The powerCauchy negativebinomial: properties and regression
 Muhammad Zubair^{1},
 Muhammad H. Tahir^{2},
 Gauss M. Cordeiro^{3}Email author,
 Ayman Alzaatreh^{4} and
 Edwin M. M. Ortega^{5}
https://doi.org/10.1186/s4048801700823
© The Author(s) 2018
Received: 24 January 2017
Accepted: 11 December 2017
Published: 8 January 2018
Abstract
We propose and study a new compounded model to extend the halfCauchy and powerCauchy distributions, which offers more flexibility in modeling lifetime data. The proposed model is analytically tractable and can be used effectively to analyze censored and uncensored data sets. Its density function can have various shapes such as reversedJ and rightskewed. It can accommodate different hazard shapes such as decreasing, upsidedown bathtub and decreasingincreasingdecreasing. Some mathematical properties of the new distribution can be determined from a linear combination for its density function such as ordinary and incomplete moments. The performance of the maximum likelihood method to estimate the model parameters is investigated by a simulation study. Further, we introduce the new logpowerCauchy negativebinomial regression model for censored data, which includes as submodels some widely known regression models that can be applied to censored data. Four real life data sets, of which one is censored, have been analyzed and the new models provide adequate fits.
Keywords
AMS Subject Classification
Introduction
respectively.
In this paper, we define a new fourparameter generalization of the PC distribution named the powerCauchy negativebinomial (PCNB) model. The new distribution is flexible to model complex positive real data sets, i.e., it can have decreasing, UBT shaped and decreasingincreasingdecreasing hazard rate functions (hrfs). It thus provides a good alternative to several wellknown life distributions.
The paper is unfolded as follows. In “The proposed model” section, we define the PCNB distribution. In “Properties of the new model” section, we obtain some of its mathematical properties including quantile function (qf), tail behaviors, a useful linear representation for its density function and some types of moments. In “Estimation” section, the model parameters are estimated by maximum likelihood and a simulation study is performed. In “Regression model” section, we present a regression model based on the PCNB distribution with censored data. In “Applications” section, the usefulness of the new distribution is illustrated by means of four real data sets where we show empirically that it outperforms some wellknown lifetime distributions. Finally, “Concluding remarks” section offers some concluding remarks.
The proposed model
General Insurance companies typically face two major problems when they want to use past or present claim amounts in forecasting future claim severity. First, they have to find an appropriate statistical distribution for their large volumes of claim amounts. Then, test how well this statistical distribution fits their claim data. Most data in general insurance problems is skewed to the right and therefore most distributions that exhibit this characteristic can be used to model the claim severity. Insurance data contains relatively large claim amounts, which may be infrequent. Hence, there is a clear need to use statistical distributions with relatively heavy tails and highly skewed like the PC distribution.
Large claims play a special role because of their importance financially. It is also hard to assess their distribution. They do not occur very often, and historical experience is therefore limited. Insurance companies may even cover claims larger than anything that has been seen before. How should such situations be tackled? The simplest would be to fit a parametric family and try to extrapolate beyond past experience. That may not be a very good idea. A generalization of the PC distribution may fit well in the central regions without being reliable at all at the extreme right tail, and such a procedure may easily underestimate big claims severely.
 (i)
If p=n=α=1, it gives the halfCauchy (HC) distribution;
 (ii)
If α=1, it reduces to the halfCauchy negative binomial (HCNB) distribution;
 (iii)
If n=1, it gives the PCgeometric distribution;
 (iv)
If α=n=1, it becomes the HCgeometric distribution;
 (v)
If p=1, it reduces to the exponentiatedPC distribution;
 (vi)
If p=n=1, it becomes the PC distribution.
Note that the special models given in (ii), (iii) and (iv) do not exist in the literature.
Properties of the new model
In this section, we provide some structural properties of the new distribution.
Quantile function and random number generation
We can easily generate PCNB random variables from (9).
Tail behaviors
where A=(2p/π)^{ n } and B=2n/(p π). For example, for fixed values of n and p, the left and right tails of the PCNB distribution are heavier when α increases. Also, for fixed values of α and p, the left tail becomes heavier when n increases.
Moments
where B_{ j }(n,p)=(n)_{ j } p^{ n }(1−p)^{ j }/j! (for j≥0) and G_{ PC }(z;α,σ) is the cdf given in Eq. (1).
where h_{n+j}(z)=h_{n+j}(z;α,σ) is the EPC density function with power parameter n+j given by Eq. (4). Equation (11) reveals that the PCNB density is a linear combination of EPC densities. So, some mathematical properties of Z can be obtained from those of the EPC distribution. Next, we provide two examples.
respectively, where a_{0}(s)=1, a_{1}(s)=s/3, a_{2}(s)=s(5s+7)/90, etc, and D_{ z }=2 π^{−1} tan−1(z/σ)^{ α }.
The first incomplete moment m_{1}(q) follows from Eq. (15) for r=1. It is useful to obtain the Bonferroni and Lorenz curves and mean deviations for the new model.
Estimation
Several approaches for parameter point estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimates (MLEs) enjoy desirable properties that can be used when constructing confidence intervals for the model parameters. Large sample theory for these estimates delivers simple approximations that work well in finite samples. The normal approximation for the MLEs in distribution theory is easily handled either analytically or numerically.
Equation (16) can be maximized either directly by using wellknown computing platforms such as the R (optim function), SAS (PROC NLMIXED) and Ox program (subroutine MaxBFGS). These scripts can be applied and executed for a wide range of initial values. This process often leads to more than one maximum. However, in these cases, we consider the MLEs corresponding to the largest value of the loglikelihood statistics. In a few cases, no maximum is identified for the selected initial values. In these cases, new initial values can be tried in order to obtain a maximum. There exist sufficient conditions for the existence of the MLEs such as compactness of the parameter space and the concavity of the loglikelihood function. These estimates can exist even when such conditions are not satisfied. For more complex models, and in particular when there is no explicit solution, it is nearly impossible to establish theoretical conditions on the existence and uniqueness of the MLEs. However, such properties can be investigated numerically for this distribution and a given data set.
For interval estimation on the model parameters, we can evaluate the estimated observed information matrix \(J(\widehat {\boldsymbol {\theta }}\)) numerically. Further, we can easily check if the fit using the PCNB model is statistically “superior” to the fits using any of its six special models. For example, for comparing the PCNB and HC distributions, i.e., testing the null hypothesis H_{0}:p=n=α=1 against H_{1}:H_{0} is false, the likelihood ratio (LR) statistic is given by \(w = 2\{\ell (\widehat {{\theta }})  \ell (\widetilde {{\theta }})\}\), where \(\widehat {{\theta }}\) and \(\widetilde {{\theta }}\) are the unrestricted and restricted estimates obtained by maximizing ℓ=ℓ(θ) under H_{1} and H_{0}, respectively. The limiting distribution of this statistic is \(\chi _{3}^{2}\) under the null hypothesis, which is rejected if w exceeds the upper 100(1−γ)% quantile of the \(\chi _{3}^{2}\) distribution.
The above loglikelihood can be maximized numerically to obtain the MLEs. We use the optim routine in the R software.
Monte Carlo simulation results: Biases, MSEs and CPs
I  II  III  

