k}\log
M_{b_{n,k}}(kt)\leq\log\left(1+\delta+\delta^{2}+\delta^{3}+...\right)$

Now letting $\delta\longrightarrow 0$ and applying the Squeeze Theorem, we
obtain

$\lim\limits_{k\to\infty}\dfrac{1}{k}\log M_{b_{n,k}}(kt)=0.$

Hence, the logarithmic moment generating function converges to 0 with speed
$k$ and by the G$\ddot{a}$rtner Ellis Theorem, $b_{n,k}$ follows a Large
Deviation Principle (LDP) with speed $k$ and a rate function $I(x)$ defined as

$\displaystyle I(x)$ $\displaystyle=\sup_{\eta\in\mathbb{R}}\left\\{\eta
x-0\right\\}=\sup_{\eta\in\mathbb{R}}\left\\{\eta x\right\\}$
$\displaystyle=\begin{cases}0\hskip 6.0pt\text{if}\hskip 6.0ptx=0,\\\
\infty\hskip 6.0pt\text{if}\hskip 6.0ptx\neq 0,\end{cases}$

where $x\in[0,\infty)$ and $\eta>0$. If $b_{n,k}$ obeys LDP with a rate
function $I(x)$, then $\exists\epsilon>0$ such that

$\displaystyle\operatorname{\mathbb{P}}\left(\sqrt{k}b_{k,n}>\epsilon\right)$
$\displaystyle=\operatorname{\mathbb{P}}\left(b_{n,k}>\frac{\epsilon}{\sqrt{k}}\right)=\operatorname{\mathbb{P}}\left(b_{n,k}>\varGamma_{k}\right)\leq
e^{-kI(x)},$

where $\varGamma_{k}=\epsilon/k$. The typical behaviour of the rate function
is when $x\neq 0$, therefore

$\lim\limits_{k\to\infty}\operatorname{\mathbb{P}}\left(\sqrt{k}b_{n,k}>\epsilon\right)\leq
0.$

Thus, $\sqrt{k}b_{n,k}\rightarrow_{p}0$ as $k\to\infty$ and this ends the
proof. ∎

###### Proof of Lemma 8.

The proof of Lemma 8 only requires that we establish that the expression holds
for at least $r=3$. Let

$\displaystyle\tilde{\mu}_{k}=\operatorname{\mathbb{E}}\left(Y_{k}|\theta_{k}=\theta\right)$
$\displaystyle=\tilde{W}_{k}(\theta)\operatorname{\mathbb{E}}(T_{k})$
$\displaystyle=\dfrac{2}{k}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)\left(1-\theta^{\alpha(k)}\dfrac{k}{k+1}\right)\mu_{k}$
(36)

It follows that

$\displaystyle\operatorname{\mathbb{E}}\left\\{\left|Y_{k}-\tilde{\mu}_{k}\right|^{3}\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left\\{\operatorname{\mathbb{E}}\left(\left|\tilde{W}_{k}(\theta)T_{k}-\tilde{\mu}_{k}\right|^{3}|\theta_{k}=\theta\right)\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left\\{\left|\tilde{W}_{k}(\theta)\right|^{3}\operatorname{\mathbb{E}}\left(\left|T_{k}-\mu_{k}\right|^{3}\right)\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left(\tilde{W}_{k}(\theta)\right)^{3}\operatorname{\mathbb{E}}\left\\{\left|T_{k}-\mu_{k}\right|^{3}\right\\}$
(37)

Now considering the first term on the right hand side of Eqn. 6, we have

$\displaystyle\operatorname{\mathbb{E}}_{\theta}\left[\left(\tilde{W}_{k}(\theta)\right)\right]=\left(\dfrac{2}{k}\right)^{3}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)^{3}\operatorname{\mathbb{E}}_{\theta}\left(1-\theta^{\alpha(k)}\frac{k}{k+1}\right)^{3}$
$\displaystyle=\dfrac{8}{k^{3}}\left(\dfrac{\frac{\alpha(k)}{k}+\frac{1}{k}}{2\frac{\alpha(k)}{k}+\frac{1}{k}}\right)^{3}\left\\{1-\frac{3}{\left(\alpha(k)+1\right)}\left(1-\frac{1}{k+1}\right)+\right.$
$\displaystyle\left.\frac{3}{\left(2\alpha(k)+1\right)}\left(1-\frac{1}{k+1}\right)^{2}-\frac{1}{\left(3\alpha(k)+1\right)}\left(1-\frac{1}{k+1}\right)^{3}\right\\}$
(38)

