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 Open Access
A new extended normal regression model: simulations and applications
 Maria C.S. Lima†^{1}Email authorView ORCID ID profile,
 Gauss M. Cordeiro†^{1},
 Edwin M.M. Ortega†^{2} and
 Abraão D.C. Nascimento†^{3}
https://doi.org/10.1186/s404880190098y
© The Author(s) 2019
 Received: 30 January 2019
 Accepted: 28 May 2019
 Published: 8 June 2019
Abstract
Various applications in natural science require models more accurate than wellknown distributions. In this context, several generators of distributions have been recently proposed. We introduce a new fourparameter extended normal (EN) distribution, which can provide better fits than the skewnormal and beta normal distributions as proved empirically in two applications to real data. We present Monte Carlo simulations to investigate the effectiveness of the EN distribution using the KullbackLeibler divergence criterion. The classical regression model is not recommended for most practical applications because it oversimplifies real world problems. We propose an EN regression model and show its usefulness in practice by comparing with other regression models. We adopt maximum likelihood method for estimating the model parameters of both proposed distribution and regression model.
Keywords
 KullbackLeibler divergence criterion
 Maximum likelihood procedures
 Monte Carlo simulation
 Normal distribution
 Regression
AMS Subject Classification
 Primary 60E05
 secondary 62N05
 62F10
Introduction
In recent years, several methods for generating new models from classic distributions have been proposed. A detailed study about “the evolution of methods for generalizing classic distributions” was made by Lee et al. (2013). A generalization of the standard normal distribution is sought because it can provide more accurate statistical models and inferential procedures. For instance, the beta normal distribution was pioneered by Eugene et al. (2002), who discussed some of its structural properties.
Additionally, the beta generalized normal (BGN) distribution was proposed by Cintra et al. (2013) to extend the beta normal distribution. They applied the BGN model to the synthetic aperture radar image processing. This paper presents a new extended normal (EN) distribution based on the family introduced by Cordeiro et al. (2013).
where a>0 and b>0 are two additional shape parameters whose role is to generate distributions with heavier/lighter tails and provide wider ranges for skewness and kurtosis. These parameters are sought as a manner to furnish a more flexible distribution.
Because of its tractable cdf (1), the EG family can be used quite effectively even if the data are censored. This family is capable to return univariate models for any type of support. Further, it allows for greater flexibility of its tails and can be widely applied in many areas such as engineering and biology.
An important advantage of the density (2) is its ability of fitting skewed data that can not be often fitted by existing distributions. Based on the cdf G(x) and pdf g(x) of any baseline G distribution, we can associate the EGG pdf (2) with two extra parameters. The EG family can be used for discriminating between the G and EGG distributions.
The baseline distribution G(x) is a special case of (2) when a=b=1. For a=1, it gives the exponentiatedG (“ExpG”) class. If b=1, we obtain the Lehmann type IIG (LTIIG) class. Eq. (2) generalizes both Lehmann types I and II alternative classes (Lehmann 1953). In fact, this equation can be defined as the exponentiated generator applied to the LTIIG class.
This paper is outlined as follows. In Section 2, we define the EN distribution and provide plots of its density function. A linear representation for the EN density function is derived in Section 3. We obtain an explicit expression for its moments in Section 4. In Section 5, we provide the maximum likelihood estimates (MLEs) of the parameters. In Section 6, we define the EN regression model and discuss the estimation of the model parameters. In Section 7, we perform some simulations and present three applications to real data sets. Finally, some concluding remarks are addressed in Section 8.
The EN distribution
Due to the analytical tractability of its pdf and its importance in asymptotic theory (such as the central limit theorem and delta mehtod), the normal distribution is the most popular model distribution in applications to real data with support in \( \mathbb {R}\).
where \(\mu \in \mathbb {R}\) is a mean parameter, σ>0 is a scale parameter and \(\phantom {\dot {i}\!}\phi (x)\,=\,(2\pi)^{1/2}\,\mathrm {e}^{x^{2}/2}\) is the standard normal pdf.
where \(\Phi (x)\,=\,\int _{\infty }^{x}\,\phi (t)\,\mathrm {d}t\) is the standard normal cdf.
Linear representation
respectively. Several properties of the exponentiated distributions have been studied by some authors recently such as those for the exponentiated Weibull (Mudholkar and Srivastava 1993) and exponentiated generalized gamma (Cordeiro et al. 2013) distributions.
Theorem 1
The proof of this theorem is given in Appendix A.
It is possible to verify using symbolic software (such as Maple) that \(\sum _{j=0}^{\infty } \,w_{j+1}=1\) as expected.
Equation (7) is the main result of this section. It reveals that the EN density is a linear combination of ExpN densities. So, several mathematical properties of the proposed distribution can then be obtained from those of the ExpN distribution using previous results given by Rêgo et al. (2012).
Moments
where \(F_{A}^{(jr)}(\cdot)\) is the Lauricella function of type A. See, for example, Exton (1978)^{1}. If n+k−j is odd, the corresponding terms in τ_{n,j} vanish.
Corollary 1
where τ_{n,j} is given by (8).
Estimation
Given a data set x_{1},…,x_{n}, the MLE of θ is determined by maximizing \( \ell _{n}(\boldsymbol {\theta })\,=\,\sum _{i=1}^{n}\,\ell (\boldsymbol {\theta };x_{i}). \)
This fact is important at least for two reasons. The estimates become the solutions of a system with three equations and three variables (say “(3,3) system”) instead of a (4,4) system. Further, Eq. (11) clarifies the relationship of \(\widehat {b}\) with \(\widehat {a}\), \(\widehat {\mu }\) and \(\widehat {\sigma }\). More details are described in the simulation section.
Additionally, in order to make inference on the model parameters, the total observed information matrix is J(θ)={−U_{rs}}, where U_{rs}=∂^{2} ℓ(θ)/∂θ_{r} ∂θ_{s}, for r,s∈{a,b,μ,σ}. By differentiating the score function, we obtain the Hessian matrix elements U_{rs} given in Appendix B.
The EN regression model
The classical normal linear regression model is usually applied in science and engineering to describe symmetrical data for which linear functions of unknown parameters are used to explain the phenomena under study. However, it is wellknown that several phenomena are not always in agreement with the classical regression model due to lack of symmetry and/or the presence of heavy and lightly tails in the empirical distribution. As an alternative to overcome this shortcoming, we propose a new regression model based on the EN distribution thus extending the normal linear regression.
where the random error Z∼EN(a,b,0,1) has the standardized EN distribution, β=(β_{1},…,β_{p})^{⊤} is the unknown vector of coefficients, σ>0 is an unknown dispersion parameter and v_{i} is the explanatory vector modeling the location parameter \(\mu _{i}=\mathbf {v}_{i}^{\top } \boldsymbol {\beta }\).
Hence, the location parameter vector μ=(μ_{1},…,μ_{n})^{⊤} of the EN regression model has the linear structure μ=Vβ, where V=[v_{1}…v_{n}]^{⊤} is a known model matrix.

