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Tnormal family of distributions: a new approach to generalize the normal distribution
Journal of Statistical Distributions and Applications volume 1, Article number: 16 (2014)
Abstract
The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the TX framework. These four families of distributions are named as Tnormal families arising from the quantile functions of (i) standard exponential, (ii) standard loglogistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the Tnormal family are studied. Four new generalized normal distributions are developed using the Tnormal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed Tnormal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized Tnormal distributions.
AMS 2010 Subject Classification
60E05; 62E15; 62P10
1 Introduction
The normal distribution is perhaps the most commonly used probability distribution in both statistical theory and applications. The normal distribution was first used by de Moivre (1733) in the literature as an approximation to the binomial distribution. However, the development of the normal distribution by Gauss (1809, 1816) became the standard used in the modern statistics. Hence, the normal distribution is also commonly known as the Gaussian distribution. Properties of the normal distribution have been well developed (e.g., see Johnson et al. 1994; Patel and Read 1996). The distribution also plays an important role in generating new distributions.
Methods for developing generalized normal distributions seemed very limited until Azzalini (1985). A random variable X_{ λ } is said to follow the skewed normal distribution S N (λ) if the probability density function (PDF) of X_{ λ }is g (xλ) = 2ϕ (x) Φ (λ x), where ϕ (x) and Φ (x) are N (0,1) PDF and cumulative distribution function (CDF) respectively. Various extensions of S N (λ) have been proposed and studied (e.g., ArellanoValle et al. 2004; Arnold and Beaver 2002; Arnold et al. 2007; Choudhury and Abdul 2011; Balakrishnan 2002; Gupta and Gupta 2004; Sharafi and Behboodian 2008; Yadegari et al. 2008). For reviews on skewed normal and its generalization, one may refer to Kotz and Vicari (2005) and Lee et al. (2013). Pourahmadi (2007) showed that the skewed normal distribution SN (λ) approaches halfnormal as λ→∞. This explains why skewed normal distribution is limited in fitting real data. In order to allow for fitting diverse magnitudes of skewness, various works have been done by introducing different methods to capture the magnitude of the skewness.
Fernández and Steel (1998) introduced a twopiece PDF as $g\left(x\right)=\left\{\begin{array}{c}\mathit{\text{cf}}(\mathit{\alpha}x),x\ge 0,\\ \mathit{\text{cf}}(x/\mathit{\alpha}),x<0.\end{array}\right.$, c > 0 and α > 0, where f is a symmetric PDF defined on ℜ, which is unimodal and symmetric around 0. When f is normal, it is a generalized skewed normal. Kotz and Vicari (2005) suggested that α and 1/α be replaced by α_{1} and α_{2} respectively, in order to have more flexibility of controlling skewness. Another general framework that introduces a skew mechanism to symmetric distributions was given by Ferreira and Steel (2006). The corresponding skew family is g (x f, q) = f (x) q [ F (x)],x ∈ ℜ. The PDF g(x f, q) is a weighted version of f(x), with the weight function given by q [ F (x)]. If q follows the uniform distribution, then, g = f. When f is normal, this is a general framework for developing skewed normal distributions.
Eugene et al. (2002) introduced the betagenerated family of distributions with CDF
where b (t) is the PDF of the beta random variable and F (x) is the CDF of any random variable. The corresponding PDF to (1.1) is given by
If F is Φ, the CDF of the normal distribution, equation (1.2) defines the betanormal distribution. If α and β are integers, (1.2) is the α^{th} order statistic of the random sample of size (α + β  1).
The betanormal distribution can be unimodal or bimodal and it has been applied to fit a variety of real data including bimodal cases (Famoye et al. 2004). The main distinction between the method of skewed normal and the betagenerated normal is that the skewed normal method introduces a skewing mechanism into the normal distribution to generate skewed normal family. The skewness of the distribution is estimated by the skewing parameter. On the other hand, the betanormal distribution is generated by adding more parameters using beta distribution as the generator. Thus, the skewness is not directly defined by a specific parameter; rather it is the combination of all shape parameters that play the role of measuring skewness. For detailed review about the methods for generating continuous distributions, including the normal distribution, one may refer to Lee et al. (2013).
Alzaatreh et al. (2013) extended the beta generated family and defined the T  X (W) family. The CDF of the T  X (W) distribution is $G\left(x\right)={\int}_{a}^{W\left(F\right(x\left)\right)}r\left(t\right)\mathit{\text{dt}}$, where r (t) is the PDF of the random variable T with support (a, b) for  ∞ ≤ a < b ≤ ∞. The function W (F (x)) is monotonic and absolutely continuous. Aljarrah et al. (2014) took W (F(x)) to be the quantile function of a random variable Y and defined the T  X {Y} family as
where Q_{ Y }(p) is the quantile function of the random variable Y. In (1.1), X is used as a random variable having CDF F (x) and then as a random variable having CDF G (x) which may be confusing. This article first gives a unified notation to redefine the T  X {Y} as T  R {Y} and proposes several different generalizations of the normal distribution using the T  R{Y} framework.
Section 2 gives the unified definition of T  R {Y} family and defines several new generalized normal families. Section 3 gives some general properties of the proposed generalized normal families. Section 4 defines some new generalized normal distributions and studies some of their properties. Section 5 provides some applications to numerical data sets and the paper ends with a short summary and conclusions.
2 Tnormal families of distributions
Let T, R and Y be random variables with CDF F_{ T }(x) = P (T ≤ x),F_{ R }(x) = P (R ≤ x) and F_{ Y }(x) = P (Y ≤ x). The corresponding quantile functions are Q_{ T }(p), Q_{ R }(p) and Q_{ Y }(p), where the quantile function is defined as Q_{ Z }(p) = inf{z : F_{ Z }(z) ≥ p}, 0 < p < 1. If densities exist, we denote them by f_{ T }(x), f_{ R }(x) and f_{ Y }(x). Now assume the random variable T ∈ (a,b) and Y ∈ (c,d), for  ∞ ≤ a < b ≤ ∞ and  ∞ ≤ c < d ≤ ∞. Following the technique proposed by Aljarrah et al. (2014), the CDF of the random variable X is defined as
Note that (2.1) is an alternative expression to (1.3) without using X in two different situations. Hereafter, the family of distributions in (2.1) will be called the T R{Y} family of distributions.
Remark 1.
If X follows the distribution in (2.1), it is easy to see that

