RUHUNA JOURNAL OF SCIENCE 

Vol 11 (2): 118-130, December 2020 
eISSN: 2536-8400                                                © Faculty of Science 
DOI: http://doi.org/10.4038/rjs.v11i2.91                                                                                   University of Ruhuna 

© Faculty of Science, University of Ruhuna 118  

Sri Lanka 

Modelling non-life insurance in Sri Lanka using Cox Hazard 

Model and classification of risky customers 

W.A.R. De Mel and W.A.P.A. Chathurangani  

 

Department of Mathematics, University of Ruhuna, Wallamadama, Matara 81000, Sri Lanka 
 

Correspondence: piumiayodyachathurangani@gmail.com;  https://orcid.org/0000-0001-8296-9218 

 

Received: 20th June 2020, Revised: 25th December 2020, Accepted: 30th December 2020 

Abstract. Some of the major factors that help the decision-making process of 

an insurance company include Time of the first claim (TFC), claim Size and 

the frequency of claims. However, in most situations researchers focus 

mainly on the second and third factors mentioned above. We hypothesize the 

importance of the TFC of an insurance contract in the decision-making 

process. Empirical evidence of motor vehicle insurance data in Sri Lanka 

suggests that nine covariates are responsible for the claim sizes. In the current 

study, our main objective is to find the key factors of those nine that are 

responsible for the TFC of the insurance contract. This study is based on the 

claim data in the whole year of 2016 of non-life insurance policies of a 

particular insurance company in Sri Lanka. Considering the TFC as right-

censored data, selected nonparametric methods, i.e., Kaplan-Meier, Nelson-

Aalen estimators, and Cox Proportional Hazard Model are used to analyze 

the data. We identified the five most influential covariates namely, vehicle 

type, log of Premium Value and that of Assured Sum, the lease type and the 

age range via fitting the Cox Model to TFC data. After a thorough residual 

analysis, the Logistic regression model has been used to identify the 

important covariates to classify future customers as risky or not.  

Key words: Classification, Cox Proportional Hazard model, Kaplan-Meier, 

Right-censored 

1   Introduction 

In medical follow-up studies, life testing, insurance, and other fields, it is impossible 

to observe the lifetime or the time of the first insurance claim of all subjects in the 

study. The reason is time itself. In most cases, it is highly likely that all the events have 

not been observed by the time one wants to analyze these lifetimes. For example, in a 

non-life insurance, not every insurance contract has a claim during its contract period.  

The individuals in the study who have no claims by the end of the study (or contract) 

period are labeled as right-censored. The only information the researcher has is the 

time between the contract initiation and the end of the study time, which is naturally 

less than the time to the first claim. The simplest kind of censoring is single censoring 

https://creativecommons.org/licenses/by-nc/4.0/
mailto:piumiayodyachathurangani@gmail.com
https://orcid.org/0000-0001-8296-9218


WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

Ruhuna Journal of Science 

Vol 11 (2): 118-130, December 2020 119 

which occurs when all subjects are censored at the same time. There are two types of 

censoring, namely Type I and Type II censoring. In Type I, the censoring time is 

predetermined whereas Type II occurs when a predetermined number of failures are 

observed, and the remaining subjects are then right censored. In many studies, subjects 

are not censored at the same time. This is called random censoring. These types of data 

are commonly referred to as survival data. The analysis of such data is important in 

many fields including reliability, engineering, biology, insurance, and medicine (De 

Mel 2014). 

Under the random censoring model, we assume that 𝑋1, 𝑋2, … , 𝑋𝑛 are independent 
nonnegative random variables with an absolutely continuous distribution 

function 𝐹(𝑡) = 𝑃(𝑋 ≤ 𝑡). The censoring variables 𝑌1, 𝑌2, … , 𝑌𝑛 are also independent 
nonnegative random variables with an absolutely continuous distribution 

function 𝐺(𝑦) = 𝑃(𝑌 ≤ 𝑦). We further assume that both random variables 𝑋 and 𝑌 are 
independent from each other. In this model, the observable random variables are 𝑍𝑖 =
min (𝑋𝑖 , 𝑌𝑖 ) and 𝛿𝑖 = 𝐼(𝑋𝑖 ≤ 𝑌𝑖 ), where 𝛿𝑖  indicates whether 𝑍𝑖 is an uncensored 
observation or not. The 𝑌𝑖 ’s right censor the 𝑋𝑖 ’s. 