For a=1, it gives the exponentiatednormal (ExpN) regression model, which has not been explored, but it can be understood as a regression under the power normal distribution pioneered by Kundu and Gupta (2013).

For b=1, it reduces to the LTIInormal (LTIIN) regression model defined as a linear model under the LTIIN distribution.

If a=b=1, it reduces to the normal linear regression.
where \(z_{i}=({x_{i}\mathbf {v}_{i}^{\top }\boldsymbol {\beta }})/\sigma \) and x_{i} is a possible outcome of X_{i}.
where j=1,…,p.
Note that a closedform expression for the MLE \(\widehat {\boldsymbol {\eta }}\) is analytically intractable and, therefore, its computation has to be performed numerically by means of a nonlinear optimization algorithm.
We can maximize the loglikelihood function (13) based on the NewtonRaphson method. In particular, we use the matrix programming language Ox (MaxBFGS function) (see Doornik 2007) to calculate \(\widehat {{\boldsymbol {\eta }}}\). Initial values for β and σ can be taken from the fit of the classical regression model (a=b=1).
Under general regularity conditions, the asymptotic distribution of \((\widehat {\boldsymbol {\eta }}{\boldsymbol {\eta }})\) is multivariate normal N_{p+3}(0,K(η)^{−1}), where K(η) is the expected information matrix. These conditions can be found in Cox and Hinkley’s Theoretical Statistics book (1974). The asymptotic covariance matrix K(η)^{−1} of \(\widehat {{\boldsymbol {\eta }}}\) can be approximated by the inverse of the (p+3)×(p+3) observed information matrix J(η) and then the inference on the parameter vector η can be based on the normal approximation N_{p+3}(0,J(η)^{−1}) for \(\widehat {{\boldsymbol {\eta }}}\).
Besides estimation of the model parameters, hypotheses tests can be considered using likelihood ratio (LR) statistics.
Numerical results
Three studies are presented in this section. First, we perform a Monte Carlo simulation study. Subsequently, two applications to real data show the potential uses of the new distribution. Third, the usefulness of the proposed regression model in Section 6 is proved empirically based on quality of life data.
7.1 Simulation study
Here, we provide a Monte Carlo simulation study in order to quantify the effectiveness of the EN distribution based on the symmetrized KullbackLeibler divergence as a goodnessoffit comparison criterion.
Initially, we provide a brief discussion on the KullbackLeibler divergence. According to Cover and Thomas (1991), this measure is the quantification of the error by assuming that the Y model is true when the data follow the X distribution. For example, it has been proposed as essential parts of test statistics and strongly applied to contexts of radar synthetic aperture image processing in both univariate (Nascimento et al. 2010) and polarimetric (or multivariate) (Nascimento et al. 2014) perspectives.
For increasing values of ε, the IntegrandKL (X,Y) has different forms. Further, IntegrandKL (X,Y)→0 when ε→0.
Figure 3b and c reveal the influence of a and b, respectively, when we employ a perturbation in each parameter under (μ,σ)=(0,1). As expected, when the value of ε increases, the distance d _{KL} also increases in both cases. However, this distance is most evident when we take smaller negative values of ε.
The KL distance between fitted and theoretical densities for n=100 and different values for a and b
b  

