Zero inflated negative binomial-generalized exponential distribution and its applications

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1 Songklanakarin J. Sci. Technol. 6 (4), , Jul. - Aug Original Article Zero inflated negative binomial-generalized eponential distribution and its applications Sirinapa Aryuyuen, Winai Bodhisuwan*, and Thidaporn Supapakorn Department of Statistics, Faculty of Science, Kasetsart University, Chatuchak, Bangkok, Thailand. Received December 01; Accepted 16 April 014 Abstract In this paper, we propose a new zero inflated distribution, namely, the zero inflated negative binomial-generalized eponential (ZINB-GE) distribution. The new distribution is used for count data with etra zeros and is an alternative for data analysis with over-dispersed count data. Some characteristics of the distribution are given, such as mean, variance, skewness, and kurtosis. Parameter estimation of the ZINB-GE distribution uses maimum likelihood estimation (LE) method. Simulated and observed data are employed to eamine this distribution. The results show that the LE method seems to he highefficiency for large sample sizes. oreover, the mean square error of parameter estimation is increased when the zero proportion is higher. For the real data sets, this new zero inflated distribution provides a better fit than the zero inflated Poisson and zero inflated negative binomial distributions. Keywords: overdispersion, zero inflated distribution, count data with etra zeros, random variate generation, LE 1. Introduction Poisson distribution provides a standard model for the analysis of count data with the assumption of equal mean and variance. However, in practice, count data often shows overdispersion, the variance is greater than the mean. In practice, negative binomial (NB) distribution was introduced to solve this problem, and it has become increasingly popular as a more fleible alternative to fit models. The NB distribution is a better fit for over-dispersed count data which is not necessarily hey-tailed (Wang, 011). For over-dispersed count data, some mied NB distributions offer a better fit when compared with the Poisson and NB distributions. Such findings are related to those of Gomez-Dèniz et al. (008), Zamani and Ismail (010), Wang (011), and Pudprommarat et al. (01). Recently, a negative binomial-generalized eponential (NB-GE) distribution was * Corresponding author. address: fsciwnb@ku.ac.th employed to fit count data. It was obtained by miing the NB distribution with a generalized eponential (GE) distribution. Also, the NB-GE distribution is more appropriate to fit count data with overdispersion and hey-tailed datasets than the Poisson and NB distributions (Aryuyuen and Bodhisuwan, 01). oreover, in practice, one frequent demonstration of overdispersion is a zero count incidence. Count data with etra zeros occur in many fields, such as public health, epidemiology, medicine, sociology, engineering and agriculture. Standard discrete distribution may fail to fit such data either because of zero inflation or over/underdispersion. Now, there is increased interest in a zero inflated distribution to account for etra zeros in data (Xie et al., 009). Count data with etra zeros create problems of violating basic assumptions implicit in the standard distribution. Failure to account for etra zeros may result in biased parameter estimates and misleading inference. Because the zero inflated distributions usually provide better statistical fit, some researchers, e.g., Lambert (199), Greene (1994), Hall (000), Famoye and Singh (00), Bodhisuwan (011) proposed these distributions.

