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A new generalization of generalized halfnormal distribution: properties and regression models
Journal of Statistical Distributions and Applications volume 5, Article number: 7 (2018)
Abstract
In this paper, a new extension of the generalized halfnormal distribution is introduced and studied. We assess the performance of the maximum likelihood estimators of the parameters of the new distribution via simulation study. The flexibility of the new model is illustrated by means of four real data sets. A new loglocation regression model based on the new distribution is also introduced and studied. It is shown that the new loglocation regression model can be useful in the analysis of survival data and provides more realistic fits than other competitive regression models.
Introduction
The generalized halfnormal (GHN) distribution has been widely modified and studied in recent years and various authors developed new generalizations of it. Following an idea due to Eugene et al. (2002), Pescim et al. (2017) introduced the beta generalized halfNormal (BGHN) distribution with applications to myelogenous leukemia data. Cordeiro et al. (2012) defined the Kumaraswamy generalized halfnormal (KwGHN) distribution for censored data. More recently, Cordeiro et al. (2013) studied some of the mathematical properties of the BGHN distribution proposed by Pescim et al. (2010b). Pescim et al. (2013) proposed the loglinear regression model based on the BGHN distribution, while Ramires et al. (2013) defined the beta generalized halfnormal geometric (BGHNG) distribution in order to achieve wider diversity among the density and failure rate functions.
The GHN density function (Cooray and Ananda 2008) with shape parameter λ>0 and scale parameter θ>0 is given (for x>0) by
and its cumulative distribution function (cdf) depends on the error function
where
and
The nth moment of the random variable X with cdf (2) is
where Γ(.) is the gamma function. The HN distribution is a submodel of GHN when λ=1.
The goal of this paper is to propose the first generalization of the generalized halfnormal distribution using the Zografos–Balakrishnan Odd LogLogisticG (“ZBOLLG” for short) family of distributions. For an arbitrary baseline cdf G(x), Cordeiro et al. (2015) proposed the probability density function (pdf) f(x) and the cdf F(x) of the ZBOLLG family of distributions with two additional shape parameters β>0 and α>0 as
and
where ξ denotes the parameter vector of the baseline distribution. We use Eqs. (1), (2) and (3) to obtain the fourparameter ZBOLLGHN pdf (for x>0)
where α>0, β>0, λ>0 are shape parameters and θ is the scale parameter. The corresponding cdf is given by
where γ(β,z)\(=\int \limits _{z}^{\infty }t^{\beta 1}\exp \left (t\right) dt\) denotes the complementary incomplete gamma function. Henceforth, we denote a random variable X with pdf (5) by X ∼ ZBOLLGHN(β,α,λ,θ). The submodels of (5) are given in Table 1.
We investigate the possible hazard rate function (hrf) and pdf shapes of ZBOLLGHN distribution. Figure 1 displays the pdf shapes of ZBOLLGHN distribution. Based on the Fig. 1, ZBOLLGHN pdf has the following shapes: leftskewed, rightskewed, symmetric and bimodal. Figure 2 displays the hrf shapes of ZBOLLGHN distribution. From Fig. 2, we conclude that the ZBOLLGHN hrf has the following shapes: increasing, decreasing, upsidedown and bathtub.
Following Cordeiro et al. (2016a), equation (6) can be expressed as
where
and Π_{w}(x;λ,θ)=[G(x;λ,θ)]^{w} denotes the cdf of the expGHN distribution with the power parameter w. The pdf (5) reduces to
where π_{w+1}(x;λ,θ)=(w+1)g(x;λ,θ)[G(x;λ,θ)]^{w} denotes the pdf of the expGHN distribution with the power parameter w+1. For the definitions of p_{j,k} and a_{w}(β,α,i,k), please see Cordeiro et al. (2016a). Equation (7) reveals that the density function of X is a linear combination of the expGHN densities. Thus, some of the structural properties of the ZBOLLGHN distribution such as ordinary and incomplete moments and generating function can be obtained from wellestablished properties of the expGHN distribution.
We are motivated to introduce the ZBOLLGHN distribution since it contains a number of aforementioned known lifetime models as illustrated in Table 1. The new distribution exhibits increasing, decreasing, upsidedown as well as bathtub hazard rates as illustrated in Fig. 2. It is shown that the new distribution can be viewed as a mixture of the twoparameter GHN model. It can also be viewed as a suitable model for fitting the leftskewed, rightskewed, symmetric and bimodal data. The ZBOLLGHN distribution outperforms several of the wellknown lifetime distributions with respect to four real data applications as illustrated in “Applications” section. The new loglocation regression model based on the ZBOLLGHN distribution provides better fits than log BGHN, log GHN and logWeibull models for volatage data set. Based on the residual analysis (martingale and modified deviance residuals) for the new loglocation regression model (log ZBOLLGHN), we conclude that none of the observed values appear as possible outliers. Thus, it is clear that the fitted model is appropriate for the voltage data set.
The rest of the paper is organized as follows. In “Estimation” section, the maximum likelihood method is used to estimate the model parameters. The performance of maximum likelihood estimators of the model parameters are investigated by means of a Monte Carlo simulation study when n is finite. A new loglocation regression model as well as residual analysis are presented in “A new loglocation regression model” section. Four applications to real data sets illustrate empirically the importance of the new model in “Applications” section. Finally, a summary is provided in “Summary” section.
Estimation
If X follows the ZBOLLGHN distribution with vector of parameters Ψ=(β,α,λ,θ)^{T}. The loglikelihood function for a single observation x of X is given by
The components of the unit score vector U=U(Ψ)=(∂β/∂ℓ,∂α/∂ℓ,∂λ/∂ℓ,∂θ/∂ℓ)^{T} are available if needed. For a random sample x=(x_{1},...,x_{n})^{T} of size n from X, the total loglikelihood is
where ℓ^{(i)}(Ψ) is the loglikelihood for the i^{th} observation. The total score function is
where U^{(i)} has the form given before. Maximization of ℓ(Ψ) (or ℓ_{n}(Ψ)) can be easely performed using wellestablished routines such as the nlm or optim in the R statistical package. Setting these equations equal to zero, U(Ψ)=0, and solving them simultaneously gives the MLE \(\widehat {\mathbf {\Psi }}\) of Ψ. These equations cannot be solved analytically and statistical software can be used to evaluate them numerically using iterative techniques such as the NewtonRaphson algorithm.
The parameter estimation procedure of ZBOLLGHN model can be summarized as follows:

The optim function of R software is used to minimize the minus loglikelihood function of GHN model by means of the NelderMead (NM) optimization method. There is no need to provide the derivatives of the objective function for NM method.

The estimated parameters of GHN distribution are used as initial values of the ZBOLLGHN model. The initial values of the additional parameters α and β are chosen as 1. Note that the ZBOLLGHN model reduces to GHN model when the parameters α=β=1. Then, the parameter estimation of ZBOLLGHN model are obtained with the optim function as given in the first step.

The inverse of estimated Hessian matrix is used to obtain the corresponding standard errors.
Simulation study
In this subsection, the performance of the maximum likelihood estimators of the ZBOLLGHN parameters are evaluated via a Monte Carlo simulation study with 10,000 replications. The coverage probabilities (CPs), mean square errors (MSES) and the bias of the parameter estimates, estimated average lengths (ALs) are calculated by means of R software. We generate N=10,000 samples of sizes n=50,55,...,500 from the ZBOLLGHN distribution with α=0.8,β=7,λ=9,θ=4. Let \(\left (\widehat {\alpha }, \widehat {\beta },\widehat {\lambda },\widehat {\theta }\right)\) be the MLEs of the new model parameters and \((s_{\widehat {\alpha }},s_{\widehat {\beta }},s_{\widehat {\lambda }},s_{\widehat {\theta }})\) be the standard errors of the MLEs. The estimated biases and MSEs are given by
and
for ε=α,β,λ,θ. The CPs and ALs are given, respectively, by
and
Figure 3 displays the numerical results for the above measures. We list below the results from these plots:

✓ The estimated biases decrease when the sample size n increases,

✓ The estimated MSEs decay toward zero as n increases,

✓ The CPs are near 0.95 and approach the nominal value when the sample size n increases,