In survival analysis, a variety of parametric and nonparametric significant tests can 

be used to identify the observed differences among the empirical survival curves. The 

most commonly used nonparametric test is based on logrank statistic (Oulidi et al. 

2010). In survival literature, the survival function is usually estimated from the 

observed data by using the Kaplan-Meier estimator (Kaplan et al. 1958). Nelson-Aalen 

proposed a nonparametric estimator (Nelson 1972) for the cumulative hazard rate 

function 𝛬(𝑡). 
In non-life insurance studies, one of the main interests is to investigate the 

association between the claim sizes or claim times of insurance contracts and related 

covariates. These covariates are sometimes referred to as risk factors. Examples of 

commonly encountered risk factors include age, sex and the health condition of the 

contract holder, vehicle type, premium value, hiring status, assured sum, lease type, 

and the brand name of a vehicle. Identifying and measuring this association helps 

insurance companies to understand how these factors are associated with the 

occurrence or nonoccurrence of an insurance claim. This in turn helps insurance 

companies to calculate the premium of a contract according to the customer’s risk. 

Since the survival data is not normally distributed and has partial information, the 

standard statistical techniques like multiple linear regressions cannot directly be 

applied to analyze such data. Cox proposed a semi-parametric multiple regression 

models for survival data called Cox’s Proportional Hazard Model (Cox 1972).  This 

model can be used to identify the important factors in the study and to compare the 

hazard rate functions among different groups.  Furthermore, contrast to parametric 

models, this method makes no assumptions about the baseline hazard function (Cox 

1972).  

In the final part of this research, we use the logistic regression model to identify a 

future customer as a risky or not. This is a classification problem with binary response 

data variable having categories, 1 for a risky customer and 0 if not risky.  In machine 
learning literature the most commonly used model for this purpose is the Logistic 

Regression model, a type of Multiple Linear Regression model. In insurance, we 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

Ruhuna Journal of Science 

Vol 11 (2): 118-130, December 2020 120 

collect the data in only the insurance policy period. Therefore, we can classify each 

customer as a risky customer or not without any difficulty. 

2 Materials and Methods  

In this section, we introduce some basic concepts and their definitions in survival 

analysis 

2.1 Survival Function 

 

Let 𝑇 be an arbitrary continuous nonnegative random variable with distribution 

function 𝐹(𝑡) = 𝑃(𝑇 ≤ 𝑡) and the density function 𝑓(𝑡) =
𝑑𝐹(𝑡)

𝑑𝑡
.  Survival function of 

 𝑇 is defined as  

𝑆(𝑡) = Pr(𝑇 > 𝑡) = 1 − 𝐹(𝑡).                                 (1) 

This measures the probability that a subject survives beyond some specific time 𝑡. The 
hazard rate function or instantaneous event rate is usually denoted by ℎ(𝑡) and it is the 
probability that an individual who is under observation has an event at time 𝑡. Define 
by   

ℎ(𝑡) =  lim
𝑑𝑡→0

Pr(𝑡 ≤ 𝑇 < 𝑡 + 𝑑𝑡| 𝑇 ≥ 𝑡)
𝑑𝑡

   =
𝑓(𝑡)

𝑆(𝑡)
.       (2) 

The function 𝛬(𝑡) = ∫ ℎ(𝑢)𝑑𝑢
𝑡

0
 is the cumulative hazard function of 𝑇 and 𝑆(𝑡) =

exp {−𝛬(𝑡)}. 

