The Zero-Adjusted Log-Symmetric Distributions: Point and Intervalar Estimation

Abstract In this paper, a new class of semi-continuous distributions called zero-adjusted log-symmetric is introduced and studied. Some properties and parameters estimation by maximum likelihood method are derived and confidence intervals (CIs) are developed. A simulation study is conducted to evaluate properties of the maximum likelihood estimators in lighter/heavier-tailed distributions. Finally, an application in a real data set is presented to illustrate the flexibility of the proposed class of distributions.


INTRODUCTION
In real situations, semi-continuous variables commonly occur in biometrics, ecology or in insurance expenditures data sets and they are characterized by the presence of true zeros and positive continuous behavior.For example, in ecology is common to find a large proportion of zeros values caused by real ecological effects like counts of abundance, proportional occupancy rates or continuous population densities and do not readily fit standard distributions like normal, Poisson, and beta distribution (Martin et al. 2005).Another example is the Medical Expenditure Panel Survey (MEPS) that contain health care expenditures from adults in USA in which many individuals record no medical expenditures (zero response) over the years.Usual methods such as transforming the response variable into logarithmic transformation cannot be used in the presence of zeros or ignoring zeros can be a bad strategy making impossible to predict the probability of zero and leading to a bad inference of the others parameters.
A strategy for modeling this kind of data with the presence of true zeros is to use a mixture distribution of two components: a continuous distribution whose support is the interval (0, ∞) and a degenerate distribution at a zero value (discrete distribution).We can see as an example, Aitchison & Brown (1957) introduced a mixture distribution between a degenerate distribution at a zero and the log-normal distribution, named delta distribution.Duan et al. (1983) proposed the two-part model.The first stage is a probit model for binary event of having zero or positive expenses and the second stage is a log-linear model for positive expenses.Heller et al. (2006) presented the zero-adjusted inverse Gaussian distribution for modelling insurance claim sizes.Rodrigues-Motta et al. (2015) proposed a framework for zero-augmented positive distributions considering the two-parameter exponential family as the continuous component, including the log-normal Weibull, gamma and inverse Gaussian

THE ZERO-ADJUSTED LOG-SYMMETRIC DISTRIBUTIONS
Let  be a random variable that follow a class of distributions called log-symmetric, whose the support is defined in the interval (0, ∞).The log-symmetric class with parameters  > 0,  > 0 and density generator (⋅) under conditions that () > 0 and ∫ ∞ 0  − 1 2 () = 1 for  > 0, has density function (PDF) given by: where ỹ =  [(   ) 1 √ ].  is a transformation for the random variable  by setting  = exp() whose distribution follows the symmetric class (Fang et al. 1990) with notation  ∼ (, , (⋅)) where −∞ <  < ∞ is the location parameter.If a random variable follows the log-symmetric class, then is denoted by  ∼ (, , (.)),where  = exp() and  are the scale and power parameters.In some cases the density generator (⋅) are indexed by an extra parameter or vector parameter denoted by  and in this work  is considered fixed.Some distributions as Birnbaum-Saunders (Birnbaum & Saunders 1969), log-normal, log-slash, log-Student-t, log-power-exponential, type-I-log-Logistic, type-II-log-Logistic, log-contaminated-normal, and generalized Birnbaum-Saunders (Díaz-Gárcia & Leiva 2005, Leiva et al. 2008) are included in this class.
When zeros occur in the data, the log-symmetric distributions are not appropriate.An alternative is to use a mixed distribution discrete-continuous.Let  be a mixture discrete-continuous distribution between two random variables, a discrete component following a Bernoulli distribution and the continuous component .In this work we propose to consider one continuous component belongs to log-symmetric class, denoted  ∼ (, , , (⋅)).The cumulative distribution function (CDF)  is given by where   () is the CDF of the log-symmetric class and 0 <  < 1 is the mixture parameter.Because  is a transformation for the random variable , then the CDF of  can be computer as   () =   ( ỹ ), where  = ( − )/√ ∼ (0, 1, (⋅)).The corresponding probability density function (PDF) is given by where   () is the PDF given in Equation (1).Members of this class are characterized by (⋅) and for each member a weight function is defined by v() = −2 ′ ( 2 )/( 2 ).Thus, the choice of (⋅) may induce also the weight function v(⋅), that it is an important function to estimate the parameters on the maximum likelihood (ML) method as will be seen later.Some properties from the zero-adjusted log-symmetric class are presented (P1) If  ∼ (, , , (⋅)) then  ∼ (, , , (⋅)) for  > 0.

