The new Poisson mixed weighted Lindley distribution with applications to insurance claims data Journal: Songklanakarin Journal of Science and Technology Manuscript ID SJST--.R Manuscript Type: Date Submitted by the Author: Original Article -Oct- Complete List of Authors: Atikankul, Yupapin; Kasetsart University Faculty of Science, Statistics; Rajamangala University of Technology Phra Nakhon, Mathematics and Statistics Thongteeraparp, Ampai; Kasetsart University, Statistics Bodhisuwan, Winai; Kasetsart University, Department of Statistics Keyword: count data, overdispersion, new weighted Lindley distribution, mixed Poisson distribution, maximum likelihood estimation
Page of Original Article The new Poisson mixed weighted Lindley distribution with applications to insurance claims data Yupapin Atikankul, Ampai Thongteeraparp, and Winai Bodhisuwan * Department of Statistics, Faculty of Science, Kasetsart University,, Thailand * Corresponding author, Email address: fsciwnb@ku.ac.th Abstract Mixed Poisson distributions have been applied for overdispersed count data analysis. In this paper, an alternative mixed Poisson distribution is proposed. The proposed distribution is derived by mixing the Poisson distribution with the new weighted Lindley distribution, named as the new Poisson mixed weighted Lindley distribution. Some special cases and several statistical properties have been derived including shape, factorial moments, probability generating function, moment generating function and moments. The maximum likelihood estimators of the parameters are obtained. Finally, two automobile insurance claims data sets are analyzed to compare the performance of the proposed distribution with some competitive distributions. Keywords: count data, overdispersion, new weighted Lindley distribution, mixed Poisson distribution, maximum likelihood estimation
Page of. Introduction Count data models are utilized in various fields such as public health, insurance, agriculture, etc. Some applications of count data have been studied, for example, the daily numbers of seizures of the patients with epilepsy (Albert, ), the number of automobile insurance claims in Germany (Tröbliger, ) and the number of roots of type of apple tree rootstock (Ridout, Demétrio, & Hinde, ). The Poisson distribution is a classic distribution for describing count data with the property of equality of the variance and mean, i.e. equidispersion. Unfortunately, in practical count data, the variance is usually larger than the mean, referred to as overdispersion, and rarely, the variance may be smaller than the mean, underdispersion. The Poisson distribution cannot be applied to account for these phenomena. Overdispersion occurs in almost all count data. An approach that has been widely applied to deal with overdispersed count data is that of mixed Poisson distributions (Karlis & Xekalaki, ; Grandell, ; Gupta & Ong, ). The best known distribution of the Poisson type is the negative binomial distribution (Greenwood & Yule, ), which it arises from a mixture of the Poisson and gamma distributions. The Lindley distribution (Lindley, ) arises from mixing the exponential distribution with the gamma distribution. Ghitany, Atieh, and Nadarajah () applied the Lindley distribution to a data set of waiting times for bank customers. The result showed that the Lindley distribution provided a better fit than the exponential distribution. Thus in various papers, the Lindley distribution and some of its modifications have been studied and developed as mixing distributions for a mixed Poisson distribution. Sankaran () proposed the discrete Poisson-Lindley distribution, which arises from the Poisson distribution and the Lindley mixing
Page of distribution (Lindley, ). Mahmoudi and Zakerzadeh () introduced a mixture of the Poisson and generalized Lindley distributions (Zakerzadeh & Dolati, ). Shanker, Sharma, and Shanker () proposed a mixed Poisson distribution with the two parameter Lindley mixing distribution (Shanker, Sharma, & Shanker, ). Shanker and Mishra () introduced the two-parameter Lindley distribution (Shanker & Mishra, ) as a mixing distribution. In the same year, the Poisson weighted Lindley distribution was proposed by El-Monsef and Sohsah (); Manesh, Hamzah, and Zamani (). Wongrin and Bodhisuwan () proposed a mixture of the Poisson and new generalized Lindley distributions (Elbatal, Merovci, & Elgarhy, ). The new weighted Lindley (NWL) distribution was proposed by Asgharzadeh, Bakouch, Nadarajah, and Sharafi (). The NWL distribution is obtained by mixing the weighted exponential distribution (Gupta & Kundu, ) with the weighted gamma distribution. The probability density function (pdf) is for x >, θ> andα >. θ (+ α ) θα x g (x) = x e x θ (+ )( ) e, () αθ (+ α ) + α (+ α ) The Lindley and weighted Lindley distributions are special cases of the NWL distribution. The pdf is log-concave and unimodal. A data set of the amount of carbon in leaves from the different mountainous areas of Navarra, Spain was modeled with the NWL distribution. The result showed that the NWL distribution is a better fit than the compared distributions. In this paper, the new Poisson mixed weighted Lindley (NPWL) distribution, a mixture of the Poisson and the new weighted Lindley distributions, is proposed. The rest of this paper is arranged as follows. In Section, the NPWL distribution is introduced. Some important statistical properties such as shape, factorial moments,
Page of probability generating function, moment generating function and moments are exhibited in Section. In Section, the steps for random variate generation are presented. In Section, the maximum likelihood estimators of the parameters are discussed. In Section, the NPWL distribution is applied to some automobile insurance claims data sets. The conclusions are presented in Section.. The new Poisson mixed weighted Lindley distribution The proposed distribution is a mixture of the Poisson distribution and the new weighted Lindley distribution (Asgharzadeh, Bakouch, Nadarajah, & Sharafi, ). Moreover; some special cases, the distribution function and the survival function of the proposed distribution are shown in this section. for Let X λ be a Poisson random variable with probability mass function (pmf) f ( x λ) = λ x e λ, () x! x =,,, K andλ >, written as X λ ~ Pois (λ). Now we assume that λ is distributed as the new weighted Lindley distribution with pdf g (λ) = θ (+ (+ λ)( e αθ(+ + α(+ forλ >, θ> andα >, written as λ ~ NWL ( θ, θαλ θλ ) e, ; then the unconditional discrete random variable X follows the new Poisson mixed weighted Lindley distribution. Proposition A discrete random variable X is distributed as the NPWL distribution, with the shape and scale parameters are θ andα, respectively, denoted as X ~ NPWL ( θ,, the pmf of X is
Page of f (x) = θ (+ x+ x αθ(+ + α(+ ( ) for x =,,, K,where θ > andα >. Proof following α x+, () x+ ( α ) + Let X λ ~ Pois ( λ ) and λ ~ NWL ( θ,, the pmf of X is derived as the f (x) = f ( x λ) g( λ) dλ = λ x e λ x! ( αθ(+ + α(+ ) θ (+ θ (+ αθ(+ + α(+ (+ λ)( e θαλ ) e θλ = ( ) x e λ (+ λ)( e ) e θ (+ αθ(+ + α(+ x! = ( ) λ x+ e ( α ) λ x! dλ λ x+ λ λ e = ( αθ (+ + α(+ ) Γ( x+ ) = Special cases Γ( x+ ) ( αθ + θ + ) θ (+ x+ θ (+ θ + x+ x αθ (+ + α(+ ( θ + ) e ( ) λ x ( α ) λ dλ+ dλ Γ( x+ ) x ( θ + ) θαλ λ e dλ θλ x ( ) λ Γ( x+ ) + x+ ( θ + ) + αθ + θ + x+. x+ ( αθ + θ + ) + dλ dλ Γ( x+ ) ( αθ + θ + ) (i) If α, the NPWL distribution becomes the Poisson-Lindley distribution (Sankaran, ). x+
Page of (ii ) If α, the NPWL distribution becomes the Poisson-weighted Lindley distribution (El-Monsef & Sohsah, ); (Manesh, Hamzah, & Zamani, ) with θ =. given by The cumulative distribution and survival functions of the NPWL distribution are F (x) = P( X x) θ (+ x+ = + + + F, x ; x ; x αθ(+ + α(+ ( ) + + + + F, x ; x ; x ( ) x+ x + ( α ) F, x+ ; x+ ; α F, x+ ; x+ ;, () x ( α ) α + S (x) = P ( X > x) = F( x), where F ( a, b; c; z) = ( a) n = Γ( a+n) Γ( a) Kotz, ). n ( a) n( b) n n= ( c) n n z, is the hypergeometric function; and! = a( a+ ) K ( a+ n), is Pochhammer s symbol (Johnson, Kemp, & Figure Some pmf plots of the NPWL distribution with different values of θ and α
Page of. Properties This section presents some statistical properties such as shape, factorial moments, probability generating function, moment generating function and moments of the NPWL ( θ, distribution.. Shape Holgate s theorem states that if g(λ) is the pdf of mixing distribution, which is unimodal and absolutely continuous distribution, then the pmf of the mixed Poisson distribution is unimodal (Holgate, ). According to this theorem, the pmf of NPWL distribution is unimodal because the pdf of NWL distribution is unimodal (Asgharzadeh, Bakouch, Nadarajah, & Sharafi, ). The NPWL distribution, with the shape parameter θ and the scale parameterα, is log-concave ( ) f ( x+ ) f ( x) <, therefore it has an increasing failure rate (Johnson, f ( x+ ) Kemp, & Kotz, ).. Factorial moments Proposition Let X ~ NPWL ( θ,, then the factorial moments of X are Proof form θ (+ Γ( k+ ) Γ( k+ ) µ [ k] = + k+ x+ αθ(+ + α(+ θ θ Γ( k+ ) ( α θ) Γ( k+ ) k+ k+ ( αθ + θ) The kth factorial moment of mixed Poisson distribution can be written in the.