2.2 Kaplan-Meier Survival Estimator 

The Kaplan-Meier (KM) (Kaplan et. al, 1958) is a nonparametric estimator of a 

survival function 𝑆(𝑡) given by  

�̂�(𝑡) = ∏ exp (−
𝑑𝑘

𝑛𝑘
) =𝑘:𝑇𝑘≤𝑡 ∏ (1 −

𝑑𝑘

𝑛𝑘
) ,𝑘:𝑇𝑘≤𝑡           (3) 

where 𝑛𝑘  is the number of subjects at risk (alive) just before 𝑇𝑘  and 𝑑𝑘 is the number 
of failures at time 𝑇𝑘. The times  𝑇1 < 𝑇2 < ⋯ < 𝑇𝐿 denote the L distinct ordered 
observed failure times in the study with n subjects. The Nelson-Aalen estimator 

(Nelson, 1972) for Λ(t) is given by �̂�(𝑡) = ∏
𝑑𝑘

𝑛𝑘
𝑘:𝑇𝑘≤𝑡

. With tied failure data, the above 

formulas need to be modified. 

 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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2.3 Cox Proportional Hazards Model 

The Cox Proportional Hazards Model (Cox 1972) is a semi-parametric model because 

it does not assume any conditions on ℎ0(𝑡).  

The Cox’s model (Cox 1972) is given by  

ℎ(𝑡| 𝑋) =  ℎ0(𝑡) exp (∑ 𝛽𝑖 𝑋𝑖 )
𝑝
𝑖=1 ,                           (4) 

where ℎ(𝑡| 𝑋) which is usually written as just ℎ(𝑡) is the hazard rate function at time 
𝑡, ℎ0(𝑡) the baseline hazard rate function, and 𝑋 = (𝑋1, 𝑋2, … , 𝑋𝑝) the covariate vector 

of size 𝑝. This model can be used to analyze survival data by regression model such as 
multiple linear regression and generalized linear models because equation (4) can be 

reduced to log(
ℎ(𝑡| 𝑋)

ℎ0(𝑡)
) = ∑ 𝛽𝑖 𝑋𝑖

𝑝
𝑖=1 . The dependent variable in the Cox’s model is the 

hazard rate function, ℎ(𝑡). Therefore, one can use this model to compare the survival 
curves in different groups by taking into account other related covariates.  

We assume that ℎ0(𝑡) is unknown and common to all subjects in the study. Most of 
the time, the coefficients 𝛽1, 𝛽2, … , 𝛽𝑝 are estimated by using the partial likelihood 

method (Cox, 1972). The partial likelihood function is given by the following equation. 

𝑃𝐿(𝛽) =  ∏ ∏ [
𝑌𝑖(𝑡)𝑒

𝑋𝑖(𝑡)𝛽

∑ 𝑌𝑖(𝑡)𝑒
𝑋𝑖(𝑡)𝛽𝑛

𝑖=1

]
𝑑𝑁𝑖(𝑡)

𝑡≥0
𝑛
𝑖=1      (5) 

where 𝑁𝑖 (𝑡) = 𝐼(𝑍𝑖 ≤ 𝑡, 𝛿𝑖 = 1) and 𝑌𝑖 (𝑡) = 𝐼(𝑋𝑖 ≥ 𝑡) are the event process and at-
risk process respectively for 𝑖𝑡ℎ  subject (Fleming et al., 1991). In most of the times, 
one cannot find the exact solutions to equation (5) but can obtain the numerical 

solutions for the coefficient vector 𝛽 using numerical method like Newton-Raphson. 

2.4 Logistic Regression Model 

To model a binary response variable 𝑌, one can use the logistic multiple regression 
model. This is given as 

log(
𝑝(𝑋)

1−𝑝(𝑋)
) = 𝛽0 + 𝛽1𝑋1 + 𝛽2𝑋2 + ⋯ + 𝛽𝑝𝑋𝑝       (6) 

where 𝑋 = (𝑋1, 𝑋2, … , 𝑋𝑝) is vector of covariate, 𝛽 = (𝛽0, 𝛽1,  𝛽2, … , 𝛽𝑝) vector of 

regression coefficients, and 𝑝(𝑋) = 𝑃(𝑌 = 1 /𝑋). With a little algebra, one can rewrite 
the equation (6) as  

𝑝(𝑋) =
exp (𝛽0+𝛽1𝑋1+𝛽2𝑋2+⋯+𝛽𝑝𝑋𝑝)

1+exp (𝛽0+𝛽1𝑋1+𝛽2𝑋2+⋯+𝛽𝑝𝑋𝑝)
                      (7) 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

Ruhuna Journal of Science 

Vol 11 (2): 118-130, December 2020 122 

In logistic regression, parameters are estimated by using maximum likelihood method. 