We observe that as $k\to\infty$,
$\operatorname{\mathbb{E}}_{\theta}\left[\left(\tilde{W}_{k}(\theta)\right)\right]\to
0$.

Furthermore, expanding the second term on the right hand sides of Eqn.6 we
have

$\displaystyle\operatorname{\mathbb{E}}\left(\left|T_{k}-\mu_{k}\right|^{3}\right)$
$\displaystyle\leq\operatorname{\mathbb{E}}\left\\{\left(|T_{k}|+|\mu_{k}|\right)^{3}\right\\}$
(39)
$\displaystyle=\operatorname{\mathbb{E}}\left\\{\left(T_{k}+\mu_{k}\right)^{3}\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}\left(T_{k}^{3}\right)+3\mu_{k}\operatorname{\mathbb{E}}\left(T_{k}^{2}\right)+3\mu_{k}^{2}\operatorname{\mathbb{E}}\left(T_{k}\right)+\mu_{k}^{3}.$

Since $T_{k}$ is exponentially distributed, the $n^{th}$ moment exists, and is
finite. The powers of $\mu_{k}$ are also finite, therefore
$\operatorname{\mathbb{E}}\left(\left|T_{k}-\mu_{k}\right|^{3}\right)<\infty$.

From Eqn. 6 and 39, we observe that
$\operatorname{\mathbb{E}}\left(|Y_{k}-\hat{\mu}_{k}|^{3}\right)\to 0$ as
$k\to\infty$. This also implies that

$\sum_{j=1}^{k}\operatorname{\mathbb{E}}\left(|Y_{j}-\hat{\mu}_{j}|^{3}\right)\to
0,\hskip 4.0ptk\to\infty$ (40)

Define $Z_{j}$ as $Z_{j}=Y_{j}-\operatorname{\mathbb{E}}(Y_{j}),$ then we have
that

$\displaystyle s_{n}^{2}=\sum_{j=1}^{k}Var(Z_{j})$
$\displaystyle=\sum_{j=1}^{k}Var\left\\{\tilde{W}_{j}(\theta)T_{j}\right\\}$
$\displaystyle=\sum_{j=1}^{k}\operatorname{\mathbb{E}}\left\\{\tilde{W}_{j}^{2}(\theta)\right\\}Var(T_{j})+\sum_{j=1}^{k}Var\left(\tilde{W}_{j}(\theta)\right)\left(\operatorname{\mathbb{E}}(T_{j})\right)^{2}$
$\displaystyle=\dfrac{4\gamma^{2}}{k}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)^{2}\left\\{1-\frac{1}{\alpha(k)+1}+\frac{(2k+1)}{6(2\alpha(k)+1)(k+1)}\right\\}$
$\displaystyle+\dfrac{4}{k^{2}}\left(\dfrac{\alpha(k)+1}{2\alpha(k)+1}\right)^{2}\sum_{j=1}^{k}\mu_{j}^{2}Var\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k+1}\right).$

Using Eqn. 40 and 6, it can be observe that

$\lim\limits_{k\rightarrow\infty}\dfrac{1}{s_{n}^{3}}\sum_{j=1}^{k}\operatorname{\mathbb{E}}\left(|Y_{j}-\hat{\mu}_{j}|^{3}\right)=0.$
(42)

This ends the proof of Lemma 8. ∎

###### Proof of Theorem 3.

Using Lemma 6 and Lemma 8, we can prove Theorem 3. It has been established in
Lemma 6 that $b_{n,k}\to_{p}0$ as $k\to\infty$. Lemma 8 also establishes that
the Lyapunov’s condition holds for Central Limit Theorem; hence, by the
Lyapunov’s Central Limit Theorem;

$\sqrt{k}\left(\hat{\gamma}_{RW}(\lambda)-\gamma\right)\longrightarrow_{d}N(\mu,\sigma^{2}),\hskip
6.0pt\text{as}\hskip 6.0ptk\to\infty$

.