2 484 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , 014 As mentioned above, the mied distribution defines one of the most important ways to obtain a new probability distribution in applied probability and operational research (Gomez-Dèniz et al., 008). For the purpose, we are looking for a new zero inflated distribution which is a more fleible alternative to fit count data with ecess zeros. The rest of this article is organized as follows. We propose the new zero inflated distribution that is a zero inflated negative binomial-generalized eponential (ZINB- GE) distribution. Some characteristics, graphs of probability mass function (pmf) and a random variate generation of ZINB-GE distribution are introduced in Section. In Section the maimum likelihood estimation (LE) method is handled to estimate the parameters of ZINB-GE distribution. Numerical eamples are provided in Section 4, including simulated data which are utilized to eamine the efficiency of LE method, and real data sets that are used to evaluate the performance of the proposed distribution. Finally, some conclusions are presented in Section 5.. New zero inflated negative binomial distribution The zero inflated (ZI) distribution can be used to fit count data with etra zeros, and assumes that the observed data are the result of a two-part process; a process that generates structural zeros and a process that generates random counts. The model of ZI can be summarized as follows: P( X ) ( ) (1 ) f ( ; ), (1) 0 where X is the count variable, is the etra proportion of zeros, f ( ; ) is the pmf of X with the parameter, and 0( ) 1 if 0; otherwise, ( ) The ZINB-GE distribution The ZINB-GE distribution is a new miture distribution, combining Bernoulli and NB-GE distributions. First, we introduce a definition and some characteristics of NB-GE distribution as follows. Definition 1 Let Y be a random variable of the NB-GE distribution. Random variable Y has the NB distribution with the parameters r and p ep( ), where is distributed as the GE distribution with the positive parameters and, i.e., y ~ NB( r, p ep( )) and ~ GE(, ). The pmf of Y is given by y r y 1 y f ( y) ( 1) ( r ), y 0,1,,, () y 0 Γ( 1)Γ 1 u / for,, 0 ( u r ) Γ u / 1 where Γ( ) is a gamma function denoted as for t 0, and () 1 Γ( ) t y t y e dy, 0 Net, the mean and variance of Y are respectively given by where is ( u) E( Y ) r( 1) and (1) Var( Y ) Y r r 1 () r r (1) 1 (1) ( u), ( 1) (1 u / ). (4) ( u / 1) Now, by using the epression (1) and Definition 1, we obtain the closed formulas for the pmf and some characteristics of ZINB-GE as follows. Definition Let X be a random variable of ZINB-GE distribution with the parameters r,, and, denoted as X~ ZINB-GE( r,,, ). The pmf of X is given by (1 ) ( r) if 0, f ( ) 1 r (1 ) ( 1) ( r ) if 1,, 0 where 0 1, r,, 0, and ( u) is defined in (). The graphs of ZINB-GE s pmf distribution with specified parameters r,, and, offered in Figure 1, shows that the distribution will be flat when parameter is increasing. However, the distribution will be flat when the parameter is decreasing. In addition, the pmf of ZINB-GE distribution can take different shapes when values of differ. Theorem 1 If X~ ZINB-GE(, r,, ) then some characteristics of X are as follows. (a) The mean and variance of X are given by E( ) 1 r r and Var( X ) X (1) where ( u) is defined in (4), and (5) X, (6) X r ( r 1) () (r 1) (1) r 1 r( (1) 1) 1. (b) The skewness and kurtosis of X are Sk( X ) r ( r 1)( r ) ( r 1) (r r 1) r 1 r 1 () () (1) (1) ( r 1) () (r 1) (1) r 1 r (1) 11 / X, (7) Ku( X ) (4) () r ( r 1)( r )( r ) ( r 1)(r 7r 6) ( r 1) (6r 1r 7) () (r 1)( r r 1) (1) r 1 4r (1) 1 1/ ( r 1)( r ) () ( r 1) () (r r 1) (1) r 1 6r (1) () (1) (1) X ( r 1) (r 1) r 1 r 1 1 /. (8)