Future observations can be classified into two groups by using 𝑝(𝑋) with some 
probability threshold, for example 𝑝(𝑋) = 0.5.  

3   Results and Discussion 

In this study, we consider a dataset consisting of motor vehicle insurance contracts 

from a leading insurance company in Sri Lanka. Customers were enrolled for a one-

year period from January 01, 2016 to December 31, 2016 therefore; the end of the 

study period was December 31, 2016. Our dependent variable is the time of the first 

claim of the contract and we also consider nine covariates, namely, vehicle type, 

premium value, hiring status, gender, age, assured sum, lease type, brand name of a 

vehicle, and class type of a vehicle. 

Our insurance dataset contains more than 300,000 non-life insurance (motor vehicle 

insurance) contracts. After removing contracts with missing values we use simple 

random sampling technique to obtain a random sample with 599 contracts. Any 

contract with no claims in the study period is treated as right censored data. In Figure 

1, we depict a part of our dataset to give some idea about the survival data. Study began 

on January 01, 2016 and was terminated on December 31, 2016. Insurance contracts 

can initiate at any time during the contract year. In Figure 1, the horizontal lines depict 

the claim time of the first claim of an insurance contract. The blue dots on these lines 

represent the actual time of the first claim and this is known as the calendar time of the 

first claim. 

 
Fig 1:  Calendar times of subjects in the study. 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

Ruhuna Journal of Science 

Vol 11 (2): 118-130, December 2020 123 

If we cannot observe the first claim during the year, these times go beyond the 

termination time of the study and they are called right censored data. For our dataset, 

we first compute the Kaplan-Meier survival estimate by using survival package in R 

software. It is depicted along with the 95% confidence band in Figure 2. 

 
Fig 2: Kaplan-Meier estimate for claim times for motor vehicle insurance claim data 

Table 1 displays the summary statistics obtained using Logrank test to compare 

significant differences among the survival curves for categories of categorical 

variables in the study. All these factors are significant at the 5% significance threshold. 

Table 1: Summary statistics for categorical variables based on logrank. 

  

 

Figure 3 displays the estimated survival curves for two age ranges.  This clearly 

displays a significant difference between two estimated curves for age range categories 

and verifies the Logrank result. Here, the two groups for age range are above and below 

35 years. But one can use more than two groups. It is clear from the Figure 3 that the 

youngers are riskier than the elders. Identifying theses risky groups helps insurance 

companies to calculate the premiums for their insurance contracts. 

Variable Name P-value 

Age range 0.0421 

Lease type 0.00441 

Vehicle type 3.69e-11 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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Vol 11 (2): 118-130, December 2020 124 

 

Fig 3: Estimated survival functions by age range 

We next fit the Cox Proportional Hazard Model (Cox 1972) including all above 

mentioned nine covariates. Results from R software are displayed below in Table 2. 

According to the output, only five variables, namely vehicle type, premium value, 

assured sum, lease type, and age range are significant at the 5% confidence threshold. 
Gender is not significant. 

After including other six predictors in the model only age range is significant but 

not gender at the significance level 0.05. We can remove the gender from the Cox 
Proportional Hazard Model (Cox 1972) since the p-value is larger than the significance 

level 0.05 but p-value for age range is less that above significance level. 

 
Table 2: Parameter estimates for Cox Proportional Hazard Model with all the variables. 

 

Variable name β (estimated 

coefficients) 

SE (est. coef.) Wald test p-value 

Vehicle type - 9.515e-02 3.341e-02 - 2.848 0.004395 

Premium value   3.731e-05 7.627e-06   4.892 1.00e-06 

Hiring Status - 4.213e-02 2.342e-01 - 0.180 0.857234 

Assured-sum - 5.978e-07 1.646e-07 - 3.632 0.000281 

Lease type   6.972e-01 1.782e-01   3.912 9.16e-05 

Gender   7.823e-02 1.965e-01   0.398 0.690582 

Age range - 5.256e-01 1.813e-01 - 2.900 0.003736 

Brand name   3.291e-03 7.609e-03   0.433 0.665334 

Class type of vehicle - 5.872e-03 6.672e-02 - 0.088 0.929873 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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After removing insignificant predictors, we fit the Cox model again with predictors, 

namely vehicle type, premium value, assured sum, lease type, and age range. The 

results from R software for the reduced Cox Proportional Hazard Model are displayed 

below in Table 3. 