Parameter  m  Bias  MSE  CP  Bias  MSE  CP  Bias  MSE  CP 
p  50  − 0.233  0.218  0.94  − 0.021  0.169  0.94  0.205  0.195  0.93 
100  − 0.209  0.200  0.96  0.016  0.154  0.98  0.154  0.147  0.96  
200  − 0.188  0.183  0.96  0.055  0.150  0.96  0.111  0.100  0.95  
500  − 0.142  0.143  0.95  0.054  0.127  0.96  0.092  0.080  0.95  
n  50  − 0.038  0.045  0.96  0.076  0.172  0.98  1.104  0.896  0.95 
100  − 0.036  0.024  0.98  − 0.016  0.022  0.96  0.983  0.521  0.95  
200  − 0.034  0.012  0.95  − 0.007  0.014  0.95  0.514  0.165  0.96  
500  − 0.027  0.006  0.95  − 0.010  0.006  0.95  − 0.024  0.013  0.95  
α  50  0.124  0.071  0.99  0.370  0.217  0.98  0.077  0.153  0.95 
100  0.072  0.025  0.98  0.172  0.162  0.98  0.005  0.062  0.97  
200  0.046  0.010  0.96  0.093  0.074  0.97  − 0.012  0.031  0.95  
500  0.028  0.004  0.95  0.061  0.034  0.95  − 0.024  0.013  0.95  
σ  50  0.312  0.573  0.93  − 0.033  0.351  0.92  0.069  1.010  1.00 
100  − 0.042  0.290  0.95  − 0.025  0.219  0.96  0.083  1.000  1.00  
200  − 0.096  0.107  0.96  − 0.018  0.168  0.96  0.101  0.902  0.99  
500  − 0.107  0.098  0.95  − 0.004  0.113  0.98  0.100  0.725  0.95 
Regression model
In many practical applications, the lifetimes are affected by explanatory variables such as the cholesterol level, blood pressure, weight and many others. Parametric models to estimate univariate survival functions and for censored data regression problems are widely used. A regression model that provides a good fit to lifetime data tends to yield more precise estimates of the quantities of interest.
In applications in the area of survival analysis, the hrf is often Ushaped or unimodal, i.e., the function is not monotonic. The regression models commonly used for survival data are the logWeibull, monotonic failure rate, loglogistic, decreasing failure rate and unimodal functions. One of the objectives of this work is to propose a new regression model, in location and scale form, called the logpowerCauchy negativebinomial (LPCNB) regression model, which presents different failure rate functional forms. The proposed model is an alternative to the traditional extreme value (or logWeibull), logistic and lognormal models, among others. One way to study the effect of these explanatory variables on the response variable Y is through a locationscale regression model, also known as a model of accelerated lifetime. These models consider that the response variable belongs to a family of distributions characterized by a location parameter and a scale parameter. Further details on this class of regression models can be found in Cox and Oakes (1984), Kalbfleisch and Prentice (2002) and Lawless (2003). In the context of survival analysis, some distributions have been used to analyze censored data. For example, more recently, Cruz et al. (2016) proposed the logodd loglogistic Weibull regression model with censored data, Lanjoni et al. (2016) defined an extended Burr XII regression model and Ortega et al. (2016) introduced the odd BirnbaumSaunders regression model with applications to lifetime data. In a similar manner, we define a locationscale regression model using the LPCNB regression model.