3 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , Figure 1. The pmf of ZINB-GE distribution with specified parameters The proof of Theorem 1 is in the Appendi.. Random variate generation of ZINB-GE distribution To generate a random variable X from the ZINB-GE (, r,, ), one can use the following algorithm: 1) Generate U from the uniform distribution, U(0,1). 1 log 1 U from the GE distribution, 1/ ) Set ~GE(, ). ) Generate Y from the NB( r, p ep( ) ) distribution. * 4) Generate U from the uniform distribution, U(0,1). 5) If * U, then set X Y ; otherwise, X = 0.. Parameter estimation of ZINB-GE distribution In this section, the parameter estimation of ZINB-GE distribution via LE procedure is provided. The likelihood function of ZINB-GE (, r,, ) is given by n L(, r,, ) I (1 ) I i1 ( i 0) ( r), r 1 i i i (1 ) ( 1) ( i 0) ( r ) i 0 then we can write the log-likelihood of the ZINB-GE (, r,, ) as log L(, r,, ) n I log (1 ) i1 ( i 0) ( r ) I log(1 ) log Γ( r ) log Γ( r) log Γ( 1) ( 0) i i i i i Γ 1 ( r ) / log Γ( 1) log ( 1) 0 Γ ( r ) / 1. By differentiating the log-likelihood function of ZINB-GE distribution, partial derivatives of the log likelihood function with respect to, r, and are given by n r r I( 0) i1 r r ( 0) Γ / 1 (1 )Γ( 1)Γ 1 / 1 i i Γ / 1 Γ( 1)Γ 1 / 1 I n (1 )Γ( 1) Γ 1 r / I( i 0) r i1 (1 ) ( r ) r Γ r / 1 1 I ( r ) ( ) (,, ) ( i 0) i r r ( r,, ) r Γ ( s) where ( s), and Γ( s) i i Γ 1 ( r ) / ( r,, ) ( 1) 0 Γ ( r ) /. 1,

4 486 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , 014 n (1 )Γ 1 r / Γ( 1) I( i 0) i1 (1 ) ( r ) Γ r / 1 1 I ( 1) (,, ) ( i r 0) ( r,, ), n (1 )Γ( 1) Γ 1 r / I( i 0) i1 (1 ) ( r) Γ r / 1 1 I (,, ) ( i r 0). ( r,, ) The LE solutions of the parameter estimates can be obtained by using numerical optimization with the nlm function in the R program (R Core Team, 01). The R code of parameter estimation of ZINB-GE distribution using the LE method is given in the Appendi. 4. Numerical study This section presents the efficiency of the LE method for parameter estimation of ZINB-GE distribution by using simulated data. In addition, we illustrate the application study of ZINB-GE distribution compared to the zero inflated Poisson (ZIP) and zero inflated negative binomial (ZINB) distributions. 4.1 Simulation study In illustrating the simulation study, the sample data generated from the ZINB-GE distribution with specified parameters (r = 10, = 10, = 10) and four values for proportion of zero (0., 0.4, 0.6, 0.8) for the sample sizes (n) as 50, 100 and 00, respectively. In each situation, the parameters are estimated from 500 replications. Hence the maimum likelihood estimators may be biased. The biased value is a difference value between the estimator and true parameter values. The sample erage of the estimated parameter, bias, variance, standard deviation (SD), and mean squared error (SE) are computed by the formulas: ˆ ˆ t1 t, Bias( ˆ ) ˆ, ˆ Var( ) ( ˆ ˆ ) t t 500 1, SD( ˆ ) Var( ˆ ) and SE( ˆ ) Var( ˆ ) ˆ Bias ( ). Table 1 illustrates statistic values from the results of studies. From these results, LE seems to he highefficiency when the sample size is large. The efficiency of LE seems to be poor for small sample sizes. In addition, the SE of parameter estimation is increasing when the zero proportion is higher as in Figure, which displays the graphs of SE for parameter estimates of ZINB-GE distribution. 4. Application study We used two data sets for this part of the analysis to illustrate the applications of the ZINB-GE distribution. The first data set has the number of hospital stays by United States residents aged 66 and over (see Flynn, 009), which is shown in Table. This data has 80.7% of zeros and the sample inde of dispersion is In addition, Table shows the number of units of consumers good purchased by households over 6 weeks (see Lindsey, 1995). This data set has 80.60% of zeros and the sample inde of dispersion is.7. These eamples he sample indices of dispersion bigger than 1 and a high percentage of zeros. For model selection in this study, we use the criteria of AIC (Akaike information criterion) and BIC (Bayesian information criterion), the goodness of fit test (chi-squared test: -test) is used to compare between observed and epected values of data. From the results in Table -, we found that the AIC and BIC values for ZINB-GE distribution are the smallest when compared with eisting models. Thus, the ZINB-GE distribution can be chosen as the best model. Also, based on the p-values of chi-squared test, the proposed distribution is appropriate to fit the data unlike the ZIP and ZINB distributions. These three criteria indicate that the ZINB-GE distribution is the best fit, while the ZIP and ZINB distributions are very poor fits. 5. Conclusions This work proposes the new zero inflated distribution, which is called the zero inflated negative binomial-general- Figure. The SE of parameter estimates of ZINB-GE distribution