Therefore, all we need to complete the proof of Theorem 3, is to specify the
parameters of the normal distribution.

Recall from (16) that

$\hat{\gamma}_{RW}(\lambda,\theta)=\sum_{j=1}^{k}\tilde{W}_{j}T_{j}-b_{n,k}\sum_{j=1}^{k}\tilde{W}_{j}C_{j}.$

Also, from Lemma 6, asymptotically, the second term on the right hand side
vanishes, so we would concentrate on the first term of the expression only.

Let

$\displaystyle
S_{k}=\sqrt{k}\left(\hat{\gamma}_{RW}(\lambda)-\gamma\right)=\sqrt{k}\left\\{\sum_{j=1}^{k}\tilde{W}_{j}T_{j}-\gamma\right\\}=\sqrt{k}\sum_{j=1}^{k}\tilde{W}_{j}-\sqrt{k}\gamma.$

The expected value of $S_{k}$ is given as

$\displaystyle\operatorname{\mathbb{E}}(S_{k})=\operatorname{\mathbb{E}}_{\theta}\left\\{\sqrt{k}\sum_{j=1}^{k}\tilde{W}_{j}\operatorname{\mathbb{E}}(T_{j})-\sqrt{k}\gamma\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left\\{\sqrt{k}\sum_{j=1}^{k}\tilde{W}_{j}\left\\{\gamma+b_{n,k}C_{j}\right\\}-\sqrt{k}\gamma\right\\}$
$\displaystyle=\operatorname{\mathbb{E}}_{\theta}\left\\{\sqrt{k}b_{n,k}\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\right\\}.$

Recall that, as $k\to\infty$ $\sqrt{k}b_{n,k}\to 0$ and
$\sum_{j=1}^{k}\tilde{W}_{j}C_{j}\to 1/(1-\rho)$; hence
$\operatorname{\mathbb{E}}(S_{k})\to\mu=0$ as $k\to\infty$. Note, the variance
of $S_{k}$ is given by

$\displaystyle Var(S_{k})$
$\displaystyle=kVar\left(\sum_{j=1}^{k}\tilde{W}_{j}T_{j}\right)$
$\displaystyle=\dfrac{4}{k}\kappa^{2}(\alpha(k))\sum_{j=1}^{k}Var\left\\{\left(1-\theta_{j}^{\alpha(k)}\dfrac{j}{k}\right)T_{j}\right\\}.$

Using the assumption that $\left(1-\theta_{j}^{\alpha(k)}\frac{j}{k+1}\right)$
and $T_{j}$ are independent, and $\theta_{j}\stackrel{{\scriptstyle
i.i.d}}{{\sim}}U(0,1)$, one can show that as $k\to\infty$,
$\sqrt{k}b_{n,k}\to_{p}0$, $\alpha(k)/k\to\varDelta<\infty$,
$Var(S_{k})\longrightarrow\sigma^{2}=\gamma^{2},$ and, this completes the
proof of Theorem 3. ∎

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# A Fast Diagnostic to Inform Screening of Discarded or Retired Batteries

Joseph A. Drallmeier  Clement Wong  Charles E. Solbrig  Jason B. Siegel  Anna
G. Stefanopoulou  Battery Controls Group, University of Michigan, Ann Arbor,
MI, 48109 USA. (e-mail: drallmei, clemwong, csolbrig, siegeljb,
annastef@umich.edu).

