# Weighted cumulative residual Entropy Generating Function and its properties

Smitha S.a, Sudheesh K. Kattumannilb and Sreedevi E. P.c
aK E College Mannanam, Kerala, India,
bIndian Statistical Institute, Chennai, India,
cCochin University of Science and Technology, Kochi, India

###### Abstract.

The study on the generating function approach to entropy become popular as it
generates several well-known entropy measures discussed in the literature. In
this work, we define the weighted cumulative residual entropy generating
function (WCREGF) and study its properties. We then introduce the dynamic
weighted cumulative residual entropy generating function (DWCREGF). It is
shown that the DWCREGF determines the distribution uniquely. We study some
characterization results using the relationship between the DWCREGF and the
hazard rate and/or the mean residual life function. Using a characterization
based on DWCREGF, we develop a new goodness fit test for Rayleigh
distribution. A Monte Carlo simulation study is conducted to evaluate the
proposed test. Finally, the test is illustrated using two real data sets.
Keywords: Entropy; Entropy generating function; Weighted cumulative residual
entropy; Rayleigh distribution; U-statistics.

##  1\. Introduction

The concept of entropy was introduced first by Shannon (1948) for measuring
the uncertainty associated with a random variable. Let $X$ be a continuous
non-negative random variable having distribution function $F(.)$, probability
density function $f(.)$ and survival function $\bar{F}(x)=P(X>x)$. The Shannon
entropy of $X$ is defined as

$H(X)=-\int_{0}^{\infty}\log f(x)f(x)dx=E(-\log f(X)),$ (1)

where “log” denotes the natural logarithm. Different entropy measures have
been introduced in the literature which is suitable for some specific
situations. The widely used measure of entropy is the cumulative residual
entropy (CRE) is given by (Rao et al. 2004)

$\mathcal{E}(X)=-\int_{0}^{\infty}\bar{F}(x)\log\bar{F}(x)dx.$

Di Crescenzo and Longobardi (2009) introduced the cumulative entropy (CE) for
estimating the uncertainty in the past lifetime of a product/subject and is
given by

$\mathcal{CE}(X)=-\int_{0}^{\infty}F(x)\log F(x)dx.$

The weighted versions of $\mathcal{E}(X)$ and $\mathcal{CE}(X)$ have been
studied in the literature as well. These are given by (Mirali et al., 2016)

$\mathcal{E}^{w}(X)=-\int_{0}^{\infty}x\bar{F}(x)\log\bar{F}(x)dx$

and (Mirali and Baratpour, 2017)

$\mathcal{CE}^{w}(X)=-\int_{0}^{\infty}xF(x)\log F(x)dx.$

For some recent development in this area, we refer to Sudheesh et al. (2022),
Balakrishnan et al. (2022) and Chakraborty and Pradhan (2023). Among these,
Sudheesh et al. (2022) defined a generalized cumulative residual entropy and
studied its properties. They show that cumulative residual entropy, weighted
cumulative residual entropy and their weighted versions are special cases of
the proposed measure.

The moment generating function (m.g.f) of a probability distribution is a
convenient tool for evaluating mean, variance and other moments of a
probability distribution. The successive derivative of the m.g.f at a point
zero gives the successive moments of the probability distribution provided
these moments exist. In information theory, generating functions have been
defined for probability density functions to determine information quantities
such as Shannon information, extropy and Kullback-Leibler divergence. Golomb
(1966) introduced the entropy generating of a probability distribution given
by

$B(s)=\int f^{s}(x)dx,\,s>0.$

The first derivative of $B(s)$ at $s=1$, gives the negative of Shannon’s
entropy in (1).

In recent times, there has been a substantial focus on information generating
functions. Clark (2020) proposed an information-generating function applicable
to point processes to compute statistics associated with entropy and relative
entropy. In a series of papers, Kharazmi and Balakrishnan (2021a, 2021b,
2021c) introduced measures such as cumulative residual information generating
and relative cumulative residual information generating, along with cumulative
residual Fisher information and relative cumulative residual Fisher
information measures, and conducted a study on their properties. Specifically,
Kharazmi and Balakrishnan (2021a) introduced two novel divergence measures and
demonstrated that notable information divergence metrics, including Jensen-
Shannon, Jensen-extropy, and Jensen-Taneja, are all specific instances of
these measures. To explore information-generating functions linked to maximum
and minimum ranked sets, as well as record values and their properties, refer
to the works of Zamani et al. (2022) and Kharazmi et al. (2021). Saha and
Kayal (2023) put forth general weighted information and relative information-
generating functions and conducted an in-depth study of their properties.
Capaldo et al.(2023) introduced the cumulative information generating function
and its suitable distortions-based extensions. Recently, Smitha et al. (2023)
proposed a cumulative residual entropy generating function and is given by

$C_{s}(F)=\int_{0}^{\infty}(\bar{F}(x))^{s}dx,\,\,s>0.$ (2)

They also proposed the dynamic version of the $C_{s}(F)$. Motivated by these
recent developments we study the properties of weighted version of $C_{s}(F).$

The rest of the paper is organized as follows. In Section 2, we define the
weighted cumulative residual entropy generating function (WCREGF) and study
some properties. In Section 3, we introduced the dynamic weighted cumulative
residual entropy generating function (DWCREGF). It is shown that the DWCREGF
determines the distribution uniquely. Also, we establish some characterization
results using the relationship of DWCREGF with hazard rate and mean residual
life function. In Section $4$, we propose a test for testing the Rayleigh
distribution using the property of DCWREGF. We conduct a Monte Carlo
Simulation to study the finite sample performance of the proposed test. Some
concluding remarks along with some open problems are given in Section 5.

##  2\. Weighted cumulative residual entropy generating function

The survival function is more useful than the probability density function in
lifetime studies. This motivates us to define WCREGF and study its properties.
Next, we define WCREGF.

######  Definition 1.

Let $X$ be a continuous non-negative random variable having survival function
$\bar{F}(x)$, then WCREGF, denoted by $C_{s}(W,F)$ is defined as

$C_{s}(W,F)=\int_{0}^{\infty}{x}(\bar{F}(x))^{s}dx,\,\,s>0.$ (3)

Differentiating $C_{s}(W,F)$ with respect to $s$, at $s=1$ and putting a
negative sign, we obtain (prime denotes the derivative)

$\left.C_{s}^{\prime}(W,F)\right|_{s=1}=-\int_{0}^{\infty}{x}\bar{F}(x)\log\bar{F}(x)dx,$

The above measure is the weighted cumulative residual entropy introduced by
Mirali et al. (2016).
In the following example, we show that two distributions can possess the same
$C_{s}(F)$ but different $C_{s}(W,F)$, which shows the importance of the
weighted generating function approach to residual entropy.

######  Example 1.

Suppose $C_{s}(F)$ and $C_{s}(G)$ are the CREGF of the random variables $X$
and $Y$, respectively. Suppose the density functions of $X$ and $Y$ are:

$f_{X}(x)=\frac{1}{2},\,0\leq x\leq 2$

and

$f_{Y}(y)=\frac{1}{2},\,2\leq y\leq 4.$

Using (2), we obtain

$C_{s}(X)=\frac{2}{s+1}\quad\text{and}\quad C_{s}(Y)=\frac{2}{s+1}.$

We also find

$C_{s}(W,F)=\frac{4}{(s+1)(s+2)}\quad\text{and}\quad
C_{s}(W,F)=\frac{4(s+3)}{(s+1)(s+2)}.$

We can see that

$C_{s}(F)=C_{s}(G).$

However, the WCREGF of $X$ and $Y$ are not identical. Even though
$C_{s}(F)=C_{s}(G)$, the WCREGF of $X$ is always smaller than that of $Y$ for
all values of $s>0$.