Table 3: Parameter estimates for the reduced Cox Proportional Hazard Model. 

Variable name β (estimated coefficients) SE (est. coef.) Wald p-value 

Vehicle type -9.91e-02   2.16e-02 -4.59 4.5e-06 

Premium value 3.78e-05   7.49e-06   5.05 4.5e-07 

Assured_sum -6.02e-07   1.61e-07 -3.75 0.00018 

Lease type 6.82e-01   1.75e-01   3.89  9.9e-05 

Age range -5.08e-01   1.79e-01  -2.84  0.00447 

According to the results in Table 3, all variables in the reduced model are significant 

at the 5% confidence threshold. Therefore, we could treat this reduced Cox 
Proportional Hazard Model with variables, namely, vehicle type, premium value, 

assured sum, lease type, an age range as our final model for the motor vehicle insurance 

data set. Before using it for prediction purposes it is necessary to carry out the model 

validation. In model validation, we check the proportional hazard assumption, 

influential observations assumption and nonlinearity assumption. 

3.1 Model validation 

Since Cox proportional hazard model fits under several assumptions it is better to 

check whether the fitted Cox model adequately explains the data. In order to check the 

three main assumptions namely, violation of the assumption of proportional hazards, 

influential data and, nonlinearity in the relationship between the log hazard and the 

covariate, the residuals method is used. Schoenfeld and Martingale residuals (Fleming 

et al. 1991) are the most frequently used residuals to check the assumptions of the Cox 

model. 

Testing the proportional hazards assumption 

To check the Proportion Hazard assumption of Cox model, we use the scaled 

Schoenfeld residuals of the final fitted model. We plot these Schoenfeld residuals with 

all the covariates in the final model, namely vehicle type, premium value, assured sum, 

lease type, and age range. Since there are no apparent patterns in any of the plots in 

Figure 4, we can assume that the proportional hazard assumption holds. 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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Fig 4: Schoenfeld residual graphs for vehicle type, premium value, assured sum, lease 

type, and age range 

In Figure 4, the solid line represents a smooth spline fit to the plot and the dashed line 

represents a ±2 standard-error band around the fit. Since there is no pattern with time, 
the assumption of proportional hazards appears to be satisfied by the above mentioned 

covariates. 

Checking for influential observations 

We use the DFBETA values to identify influential observations or outliers. Here 

DFBETA measures the difference in each parameter estimate with and without the 

influential point (Montgomery et al. 2006). Figure 5 displays these DFBETA values 

from the final fitted model. We use the usual cutoff value,|𝐷𝐹𝐵𝐸𝑇𝐴| > 2/√𝑛 to 
identify an influential observation. Here 𝑛 is the sample data value in the training 
dataset which is used for training the Cox model. This cutoff is 0.08 for our study. 

There are only a few DFBETA values that correspond to large premium values and 

assured sum. But we can neglect these points in the dataset. 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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Fig 5: Index plots of DFBETA values for the fitted Cox regression versus age range, 

premium, lease type, assured sum, and vehicle type. 

Detecting nonlinearity 

Nonlinearity, an incorrectly specified functional form in the parametric part of the 

model, is a potential problem in Cox regression like in linear and generalized linear 

models. The martingale residuals may be plotted against covariates to detect 

nonlinearity. For nonlinear assumption for Cox’s model, continuous type covariates 

should be linear. In our study, this assumption is checked only for the predictors, 

namely, assured sum and premium value. Figure 6 displays these two residual plots 

which are clearly not linear. 



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Fig 6: Martingale residual graph for covariates, Assured sum and Premium. 

In Figure 6, we plot the martingale residuals against the two covariates, Assured-sum 

and Premium. We also fit the LOWESS smooth based on a span 0.2. Here, null Cox 

model means the model with all continuous type variables. To correct this nonlinear 

violation, we use logarithm transformations of both above variables in the fit. The 

results are displayed in Table 4. All predictors in this model fit are significant at the 

5% significant threshold and all residual plots seem to be good. But we do not report 
these plots here because they are very similar to the above residual plots. Therefore, 

we can treat this fitted model as our final model for the motor vehicle insurance data. 