Let Z∼PCNB(n,p,α,σ) be a random variable having the density (7). A class of regression models for location and scale is characterized by the fact that the random variable Y= log(Z) has a distribution with location parameter μ(v), which depends only on the explanatory variable vector, and a scale parameter a. Then, we can write Y=μ(v)+aW, where a>0 and the distribution of W does not depend on v.
where n>0 and p∈(0,1) are shape parameters, \(\mu \in \mathbb {R}\) is the location parameter and a>0 is the scale parameter.
We refer to Eq. (17) as the LPCNB distribution, say Y∼LPCNB(n,p,μ,a). If Z∼PCNB(n,p,α,σ), then Y= log(Z)∼LPCNB(n,p,μ,a).
where the random error w_{ i } has density function (19), τ=(τ_{1},…,τ_{ p })^{ T }, a>0, n>0 and p∈(0,1) are unknown parameters. The parameter \(\phi _{i}=\mathbf {v}_{i}^{T} \tau \) is the location of y_{ i }. The location parameter vector ϕ=(ϕ_{1},…,ϕ_{ m })^{ T } is represented by a linear model ϕ=vτ, where V=(v_{1},…,v_{ m })^{ T } is a known model matrix. The LPCNB model (20) opens new possibilities for fitting many different types of data.
where q is the number of uncensored observations (failures) and \(w_{i}={\left (y_{i}\textbf {v}_{i}^{T} {\boldsymbol {\tau }}\right)}/{a}\). The MLE \(\widehat {\boldsymbol {\theta }}\) of θ can be evaluated by maximizing the loglikelihood (21). We use the procedure NLMixed in SAS to calculate \(\widehat {\boldsymbol {\theta }}\). Initial values for τ and a are taken from the fit of the LPC regression model with p=n=1.
The elements of the (p+3)×(p+3) observed information matrix J(θ), namely J_{ pp },J_{ pn }, \(\phantom {\dot {i}\!}J_{pa},J_{p\tau _{j}},J_{nn},J_{na}, J_{n \tau _{j}},J_{aa},J_{a \tau _{j}}\) and \(J_{\tau _{j}\tau _{s}}\phantom {\dot {i}\!}\) (for j,s=1,…,p), can be evaluated numerically. Inference on θ can be conducted in the classical way based on the approximate multivariate normal \(N_{p+3}\left (0,J(\widehat {\boldsymbol {\theta }})^{1}\right)\) distribution for \(\widehat {\boldsymbol {\theta }}\).
We can use the likelihood ratio (LR) statistic for comparing some special models with the LPCNB regression model. We consider the partition \(\boldsymbol {\theta }=\left (\boldsymbol {\theta }_{1}^{T},\boldsymbol {\theta }_{2}^{T}\right)^{T}\), where θ_{1} is a subset of parameters of interest and θ_{2} is a subset of remaining parameters. The LR statistic for testing the null hypothesis \(H_{0}:{\boldsymbol {\theta }}_{1} ={\boldsymbol {\theta }}_{1}^{(0)}\) versus the alternative hypothesis \(H_{1}:{\boldsymbol {\theta }}_{1} \neq {\boldsymbol {\theta }}_{1}^{(0)}\) is given by \(w= 2\{\ell (\widehat {\boldsymbol {\theta }})\ell (\widetilde {\boldsymbol {\theta }})\}\), where \(\widetilde {\boldsymbol {\theta }}\) and \(\widehat {\boldsymbol {\theta }}\) are the estimates under the null and alternative hypotheses, respectively. The statistic w is asymptotically (as n→∞) distributed as \(\chi _{q}^{2}\), where q is the dimension of the subset of parameters θ_{1} of interest.
Applications
respectively, where K_{1}=2^{ a }/[σ π^{ a } B(a,b)] and K_{2}=a b 2^{ a }/(σ π^{ a }).
MLEs, their SEs (in parentheses) and goodnessoffit measures for the first data set
Distribution  Estimates  AIC  KS  pvalue  