✓ The ALs decrease for all parameters when the sample size n increases.
These results reveal the consistency property of the MLEs.
A new loglocation regression model
Let X denote a random variable following the ZBOLLGHN distribution (5) and let Y=log(X). The density function of Y (for \(y\in \mathfrak {R} \)) and replacing μ= log(θ), \(\sigma =\sqrt {2}/{2\lambda }\) can be expressed as
where μ∈ℜ is the location parameter, σ>0 is the scale parameter and α>0 and β>0 are the shape parameters. We refer to Eq. (8) as the pdf of LZBOLLGHN distribution, say Y∼LZBOLLGHN(α,β,μ,σ). The survival function corresponding to (8) is given by
The hrf is simply h(y)=f(y)/S(y). The standardized random variable Z=(Y−μ)/σ has density function
Figure 4 provides some plots of the density function (8) for selected parameter values. They reveal that this distribution is a good candidate to model left skewed and symmetric data sets.
Based on the LZBOLLGHN density, we propose a linear locationscale regression model linking the response variable y_{i} and the explanatory variable vector \(\mathbf {v}_{i}^{\intercal }=(v_{i1},\ldots,v_{ip})\) given by
where the random error z_{i} has density function (10), \(\boldmath {\beta }=(\beta _{1},\ldots,\beta _{p})^{\intercal }\), and σ>0, α>0 and β>0 are unknown parameters. The parameter \(\mu _{i}=\mathbf {v}_{i}^{\intercal } \boldmath {\beta }\) is the location of y_{i}. The location parameter vector \({\boldmath {\mu }}=(\mu _{1},\ldots,\mu _{n})^{\intercal }\) is represented by a linear model μ=Vβ, where \(\mathbf {V}=(\mathbf {v}_{1},\ldots,\mathbf {v}_{n})^{\intercal }\) is a known model matrix.
The LZBOLLGHN model (11) provides new opportunities for modeling several types of data sets. This model contains two important regression models as its submodels: (i) for β=1, the LZBOLLGHN model reduces to logOLLGHN regression model introduced by Pescim et al. (2017); (ii) for α=β=1, the LZBOLLGHN model reduces to logGHN regression model.
Let F and C be the sets of individuals for which y_{i} is the loglifetime or logcensoring, respectively. Assume that the observed lifetimes and censoring times are independent. The loglikelihood function for the vector of parameters \(\Theta =(\alpha,\beta,\sigma,\boldmath {\beta }^{\intercal })^{\intercal }\) from model (11) is given by \(l(\Theta)=\sum \limits _{i \in F}l_{i}(\Theta)+\sum \limits _{i \in C}l_{i}^{(c)}(\Theta)\), where l_{i}(Θ)= log[f(y_{i})], \(l_{i}^{(c)}(\Theta)=\log [S(y_{i})]\). The f(y_{i}) and S(y_{i}) are defined in(8) and (9), respectively. The total loglikelihood function for Θ is given by
where \({u_{i}}=2\Phi [\exp (z_{i}\sqrt {2}/2)]\), z_{i}=(y_{i}−μ_{i})/σ, and r is the number of uncensored observations (failures). The MLE \(\widehat {\Theta }\) of the vector of unknown parameters can be evaluated by maximizing the loglikelihood (12). The R software is used to estimate unknown parameters of LZBOLLGHN regression model
The likelihood ratio (LR) statistic can be used for comparing some submodels of LZBOLLGHN regression model. For example, the LR statistic can be used to discriminate between the LZBOLLGHN and LGHN regression models since they are nested models, or equivalently to test H_{0}:α=β=1. The LR statistic reduces to \(w=2\left [\ell (\hat {\alpha },\hat {\beta },\hat {\sigma },\boldsymbol {\hat {\beta }})\ell (1,1,\tilde {\sigma },\boldsymbol {\tilde {\beta }})\right ]\), where \(\left (\hat {\alpha },\hat {\beta },\hat {\sigma },\boldsymbol {\hat {\beta }}\right)\) are the unrestricted MLEs and \((1,1,\tilde {\sigma },\boldsymbol {\tilde {\beta }})\) are the restricted estimates under H_{0}. The statistic w is asymptotically (as n→∞) distributed as \(\chi _{k}^{2}\), where k is difference of two parameter vectors of nested models. For example, take k=2 for the above hypothesis test.
Residual analysis
Residual analysis has critical role to check the adequacy of the fitted model. In order to analyze departures from error assumption, two types of residuals are considered: martingale and modified deviance residuals.
Martingale residual
The martingale residuals is defined in counting process and takes values between +1 and −∞ (see for details, Fleming and Harrington (1994)). The martingale residuals for LZBOLLGHN model is,
where \(u_{i}=2\Phi \left [\exp \left (z_{i}\sqrt {2}/2\right)\right ]\) and z_{i}=(y_{i}−μ_{i})/σ.