So we can use this model for future predictions. 

Table 4: Final fitted Cox model 

Variable name β (estimated 

coefficients) 

SE (coef) Wald p-value 

Vehicle type - 0.07102    0.02604 - 2.728  0.006378 

Log (Premium value)   1.28256    0.30479     4.208  2.58e-05 

Log (Assured sum) - 0.69764    0.24308  - 2.870  0.004105 

Lease type   0.62199    0.17238     3.608  0.000308 

Age range - 0.47701    0.18084  - 2.638  0.008344 

3.2 Classification of the risky customers using logistic regression model 

In this subsection, we fit the logistic regression model to the response variable with the 

above nine predictors in the motor vehicle claim dataset. Here, we choose two levels, 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

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namely, a risky customer and a not risky customer as our response variable and it is a 

qualitative variable with two levels. In this study, we treat a customer as a risk if his or 

her first claim size is above 30,000 during the contract duration; otherwise we consider 
him or her as not risky. To fit this model, we randomly select  500 data points from 
our original sample for training data and we treat the remaining 99 data points as the 

test data. Table 5 shows the coefficient estimates for a logistic regression model that 

uses all of the above predictor variables, but we present only significant variables at 

the 5% significant level. They are vehicle type, premium value, lease type, and age 
range.  

Table 5: Parameter estimates in fitted Logistic regression model. 

 Estimate Std. Error Z value 

(Intercept) 1.94E+00 5.55E-01 -3.498 0.000468 

Premium value 1.41E-05 5.91E-06  2.386 0.017011 

Lease 2.46E-01 3.91E-01  2.63 0.005287 

Vehicle -1.44E-01 5.31E-02 -2.718 0.006576 

Age range -8.04E-01 3.89E-01 -2.064 0.039042 

Table 6: Confusion matrix for the test dataset 

Next, we use the fitted logistic regression model with five significant predictors, 

namely, vehicle type, premium value, lease type, and age range to predict the 

customers in the test dataset as a risky or not risky customer. The results are displayed 

in the following confusion matrix for the test dataset in Table 6.  

For the test dataset, the logistic regression model makes 83 correct classifications out 

of 99 sample data and the percentage accuracy is 0.8384. This achieves high accuracy. 

But logistic regression model makes only 16 misclassifications, and the percentage 

misclassifications is 0.1616.  

The final goal of any insurance is to create profitable portfolio by assessing risk 

factors. As we mentioned, many of the research work conducted earlier related to 

motor insurance contracts analyzed the claim sizes with Cox proportional hazard 

models without conducting a proper residual analysis. In our study, we analyzed the 

time of the first claim of motor vehicle insurance contracts by using Cox’s proportional 

hazard model. We identified five predictors as the most influential to the time of the 

first claim. These findings will help the company to take the decisions regarding the 

premium size of the contract for the different age groups, for example, according to 

their risk. The survival estimates of the time of the first claim obtained by using 

 
True default status 

No Yes Total 

Predicted risk value No 82 3 85 

Yes 13 1 14 

Total 95 4 99 



WAR De Mel and WAPA Chathurangani  Modelling non-life insurance in Sri Lanka 

Ruhuna Journal of Science 

Vol 11 (2): 118-130, December 2020 130 

Kaplan-Meier estimator will give the idea of the distribution of these claim time data. 

Since one contract may have more than one claim in its contract period, in future 

studies, we can analyze this type of data by treating them as recurrent event data. By 

conducting a thorough residual analysis, we found the optimal transformation of some 

selected covariates. Finally, we obtained a simple way to identify the risky customers 

by applying the logistic regression model. In future studies, one can apply more 

sophisticated methods like linear or quadratic discriminant analysis and Bayesian 

classifier methods to high accuracy. 

 

Supplementary Material 

Supplementary material for this paper is available separately. 

Acknowledgments 

Two anonymous reviewers are acknowledged for valuable comments on the initial draft of the 

manuscript.   

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