Zero-adjusted type-I-log-Logistic
Zero-adjusted type-II-log-Logistic Zero-adjusted log-contaminated-normal where where  1 follows a log-normal distribution with parameters (, / 2 ) t and  1 follows a log-normal distribution with parameters (, ) t .

SIMULATION STUDY AND RESULTS
To evaluate the performance of the MLE for the parameters of the ZALS class, a Monte Carlo simulation study is conduced.In this study we considered some distributions with heavy/light tails, for example zero-adjusted log-Student-t (ZALSt) with  = 4 and the zero-adjusted log-power-exponential (ZALPE) with  = −0.5.In each scenario we consider sample size  = 10, 30 and 50, the skewness parameter for the continuous component (or relative dispersion)  ∈ {1, 2, 3} and the proportion parameter  ∈ {0.1, 0.2, 0.3, 0.4} .The parameter  is fixed at 2 and we consider 5000 Monte Carlo replications.To generate samples the inverse CDF method is used by (P2).All the simulations were performed using the gamlss package in R language (R Core Team) .All codes were developed by authors and it can be obtained upon request by authors.
The Tables II and III present descriptive measures : empirical Mean, Median, Standard Deviation (SD), Mean Square Error (MSE), √MSE and bias of θ of the simulations results for the ZALSt and ZALPE distributions.We can be observe in the Table II from the ZALSt distribution that the empirical Mean of η is affected by increasing  or , or on both parameters, but for the case of the Median, it is not affected by increasing , but this occurs when we increase the value of .Similar behavior can be observed for ZALPE distribution (see Table III).For the two distributions and for each scenario considered, when the sample size increases, the empirical Mean and the Median close to true value.In particular, the lowest bias of η is presented based on ZALPE distribution.For the behavior of φ , the empirical Mean and the Median are affected by increasing the value of .In addition when  increase the φ tend to true value.For the case of the MLE of , the empirical Mean and the Median are closer to true value in each scenario regardless of the sample size , this means that the for all values of  considered, the results are similar.The bias of the MLE for  overestimates the true value in all scenarios for the ZALSt and ZALPE distributions.As expected, the bias decreases as the sample size increases.Note that for the case of the ZALPE distribution, presents better properties in relation to the bias of .In the Table III, the bias of the MLE of  underestimates the true value but it did not happen in the case of the ZALSt distribution, and it can be observed that is less biased regardless of the parameter .For the ZALPE distribution, the bias is decreasing as the value of  increases but it increases as the sample size increases, this means that it less biased as the value  increases.In the case of the bias for the MLE of , it is little biased and also underestimates in the majority of the scenarios for each distribution.Respect to the values of SD and the √MSE, the distribution ZALPE had better properties than the ZALSt distribution for the cases of the MLE of  and .As expected, these values are increasing as the parameter  or  or on both parameters increase, but also decrease considering a larger sample size .
Tables IV and V provide the lower (LC), upper (UC) and the coverage (CP) probabilities in % of the CIs for the parameters log(), log() and logit() with (log( η )) = φ /{  (1 − π )}, (log( φ )) = 4/{(  − 1)(1 − π )} and (logit( π )) = 1/{ π (1 − π )} by taken the 0.90, 0.95 and 0.99 confidence levels for the ZALSt and ZALPE distribution, respectively.For the cases of the parameters log() and log(), we can observe that the coverage probabilities of the CI for the three confidence levels from the ZALPE distribution have smaller coverage probability than the ZALSt distribution.Also, as the value of  increases, the coverage probabilities on these two parameters are affected in the sense that it is getting smaller than the indicated confidence level.In the case of logit(), the coverage probabilities have similar properties regardless of the distribution, this because the MLE of logit() is the same for each distribution.As expected, the coverage probabilities increase as  increases.Note that the confidence intervals for log() are balanced, this means, the values of LC and UC are similar.The same behavior can be observed in the intervals for logit() as the value of  increases.An unbalanced behavior is noted in the confidence intervals for log().This occurs for the three distributions and also for the confidence levels considered.
Next, in the Figures 2 and 3 display the empirical(histogram) and asymptotic distribution of log( η ), log( φ ) and logit( π ) for each sample size by considering the scenario  = (2, 4, 0.4) t for the ZALSt and ZALPE distribution, respectively.Additionally, a straight-line segments represent the asymptotic CI computed, and the vertical line represent the true value of the parameter.The empirical distribution for log( η ) is apparently symmetric in all three distributions considered, but for the case of log( φ ) and logit( π ) present a little skewness.