5 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , Table 1. The parameter estimates of ZINB-GE distribution with different parameter n = 50 n = 100 n = 00 Parameter Estimate SE Estimate SE Estimate SE (SD) (SD) (SD) = (0.08) (0.07) (0.05) r (7.6) (5.04) (.87) (8.06) (5.54) (4.47) (9.1) (5.98) (4.01) = (0.09) (0.07) (0.05) r (8.49) (5.55) (4.01) (9.0) (6.67) (4.85) (11.0) (6.58 ) (4.57) = (0.09) (0.06) (0.05) r (10.85) (6.75) (4.07) (1.) (7.6) (5.15) (14.87) (8.4) (4.7) = (0.08) (0.05) (0.0) r (11.86) (7.7) (5.58) (1.9) (1.74) (5.58) (16.9) (8.9) (6.1) ized eponential distribution. In particular, the closed form and some characteristics of the proposed distribution are introduced. Parameter estimation is also implemented by using LE, and the usefulness of the ZINB-GE distribution is illustrated by real and generated data. Based on the results, the proposed distribution is the best fit while the ZIP and ZINB distributions are very poor fits for count data with etra zeros. In conclusion, the ZINB-GE distribution is a fleible alternative for analysis of count data characterized by etra zeros. Also, the LE method seems to he high-efficiency for large sample sizes, and the SE of parameter estimation increases when the zero proportion is higher. Acknowledgements The authors wish to gratefully acknowledge to the referee of this paper who helped to clarify and improve the presentation. The first author would like to thank the Research Professional Development Proect under the Science Achievement Scholarship of Thailand (SAST) for funding. References Aryuyuen, S. and Bodhisuwan, W. 01. The negative binomial-generalized eponential (NB-GE) distribution. Applied athematical Sciences. 7,

6 488 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , 014 Table. Observed and epected frequencies of the number of hospital stays of United States residents aged 66 and over Number of hospital stays Observed value Epected value by fitting distribution ZIP ZINB ZINB-GE Parameter estimates ˆ ˆ ˆ ˆ ˆr.968 ˆr.040 ˆp ˆ 1.01 ˆ 7.57 AIC BIC Degree of freedom p-value < < Table. Observed and epected frequencies of the number of units of consumers good purchased by households over 6 weeks Units of consumers good Households/ Observed value Epected value by fitting distribution ZIP ZINB ZINB-GE Parameter estimates ˆ ˆ ˆ ˆ.0609 ˆr ˆr ˆp 0.6 ˆ 0.45 ˆ AIC BIC Degree of freedom p-value < <

7 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , Bodhisuwan, W Zero inflated Waring distribution and its application. Procedings of the 7th Congress on Science and Technoloby of Thailand, Thailand. Famoye, F. and Singh, K.P. 00. On inated generalized Poisson regression models. Advances and Applications in Statistics., Flynn, ore fleible GLs zero-inflated models and hybrid models. Casualty Actuarial Society E- Forum, Winter, U.S.A., pp Gomez-Dèniz E., Sarabia, J.. and Calderín-Oeda, E Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications. Insurance athematics and Economics, 4, Greene, W.H Accounting for ecess zeros and sample selection in Poisson and negative binomial regression models. Working paper No , Department of Econometrics, Stern School of Business, New York University, New York, U.S.A. Hall, D.B Zero-inated Poisson and binomial regression with random effects: a case study. Biometrics. 56, Lambert, D Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics. 4, Lindsey, J. K odelling Frequency and Count Data. Oford science publications, Clarendon Press, UK., p Pudprommarat, C., Bodhisuwan, W. and Zeephongsekul, P. 01. A new mied negative binomial distribution. Journal of Applied Sciences. 17, R Core Team, 01. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Wang, Z One mied negative binomial distribution with application. Statistical Planning and Inference. 141, Xie, F., Wei B. and Lin, J Score tests for zero-inflated generalized Poisson mied regression models. Computational Statistics and Data Analysis. 5, Zamani, H. and Ismail, N Negative binomial-lindley distribution and its application. Journal of athematics and Statistics. 1, 4-9.