Page of µ = E [ X ) ] [ k] ( k = λ g( λ) dλ. If X ~ NPWL ( θ,, then the factorial moments of X is given by µ [ k] = k θ (+ θαλ λ (+ λ)( e ) e αθ(+ + α(+ θ (+ αθ(+ + α(+ k θλ k θλ k αθ θ λ k αθ θ λ + + ( + ) ( + ) = ( λ e + λ e λ e λ e ) dλ θ (+ Γ( k+ ) Γ( k+ ) Γ( k+ ) = + k+ x+ k+ αθ(+ + α(+ θ θ ( α θ). Probability generating function θλ k dλ Γ( k+ ) k+ ( αθ + θ) Proposition Let X ~ NPWL ( θ,, then the probability generating function (pgf) of X is Proof G (s) = θ (+ θ s+ αθ (+ + α(+ ( θ s+ ) αθ + θ s+ ( αθ + θ s+ ) The pgf of mixed Poisson distribution can be defined as G (s) = E ( s X ) = ( ) e s g( λ) dλ. If X ~ NPWL ( θ,, then the pgf of X is obtained by G (s) = ( s) λ θ (+ θαλ e (+ λ)( e ) e αθ(+ + α(+ θ (+ αθ(+ + α(+ λ θλ dλ., s < θ +. θ s λ θ s λ αθ θ s λ αθ θ s λ ( + ) ( + ) ( + + ) ( + + ) = ( λe + e λe e ) dλ
Page of = = θ (+ αθ (+ + α(+ ( θ s+ ) θ (+ θ s+ αθ (+ + α(+ ( θ s+ ). Moment generating function + θ s+ ( αθ + θ s+ ) αθ + θ s+ ( αθ + θ s+ ) αθ + θ s+, s < θ +. Proposition Let X ~ NPWL ( θ,, then the moment generating function (mgf) of X is M X (t) =. Moments t θ (+ θ e + (+ ) + (+ ) t αθ α α α ( θ e + ) αθ + θ e t ( αθ + θ e The mgf is obtained trivially from the pgf as M X (t) = G (e t ). t + + ), t < log( θ + ). The kth raw moment is obtained by taking the kth derivative of mgf with respect to t and setting t to zero. Let δ = θ (+ αθ(+ + α(+ δ =, NPWL ( θ, then the first four raw moments of X are µ = δ µ = µ = µ = δ, θ + δ + + δ δ, θ δ α α θ and δ = θ ( α θ) θ + θ + δ + + + δ δ δ, θ δ θ + θ + θ + δ + + + + δ δ δ δ. θ δ, if X ~
Page of The central moments of X can be written in term of raw moments as µ k = k E ( X µ ) = Thus, the first four central moments of X are µ =, µ = µ = µ = θ + θ + δ + + δ δ δ δ, θ δ k k j ) µ j j= k j ( µ. θ + θ + δ + + + δ δ δ θ δ ( θ + ) ( δ + + δ ) δ δ + + δ δ, θ δ + θ + θ + θ + δ + + + + δ δ δ δ θ δ θ + θ + δ + + + δ δ δ θ δ δ θ + δ + δ+ + δ δ ( δ δ ) θ δ. The skewness, kurtosis and index of dispersion (ID) of X ~ NPWL ( θ, are µ skewness (X ) = /, µ µ kurtosis (X ) =, µ ID (X ) = = V ( X ) E( X ) δ δδ µ µ = δ δ θ δ δ θ θ θ θ + θ + θδ + δ δ.