###### Abstract

With the increased pervasiveness of Lithium-ion batteries, there is growing
concern for the amount of retired batteries that will be entering the waste
stream. Although these batteries no longer meet the demands of their first
application, many still have a significant portion of their initial capacity
remaining for use in secondary applications. Yet, direct repurposing is
generally not possible and each cell in a battery must be evaluated,
increasing the cost of the repurposed packs due to the time intensive
screening process. In this paper, a rapid assessment of the internal
resistance of a cell is proposed. First, this method of measuring the
resistance is completed on cells from twelve retired battery packs and one
fresh pack using a hybrid pulse power characterization (HPPC) test as a
benchmark for the analysis. Results from these tests show relatively constant
resistance measurements across mid to high terminal voltages, allowing this
metric to be independent of state of charge (SOC). Then, the relation between
internal resistance and capacity across the various packs is discussed.
Initial experimental results from this study show a correlation between
internal resistance and capacity which can be approximated with a linear fit,
suggesting internal resistance measurements taken above a threshold cell
terminal voltage may be a suitable initial screening metric for the capacity
of retired cells without knowledge of the SOC.

###### keywords:

Diagnostics, Batteries, State of Health, Fault diagnosis, Monitoring

##  1 Introduction

The current exponential increase in the production of lithium-ion batteries
for clean energy and electric vehicle applications will inevitably result in
the presence of a large volume of “retired” batteries soon. Estimates from IEA
(2020) suggest 100-120 GWh of EV batteries will be retired by 2030. Batteries
are retired when they are unable to meet the original application’s
performance requirements. A battery pack for an electric vehicle, for
instance, is under warrantied to maintain 80% of its original capacity
(Groenewald et al. (2017), Cusenza et al. (2019)) within ten years or 100,000
miles travelled with the battery. Retired batteries, while no longer able to
meet performance standards of the the original application, still retain much
of their performance capabilities in most cases and are sufficient to be
applied in less-demanding applications (Zhu et al. (2021)). As such, to
maximize the environmental and economic benefits of lithium-ion batteries,
remanufacturing and repurposing is a key step in a battery’s life-cycle (Hua
et al. (2020)).

However, the cost of testing and refurbishing batteries is high and may
prevent applications of retired batteries for second life from being
economically viable (Rallo et al. (2020), Martinez-Laserna et al. (2018)).
Neubauer et al. (2015) indicates the greatest portion of the testing and
refurbishing cost is the laborer costs, much of which is due to the time spent
disassembling and testing a battery. A significant contributor to long test
times is the assessment of a battery’s state-of-health (SOH), which is a
measure of a battery’s current performance conditions compared to its original
state. For instance, assessing battery capacity fade, a key metric in SOH, is
conventionally completed by evaluating changes in the open circuit voltage
(OCV) with respect to state of charge (SOC) (Roscher et al. (2011)). However,
obtaining the OCV curves takes on the order of $20+$ hours. Further, direct
testing and repurposing batteries at the pack level typically depends on the
active balancing and more time intensive measures of evaluating individual
cells is required. As noted by Zhou et al. (2017), variations in individual
cells making up a battery pack increase with aging of the pack and a few
individual cells can reduce the performance of the entire pack. Thus,
developing a strategy that reduces testing time while providing an acceptable
estimate of a cell’s capacity could significantly reduce cost of testing
batteries and make applications of retired batteries more economically
feasible.

In (Zhou et al. (2020)), the current methods for evaluating retired or aged
battery packs are segregated into two methodologies. The first focuses on the
effectiveness and consistency of the evaluation process while neglecting time
and effort considerations. An example of this is presented by Chung (2021) in
which the standard method of evaluating the safety and performance of a
retired battery is completed, as outlined by the Underwriters Laboratories
(UL), a global safety certification company, in UL 1974. The second
methodology, which is where this study aims to contribute, focuses on a
surrogate the OCV that can be aquired faster than the current state of the art
of $20+$ hours while still maintaining an acceptable level of consistency.
Zhou et al. (2020) used incremental capacity curves obtained with high
charging current rates which are then used to extract approximations of
capacity and internal resistance of cells. However, a full charge of the cells
is still required despite occurring at a faster rate. Weng et al. (2013)
proposes a battery’s capacity estimation scheme that only needs charging data
from 60%–85% SOC to estimate the battery capacity and avoids the need for a
time-consuming full charge and discharge. Recent work by Mohtat et al. (2022)
shows that this range of SOC provides critical information that correlates
well with a cell’s SOH, yet a full charge for one cell is typically required
to reset SOC evaluation when the initial SOC is unknown.