8 490 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , 014 Proof of Theorem 1: Appendi From Definition 1, let Y~ NB-GE( r,, ) be a random variable of the NB-GE distribution and some epected values of Y are E( Y ) ( r r) ( r r) r, () (1) E( Y ) ( r r r) (r 6r r) (r r r) r, () () (1) E( Y ) ( r 6r 11r 6 r) (4r 18r 6r 1 r) (4) () (6r 18r 19r 7 r) (4r 6r 4 r r) r If X~ ZINB-GE(, r,, ), then the epected value of X is given by E( X ) f ( ) 1 (0) (1 r ) ( ) (1 r ) ( 1) ( r ) y r y 1 y From E( Y ) y f ( y) y ( 1) ( r ) r( (1) 1) y0 y1 y 0., we he E( X ) (1 )E( Y ) (1 ) r (1) 1. Consequently, we obtained the epected values of X as follows E( X ) E( X ) 4 E( X ) r 1 (1 ) ( 1) ( r ) 1 0 Y r (1 ) ( 1) r 1 ( r ) 4 (1 ) ( 1) ( r ) 1 0 From Var( ) E( X ) E( X ), the variance of X is X (1 )E( ), (1 )E( Y ), 4 (1 )E( Y ). Var( ) r ( r 1) () (r 1) (1) r 1 r( (1) 1) 1. The skewness and kurtosis of X are X Sk( X ) [E( X ) E( X ) E( X ) E( X ) ] / X, Ku( X ) [E( X ) 4E( X )E( X ) 6E( X ) E( X ) E( X ) ] / X. Now, by using the epression as below, the skewness and kurtosis of X can be written in (7) and (8), respectively.

9 S. Aryuyuen et al. / Songklanakarin J. Sci. Technol. 6 (4), , The R code of the parameter estimation of the ZINB-GE distribution with the LE method: mlogl<-function(theta,){ fzinbge<-function(theta,){ mm<-length(); k<-numeric(mm); zinbge<-function(theta,){ if(==0){ p<-(-log(theta[4]+((1-theta[4])*((gamma(theta[]+1) *gamma(1+theta[1]/theta[]))/gamma(theta[]+theta[1] /theta[]+1))))); }else if(>0){ pp1<-(gamma(theta[]+1)*(gamma(1+theta[1]/theta[]))) /(gamma(theta[]+theta[1]/theta[]+1)); for( in 1:){ p1<-((factorial()/(factorial()*factorial(-))) *(-1)^)*((gamma(theta[]+1)*gamma(1+(theta[1]+) /theta[]))/gamma(theta[]+(theta[1]+)/theta[]+1)); pp1<-pp1+p1; } p<-(-log(1-theta[4]))-log(factorial(theta[1]+-1)) +log(factorial(theta[1]-1))+log(factorial())-log(pp1); }p} for(i in 1 : length()){ k[i]<-zinbge(theta,[i])}k} sum(fzinbge(theta,))} theta.start<-c(start_r,start_alpha,start_beta,start_phi) out<-nlm(mlogl,theta.start, = ) r_le<-out$estimate[1] alpha_le<-out$estimate[] beta_le<-out$estimate[] phi_le<-out$estimate[4]