Robust estimates of insurance misrepresentation through kernel quantile regression mixtures

• Published date: 01 September 2021
• Author: Hong Li,Qifan Song,Jianxi Su
• Date: 01 September 2021
• DOI: http://doi.org/10.1111/jori.12358

J Risk Insur. 2021;88:625–663. wileyonlinelibrary.com/journal/jori

|

625

Received: 15 September 2020

|

Revised: 13 May 2021

|

Accepted: 28 June 2021

DOI: 10.1111/jori.12358

ORIGINAL ARTICLE

Robust estimates of insurance

misrepresentation through kernel quantile

regression mixtures

Hong Li

1

|Qifan Song

2

|Jianxi Su

2

1

Department of Economics and Finance,

University of Guelph, Guelph, Ontario,

Canada

2

Department of Statistics, Purdue

University, West Lafayette, Indiana, USA

Correspondence

Hong Li, Department of Economics and

Finance, University of Guelph, 50 Stone

Road East, Guelph, ON N1G 2W1,

Canada.

Email: lihong@uoguelph.ca

Funding information

National Science Foundation,

Grant/Award Number: DMS‐1811812;

Natural Sciences and Engineering

Research Council of Canada,

Grant/Award Numbers: DGECR‐2020‐

00347, RGPIN‐2020‐05387

Abstract

This paper pertains to a class of nonparametric meth-

ods for studying the misrepresentation issue in in-

surance applications. For this purpose, mixture models

based on quantile regression in reproducing kernel

Hilbert spaces are employed. Compared with the ex-

isting parametric approaches, the proposed framework

features a more flexible statistics structure which could

alleviate the risk of model misspecification, and is in

the meantime more robust to outliers in the data. The

proposed framework can not only estimate the pre-

valence of misrepresentation in the data, but also help

identify the most suspicious individuals for the vali-

dation purpose. Through embedding state‐of‐the‐art

machine learning techniques, we present a novel sta-

tistics procedure to efficiently estimate the proposed

misrepresentation model in the presence of massive

data. The proposed methodology is applied to study the

Medical Expenditure Panel Survey data, and a sig-

nificant degree of misrepresentation activity is found

on the self‐reported insurance status.

KEYWORDS

big data, insurance claim models, misrepresenter identification,

misrepresentation risk assessment, nonparametric regression

mixtures

© 2021 American Risk and Insurance Association

1|INTRODUCTION

Insurance companies collect policyholders' information—summarized by the so‐called in-

surance rating factors—to calculate risk‐adjusted premiums. In the real world practice, it often

happens that policyholders intentionally make untrue statements on some key rating factors so

as to alter the insurance eligibility and/or lower the insurance premiums. In actuarial parlance,

this type of fraudulent behaviors are best referred to as the insurance misrepresentation. Rating

factors subject to misrepresentation are typically self‐reported. Examples include the smoking

status in health insurance, the millage, and use of vehicles in auto insurance. Misrepresentation

phenomena can be also found in insurance‐related survey data when revealing a particular

level of true information is associated with a type of costs, such as social desirability or financial

cost. Shutting eyes to misrepresentation activities may degrade the quality of actuarial models,

leading to infelicitous business decisions and/or unfair premium schemes. Misrepresentation

risk is also of particular interest to insurance regulators who attempt to understand how the

presence of fraudulent behaviors by a group of insured individuals might affect the welfare of

the others (Gabaldón, 2014). Hence, assessing and managing the misrepresentation risk is an

indispensable component in modern insurance practice.

There are two strands of literature in studying the misrepresentation risk. The first strand of

literature focuses on the qualitative research of misrepresentation, attempting to propose su-

perior policy designs and proactive overseeing processes for deterring the misrepresentation

behaviors (see, Hamilton, 2009, for a comprehensive review). The other strand of literature

focuses on the quantitative aspect of misrepresentation management and aims to quantify the

extent of misrepresentation risk using statistical models. Our paper sits squarely in the latter

strand.

Arguably, modeling insurance misrepresentation is a challenging task. What complicates

the task, from the statistical standpoint, is the unobservable nature of fraudulent activities.