Page of The mean, ID, skewness and kurtosis plots of the NPWL distribution are shown in Figure. The Figure illustrates that the mean is decreasing as θ and α increase. For fixed θ, as α increases, the ID is increasing but the ID is decreasing as θ increases, for fixed α. Moreover, we can see that the ID is greater than one, thus the NPWL distribution is overdispersed. While the skewness and kurtosis are increasing as θ and α increase. Figure Mean, ID, skewness and kurtosis plots of the NPWL distribution. Random variate generation In this section, a NPWL random variate generation is indicated. Let X ~ Pois ( λ ) and λ ~ NWL ( θ, α ), then random variables from the NPWL ( θ, α ) distribution can be generated by the following algorithm.. Generate λ i by using rnwlindley function of LindleyR package (Mazucheli, Fernandes and de Oliveira, ) in the R programming language (R Core Team, ).. Generate X i from Pois λ ), i =,,, n.. Parameter Estimation ( i In this section, the parameter estimates of the NPWL distribution are obtained by using maximum likelihood. Let X, X, K, X be independent and identically distributed as NPWL n T distribution with Θ = ( θ,, the parameter vector. Then the likelihood function from () is
Page of and L (Θ) n θ (+ θ + x+ αθ + θ + x+ = + = +. x x i αθ (+ + α(+ ( θ + ) ( αθ + θ + ) The associated log-likelihood function can be expressed as: l (Θ) = n log( θ) + nlog(+ nlog( nlog( θ(+ + + + n i= n xi + n log( ) i= xi + n log( α ) i= n x + + + + + + i x θ x αθ θ α x + i ( )( ) ( )( ) ) log. The score functions are found to be l(θ) = θ l(θ ) = α n n(+ θ θ(+ + + α i xi + n xi + n i= ( α+ ) α i= + n i= n x + + + i ( xi )( θ ) ( α xi + ) ( α+ )( ) xi+ ( xi + )( α ) ( α xi + )( ) n i x i + x i + x + ( α ) + ( α+ )( x + )( + + )( + + ) i i θ xi αθ θ + xi+ xi+ ( xi + )( α ) ( α xi + )( ) n n α+ + ( ) xi + n θ(+ + (+ i= θ α( θ(+ + + αθ + θ + n i= θ( x + )( x ( x i i + )( α ) i xi+ n + )( α ) xi+ ( α x θ( ) i xi+ + )( ) xi+ xi+ The score functions are set equal to zero in order to obtain the parameter estimates of the NPWL distribution. Although these equations are non-linear, the maximum likelihood estimates can be solved by the numerical methods. In this paper,,.
Page of the method of moment estimators are given as the initial values for the BFGS method in the optimx function of optimx package (Nash & Varadhan, ) in the R programing language (R Core Team, ).. Applications We consider the application of the NPWL distribution to two automobile insurance claims data sets. These data sets are overdispersed count data with excess zeros. The competitive distributions are as follows: (i) The Poisson distribution (Poisson, ), with pmf f (x) = λ x e λ, λ > x! (ii) The Poisson-Lindley distribution (Sankaran, ), with pmf θ ( x+ θ + ) f (x) =, θ > x+ ( θ + ) (iii) The Poisson-weighted Lindley distribution (El-Monsef & Sohsah, ; Manesh, Hamzah, & Zamani, ), with pmf + c Γ( x+ c) θ (+ θ + x+ c) f (x) =, c > and θ >. + x+ c x! Γ( c) ( θ + c) ( θ + ) The mixed Poisson distributions have proportion of zeros higher than the Poisson distribution. Thus, they have been applied to overdispersed and zero-inflated data. Puig () proposed the zero-inflation index, which is a measure for detecting zero-inflation. If X is a non-negative integer random variable with mean and proportion of zeros are µ and p, respectively. The zero-inflation index is zi (X ) = log( p ) +. () µ
Page of If X is a Poisson random variable, the zero-inflation index is equal to but if X is a zero-inflated, then the zero-inflation index is greater than. Figure shows the zeroinflation index versus the index of dispersion for the PL, PWL and NPWL distributions. The zero-inflation index of the three distributions are similar. That is, the zero-inflation index is increasing as the index of dispersion increases. Figure Zero-inflation index versus index of dispersion In this paper, the distribution with minimum the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), the negative log-likelihood (-LL) and maximum the p-value based on the Anderson-Darling (AD) goodness of fit test for a discrete distribution (Choulakian, Lockhart, & Stephens, ) is recommended as the most appropriate distribution for fitting overdispersed count data. Moreover, the likelihood ratio (LR) test is employed to compare special cases of the NPWL distribution. Application Lemaire () presented the number of claims from third-party automobile liability in Belgium. This data set is overdispersed with the mean., the variance., the ID. and the zi.. Table Distribution of number of claims from third-party automobile liability in Belgium