In Table 1, we derive the expression for $C_{s}(W,F)$ for some well-known
lifetime distributions.

Table 1. Expression of $C_{s}(W,F)$ for some well-known distributions Distribution | $\bar{F}(x)$ | $C_{s}(W,F)$
---|---|---
Uniform | $\frac{b-x}{b-a}$ | $\frac{(b-a)(as+a+b)}{(s+1)(s+2)}$
Exponential | $\ e^{-\lambda x}$ | $\frac{1}{{s^{2}}{\lambda^{2}}}$
Pareto | $\frac{1}{x^{\lambda}}$ | $\frac{1}{{\lambda s}-2};{{\lambda s}>2}$
Lomax | $(\frac{m}{m+x})^{n}$ | $\frac{m^{2}}{(2-ns)(1-ns)}$
Rayleigh | $\ e^{\frac{-x^{2}}{2{\sigma}^{2}}}$ | $\frac{\sigma^{2}}{s}$

Next, we study some properties of $C_{s}(W,F)$. Suppose, for the random
variable $X$ having distribution function $F$, let $Y=aX+b$ with $a>0$ and
$b\geq 0$, then we can observe that $C_{s}(W,Y)$ is the sum of ${a^{2}}$ times
WCREGF of $X$ and $ab$ times CREGF of $X$. That is

$C_{s}(W,Y)={a^{2}}C_{s}(W,F)+abC_{s}(X).$

The proportional hazards model (Cox, 1972) estimates the effects of different
covariates influencing the failure time of a system/subject. For more details
on the theory and applications of the proportional hazards model, one may
refer to Kalbfleisch and Prentice (2002). Let $X^{*}$ be a continuous non-
negative random variable with survival function (proportional hazards model)

$\bar{F}^{*}(t)=[\bar{F}(t)]^{\theta},\,\theta>1,$

where $\bar{F}^{*}$ is the survival function of $X^{*}$. We can observe that
the WCREGF of $X$ and $X^{*}$ are related by

$C_{s}(W,F^{*})=C_{s\theta}(W,F).$

For a series system with $n$ independent and identical components each having
lifetime $X_{i},\,i=1,\ldots,n$, the lifetime of the system is given by
$\min(X_{1},\ldots,X_{n})$. Denote $\bar{F}_{s}$ is the survival function of
the random variable $\min(X_{1},\ldots,X_{n})$. Let $X_{(1)}$ be the first
order statistic based on a random sample $X_{1},...X_{n}$ from F. We have
$\bar{F}_{s}(x)=[\bar{F}(x)]^{n}$, then

$C_{s}(W,F_{s})=C_{ns}(W,F).$

In the following theorem, we obtain the relationship between $C_{s}(W,F)$ and
Shannon’s entropy measure.

######  Theorem 1.

Let $X$ be continuous non-negative random variable having Shannon’s entropy
$H(X)$ in (1), then

$C_{s}(W,F)\geq\exp[H(X)+E(logX)-s].$

Proof: By using log-sum inequality, we obtain

$\displaystyle\int_{0}^{\infty}f(x)\log\left(\frac{f(x)}{x(\bar{F}(x))^{s}}\right)dx$
$\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}f(x)\log\left(\frac{\int_{0}^{\infty}f(x)dx}{\int_{0}^{\infty}x(\bar{F}(x))^{s}dx}\right)dx.$
$\displaystyle\geq$
$\displaystyle-\log\left(\int_{0}^{\infty}x(\bar{F}(x))^{s}dx\right)$
$\displaystyle=$ $\displaystyle-\log(C_{s}(W,F)).$

From the above expression, we have the inequality stated in the theorem.

##  3\. Dynamic weighted cumulative residual entropy generating function

In various fields such as reliability, survival analysis, economics, business,
etc., the duration of a study period is a significant variable. In these
instances, information-generating functions become dynamic, incorporating a
time dependency. Inspired by the dynamic nature associated with time-dependent
functions, this section introduces the dynamic version of WCREGF, which is
known as the dynamic weighted cumulative residual entropy generating function
(DWCREGF). The DWCREGF is defined as

$C_{s}(W,X;t)=\int_{t}^{\infty}x\left(\frac{\bar{F}(x)}{\bar{F}(t)}\right)^{s}dx,\,s>0.$

######  Remark 1.

Clearly $C_{s}(W,X;0)=C_{s}(W,F)$, which is WCREGF.

In Table 2, we give the expression for $C_{s}(W,F,t)$ for some well-known
lifetime distributions.

Table 2. Expression of $C_{s}(W,F,t)$ for some well-known distributions Distribution | $\bar{F}(x)$ | $C_{s}(W,F,t)$
---|---|---
Uniform | $\frac{b-x}{b-a}$ | $\frac{(b-t)[t(s+2)+(b-t)]}{(s+1)(s+2)}$
Exponential | $\ e^{-\lambda x}$ | $\frac{(1+\lambda st)}{(s\lambda)^{2}}$
Pareto | $\frac{1}{x^{\lambda}}$ | $\frac{t^{2}}{{\lambda s}-2};{{\lambda s}>2}$
Lomax | $(\frac{m}{m+x})^{n}$ | $\frac{(m+t)}{(ns-2)}[t+\frac{(m+t)}{(ns-2)}];ns-2>0$
Rayleigh | $\ e^{\frac{-x^{2}}{2{\sigma}^{2}}}$ | $\frac{\sigma^{2}}{s}$

######  Definition 2.

Let $X$ be a non-negative random variable with survival function $\bar{F}(x)$,
then the weighted mean residual lifetime (WMRL) is defined as

$m_{F}(W;t)=\int_{t}^{\infty}x\frac{\bar{F}(x)}{\bar{F}(t)}dx.$

In particular $m_{F}(W;0)=\int_{0}^{\infty}x\bar{F}(x)dx=\frac{1}{2}E(X^{2}).$

######  Theorem 2.

Let $X$ be a non-negative random variable and let $C_{s}(W,F)<\infty,$ then

$C_{s}(W,F)<m_{F}(W;0),s>0.$

Proof: The result follows from the definition of $C_{s}(W,F)$ and using the
inequality $(\bar{F}(x))^{s}<\bar{F}(t)$, $t>0,s>0$.

######  Theorem 3.

For a non-negative random variable $X$ with survival function $\bar{F}(x)$,
then

$C_{s}(W,X;t)\leq m_{F}(W;t),s>0.$

Proof: For all $x\geq t$, we have $\bar{F}(x)\leq\bar{F}(t)$. Hence using the
definitions of $C_{s}(W,X;t)$ and $m_{F}(W;t)$, we obtain the desired result.

Next, we obtain the relationship between hazard rate and DWCREGF. By
differentiating $C_{s}(W,X;t)$ with respect to $t$, we obtain the following
result:

$h(t)=\frac{t+C^{\prime}_{s}(W,X;t)}{s.C_{s}(W,X;t)}.$ (4)

The following theorem shows that the dynamic weighted cumulative residual
entropy generating function determines the distribution of $X$ uniquely.

######  Theorem 4.

Let $X$ be non-negative random variable with density function $f(x)$, the
survival function $\bar{F}(x)$ and the hazard rate $h(x)$. Then $C_{s}(W,X;t)$
uniquely determines the distribution of $X$.