Namely, knowledge of whether misrepresentation actually occurred cannot be discovered until

a formal investigation is undertaken. Thereby, traditional statistics methods (e.g., discriminant

analysis and logistic regression) which require access to a sample frame containing the random

variable of concern (i.e., the misrepresentation status under the individual level), may not be

directly used to study misrepresentation. Consequently, a new set of statistics tools are natu-

rally called upon. In particular, we have a keen interest in developing a rigorous statistics

framework to deliver scientifically sound answers to the following two questions, which are of

fundamental importance in quantitative misrepresentation risk management:

Q1.Based on a given set of insurance claims data, how to assess the level of misrepresentation

activities in the data, before actually observing the fraudulent behaviors?

Q2.If a significant level of misrepresentation activities are discovered, how to select the most

suspicious individuals for the validation purpose?

Despite the practical importance, modeling misrepresentation has not received much at-

tention from insurance researchers until recently, and thus the related literature is considerably

limited. In particular, among the existing misrepresentation models, parametric regressions are

always used to model the insurance claim records. Misspecifying the parametric structure to

the claim data may significantly contaminate the accuracy of the misrepresentation assessment.

In this paper, we aim to advance the present parametric misrepresentation models to a non-

parametric framework via deploying kernel quantile regression (KQR). The research problem is

626

|

LI ET AL.

of great relevance to the insurance practice in its own right. In addition to the methodological

contribution, our careful numerical investigation reveals that improper use of parametric

misrepresentation models may seriously misestimate the misrepresentation behavior. In the

simulation examples, we find that misspecified parametric misrepresentation models may

underestimate the true misrepresentation rate by 90% and also cause excessively large error

rates in the misrepresenter identification process. Furthermore, in a real data application based

on the Medical Expenditure Panel Survey (MEPS) data, we identify a 50% difference in the

estimated misrepresentation rate by using the proposed nonparametric model compared with

the parametric result reported in the earlier literature. Thereby, a practical message which we

aim to convey to the insurance analysts is that, when modeling insurance misrepresentation,

parametric models must be used with extra cautions. If there is no prior knowledge about the

true regression structure of the data generating process (DGP), then the insurance analysts

should at least try the nonparametric framework proposed in this current paper.

The rest of this paper proceeds as follows. Starting off by a literature review in Section 2,we

will argue that there are two major limitations inherent in the parametric misrepresentation

models. Then in Section 3, we introduce a nonparametric alternative to address the issues. In

Section 4, an innovative learning algorithm is suggested to implement the proposed mis-

representation model. The algorithm not only finds nonparametric estimates of mis-

representation probability (which answers question Q1) as well as the regression structure

between insurance claims and relevant covariates, but also helps identify potential mis-

representers (which answers question Q2). Based on extensive simulation studies in Section 5,

we show that the proposed framework is able to accurately estimate the misrepresentation

probability under various DGP's, ranging from the univariate linear models to multivariate

nonlinear models. Moreover, misrepresenters are successfully detected with very low error

rates. Therefore, the proposed misrepresentation framework seems to be able to deliver a

satisfactory performance under different challenging situations. Moreover, another numerical

example based on a synthetic nonlife insurance portfolio is included in Section 6.1 to illustrate

the big data application of the proposed model. Finally, in Section 6.2 the proposed model is

applied to study the MEPS data. In line with the results of Akakpo et al. (2019), our model

suggests a significant percentage of respondents misrepresented on the self‐reported insurance

status in 2014, to avoid the potential tax penalty. However, our estimated percentage is sub-

stantially lower than that reported in Akakpo et al. (2019). This discrepancy may be attributed

to the fact that the proposed KQR method could better cope with the nonlinear dependence

structure inherent in the MEPS data, and thus leads to a more reliable estimation of the

misrepresentation probability.

2|PRESENT LITERATURE AND MOTIVATION

To the best of our knowledge, there are only a few existing works on misrepresentation

modeling, including Akakpo et al. (2019), Xia (2018), Xia and Gustafson (2016,2018), and Xia

et al. (2018). In particular, one of the most recent attempts made in Akakpo et al. (2019)

significantly inspires our undertakings. In that paper, the authors built on the theoretical

groundwork established by Xia and Gustafson (2016) and proposed a mixture of parametric

regression to model insurance misrepresentation. Before placing the present paper into per-

spective, the following paragraphs give a coarse overview of the misrepresentation model in

Akakpo et al. (2019). Some standard notations for describing the misrepresentation problem of

LI ET AL.

|

627