APPLICATION
We consider the allowances and expenses dataset from elected councilors including the City Mayor in Leicester City, UK, from the period 2012/2013.The data were extracted from the Leicester City Council, database available at https://data.leicester.gov.uk/pages/home/.We are interested in the Special Responsibility Allowance (£) that consist in additional allowance for specific responsibilities such as Committee Chair received by some councilors.The sample size contains 18 zeros values, which represents 32.73% of the observations.VI presents some such as as the sample size (n), minimum and maximum values, mean ( ȳ ), median, standard deviation (SD), first quartile (Q1), third quartile (Q3), interquatile range (IQR), coefficient of variation(CV), and coefficient of kurtosis, coefficient of skewness (CS), and coefficient of kurtosis (CK).Note that data has positive skewness (asymmetry) and large kurtosis.Also, Figure 4 presents the histogram considering the zero values and an adjusted-boxplot considering only the positive values.We can observe a positive skewness and the presence of two outliers in Figure 4. Therefore, we fit three distribution of ZALS class such as zero-adjusted log-normal (ZALN), zero-adjusted Student-t (ZALSt) and zero-adjusted Power-exponential (ZALPE) and we compared with zero-adjusted inverse Gaussian (ZAIG), zero-adjusted Gamma (ZAG) and the zero-adjusted reparameterized Birnbaum-Saunders (ZARBS) distribution.The extra parameter for ZALSt and ZALPE distributions was chosen by minimizing the AIC in a grid of value grid  = [4, 10],  = (±0.3,±0.2, ±0.1), respectively.Based on the criteria, the extra parameter are  = 4 and  = 0.3 for ZALSt and ZALPE distribution, respectively.
Table VII presents the MLEs of the parameters for fitted distributions standard errors (in parentheses), the −2log( L ), the Akaike Information Criterion (AIC= 2 − 2log( L )) and the Bayesian information criterion (BIC= log() − 2log( L )), were  is the number of observations,  is the number of estimated parameters and L = ( θ ) is the likelihood evaluated at the estimated parameters.Additionally the Kolmogorov-Smirnov (KS) test is computed as the goodness-of-fit.We can observe the fitted ZALSt distribution presents the lower value of AIC and BIC.In addition, the KS test indicates that the three fitted ZALS distributions and the fitted Gamma distribution not have enough evidence to reject that is drawn from the reference distribution.With respect fitted the reparameterized Birnbaum-Saunders and fitted Inverse Gaussian distribution we do not same conclusion.Additionally hazard function from the selected continuous fitted distribution is displayed in Figure 4 has a decreasing behavior.Finally, fitted densities and Q-Q plots for each distribution are displayed in Figure 5.

CONCLUSIONS
We have proposed a new distribution class for asymmetric positive data with light/heavy tails that contains true zeros named Zero-adjusted log-symmetric (ZALS).Some properties are presented, for example, moments, quantily function, Shannon entropy among others.Maximum likelihood method was used to estimate the parameters of the ZALS class and asymptotic confidence intervals are developed.The Monte Carlo simulation study is performed to examine the properties of MLEs of the parameters considering the ZALPE and ZALSt distributions.The MLEs have a good performance.The empirical distributions of the MLEs close to normal distribution.Finally, an application using real data set is presented in order to show the flexibility and variety of the ZALS class.Fitted ZALSt distribution has better performance than competitive distributions based on AIC and BIC criteria.

Figure 4 .
Figure 4. Histogram, adjusted-boxplot and fitted log-Student-t distribution hazard function for the Special Responsibility Allowance of councilors in Leicester City, period 2012/2013.

Table I .
Values of   and   for some symmetric class distributions .

Table II .
Empirical statistics for the MLE estimator for  and  with  = 2 and  = 10, 30, 50 for the zero-adjusted log-Student-t distribution with  = 4 .

Table VI .
Some statistics for the Special Responsibility Allowance of councillors in Leicester City, period 2012/2013 dataset.

Table VII .
Estimated parameters and statistics for the considered fitted distributions for the Special Responsibility Allowance of councilors in Leicester City, period 2012/2013 dataset.