Advancing this state of the art, the following study investigates relation
between an estimated internal resistance parameter, $R_{s}$ and the measured
capacity of retired batteries. While internal resistance is only one of many
parameters to influence the performance of a battery (Zhou et al. (2017)), it
may be a rapid metric with which to initially screen retired batteries. In the
following sections, the testing procedures used to find the cell capacity as
well as to rapidly estimate internal resistance through charge interrupt are
provided. Additionally, a hybrid pulse power characterization (HPPC) test is
completed to substantiate the internal resistance estimates. Then, these tests
were performed on discarded DeWALT lithium-ion power tool battery packs and a
discussion of the $R_{s}$ characteristics is provided. From these results, a
clear correlation between cell capacity and resistance is observed. Further,
these results hold across mid to high range terminal voltage values. This
evaluation technique can then be used once surpassing a threshold cell
terminal voltage without any knowledge of SOC, providing a fast initial
screening test for aged Lithium-ion batteries.

##  2 Battery Packs

Figure 1: Schematic of the 2P5S test setup along with an image of the battery
pack and pack terminals.

The battery packs used in this study were DeWalt 20 V, 5 A-h model DCB205
lithium ion power tool battery packs, with a total of thirteen battery packs
analyzed. An image of the pack as well as a schematic of the 2P5S cell
arrangement within the pack is provided in Fig. 1. Each cell is an 18650
cylindrical cell with a capacity of 2.5Ah. It should be noted here that
throughout the rest of the paper, references made to cells 1 through 5 denote
the cell pairs in parallel, as shown in Fig. 1.

Twelve of the packs are considered aged and were acquired from a recycling
center after consumers discarded them, presumably when deemed past their
usable lifespan. The use and cycling history of these batteries are unknown. A
fresh pack was purchased to act as a control test. The aged packs were grouped
according to the two primary ID codes found, being N330105, which was
determined to contain Samsung INR18650-25R cells with an NCA chemistry, and
N437615 which contained LG LGABHE21865 cells with an NMC chemistry.
Additionally, the new pack had an ID code of N522573 which also contains cells
manufactured by LG with an NMC chemistry. In the following discussions,
letters are used to denote the various packs. Pack A denotes the fresh pack
while all other letters are the retired packs.

##  3 Experimental Setup

The tests sequences and data acquisition were completed with an A&D iTest Test
Cell System. The iTest System has an analog input module used for acquisition
of the individual cell pair data voltage. A Bitrode FTV-1 Power Module with 3
current ranges (2A,20A,200A) controlled by the iTest system was the source of
the voltage and current profiles.

The following sections outline the main testing procedures for the battery
packs followed by an additional HPPC sequence. The estimated resistance
parameters from the discharge interrupt, charge interrupt, and HPPC are
compared to demonstrate the validity of the more rapid diagnostic procedure of
interrupting a charge or discharge without providing that battery an extended
rest period. The parameter of interest here is the ohmic resistance, $R_{s}$,
which is responsible for the instantaneous voltage change when the cell is
under load.

###  3.1 Testing Procedure

Figure 2: Cell Voltage and pack current profiles used for full
characterization of each battery pack. Test sequences (TS) 1-6 are described
in Tab. 1 and the discharge interrupt of TS2, charge interrupt of TS4, and
HPPC of TS7 are detailed in section 3.1.