Proof: Using the relationship between hazard rate and dynamic weighted
cumulative residual entropy generating function in (4), we can write

$\frac{-d\log\bar{F}(t)}{dt}=\frac{t+C^{\prime}_{s}(W,X;t)}{s.C_{s}(W,X;t)}.$

Integrating for $t$ over the interval $(0,x)$, we have

$\bar{F}(x)=\exp\left[{-\int_{0}^{x}\frac{t+C^{\prime}_{s}(W,X;t)}{s.C_{s}(W,X;t)}}dt\right].$

This shows that the knowledge of $C_{s}(W,X;t)$ enables us to determine the
distribution of $X$. Now, suppose that $F(.)$ and $G(.)$ be two distribution
functions such that

$C{s}(W,F;t)=C_{s}(W,G;t).$ (5)

Differentiating both side of the above equation with respect to $t$, we obtain

$-t+sh_{1}(t)C_{s}(W,F;t)=-t+sh_{2}(t)C_{s}(W,G;t),$

where $h_{1}(t)$ and $h_{2}(t)$ are hazard rates corresponding to $F$ and $G$,
respectively.
In view of equation (5), we have $h_{1}(t)=h_{2}(t)$. This implies that
$C_{s}(W,F;t)$ determines the distribution of $X$ uniquely.

The next theorem shows that the dynamic weighted residual entropy generating
function is independent of $t$ if and only $X$ has Rayleigh distribution.

######  Theorem 5.

If $X$ is a non-negative continuous random variable having distribution
function $F(x)$, then the dynamic weighted cumulative residual entropy
generating function is independent of $t$ if and only if $X$ has Rayleigh
distribution.

Proof. Let $C_{s}(W,F;t)=k$, where $k$ is a positive constant. Then

$C_{s}^{\prime}(W,F;t)=0.$

From Equation (4), we obtain

$sk.h(t)=t.$

Or

$h(t)=\frac{t}{sk}=at,$

where $a$ is a constant. And $h(t)$ is the hazard rate corresponding to the
Rayleigh distribution.

Conversely assume that, $X$ has Rayleigh distribution where the survival
function is given by

$\bar{F}(x)=\exp(\frac{-x^{2}}{2b^{2}}).$

Therefore,

$\displaystyle C_{s}(W,F;t)$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\frac{-st^{2}}{2b^{2}}}}\int_{t}^{\infty}x{e^{\frac{-sx^{2}}{2b^{2}}}}dx$
$\displaystyle=$ $\displaystyle\frac{b^{2}}{s}=k,$

which is a constant. Accordingly, WDCREGF is independent of $t$ if and only if
$X$ has Rayleigh distribution.

We use this characterization property to develop goodness of fit test for
Rayleigh distribution.

######  Theorem 6.

If $m_{F}(W;t)$ is the weighted mean residual life of $X$, then the relation

$sC_{s}(W,X;t)=m_{F}(W;t)$ (6)

holds if and only if $X$ has the survival function
$\bar{F}(t)=\exp(\frac{-t^{2}}{2b^{2}})$.

Proof: Let $X$ has Rayleigh distribution with survival function

$\bar{F}(t)=\exp(\frac{-t^{2}}{2b^{2}}).$

Then $m_{F}(W;t)=b^{2}$ and $C_{s}(W,X;t)=\frac{b^{2}}{s}$ and the if part of
the theorem holds.
To prove the converse, assume that equation (6) holds. Differentiating both
sides of equation (6) with respect to $t$, and using the definition of the
hazard rate, we obtain

$h(t).m_{F}(W;t)=t.$ (7)

But

$\frac{d}{dt}m_{F}(W;t)=h(t).m_{F}(W;t)-t$

Therefore, using the above two equations, we obtain

$\frac{d}{dt}m_{F}(W;t)=0.$

Hence $m_{F}(W;t)$ is constant. Therefore equation (7) becomes

$h(t)=\frac{t}{k}.$

This gives

$\bar{F}(x)=\exp(\frac{-t^{2}}{2k}),$

which is the survival function of Rayleigh distribution and hence the proof of
the theorem.

##  4\. Test for Rayleigh distribution

In the previous section, we proved that the WDCREGF characterizes the
distribution of $X$. Also, we proved that constant WDCREGF is a
characterization property of Rayleigh distribution. We develop a test for
Rayleigh distribution against the decreasing DCREGF class.

Let $X_{1},\ldots X_{n}$ be a random sample of size $n$ from $F$. We are
interested in testing the null hypothesis

$H_{0}:X\text{ has Rayleigh distribution}$

against the alternative hypothesi

$H_{1}:X\text{ has decreasing DCREGF and not Rayleigh.}$

For testing the above hypothesis first we define a departure measure that
discriminates between null and alternative hypotheses. Note that
$C_{s}(W,X;t)$ is decreasing in $t$ if $C_{s}^{{}^{\prime}}(W,X;t)\leq 0.$
That is,

$\frac{sf(t)}{\bar{F}^{s+1}(t)}\int_{t}^{\infty}x\bar{F}^{s}(x)dx-t\leq 0$

or

${t\bar{F}^{s+1}(t)}-{sf(t)}\int_{t}^{\infty}x\bar{F}^{s}(x)dx\geq 0.$

Hence, we consider a measure of departure $\Delta(F)$ given by

$\displaystyle\Delta(F)=\int_{0}^{\infty}\left({t\bar{F}^{s+1}(t)}-{sf(t)}\int_{t}^{\infty}x\bar{F}^{s}(x)dx\right)dt.$
(8)

Clearly, $\Delta(F)$ is zero under $H_{0}$ and positive under $H_{1}$.
Accordingly, $\Delta(F)$ can be considered as a measure of departure from
$H_{0}$ towards $H_{1}$. As the proposed test is based on U-statistics, first
we express $\Delta(F)$ in terms of expectation of the function of random
variables. Observe that $\bar{F}^{n}(x)$ is the survival function of
$\min(X_{1},\ldots,X_{n})$. For a non-negative random variable
$E(\min(X_{1},\ldots,X_{n}))=\int_{0}^{\infty}\bar{F}^{n}(x)dx$ and
$E(\min(X_{1},\ldots,X_{n})^{2})=\int_{0}^{\infty}2x\bar{F}^{n}(x)dx.$
Consider

$\displaystyle\Delta(F)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left({t\bar{F}^{s+1}(t)}-{sf(t)}\int_{t}^{\infty}x\bar{F}^{s}(x)dx\right)dt$
(9) $\displaystyle=$
$\displaystyle\frac{1}{2}E(\min(X_{1},\ldots,X_{s+1})^{2})-\int_{0}^{\infty}{sf(t)}\int_{t}^{\infty}x\bar{F}^{s}(x)dxdt.$

Changing the order of integration, from (9) we have

$\displaystyle\Delta(F)$ $\displaystyle=$
$\displaystyle\frac{1}{2}E(\min(X_{1},\ldots,X_{s+1})^{2})-s\int_{0}^{\infty}x\bar{F}^{s}(x)\int_{0}^{x}f(t)dtdx$
$\displaystyle=$
$\displaystyle\frac{1}{2}E(\min(X_{1},\ldots,X_{s+1})^{2})-\frac{s}{2}\int_{0}^{\infty}x2\bar{F}^{s}(x)(1-\bar{F}(x))dx$
$\displaystyle=$
$\displaystyle\frac{(s+1)}{2}E(\min(X_{1},\ldots,X_{s+1}){{}^{2}})-\frac{s}{2}E(\min(X_{1},\ldots,X_{s})^{2}).$

We find the test statistic using the theory of U-statistics. Consider a
symmetric kernel

$\displaystyle
h_{1}(X_{1},\ldots,X_{s+1})=\frac{(s+1)}{2}\min(X_{1},\ldots,X_{s+1})^{2}-\frac{1}{2(s+1)}\sum_{C_{s}}s\min(X_{i_{1}},\ldots,X_{i_{s}})^{2},$

where summation is over the set $C_{s}$ of all combination of $s$ integers
$i_{1}<i_{2}<\ldots<i_{s}$ chosen from the set $(1,\ldots,s+1)$. Then
$E(h_{1}(X_{1},\ldots,X_{s+1}))=\Delta(F)$. Hence a U-statistic based test
statistic is given by

$\widehat{\Delta}=\frac{1}{C_{m,n}}\sum\limits_{C_{m,n}}h{(X_{i_{1}},X_{i_{2}},\cdots,X_{i_{s+1}})},$

where the summations is over the set $C_{m,n}$ of all combinations of $(s+1)$
distinct elements $\\{i_{1},i_{2},\cdots,i_{s+1}\\}$ chosen from
$\\{1,2,\cdots,n\\}$. We reject the null hypothesis $H_{0}$ against the
alternative $H_{1}$ for large value of $\widehat{\Delta}$. We obtain a
critical region of the proposed test using the asymptotic distribution of
$\widehat{\Delta}$.