a  0.5  0.8  1.0  1.5  2.0  2.5 
0.5  6.837  5.984  5.662  4.064  2.690  0.545 
0.8  4.259  5.628  4.636  4.497  3.581  1.263 
1.0  4.663  5.149  4.999  4.834  4.242  2.135 
1.5  1.988  3.450  4.349  5.791  5.549  4.301 
2.0  1.527  2.884  3.687  6.786  7.075  7.517 
2.5  1.435  2.728  3.837  6.590  7.287  10.148 
7.2 Two applications to real data
Here, we perform two applications to real data sets. First, we consider the data the strengths of glass fibres analyzed by Jones and Faddy (2004). These data were obtained at the National Physical Laboratory (UK) to explain the breaking strength of sixty three glass fibres having length 1.5 cm.
As a second application, we consider the fatigue life data (Meeker and Escobar 1998) for sixty seven specimens of Alloy T7987 that failed before having accumulated three hundred thousand cycles of testing. The data set was rounded to the nearest thousand cycles.
We prove empirically the efficiency of the EN distribution versus the normal, skewnormal (SN) (Azzalini 1984) and beta normal (BN) (Eugene et al. 2002) distributions.
The GoF measures of the fitted EN, SN, normal (N) and beta normal (BN) distributions to two real data sets
Models  

Measures  EN  SN  N  BN 
First application  
\(\hat {\theta }_{i}\)  (0.2490, 3.2023, 1.7709, 0.9768)  (4.746, 1.109, 0.016)  (4.7619, 1.1094)  (991.2293, 54.4065, 23.4249, 17.3049) 
\(\text {Sd}(\hat {\theta }_{i})\)  (<0.0001, 0.0005, <0.0001, <0.0001)  (1.029, 0.039, 0.004)  (0.2420, 0.1711)  (1.1628, 1.3819, 0.4080, 0.1533) 
W ^{∗}  0 . 0 6 3 0  1.5672  0.0800  0.4296 
A ^{∗}  0 . 3 9 1 8  7.5418  0.4758  2.3953 
KS  0 . 1 4 8 4  1.0000  0.1452  1.0000 
Second application  
\(\hat {\theta }_{i}\)  (0.4804, 100.0000, 141.4568, 75.5640)  (0.0009, 166.0232, 32.7894)  (166.0746, 46.3746)  (1001.6514, 4.5535, 565.4361, 276.6320) 
\(\text {Sd}(\hat {\theta }_{i})\)  (0.0126, 0.0005, 0.0002, 0.0001)  (0.1082, 0.0001, <0.001)  (5.6655, 4.0061)  (149.8685, 0.6375, 0.2980, 0.1719) 
W ^{∗}  0 . 0 2 5 9  0.1037  0.1185  0.3018 
A ^{∗}  0 . 2 4 2 8  0.8134  0.8942  1.8874 
KS  0 . 0 5 3 2  0.8519  0.0835  1.000 
The GoF’s measures for the EN distribution correspond to the lowest values among the discrimination criteria (highlighted in Table 2). These results provide evidence that the EN distribution is the most suitable model (among those considered) to describe both data sets.
7.3 Application for regression models
We assess changes on the oral healthrelated quality of life (OHRQL) of schoolchildren. To that end, a followup exam of three years was made to evaluate the impact of caries incidence on the OHRQL of adolescents. The data were obtained from a study (for more details, see Paula et al. 2012) developed by the Department of Community Dentistry, Division of Health Education and Health Promotion, Piracicaba Dental School, University of CampinasUNICAMP.