The cell voltage and pack current profile initially used for evaluation of
each battery pack is shown in Fig. 2. The profile can be separated into seven
distinct test sequences (TS), labeled above the figure, with a 30 minute rest
period between each individual test. The parameters for each sequence are
listed in Tab. 1. The charge and discharge rate for each sequence are listed
as C-rates, with the units of $h^{-1}$ and defined as $C\equiv\frac{I}{Q_{p}}$
where $I$ is the current required to charge or discharge the pack ampere hour
capacity $Q_{p}$ in the specified time. In TS1, an initial charge and
discharge is completed for initial cycling of the pack as the initial state of
charge (SOC) is unknown. The TS2 contains two current interrupts, which will
be reffered to as discharge interrupts, occuring at 4V and 3.2V, to provide a
means of calculating $R_{s,DI}$ from the instantaneous voltage change as

$R_{s}=\frac{\Delta V_{s}}{\Delta I}$ (1)

where $V_{s}$ is considered to be the instantaneous voltage change when
current is interrupted. During these discharge interrupts, the sampling rate
is 1 Hz. The following TS3 simply provides an additional charge and discharge
to isolate TS2 from TS4. In TS4, a charge interrupt procedure is completed
across a range of voltages with a sampling rate of 10 Hz. A charging rate of
$C/3$ is interrupted at 560 second intervals during which the current is set
to 0 for 20 seconds. This allows $R_{s,CI}$ to be calculated over a range of
cell voltages again using (1).

Table 1: Key details for each test sequence. | Charge Rate | Discharge Rate
---|---|---
TS1 | C/3 | C/3
TS2 | C/2 | C/2.5
TS3 | C/3 | C/2
TS4 | C/3 w/ interrupt | N.A.
TS5 | N.A | C/20
TS6 | C/20 | N.A.
TS7 | N.A. | C/3.33 w/ interrupt

The purpose of TS5 is to find the capacity of each cell. Ideally, each
charging profile is meant to follow a constant current, constant voltage
(CCCV) charging strategy. From this fully charged initialization point in TS5,
coulomb counting could be utilized to find cell capacity during the slow
discharge of TS5. This slow discharge reduces the impact of internal
resistance and concentration gradients on the measured terminal voltage,
allowing for an accurate estimate of capacity by assuming the measured
terminal voltage approximates the open circuit voltage (OCV) of the battery.
However, due to imbalances in the individual cell voltages as the battery pack
relies on an external balancing circuit, constant voltage charging could not
always be achieved. Therefore, capacity is still measured using coulomb
counting, but only between voltage set points of 4.1V and 2.95V.

Figure 3: Comparison of measured cell terminal voltage during different
discharge rates. When estimating capacity between two voltage points, the
voltage drop due to resistance can heavily impact estimates with high
discharge rates, increasing the estimated capacity.

Using this method of quantifying capacity between voltage limits introduces a
non-intuitive consequence of increasing the measured capacity for higher
discharge rates. As discharge rate increases, measured capacity becomes more
dependent on Rs due to the initial drop in terminal voltage, as shown in Fig.
3. However, by using the slow discharge rate, this relation is small and can
be neglected. Therefore, a slow discharge rate is imperative for a reliable
estimation of capacity, especially when using this method of measuring between
voltage limits.

All packs also underwent an HPPC testing sequence, TS7, where each cell was
subjected to series of 10-minute C/3.33 constant-current discharge pulses,
followed by a 60-minute rest period. This test was chosen as it provides a
benchmark industry standard test procedure for characterizing the batteries
internal resistance parameters (Zhang et al. (2011)). Figure 4 provides a
cell’s voltage response to the current pulses of the HPPC testing sequence
which was recorded at a rate of 10 Hz. At every discharge pulse, $R_{s,HPPC}$
was calculated using (1). The response of the voltage to a step change in
current during the HPPC test is shown in Fig. 4, with the change in voltage
due to the ohmic resistance denoted as $\Delta V_{s}$.

The methods utilized to calculate $R_{s,DI}$, $R_{s,CI}$, and $R_{s,HPPC}$
from a step change in current are identical. Yet, should be noted that while
these methods attempt to capture the instantaneous voltage change, it is
unavoidable that time will elapse between the discrete sampling points,
therefore capturing some of the more rapid diffusion dynamics in the $R_{s}$
value. As the sampling times when finding $R_{s,CI}$ and $R_{s,HPPC}$ are
identical, a direct comparison of these results is valid. However, the larger
sample time used when calculating $R_{s,DI}$ increases the resistance value.