In the next theorem, we state the asymptotic distribution of
$\widehat{\Delta}$.

######  Theorem 7.

As $n\rightarrow\infty$, $\sqrt{n}(\widehat{\Delta}-\Delta(F))$ converges in
distribution to normal random variable with mean zero and variance
$(s+1)^{2}\sigma^{2}$, where $\sigma^{2}$ is given by

$\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}Var\Big{(}(s+1)X^{2}{{\bar{F}}^{s}}(X)+s(s+1)\int_{0}^{X}{{{y^{2}\bar{F}}^{s-1}}(y)}d{F}(y)$
(10)
$\displaystyle\quad-\frac{s^{2}}{(s+1)}X^{2}{{\bar{F}}^{s-1}}(X)-\frac{(s-1)s^{2}}{(s+1)}\int_{0}^{X}{{{y^{2}\bar{F}}^{s-2}}(y)}d{F}(y)\Big{)}.$

Proof: By the central limit theorem for U-statistics, asymptotic distribution
of $\widehat{\Delta}$ is normal with mean $\Delta(F)$. The asymptotic variance
is $(s+1)^{2}\sigma_{1}^{2}$ where $\sigma_{1}^{2}$ is given by (Lee, 2019)

$\sigma_{1}^{2}=Var\left[E\left(h(X_{1},\ldots,X_{(s+1)})|X_{1}\right)\right].$
(11)

Denote $Z=\min(X_{2},X_{3},...,X_{s})$, then the distribution of $Z$ is given
by $1-\bar{F}^{s-1}(x)$, where $\bar{F}(x)=1-F(x)$. Consider

$\displaystyle
E\left[{\min\left({x,{X_{2}},{X_{3}},...,{X_{s}}}\right)}\right]^{2}$
$\displaystyle=$ $\displaystyle
E\left[{x^{2}I(Z>x)}\right]+E\left[{Z^{2}I(Z\leq x)}\right]$ $\displaystyle=$
$\displaystyle
x^{2}{{\bar{F}}^{s-1}}(x)+(s-1)\int_{0}^{x}{{{y^{2}\bar{F}}^{s-2}}(y)}d{F}(y).$

Similarly, we have

$\displaystyle
E\left[{\min\left({x,{X_{2}},{X_{3}},...,{X_{s+1}}}\right)}\right]^{2}$
$\displaystyle=$ $\displaystyle
x^{2}{{\bar{F}}^{s}}(x)+s\int_{0}^{x}{{{y^{2}\bar{F}}^{s-1}}(y)}d{F}(y).$

Hence

$\displaystyle\hskip-72.26999ptE\left(h\big{(}{{X_{1}},{X_{2}},...,{X_{s}}}|X_{1}=x\big{)}\right)$
$\displaystyle=$
$\displaystyle(s+1)x^{2}{{\bar{F}}^{s}}(x)+s(s+1)\int_{0}^{x}{{{y^{2}\bar{F}}^{s-1}}(y)}d{F}(y)$
$\displaystyle-\frac{s^{2}}{(s+1)}x^{2}{{\bar{F}}^{s-1}}(x)-\frac{(s-1)s^{2}}{(s+1)}\int_{0}^{x}{{{y^{2}\bar{F}}^{s-2}}(y)}d{F}(y)+\frac{sk}{2(s+1)},$

where $k=E[\min(X_{2},\ldots,X_{s+1})]^{2}$, a constant. Therefore, from (11)
we obtain the variance expression specified in the statement of the theorem.

Under the null hypothesis $H_{0}$, $\Delta{(F)}=0$. Hence we have the
following corollary.

######  Corollary 1.

Under $H_{0}$, as $n\rightarrow\infty$, $\sqrt{n}\widehat{\Delta}$ converges
in distribution to a Gaussian random variable with mean zero and variance
$\sigma_{0}^{2}$, where $\sigma_{0}^{2}$ is the value of the $\sigma^{2}$
evaluated under $H_{0}$.

An asymptotic critical region of the test can be obtained using Corollary 1.
Let $\widehat{\sigma}_{0}^{2}$ be a consistent estimator of the asymptotic
null variance $\sigma_{0}^{2}$. We reject the null hypothesis $H_{0}$ against
the alternative hypothesis $H_{1}$ at an approximate significance level
$\alpha$, if

$\frac{\sqrt{n}\widehat{\Delta}}{{\widehat{\sigma}_{0}}}>Z_{\alpha},$

where $Z_{\alpha}$ is the upper $\alpha$-percentile point of the standard
normal distribution.

##  5\. Simulation Studies

We conduct an extensive Monte Carlo simulation study to assess the finite
sample performance of the proposed test using the empirical type I error and
power of the test. The simulation is done using the R software and repeated
ten thousand times using different sample sizes.

The empirical type I error and power of test are compared with the classical
goodness of fit tests including the Anderson Darling (AD) test, Cramer von
Mises (CvM) test and Kolmogrv-Sminrnov (KS) test and other goodness of fit
tests, specifically developed for Rayleigh distribution. The proposed test can
be performed with different choices of the parameter $s.$ In simulation
studies, we observe that the performance of the proposed test is optimum for
the choice $s=1.$ Hence for the comparison purpose with other competent tests,
we choose $s=1.$ We generate lifetimes of different sample sizes,
($n=10,20,30,40,50$) to calculate the empirical type I error and power of the
proposed test and the other competent tests.

Finding a consistent estimator of the null variance $\sigma_{0}^{2}$ is
difficult. Hence we find the critical region of the proposed test using Monte
Carlo simulation. We determine $\alpha$-th quantiles $c_{1}$ in such a way
that $P(\widehat{\Delta}>c_{1})=\alpha$. The algorithm used to find the
empirical power of the proposed test is summarised as follows.

  1. (1)

Generate lifetime data from the desired alternative and calculate the value of
$\widehat{\Delta}$.

  2. (2)

Generate lifetime data from the Rayleigh distribution and calculate the value
of $\widehat{\Delta}$.

  3. (3)

Repeat Step 2 10000 times and determine the critical point.

  4. (4)

Repeat Steps 1-3 10000 times and calculate empirical power as the proportion
of significant test statistics.