Modified deviance residual
The main drawback of martingale residual is that when the fitted model is correct, it is not symmetrically distributed about zero. To overcome this problem, modified deviance residual was proposed by Therneau et al. (1990). The modified deviance residual for LZBOLLGHN model is,
where \(\hat r_{M_{i}}\) is the martingale residual.
Applications
In this section, four real data sets are used to compare ZBOLLGHN model with its submodels and betaGHN model introduced by Pescim et al. (2013). The first three data sets are used to demonstrate the univariate data fitting performance of ZBOLLGHN distribution. The fourth data set is used to investigate the usefulness of the proposed distribution in survival analysis. The optim function is used to estimate the unknown model parameters. The MLEs and corresponding standard errors, estimated −ℓ, KolmogorovSmirnov (KS) statistic and corresponding pvalue, Cramérvon Mises (W*), AndersonDarling (A*) statistics and Akaike Information Criteria (AIC) are reported in Tables 2, 4 and 6. The lower the values of these criteria show the better fitted model on the data sets. The histograms with fitted pdfs are provided for visual comparison of the fitted distribution functions. Moreover, fitted hrfs and PP plots of the best fitted models are displayed in Figs. 5, 7 and 9.
Lifetime of device data
The first data set is given by Sylwia (2007) on the lifetime of a certain device. Table 2 shows the estimated parameters and their standard errors, −ℓ, A*, W*, KS and its corresponding pvalue and AIC values. Based on the figures in Table 2, it is clear that ZBOLLGHN model provides the best fit for this data set. Figure 5a displays the estimated pdfs of the fitted models. Figure 5b displays the PP plot of ZBOLLGHN distribution and its fitted hrf. Figure 5 shows that ZBOLLGHN distribution provides superior fit to the leftskewed data set.
Table 3 shows the LR statistics and the corresponding pvalues for the first data set. From Table 3, the computed pvalues are smaller than 0.05, so the null hypotheses are rejected for all submodels. We conclude that the ZBOLLGHN model fits the first data better than its submodels according to the LR test results.
In addition, the profile loglikelihood functions of the ZBOLLGHN distribution are plotted in Fig. 6. These plots reveal that the likelihood functions of the ZBOLLGHN distribution have solutions that are maximizers.
Failure times of windshield data
The second data set represents the failure times for a particular windshield model including 85 observations that are classified as failed times of windshields (Murthy et al. 2004). Table 4 shows the estimated parameters and their standard errors, −ℓ and AIC values. Based on the figures in Table 4, ZBOLLGHN distribution provides the best fit among others. Figure 7a displays the histogram with fitted pdfs and Fig. 7b displays the fitted hrf and PP plot of ZBOLLGHN distribution. These figures reveal that ZBOLLGHN model provides superior fit to the second data set.
Table 5 shows the LR statistics and the corresponding pvalues for the second data set. From Table 5, the computed pvalues are smaller than 0.05, so the null hypotheses are rejected for all submodels. We conclude that the ZBOLLGHN model fits the first data better than its submodels according to the LR test results.
The profile loglikelihood functions of the ZBOLLGHN distribution are plotted but not included here. These plots reveal that the likelihood functions of the ZBOLLGHN distribution have solutions that are maximizers.
Strengths of glass fibres data
The third data set obtained from Smith and Naylor (1987) represents the strengths of 1.5 cm glass fibres, measured at the National Physical Laboratory, England. Unfortunately, the units of measurement are not given in the paper. This data set have been analyzed recently with the beta generalized exponential distribution, which was introduced and studied by BarretoSouza et al. (2010). Table 6 shows the estimated parameters and their standard errors, −ℓ and AIC values. Based on the figures in Table 6, ZBOLLGHN distribution provides the best fit among others. Figure 8a displays the histogram with fitted pdfs and Fig. 8b displays the fitted hrf and PP plot of ZBOLLGHN distribution. These figures reveal that ZBOLLGHN model provides superior fit to the third data set.