Figure 4: Profile of the cell voltage during the HPPC test sequence. During
each interupt of the discharge current, the instantaneous as well as total
voltage change of the cell is recorded to calculate the internal resistance
parameters.

##  4 Results and Discussion

The following sections present the $R_{s}$ parameters calculated from the
introduced test sequences and outline the possible efficacy of the $R_{s}$
parameter in the estimation of the remaining capacity of a lithium-ion
battery. The characteristics of $R_{s}$ over a range of terminal voltages is
discussed along with a comparison of the $R_{s}$ parameters calculated by the
charge interrupt and HPPC test sequences as well as a brief discussion of the
discharge interupt results to convey the importance of a standardized sampling
time when implementing this method. Additionally, the apparent linear relation
between $R_{s}$ and battery capacity, a key factor of the SOH the battery, is
introduced despite the relatively small sample set of aged batteries in this
study.

###  4.1 Series Resistance Parameters

As the HPPC test sequence provides a benchmark to the industry standard
practice for parameterizing internal resistance in batteries, a comparison
between the $R_{s,HPPC}$ calculated from the HPPC test and the $R_{s,CI}$
calculated from the charge interrupt is provided in Fig. 5 for cell 4 in three
different aged battery packs. The packs C and J were chosen as pack C contains
cells manufactured by LG, while pack J contains cells manufactured by Samsung.
In addition, pack H was included as this pack showed much more significant
signs of aging as compared to the rest of the packs studies, as will be
discussed later.

Figure 5: Results of the $R_{s}$ calculations for the charge interrupt (CI)
and HPPC test sequences over a range of terminal voltages for cell 4 in
battery packs C, J, and H.

From Fig. 5 it is apparent that while there is not perfect alignment of the
results from the charge interrupt and HPPC tests, the trend in $R_{s}$ between
the different packs is maintained. The resistance calculated in the cell from
pack H is considerably larger than that of packs C and J. It should also be
noted that while each testing sequence calculates $R_{s}$ in a similar manner,
the charge interrupt was completed during charging of the battery while the
HPPC test was completed during a discharge, which may account for some of the
difference in $R_{s}$ values. Furthermore, as the HPPC test attempts to
characterize not only the ohmic resistance $R_{s}$ but also the dynamic
resistance, an extended rest period is required, resulting in a test lasting
approximately 30 hours in this case. The charge interrupt test lasted only 3
hours, an order of magnitude reduction in time required to complete the test.

Figure 6: Values of $R_{s}$ for each of the five cells in a battery pack.
Values are taken from charge interrupt test for all thirteen pack analyzed
with the fresh pack denoted by A. Cells 1 and 5 are on the exterior of the
pack while cells 2, 3, and 4 are on the interior.

Furthermore, the $R_{s}$ values are relatively constant at mid and high values
of the cell terminal voltage. While there is a gradual increase at low
voltages, typically around 3.6V or lower, this lower voltage range makes up a
significantly smaller portion of the operating range due to the rapid voltage
drop near low SOC, as shown in Fig. 3. As such, single charge interrupts after
charging to a voltage threshold of approximately 3.6V can provide a rapid
approximation for the $R_{s}$ parameter of a cell.

In Fig. 6, the $R_{s,CI}$ values from the charge interrupt tests are provided
for all five cells in the packs analyzed. The trend of increase $R_{s,CI}$ at
lower is observed for all packs, including the fresh pack denoted as A.
Further, the increase of $R_{s,CI}$ in cells 1 and 5 show a significant
departure from the resistance values measured in cells 2, 3, and 4. However,
referring back to the pack schematic provided in Fig. 1, it is assumed that
the increase in resistance is not entirely due to internal resistance of cells
1 and 5. Rather, the external resistance of the tabs denoted by $B+$ and $B-$
in Fig. 1 may be responsible for a significant portion of the resistance
increase. The interior cells 2, 3, and 4 are isolated from this issue as the
tabs used to measure the terminal voltage are not current carrying. Further
experiments are required to isolate the true internal resistance of cells 1
and 5 from the external tab resistance. All of the following discussion will
focus on the results from cells 2, 3, and 4.