x_{i}: overall score of the OHRQL at time of follow up;

v_{i1}: number of teeth decayed, missing and filled (TDMF)
(0=without TDMF increment; 1=with TDMF increment).
MLEs, SEs in (·) and pvalues between [·] for the EN, ExpN, LTIIN, normal and GN regressions fitted to the OHRQL data
Estimates  

Model  a  b  σ  β _{0}  β _{1} 
EN  0.1058  0.1624  7.0973  30.8602  10.3707 
(0.0156)  (0.0139)  (0.1076)  (0.5989)  (0.5728)  
[ <0.001]  [ <0.001]  
ExpN  1  0.0489  9.6059  64.4529  14.2817 
(0.0024)  (0.1431)  (2.2873)  (1.3420)  
[<0.001]  [ <0.001]  
LTIIN  0.1900  1  10.5286  4.6992  7.4972 
(0.0129)  (0.2224)  (1.5689)  (0.8623)  
[0.0030]  [<0.001]  
Normal  1  1  19.3853  25.6139  7.2402 
(0.8035)  (1.9289)  (2.3872)  
[<0.0001]  [0.0026]  
GN  5.1249  23.5382  31.2214  7.0051  
(2.9739)  (1.4877)  (28.4676)  (2.2982)  
[0.2737]  [0.0025] 
Iterative maximization of the loglikelihood function (13) starts with initial values for β and σ taken from the fit of the classical regression model (a=b=1). In general, all fitted regression models reveal that v_{1} is significant at a 1% level of significance and that there is a significant difference between the levels of the numbers of teeth decayed, missing and filled. As expected, we find reciprocal relations between \(\mu _{i}=\mathbb {E}(X_{i})\) and v_{1i} in the EN, LTIIN, GN and normal regression models, except for the ExpN regression (whichalthough well adjusteddoes not seem to be a coherent model). On the other hand, based on the estimates of σ, the EN regression model reveals advantages in relation to the other models.
AIC, CAIC and BIC Statistics
Statistic  

Model  AIC  CAIC  BIC 
EN  2515.4  2515.6  2533.7 
ExpN  2725.8  2725.9  2740.5 
LTIIN  2523.6  2523.8  2538.3 
Normal  2557.2  2557.3  2568.3 
GN  2545.0  2545.1  2559.7 
LR statistics
Model  Hypotheses  λ  pvalue 

EN vs ExpN  H_{0}:a=1 vs H_{1}:H_{0} is false  212.4  <0.001 
EN vs LTIIN  H_{0}:b=1 vs H_{1}:H_{0} is false  10.2  0.0014 
EN vs Normal  H_{0}:a=b=1 vs H_{1}:H_{0} is false  45.8  <0.001 
The figures in this table, specially the pvalues, indicate that the EN regression model yields a better fit to these data than the other submodels.
Conclusions
Flexible statistical distributions have been sought for describing data from practical situations in which the use of classical ones is not recommended. In this paper, we propose an extension of the normal distribution based on the exponentiated generalized family defined by Cordeiro et al. (2011), which adds two extra shape parameters to a baseline distribution. We provide some structural properties of the new extended normal (EN) distribution. The model parameters are estimated by maximum likelihood. The efficiency of this distribution is illustrated by means of two applications to real data sets. There is a clear evidence that the EN distribution outperforms the skewnormal distribution and can be a competitive alternative to the beta normal distribution. The classical regression model does not produce good results in many real problems, and for this reason several extensions have arisen in recent years. We propose a new regression model based on the EN distribution and prove its importance in real applications. This new regression model opens a wide range of research topics following the basic inference concepts of the normal linear regression model.
Appendix A: Proof for the Theorem 3.1
where \(w_{j+1}= \sum _{m=1}^{\infty } (1)^{j+m+1}\,{b \choose m}\, {m\,a \choose j+1}\) and H_{j+1}(x) is the ExpG cdf with power parameter j+1. By differentiating the last equation, we obtain (7).
Appendix B: The Hessian matrix
Exton H. Handbook of hypergeometric integrals: theory, applications, tables, computer programs, 1978
Notes
Declarations
Acknowledgements
The authors would like to thank the financial support of CNPq and FACEPE, Brazil.
Funding
Not applicable.
Authors’ contributions
The authors, viz MCSL, GMC, EMMO and ADCN 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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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