Table 3.  Comparison of empirical power for different alternatives ($\alpha=0.01$)  | $n$ | SSS | KS | CvM | AD | AH | MI | JH | VTS
---|---|---|---|---|---|---|---|---|---
Rayleigh(1) | 10 | 0.0121 | 0.0175 | 0.0193 | 0.0119 | 0.0114 | 0.0118 | 0.0086 | 0.0115
| 20 | 0.0115 | 0.0142 | 0.0161 | 0.0110 | 0.0109 | 0.0112 | 0.0090 | 0.0111
| 30 | 0.0110 | 0.0128 | 0.0127 | 0.0105 | 0.0092 | 0.0107 | 0.0108 | 0.0106
| 40 | 0.0094 | 0.0116 | 0.0119 | 0.0095 | 0.0095 | 0.0094 | 0.0105 | 0.0095
| 50 | 0.0096 | 0.0111 | 0.0114 | 0.0102 | 0.0104 | 0.0096 | 0.0105 | 0.0098
Weibull (2, 2) | 10 | 0.3589 | 0.2456 | 0.1892 | 0.5673 | 0.1298 | 0.3974 | 0.1988 | 0.2387
| 20 | 0.6586 | 0.5784 | 0.5181 | 0.7172 | 0.3912 | 0.6313 | 0.2987 | 0.5218
| 30 | 0.8287 | 0.8241 | 0.8293 | 0.8518 | 0.4394 | 0.7052 | 0.4688 | 0.6959
| 40 | 0.9461 | 0.9396 | 0.9422 | 0.9219 | 0.5865 | 0.7366 | 0.6551 | 0.8277
| 50 | 0.9869 | 0.9797 | 0.9835 | 0.9723 | 0.7893 | 0.8248 | 0.8102 | 0.8978
Pareto (2,2) | 10 | 0.9237 | 0.9704 | 1.0000 | 0.9968 | 0.2714 | 0.5288 | 0.2981 | 0.5593
| 20 | 0.9692 | 1.0000 | 1.0000 | 1.0000 | 0.3892 | 0.5620 | 0.6118 | 0.7785
| 30 | 0.9814 | 1.0000 | 1.0000 | 1.0000 | 0.5998 | 0.8229 | 0.8367 | 0.8387
| 40 | 0.9843 | 1.0000 | 1.0000 | 1.0000 | 0.7461 | 0.8736 | 0.8848 | 0.9166
| 50 | 0.9885 | 1.0000 | 1.0000 | 1.0000 | 0.8823 | 0.9159 | 0.9285 | 0.9459
Log-normal (1,1) | 10 | 0.6927 | 0.9755 | 0.9812 | 0.9854 | 0.3414 | 0.3563 | 0.2856 | 0.5470
| 20 | 0.9418 | 1.0000 | 1.0000 | 1.0000 | 0.6534 | 0.6386 | 0.4196 | 0.8733
| 30 | 0.9816 | 1.0000 | 1.0000 | 1.0000 | 0.8275 | 0.8042 | 0.6954 | 0.9671
| 40 | 0.9924 | 1.0000 | 1.0000 | 1.0000 | 0.9266 | 0.9151 | 0.8566 | 0.9935
| 50 | 0.9962 | 1.0000 | 1.0000 | 1.0000 | 0.9584 | 0.9554 | 0.9532 | 0.9985
Half-normal (0.5) | 10 | 0.3727 | 0.5036 | 0.6457 | 0.5854 | 0.1987 | 0.2795 | 0.1643 | 0.3654
| 20 | 0.6463 | 0.8925 | 0.9382 | 0.8361 | 0.3936 | 0.3317 | 0.2666 | 0.5646
| 30 | 0.8241 | 0.9831 | 0.9934 | 0.9453 | 0.5404 | 0.4218 | 0.4097 | 0.7438
| 40 | 0.9102 | 0.9980 | 0.9994 | 0.9842 | 0.7415 | 0.5762 | 0.5924 | 0.8589
| 50 | 0.9598 | 0.9995 | 1.0000 | 0.9928 | 0.7845 | 0.6517 | 0.7215 | 0.9283
LFR (0.5) | 10 | 0.8779 | 0.7316 | 0.8175 | 0.7597 | 0.1911 | 0.1939 | 0.2841 | 0.7658
| 20 | 0.9956 | 0.9874 | 0.9952 | 0.9635 | 0.2625 | 0.3543 | 0.3184 | 0.8279
| 30 | 1.0000 | 1.0000 | 1.0000 | 0.9898 | 0.5421 | 0.4145 | 0.6478 | 0.8811
| 40 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.7132 | 0.6545 | 0.7933 | 0.9221
| 50 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.7818 | 0.7561 | 0.8882 | 0.9512
Table 4. Comparison of empirical power for different alternatives ($\alpha=0.05$) | $n$ | SSS | KS | CvM | AD | AH | MI | JH | VTS
---|---|---|---|---|---|---|---|---|---
Rayleigh (1) | 10 | 0.0479 | 0.0563 | 0.0575 | 0.0529 | 0.0442 | 0.0525 | 0.0521 | 0.0532
| 20 | 0.0488 | 0.0551 | 0.0561 | 0.0518 | 0.0471 | 0.0518 | 0.0516 | 0.0511
| 30 | 0.0512 | 0.0526 | 0.0531 | 0.0478 | 0.0482 | 0.0487 | 0.0515 | 0.0509
| 40 | 0.0509 | 0.0512 | 0.0519 | 0.0489 | 0.0484 | 0.0492 | 0.0478 | 0.0489
| 50 | 0.0503 | 0.0491 | 0.0489 | 0.0490 | 0.0489 | 0.0505 | 0.0494 | 0.0495
Weibull (2, 2) | 10 | 0.5758 | 0.3204 | 0.2616 | 0.6567 | 0.2041 | 0.4197 | 0.2292 | 0.3528
| 20 | 0.8399 | 0.6178 | 0.5718 | 0.7728 | 0.4109 | 0.6731 | 0.3206 | 0.6521
| 30 | 0.9393 | 0.8424 | 0.8629 | 0.8846 | 0.5199 | 0.7405 | 0.5368 | 0.7695
| 40 | 0.9771 | 0.9539 | 0.9642 | 0.9423 | 0.286 | 0.7836 | 0.6904 | 0.8921
| 50 | 0.9935 | 0.9979 | 0.9983 | 0.9972 | 0.8008 | 0.8824 | 0.8567 | 0.9487
Pareto (2, 2) | 10 | 0.9395 | 1.0000 | 1.0000 | 1.0000 | 0.3109 | 0.6351 | 0.3326 | 0.6571
| 20 | 0.9704 | 1.0000 | 1.0000 | 1.0000 | 0.4257 | 0.7512 | 0.6596 | 0.8043
| 30 | 0.9904 | 1.0000 | 1.0000 | 1.0000 | 0.6323 | 0.7599 | 0.8672 | 0.8976
| 40 | 0.9984 | 1.0000 | 1.0000 | 1.0000 | 0.8256 | 0.8226 | 0.9098 | 0.9419
| 50 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9382 | 0.8641 | 0.9695 | 0.9763
Log-normal (1,1) | 10 | 0.7438 | 0.9934 | 0.9947 | 0.9951 | 0.5675 | 0.4399 | 0.3934 | 0.7367
| 20 | 0.9511 | 1.0000 | 1.0000 | 1.0000 | 0.7763 | 0.6845 | 0.6445 | 0.9261
| 30 | 0.9891 | 1.0000 | 1.0000 | 1.0000 | 0.8490 | 0.8516 | 0.8218 | 0.9853
| 40 | 0.9972 | 1.0000 | 1.0000 | 1.0000 | 0.9365 | 0.9494 | 0.9051 | 0.9976
| 50 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9876 | 0.9794 | 0.9739 | 1.0000
Half-normal (0.5) | 10 | 0.5075 | 0.7254 | 0.7863 | 0.7867 | 0.3291 | 0.3214 | 0.2804 | 0.4712
| 20 | 0.7888 | 0.9569 | 0.9684 | 0.9608 | 0.4505 | 0.4379 | 0.3898 | 0.6529
| 30 | 0.9073 | 0.9971 | 0.9988 | 0.9895 | 0.6446 | 0.5355 | 0.5378 | 0.7986
| 40 | 0.9545 | 0.9998 | 1.0000 | 0.9998 | 0.7935 | 0.6472 | 0.6489 | 0.8921
| 50 | 0.9855 | 1.0000 | 1.0000 | 1.0000 | 0.8528 | 0.7828 | 0.7942 | 0.9883
LFR (0.5) | 10 | 0.9521 | 0.9097 | 0.9373 | 0.9584 | 0.2871 | 0.2448 | 0.3109 | 0.8201
| 20 | 0.9998 | 0.9958 | 0.9982 | 0.9972 | 0.4291 | 0.5348 | 0.5621 | 0.8827
| 30 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.6532 | 0.6689 | 0.7818 | 0.9237
| 40 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.7881 | 0.7387 | 0.8356 | 0.9459
| 50 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.8289 | 0.8636 | 0.8902 | 0.9752