PCNB(α, p, n, σ)  1.8529  0.0029  0.3905  3.5171  487.1582  0.0791  0.9131 
(0.3737)  (0.0015)  (0.1375)  (1.6477)  
BW(a, b, c, λ)  8.9783  0.10422  0.5264  0.3086  489.5797  0.1028  0.6655 
(3.9535)  (0.0224)  (0.0250)  (0.0033)  
PCG(α, p, σ)  1.182  71.1917  0.9153  492.4754  0.0911  0.8009  
(0.1415)  (6.7322)  (0.9334)  
BHC(a, b, σ)  1.5514  0.9514  11.1816  499.1124  0.1256  0.4096  
(0.6308)  (0.3152)  (9.3295)  
KHC(a, b, σ)  1.3321  0.9188  14.3689  497.3825  0.1528  0.1936  
(0.8090)  (0.3935)  (7.9620)  
PC(α, σ)  1.0127  25.1088  492.7543  0.0877  0.8360  
(0.1271)  (5.2807) 
MLEs, their SEs (in parentheses) and goodnessoffit measures for the second data set
Distribution  Estimates  AIC  KS  pvalue  

PCNB(α, p, n, σ)  1.6228  0.0035  0.7221  2.8888  2364.093  0.046  0.7588 
(0.1972)  (0.0009)  (0.1969)  (1.7041)  
BW(a, b, c, λ)  7.6164  0.1194  0.568  1.1134  2367.086  0.0767  0.1632 
(1.5171)  (0.0088)  0.0025)  (0.0033)  
PCG(α, p, σ)  1.3668  76.7098  2.9658  2368.893  0.0483  0.6507  
(0.0905)  (131.7613)  (3.3747)  
BHC(a, b, σ)  1.7824  0.9400  22.1049  2388.513  0.1029  0.0219  
(0.2498)  (0.1014)  (4.6448)  
KHC(a, b, σa)  1.4395  1.1559  36.2886  2375.904  0.0837  0.1013  
(0.1744)  (0.1462)  (7.4700)  
PC(α, σ)  1.1812  53.1034  2369.847  0.0524  0.6033  
(0.0723)  (4.5017) 
In conclusion, the PCNB model is certainly an appropriate model for fitting the first two data sets.
MLEs, their SEs (in parentheses) and goodnessoffit measures for the third data set
Distribution  Estimates  AIC  BIC  