Page of Table illustrates that the p-values based on the AD test for the PL and Poisson distributions are less than the % significance level, hence this data set cannot be described by the PL and Poisson distributions. The mean of all distributions are equal to the mean of the data set but the variance, ID and zi of the PWL and NPWL distributions are nearest the variance, ID and zi of the data set. However, the NPWL distribution provides the lowest negative log-likelihood, AIC, BIC and the highest p-value based on the AD test for a discrete distribution. The LR statistic for testing H : PWL vs H : NPWL is. with a p-value.. Thus, there is no statistically significant difference between the PWL and NPWL distributions. Application Tröbliger () as cited in Klugman, Panjer and Willmot () presented the count of motor vehicle insurance claims per policy in Germany during. It has the mean of., the variance of., the ID of. and the zi of.; therefore, this data set is overdispersed. Table Distribution of number of automobile insurance claims in Germany Table shows that the Poisson distribution also cannot describe this data set because its p-value based on the AD test for a discrete distribution is less than the % significance level. The mean of all distributions are also equal to the mean of the data set. The variance and ID of the PL distribution and the data set are equal but the zi of the PWL distribution is closest the zi of the data set. The NPWL distribution also provides the lowest negative log-likelihood and the highest p-value based on the AD
Page of test for a discrete distribution but the PL distribution provides the lowest AIC and BIC. However, the number of parameters of the PL distribution is less than the NPWL distribution. In order to test H : PL vs H : NPWL, the LR statistic is. with a p-value.. Hence, there is no statistically significant difference between the PWL and NPWL distributions. The LR statistic for testing H : PWL vs H : NPWL is. with a p-value.. There is also no statistically significant difference between the PWL and NPWL distributions. The log expected values of the distributions and the log observed values are shown in Figure. The Figure shows that the log expected values of the NPWL and PWL distributions are close to the observed values in both data sets. However, the log expected values of the Poisson distribution are far from observed values. Thus, the Poisson distribution cannot be applied to describe the overdispersed count data. Figure Log plots of the expected values and the observed values. Conclusions In this paper, an alternative mixed Poisson distribution, namely the new Poisson mixed weighted Lindley (NPWL) distribution, is introduced. The NPWL distribution is derived from the Poisson distribution where the parameter follows the new weighted Lindley distribution. The PL and PWL distributions are special cases of the NPWL distribution. Also, the pmf of NPWL distribution is log-concave and unimodal. Some statistical properties were studied such as shape, factorial moments, probability generating function, moment generating function and moments. Parameter estimation
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Page of Table Distribution of number of claims from third-party automobile liability in Belgium Number of Observed claims frequencies Estimated parameters Mean Variance ID zi -LL AIC BIC AD statistic p-value Expected frequencies Poisson PL PWL NPWL........................ λˆ =....... <. θˆ =.......... ĉ =. θˆ =.......... θˆ =. αˆ =..........
Page of Table Distribution of number of automobile insurance claims in Germany Number of Observed claims frequencies Estimated parameters Mean Variance ID zi -LL AIC BIC AD statistic p-value Expected frequencies Poisson PL PWL NPWL............................. λˆ =........ <. θˆ=.......... ĉ =. θˆ=.......... θˆ =. αˆ =..........
Page of f(x)..... θ =., α =. x f(x)..... θ =, α =. x (a) ID =., skewness =., kurtosis =. (b) ID =., skewness =., kurtosis =. f(x)..... θ =., α =. x x f(x)..... θ =, α = (c) ID =., skewness =., kurtosis =. (d) ID =., skewness =., kurtosis =. f(x)..... θ =, α =. x f(x)..... θ =, α = x (e) ID =., skewness =., kurtosis =. (f) ID =., skewness =., kurtosis =. Figure Some pmf plots of the NPWL distribution with different values of θ and α
Page of mean alpha alpha mean theta skewness skewness theta kurtosis ID ID alpha alpha theta kurtosis Figure Mean, ID, skewness and kurtosis plots of the NPWL distribution theta
Page of zero-inflation index..... ID NPWL PWL PL Figure Zero-inflation index versus index of dispersion
Page of log (frequency) Data set x observed NPWL PWL PL POISSON log (frequency) Data set Figure Log plots of the expected values and the observed values x observed NPWL PWL PL POISSON