To find the empirical power, lifetime random variables are generated from
different choices of alternatives including Weibull, log-normal, Pareto, half-
normal and linear failure rate (LFR) distributions, where the distribution
functions are:

  * •

Weibull distribution: $F_{1}(x)=1-e^{(-x/\lambda)^{k}}$, $x>0$,
$k,\,\lambda>0$.

  * •

Pareto distributions: $F_{2}(x)=(\lambda/x)^{\alpha}$ $x>0$,
$\alpha,\,\lambda>0$.

  * •

Lognormal distribution: $F_{3}(x)=\Phi(\frac{\ln x-\mu}{\sigma})$, $x>0$,
$-\infty<\mu<\infty,\,\sigma^{2}>0$ where $\Phi(x)$ is the cumulative
distribution function of the standard normal random variable.

  * •

Half-normal distribution: $F_{4}(x)=erf(\frac{x}{\sigma\sqrt{2}})$,
$\sigma>0$, where erf is the error function.

  * •

LFR distribution: $F_{5}(x)=1-\exp(-\frac{-\lambda x^{2}}{2})$ $x\geq 0$,
$\lambda>0$.

We compare the performance of our test (denoted by SSS) with goodness of fit
test for Rayleigh distribution proposed by Ahrari et al. (2022) (AH),
Meintanis and Illipoulos (2003) (MI), Jahanshahi et al. (2016) (JH) and
Vaishakh et al. (2022) (VTS) as well as with the classical tests Kolmogorov
Smirnov (KS), Cramer von Mises (CvM) and Anderson-Darling (AD) test.

We report the results of the empirical power comparison in Tables 1 and 2.
Table 1 gives the power of $\widehat{\Delta}$ for different alternatives at
significance level $\alpha=0.01$ and Table 2 presents the same when
$\alpha=0.05$. It is evident from Tables 1 and 2 that, the empirical type I
error of the test approaches the chosen significance level as sample size
increases. Also, the proposed test exhibits good power against all choices of
alternatives which increases with sample size. The results from the simulation
study ensure that the proposed test is superior to other competent tests.

##  6\. Data Analysis

We illustrate the proposed testing procedure with two real data sets. To find
the p-values, we use the following parametric bootstrap procedure.

  1. (1)

Obtain the moment estimate of the parameter of Rayleigh distribution from the
observed data.

  2. (2)

Generate 10000 bootstrap samples from Rayleigh distribution having parameter
estimated in Step 1.

  3. (3)

Calculate the test static value for each sample generated in the previous
step.

  4. (4)

Calculate the bootstrap p-value as the proportion of the test static value
calculated in Step 3 that is greater than the test static estimated for the
original data.

Data 1 : We analyze the data, given in Caroni (2002), which represents the
failure times of 23 ball bearings, to illustrate the proposed test. These
failure times are: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.48, 51.84,
51.96, 54.12, 55.56, 67.80, 67.80, 67.80, 68.88, 84.12, 93.12, 98.64, 105.12,
105.84, 127.92, 128.04, 173.40. This data set originally used by Lieblein and
Zelen (1956), is also discussed in Caroni (2002). The observations are the
number of million revolutions before failure for each 23 ball bearings ordered
according to life endurance. Later, the data is analysed by many authors
including Kim and Han (2009), Dey and Dey (2014) and Vaisakh et al. (2023) in
studying Rayleigh distribution. The bootstrap p-value, using the above-
described computational algorithm is obtained as 0.6425. Accordingly, we can
conclude that, the hypothesis that the data follows Rayleigh distribution can
be accepted for this data. We can note that this result agrees with the
results from previous studies on Rayleigh distribution.

Data 2 : To illustrate the performance of the proposed test, we consider the
data on survival times in weeks for 20 male rats that were exposed to a high
level of radiation. The data are due to Furth et al. (1959) and have been
studied by Lawless (2003). The complete data is given in Example 4.2.1 (Page
168) of Lawless (2003). The bootstrap p-value is obtained as 0.0152. Hence we
reject the null hypothesis that the data follows Rayleigh distribution at a 5%
level of significance. Vaisakh et al. (2023) showed that survival time data
follow a gamma distribution.

##  7\. Conclusion

The generating function approach to entropy become popular in recent times. In
this paper, we defined the weighted cumulative residual entropy generating
function and studied its properties. The dynamic version of the proposed
generating function is also given. We proved that the dynamic weighted
cumulative residual entropy generating function (DWCREGF) determines the
distribution uniquely. We proved that the constant DWCREGF characterizes the
Rayleigh distribution. Using this characterization result, we developed a
goodness of fit test for Rayleigh distribution and studied its properties. A
Monte Carlo simulation study showed that the proposed test has a well-
controlled error rate and has good power for various alternatives. The
proposed test is illustrated using two real data sets.

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# Improving Query-Focused Meeting Summarization with
Query-Relevant Knowledge

Tiezheng Yu , Ziwei Ji, Pascale Fung
Center for Artificial Intelligence Research (CAiRE)
Department of Electronic and Computer Engineering
The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Query-Focused Meeting Summarization (QFMS) aims to generate a summary of a
given meeting transcript conditioned upon a query. The main challenges for
QFMS are the long input text length and sparse query-relevant information in
the meeting transcript. In this paper, we propose a knowledge-enhanced two-
stage framework called Knowledge-Aware Summarizer (KAS) to tackle the
challenges. In the first stage, we introduce knowledge-aware scores to improve
the query-relevant segment extraction. In the second stage, we incorporate
query-relevant knowledge in the summary generation. Experimental results on
the QMSum dataset show that our approach achieves state-of-the-art
performance. Further analysis proves the competency of our methods in
generating relevant and faithful summaries.

##  1 Introduction

Meetings are an essential part of human collaboration and communication.
Especially in recent years, the outbreak of Covid-19 has led people to meet
online, where most meetings are automatically recorded and transcribed. Query-
Focused Meeting Summarization (QFMS) Zhong et al. (2021) aims to summarize the
given meeting transcript conditioned upon a query, which helps people
efficiently catch up to the specific part of the meeting they want to know.

There are two main challenges for QFMS: Firstly, meeting transcripts can be so
long that current deep learning models cannot encode them at once. Even for
models Beltagy et al. (2020); Xiong et al. (2022) that accept long text input,
the cost of computational complexity is enormous. Secondly, query-relevant
content is sparsely scattered in the meeting transcripts, meaning a
significant part of the transcripts is noisy information when given a
particular query. Therefore, the models need to reduce the noisy information’s
impact effectively.