Table 7 shows the LR statistics and the corresponding pvalues for the third data set. From Table 7, the computed pvalues are smaller than 0.05, so the null hypotheses are rejected for all submodels. We conclude that the ZBOLLGHN model fits the first data better than its submodels according to the LR test results.
The profile loglikelihood functions of the ZBOLLGHN distribution are plotted but not included here. These plots reveal that the likelihood functions of the ZBOLLGHN distribution have solutions that are maximizers (Fig. 8).
Voltage data
Lawless (2003) reported an experiment in which specimens of solid epoxy electricalinsulation were studied in an accelerated voltage life test. The sample size is n=60, the percentage of censored observations is 10% and there are three levels of voltage: 52.5, 55.0 and 57.5. The variables involved in the study are: x_{i} failure times for epoxy insulation specimens (in min); c_{i}  censoring indicator (0 =censoring, 1 =lifetime observed); v_{i1}  voltage (kV).
The data set was used by Pescim et al. (2013) for illustrating the logBGHN (LBGHN) regression model. Pescim et al. (2013) compared the logBGHN (LBGHN) regression model with LOLLGHN and logGHN (LGHN) models. In this section we compare the LZBOLLGHN regression model with models reported in Pescim et al. (2013). The regression model fitted to the voltage data set is given by
where the random variable y_{i} follows the LZBOLLGHN distribution given in (8). The results are presented in Table 8. The MLEs of the model parameters and their SEs and the values of the AIC and BIC statistics are listed in Table 8.
Based on the figures in Table 8, we conclude that the fitted LZBOLLGHN regression model has the lowest AIC and BIC values. Figure 9 provides the plots of the empirical and estimated survival function for the LZBOLLGHN regression model. We can conclude from these plots that LZBOLLGHN regression model provides a good fit to the data.
Residual Analysis of LZBOLLGHN model
Figure 10 displays the index plot of the modified deviance residuals and its QQ plot against N(0,1) quantiles. Based on the Figure 10, we conclude that none of the observed values appears as a possible outlier. Thus, it is clear that the fitted model is appropriate for these data set (Fig. 10).
Summary
A new model called ZografosBalarkishnan odd loglogistic generalized halfnormal is introduced and studied. We assess the performance of the maximum likelihood estimators of the parameters of the new distribution with respect to the sample size n. The assessment is based on a graphical simulation study. The flexibility of the new model is illustrated by means of the three real data sets. The new model performs much better than beta generalized halfnormal, generalized halfnormal, odd loglogistic generalized halfnormal and the generalized halfnormal models. Additionally, a new loglocation regression model based on the new distribution is introduced and studied. The martingale residual and the modified deviance residuals to detect outliers and evaluate the model assumptions are defined. We demonstrate that the new loglocation regression model can be very useful in the analysis of real data and provide more realistic fits than other regression models such as the log beta generalized halfnormal, the log generalized halfnormal and the logWeibull regression models. The potentiality of the new regression model is illustrated by means of a real data.
Abbreviations
 ALs:

Average lengths
 BGHN:

Beta generalized halfnormal
 BGHNG:

Beta generalized halfnormal geometric
 CPs:

Coverage probabilities
 GHN:

Generalized halfnormal
 HN:

Halfnormal
 KwGHN:

Kumaraswamy generalized halfnormal
 LZBOLLGHN:

LogZografosBalarkishnan odd loglogistic generalized halfnormal
 MLEs:

Maximum likelihood estimates
 MSEs:

Means square errors
 OLLGHN:

odd loglogistic generalized halfnormal
 ZBOLLG:

ZografosBalarkishnan odd loglogisticG
 ZBOLLGHN:

ZografosBalarkishnan odd loglogistic generalized halfnormal
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EA, HMY and GGH have contributed jointly to all of the sections of the paper. All authors read and approved the final manuscript.
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Correspondence to Emrah Altun.
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Keywords
 Regression
 Residuals
 Simulation
AMS 2010 Subject Classification
 60E05
 62J05