From Fig.5 and 6, pack H has a noticeably higher resistance than most packs,
followed by packs C and G. The other aged packs are more densely populated
around the fresh pack A. As the objective of this study is to analyse the
ability of $R_{s}$ to provide an indication of the capacity, a defining metric
of SOH, these differences in $R_{s}$ are next presented with respect to the
measured capacity of each pack.

###  4.2 Capacity and Series Resistance

Figure 7: Capacity and resistance for each pack for cell 4. Capacity was
measured across a voltage range of 4.1V to 2.95V for C-rate of C/20 and
resistance was measured through the use of charge interrupts.

As outlined in section 3.1, the capacity of the battery packs are calculated
between the voltage limits of 2.95V and 4.1V during the the C/20 discharge in
TS5, which takes upwards of 20 hours for each pack. In Fig. 7, the capacity of
cell 4 is shown with respect to the corresponding ohmic resistance $R_{s,CI}$.
The $R_{s,CI}$ value used from each pack was the sample point closest to a
terminal voltage of 4V. While there is a limited sample size provided for this
comparison, the overall relation between capacity and $R_{s,CI}$ can initially
be approximated by a linear fit. As mentioned previously, pack H has the
highest measured resistance, and this corresponds to the lowest measured
capacity for all of the cells. Similarly, pack E, L and N have some of the
lowest resistances and highest measured capacities, even higher than that of
the fresh pack. However, pack M displays the second lowest capacity while
having a resistance almost identical to that of fresh pack A, limiting
somewhat the linear relationship between capacity and resistance, along with
pack D.

The proposed linear relationship between $R_{s}$ and capacity can potentially
be explained from a lithium inventory loss perspective as detailed by Prasad
and Rahn (2013). The solid electrolyte interphase (SEI) layer is an
electrically insulating film on the negative graphite electrode in a lithium-
ion cell, protecting the electrode from the solvents in the electrolyte. While
this layer is largely permeable to lithium ions, the SEI layer will slowly
consume lithium as the cell cycles and is considered a major factor in
capacity loss for lithium ion batteries. As the SEI layer grows and consumes
more lithium, the SEI thickness increases, increasing the internal resistance
of the cell and causing more power to be lost (Plett (2015)).

A linear fit including all of the packs obtains an $R^{2}$ value of 0.50,
which means 50% of the variance in the data can be explained by the linear
fit. While this is a relatively low metric for the goodness of fit, it should
be noted that none of the aged pack meet the 80% capacity metric typically
considered for classifying automotive battery packs as below the range of the
warranty. Further, if we eliminate pack H from the fit, the general slope of
the line is maintained, as shown in Fig. 7, but the $R^{2}$ value drops to
0.26 due to the limited range of pack capacity. This suggest further analysis
with a wide range of aged cells are required to experimentally substantiate
the capacity-resistance relationship.

##  5 Conclusions

This paper introduces the methodology for a rapid measurement of ohmic
resistance, $R_{s}$, in a lithium-ion battery and discusses the potential
efficacy of using $R_{s}$ as a metric to estimate the remaining capacity of a
cell. From the thirteen battery packs analysed, $R_{s,CI}$ provides an
approximately linear indication of capacitance without the necessity
evaluating the OVC curve with respect to SOC as shown in Fig. 7. Further,
$R_{s,CI}$ remained constant for mid to high range terminal voltage values,
eliminating the need for an exact knowledge of SOC when utilizing $R_{s,CI}$
as a capacity metric.

Further experimentation with more cells, and in particular more heavily aged
cells, is needed to provide a statistical analysis of the $R_{s,CI}$ and
capacity correlation of discarded cells.

The authors would like to acknowledge the support of Paul Hernley from Battery
Solutions, LLC who provided the used battery packs from their recycling
facility.

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