In this paper, we focus on the two-stage framework, extracting query-relevant
segments from the meeting transcripts and then generating the summary based on
the selected content. Compared to the end-to-end approaches Zhu et al. (2020);
Pagnoni et al. (2022) that directly encode the entire meeting transcripts, the
two-stage framework is better at keeping computational efficiency and easier
to scale up to longer inputs. Specifically, we propose Knowledge-Aware
Summarizer (KAS) that incorporates query-relevant knowledge in both stages. In
the first stage, we extract knowledge triples from the text segments and
introduce knowledge-aware scores to improve segment ranking. In the second
stage, the extracted query-relevant knowledge triples are utilized as extra
input for the summary generation. We conduct experiments on a QFMS dataset
named QMSum Zhong et al. (2021) and achieve state-of-the-art performance. We
further investigate how the different numbers of extracted segments affect the
final performance. In addition, we manually evaluate the generation quality
regarding fluency, relevance and factual correctness.

Figure 1: An example of query-relevant knowledge triple extracted from meeting
transcripts. The knowledge triple can be used in query-relevant segment
extraction as well as summary generation. Figure 2: An overview of our
proposed framework. We first extract query-relevant knowledge triples based on
the query and meeting transcripts and select top-$k$ segments through
knowledge-aware ranking. Then we generate the query-focused meeting summary by
FiD-BART from query, knowledge and selected meeting transcripts.

Our contributions in this work are threefold: (1) Our work demonstrates the
effectiveness of leveraging query-relevant knowledge in QFMS. (2) We propose
KAS, a two-stage framework incorporating query-relevant knowledge for QFMS.
(3) Experimental results show that our approach achieves state-of-the-art
performance on a QFMS dataset (QMSum). Further analysis and human evaluation
indicate the advantage of our method.

##  2 Related Works

Existing QFMS methods can be divided into two categories Vig et al. (2022):
two-stage approaches and end-to-end approaches. Two-stage approaches Zhong et
al. (2021); Vig et al. (2022); Zhang et al. (2022) first extract query-
relevant snippets and then generate the summary upon the extracted snippets,
while end-to-end approaches Zhu et al. (2020); Zhong et al. (2022); Pagnoni et
al. (2022) directly generate summaries from the whole meeting transcripts.
However, both types of methods have their disadvantages. For example, most of
the two-stage approaches select query-relevant content based on utterance,
which ignores the contextual information between multiple utterances. As for
the end-to-end approaches, the computational and memory requirements will
increase rapidly when the input text becomes longer, making models challenging
to adapt to long meeting transcripts. To our knowledge, we are the first to
incorporate query-relevant knowledge in QFMS and demonstrate its
effectiveness. Besides, we include multiple utterances in each segment to
preserve the contextual information, and our approach can easily extend to
process long input text.

Model ROUGE-1 ROUGE-2 ROUGE-L Average Two-stage MaRGE Xu and Lapata (2021)
31.99 8.97 27.93 22.96 DYLE Mao et al. (2022) 34.42 9.71 30.10 24.74 SUMMN
Zhang et al. (2022) 34.03 9.28 29.48 24.26 RelReg Vig et al. (2022) 34.91
11.91 30.73 25.85 RelReg-W Vig et al. (2022) 36.45 12.81 32.28 27.18 End-to-
end LED Beltagy et al. (2020) 31.60 7.80 20.50 19.97 DialogLM Zhong et al.
(2022) 34.50 9.92 30.27 24.90 SegEnc-W Vig et al. (2022) 37.80 13.43 33.38
28.20 BART-LS Xiong et al. (2022) 37.90 12.10 33.10 27.70 SOCRATIC Pagnoni et
al. (2022) 38.06 13.74 33.51 28.44 KAS (Ours) 38.80 14.01 34.26 29.02

Table 1: Results on QMSum test set. The previous works can be divided into
two-stage and end-to-end approaches.

##  3 Methodology

This section presents our two-stage framework. Firstly, we introduce the
extractor, which extracts query-relevant segments and knowledge from the
source document. Then, a generative model synthesizes the query, extracted
segments, and knowledge into the final summary.

###  3.1 Knowledge-aware Extractor

#### Meeting Transcripts Segmentation

To preserve the contextual information between utterances, we split the
meeting transcripts into segments, and each segment could contain multiple
utterances. To do so, given an input meeting transcripts $T$, we will separate
it into $n$ segments $S=\\{S_{1},S_{2},...,S_{n}\\}$ where each $S_{i}$ is
fewer than $l$ tokens. Specifically, we feed meeting transcripts to the
segment utterance by utterance until it reaches $l$.

#### Knowledge-aware Ranking

The knowledge-aware ranking approach will select top-$k$ segments according to
a combination of semantic search scores and knowledge-aware scores. We apply
Multi-QA MPNet Song et al. (2020) to calculate semantic search scores. Multi-
QA MPNet is trained on 215M question-answer pairs from various sources and
domains, including Stack Exchange, MS MARCO Nguyen et al. (2016), WikiAnswers
Fader et al. (2014) and many more. Given the query and the segments, the model
outputs 768-dimension vectors to represent them. The semantic search score for
each segment is computed according to the cosine similarity:

$\displaystyle Score_{se}=\frac{MPNet(Q)\cdot
MPNet(S_{i})}{\left\|MPNet(Q)\right\|\left\|MPNet(S_{i})\right\|}$ (1)

To compute the knowledge-aware score for each segment, we first use OpenIE
Angeli et al. (2015) to extractive knowledge triples. Then, to filter out the
triples irrelevant to the query, we only keep the triples that contain
overlapping words with the query. The knowledge-aware score is obtained by L2
normalizing the number of the remaining triples $m_{i}$ in each segment:

$\displaystyle Score_{ka}=\frac{m_{i}}{\sqrt{\sum_{i=1}^{n}{|m_{i}|^{2}}}}$
(2)

Finally, we calculate the ranking score by summing the semantic search score
and the knowledge-aware score:

$\displaystyle Score_{rank}=Score_{se}+Score_{ka}$ (3)

The segment with the top-$k$ ranking score will be selected for the next stage
of summary generation. We denote the remaining segments as
$S=\\{S_{1},S_{2},...,S_{k}\\}$.

###  3.2 Generator

We choose BART-large Lewis et al. (2020), a transformer-based Vaswani et al.
(2017) generative pre-trained language model, as our backbone model for the
generator because of its remarkable performance on text summarization
benchmarks. Following the idea of Fusion-in-Decoder (FiD) and its applications
in generation tasks Izacard and Grave (2021); Su et al. (2022); Vig et al.
(2022), we employ FiD-BART for encoding multiple segments independently in the
encoder and fuse information from all segments in the decoder jointly through
the encoder-decoder attention.

To incorporate the extracted knowledge in the summary generation, we use the
knowledge as extra inputs other than the query and segments. In detail, we
remove the stop words in the knowledge triples and then merge all the
remaining words of knowledge triples in each segment as a set of knowledge
phrases. The concatenation of each segment $S_{i}$, knowledge phrases $K_{i}$
and the query $Q$ will be processed by FiD-BART encoder:

$\displaystyle h_{enc}^{i}=Encoder(Q\oplus K_{i}\oplus S_{i})$ (4)

Finally, the decoder performs encoder-decoder attention over the concatenation
of all segments’ encoder outputs. In this way, the computational complexity
grows linearly with the number of segments rather than quadratically, while
jointly processing all segments in the decoder enables the model to aggregate
information from multiple segments.

##  4 Experiments

###  4.1 Experimental Setup

We choose the top 12 text segments in the segment selection, with each fewer
than 512 tokens. For the summary generator, inspired by the effectiveness of
pre-finetuning models on relevant datasets to transfer task-related knowledge
in the abstractive summarization Yu et al. (2021); Vig et al. (2022), we
initialize our BART-large model using the checkpoint
111https://github.com/salesforce/query-focused-sum pre-finetuned on WikiSum
Liu et al. (2018). See Appendix A and B for more details of the experimental
setup and baselines.