PCNB(α, p, n, σ)  1.2865  0.0070  1.6620  4.2398  592.1857  599.9130 
(0.2690)  (0.0055)  (1.0335)  (6.2151)  
BW(a, b, c, λ)  17.8517  0.7694  0.3143  4.0437  594.1023  601.8296 
(59.5446)  (1.8538)  (0.4824)  (25.6600)  
PCG(α, p, σ)  1.4738  0.0053  9.2046  599.0166  604.8121  
(0.1888)  (0.0041)  (6.0303)  
BHC(a, b, σ)  1.9480  1.0755  127.2517  598.7476  604.5431  
(0.6174)  (0.2655)  (54.0490)  
KHC(a, b, σ)  1.9059  1.0921  130.6862  598.7297  604.5252  
(0.6031)  (0.2780)  (55.0280)  
PC(α, σ)  0.5788  45.3099  649.3128  653.1764  
(0.1101)  (12.3464) 
Regression model example : Entomology data. First, we use the data from a study carried out at the Department of Entomology of the Luiz de Queiroz School of Agriculture, University of São Paulo, which aims to assess the longevity of the Mediterranean fruit fly (ceratitis capitata). The need for this fly to seek food just after emerging from the larval stage has permitted the use of toxic baits for its management in Brazilian orchards for at least fifty years. This pest control technique consists of using small portions of food laced with an insecticide, generally an organophosphate, that quickly kills the flies, instead of using an insecticide alone. Recently, there have been reports of the insecticidal effect of extracts of the neem tree leading to proposals to adopt various extracts (aqueous extract of the seeds, methanol extract of the leaves and dichloromethane extract of the branches) to control pests such as the Mediterranean fruit fly. The experiment was completely randomized with eleven treatments, consisting of different extracts of the neem tree, at concentrations of 39, 225 and 888 ppm.

Group 1: Control 1 (deionized water); Control 2 (acetone  5%); aqueous extract of seeds (AES) (39 ppm); AES (225 ppm); AES (888 ppm); methanol extract of leaves (MEL) (225 ppm); MEL (888 ppm); and dichloromethane extract of branches (DMB) (39 ppm).

Group 2: MEL (39 ppm); DMB (225ppm) and DMB (888 ppm).
The response variable in the experiment is the lifetime of the adult flies in days after exposure to the treatments. The experimental period was set at 51 days, so that the numbers of larvae that survived beyond this period were considered as censored data. The total sample size is n=72, because four observations were lost. Therefore, the variables used in this study are: z_{ i }lifetime of ceratitis capitata adults in days, v_{i1}sex of the larvae and v_{i2}group (0=group 1, 1=group 2). We start the analysis of these data considering only failure (z_{ i }) and censoring (c_{ i }) data and an appropriate model for fitting the data could be the LPCNB and LPC distributions.
MLEs of the parameters from the LPCNB regression model fitted to the entomology data set, the corresponding SEs (given in parentheses), pvalues in [ ·]
Model  a  n  p  τ _{0}  τ _{1}  τ _{2} 

LPCNB  0.2514  0.4118  0.1496  2.9793  0.0188  0.2787 
(0.0383)  (0.0897)  (0.1470)  (0.1471)  (0.0779)  (0.0854)  
[ < 0.001]  [0.8098]  [0.0013]  
LPC  0.4100  1  1  3.0781  0.0207  0.2779 
(0.0293)  (0.0617)  (0.0832)  (0.0939)  
[ < 0.001]  [0.8038]  [0.0035] 
AIC, CAIC and BIC statistics for comparing the LPCNB and LPC regression models
Model  AIC  CAIC  BIC 

LPCNB  332.8  333.4  351.7 
LPC  346.0  346.0  358.6 
MLEs of the parameters from the fitted LPCNB regression model to the entomology data
Model  a  n  p  τ _{0}  τ _{2} 

LPCNB  0.2532  0.4164  0.1569  2.9917  0.2771 
(0.0375)  (0.0879)  (0.1496)  (0.1391)  (0.0846)  
[ < 0.001]  [0.0013] 
Concluding remarks
We consider a lifetime model in the context of insurance claims where the claim sizes follow a power Cauchy and the number of claims is negative binomial distributed. In these terms, we propose a new model by compounding the powerCauchy and negativebinomial distributions called the powerCauchy negativebinomial (PCNB) distribution. We provide a useful linear representation for its density, which allows to obtain some properties for the proposed distribution. We use the maximum likelihood method for estimating the model parameters. The suitability of these estimates is investigated by a simulation study. We fit the proposed distribution to three real data sets to show empirically its flexibility. We proposed a new class of regression models for location and scale based on the logarithm of the PCNB random variable. Estimation and inference on the regression coefficients are discussed and an application to real data in Entomology is addressed. Various future studies can be conducted, such as employing other estimation techniques (bootstrap and Bayesian methods) and investigating the sensitivity of the LPCNB regression model using diagnosis and analysis of residuals. which led to this improved version.
Declarations
Acknowledgement
The authors are grateful to the EditorinChief, the Associate Editor and anonymous referees for their many helpful comments and suggestions on an earlier version of this paper which led to this improved version.
Authors’ contributions
The authors, viz MZ, MHT, GMC, AA and EMMO with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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