We evaluate the models on the QMSum Zhong et al. (2021) dataset, which
consists of 1,808 query-summary pairs over 232 meetings from product design,
academic, and political committee domains. We report ROUGE
222https://github.com/pltrdy/files2rouge Lin (2004) as the automatic
evaluation results.

model ROUGE-1 ROUGE-2 ROUGE-L KAS ($k$=4) 36.46 12.06 31.75 KAS ($k$=8) 37.86
13.75 33.30 KAS ($k$=12) 38.80 14.01 34.26

Table 2: Results Our KAS on the QMSum test set with differen number of $k$.
$k$ denotes the number of segments that are selected in the first stage.

model R-1 R-2 R-L Entity F-1 KAS 38.80 14.01 34.26 35.59 w/o KA in ranking
37.71 12.96 32.94 34.03 w/o KA in generator 37.52 13.57 33.29 35.06

Table 3: Ablation results on the QMSum test set without knowledge-Aware (KA).
Our KA modules improve the performance of KAS on both stages.

###  4.2 Main Results

We compare our proposed model with strong baselines and previous state-of-the-
art models. As shown in Table 1, our approach outperforms both the two-stage
and end-to-end methods by a large margin on all evaluation metrics. We also
investigate how the number of selected segments in the first stage affects the
final performance. As shown in Table 2, when the number of selected segments
grows, the quality of the final summary also improves. Previous end-to-end
approaches usually outperform two-stage approaches because they encode the
entire meeting transcripts, which conduct lots of computing resources. Our
approach performs better by encoding 12 segments of input (6,144 tokens) in
the second stage generation than the state-of-the-art end-to-end approach that
directly encodes 16384 tokens. Therefore, we strike a balance between
performance and efficiency, which is essential in real-world applications.
Besides, our approach can easily extend to processing long input text since
the ranking method is unsupervised.

###  4.3 Ablation Study

We conduct an ablation study to investigate the contribution of the knowledge-
aware modules by removing the knowledge-based scoring in the ranking (w/o KA
in ranking) and the knowledge input in the summary generation (w/o KA in
generator), respectively. Besides, we evaluate the entity-level factual
consistency Nan et al. (2021) of the summary to test the effectiveness of our
knowledge-aware modules in keeping the knowledge entities in generated
summaries. Specifically, we report the F-1 scores of the entity overlap
(Entity F-1) between the source and the generated summary. Table 3 presents
that the model’s performance decreases in both metrics, especially the Entity
F-1, without the knowledge-aware module, which indicates the effectiveness of
our method.

model Fluency Relevance Factual Correctness SegEnc-W 4.68 4.60 3.76 KAS 4.79
4.68 4.47 Ground truth 4.92 4.82 4.60

Table 4: Human evaluation results on QMSum test set.

###  4.4 Human Evaluation

We further conduct a human evaluation to assess the models on fluency,
relevance, and factual correctness. We randomly select 50 samples from the
QMSum test set and ask three annotators to score the summary from one to five,
with higher scores being better. We compare SegEnc-W because it is the best
publicly available model. Table 4 illustrates our approach slightly
outperforms SegEnc-W according to fluency and relevance and achieves a
significantly better score than SegEnc-W on factual correctness with
p-value$<$0.05. Since SegEnc-W and our methods both use BART based generators,
it is in line with our expectation that we reach similar fluency and relevance
scores. Meanwhile, our proposed query-relevant knowledge can help reduce
hallucination and generate more factual correctness summaries. More details of
the human evaluation setup are in Appendix C.

##  5 Conclusion

In this paper, we propose a two-stage framework named KAS that incorporates
query-relevant knowledge in both stages. Extensive experimental results on the
QMSum dataset show the effectiveness of our method. We further conduct
detailed analysis and human evaluation to prove our method’s capacity to
generate fluency, relevant and faithful summaries.

##  6 Limitations

Our method is trained and tested on the only publicly available QFMS dataset
named QMSum. QMSum consists of three different domains (Academic, Committee
and Product), which makes the evaluation robust. However, QMSum only contains
1,808 samples, which is relatively small. We hope larger QFMS datasets be
proposed to accelerate the development of this field.

In the first stage of our approach, we extract a fixed length (6,144 tokens)
of the meeting transcripts as the second stage input text. Therefore, the
model’s performance could be affected since some query-relevant information
could be cut off in the first stage.

##  7 Ethics Statement

Although our approach can generate query-focused meeting summaries and achieve
a much better factual correctness score than other models, we can not avoid
generating hallucinated content from the generative models. We recommend that
when our approach is deployed in real-world applications, additional post-
processing should be carried out to remove unreliable summaries. In addition,
we will indicate that the summary we provide is for reference only, and users
need to check the original meeting transcripts to get accurate information.

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##  Appendix A Experimental Details

We use the off-the-shelf model Multi-QA MPNet
333https://huggingface.co/sentence-transformers/multi-qa-mpnet-base-cos-v1 for
semantic searching. We initialize the generator from the off-the-shelf BART-
large model through Hugging Face. We use learning rates $6e^{-5}$ following
Lewis et al. (2020) and Adam optimizer Kingma and Ba (2014) to fine-tune KAS.
We use a batch size of one and train all models on one RTX 3090 for ten
epochs. During decoding, we use beam search with a beam size of five and
decode until an end-of-sequence token is emitted. The final results are the
average test set performance on the best three checkpoints in the validation
set. In addition, the QMSum dataset has MIT License, so we can freely use it.

##  Appendix B Baselines Details

We conduct both two-stage and end-to-end baselines for comparison. MaRGE Xu
and Lapata (2021) used Masked ROUGE extractor in the first stage. DYLE Mao et
al. (2022) is a dynamic latent extraction approach for abstractive long-input
summarization. SUMMN Zhang et al. (2022) is a multi-stage summarization
framework for long input dialogues and documents. RelReg (RELevance
REGression) Vig et al. (2022) is similar to MaRGE, which trains a relevance
prediction model directly on QFS data using the original, non-masked query.
RelReg-W Vig et al. (2022) uses the same framework as RelReg but initializes
the generator from the WikiSum pre-trained checkpoint. LED Beltagy et al.
(2020) is an encoder-decoder transformer-based model that employs efficient
attention to process long input text. DialogLM Zhong et al. (2022) is a pre-
trained neural encoder-decoder model for long dialogue understanding and
summarization, and it uses a hybrid attention approach by combining sparse
local attention with dense global attention. SegEnc-W Vig et al. (2022) is a
fusion-in-decoder method that initialized the model from WikiSum Liu et al.
(2018). BART-LS Xiong et al. (2022) used pooling-augmented blockwise attention
to improve the efficiency and pre-trains the model with a masked-span
prediction task with spans of varying lengths. SOCRATIC Pagnoni et al. (2022)
uses a question-driven, unsupervised pretraining objective to improve
controllability in summarization tasks, and it follows the SegEnc framework.

##  Appendix C Human Evaluation

In Table 4, we conduct a human evaluation of the fluency, relevancy and
factual correctness of the generated summaries from the QMSum dataset. In
detail, we randomly sample 50 summaries from KAS, SegEnc-W and the ground
truth summary from the same meeting transcripts for comparison. Assessments
are scored on a scale of one to five, with higher scores being better. Fluency
means the summary is grammatically correct and contextually coherent.
Relevancy means the summary is relevant to the query. Factual correctness
means the summary is faithful according to the information given in the
meeting transcripts and does not contain hallucinations. We assign each
summary to three annotators and take the average score as the final result. In
total, we used three annotators from Hong Kong, and all annotators voluntarily
participated in the human evaluation. All annotators agree to have their
evaluation results included as part of this paper’s results.