(i)
$$X\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}{Q}_{R}\left({F}_{Y}\right(T\left)\right)$$
,

(ii)
Q _{ X }(p) = Q _{ R }(F _{ Y }(Q _{ T }(p))),

(iii)
If $T\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}Y$ then $X\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}R$ and

(iv)
If $Y\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}R$ then $X\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}T$.
The corresponding PDF associated with (2.1) is
where ${{Q}^{\prime}}_{Y}\left({F}_{R}\right)=\frac{d}{d{F}_{R}}{Q}_{Y}\left({F}_{R}\right)$. Using the fact that Q_{ Y }(F_{ Y }(x)) = x, it follows that Q^{′}_{ Y }(F_{ Y }(x)) × f_{ Y }(x) = 1 so that Q^{′}_{ Y }(p) = 1/f_{ Y }(Q_{ Y }(p)). By taking p = F_{ R }(x), (2.2) reduces to
From (2.1) and (2.3), the hazard function of the random variable X can be written as
One can see from (2.3) and (2.4) that
Some general properties of the T  R {Y} family were recently studied in the literature, for more details see Aljarrah et al. (2014). Equivalent expressions to (2.2)  (2.4) are given in Aljarrah et al. (2014) by using the T  X {Y} notation. Table 1 gives some distributions of the T  R {Y} families based on quantile functions of some standard forms of distribution and some commonly used random variables T. The explicit expression of a T  R {Y} family can be obtained using (2.3) for different combinations of random variables T, R, and Y.
Several extensions from Table 1 can be made. First, one can use the quantile function of nonstandard distributions, such as nonstandard exponential, loglogistic, logistic, extreme value, and Weibull. For example, the quantile function of loglogistic is Q_{ Y }(p) = α (p/(1p))^{1/ β}, α, β > 0. By using this Q_{ Y } function, two additional parameters corresponding to the loglogistic distribution may be added to the T  R{loglogistic} family. Aljarrah et al. (2014) gave a more detailed list of T  R {Y} distributions based on quantile functions of nonstandard distributions. Secondly, one can introduce exponentiated and scale parameters by replacing F_{ T }(x) by ${F}_{T}^{\delta}\left(\mathrm{\alpha x}\right),\phantom{\rule{0.3em}{0ex}}\mathit{\alpha},\delta >0$ as well as for F_{ R }(x).
If R is a normal random variable with PDF f_{ R }(x) = ϕ (x) and CDF F_{ R }(x) = Φ (x), then (2.1) gives the T  normal{Y} family of distributions as
The corresponding PDF associated with (2.5) is
The hazard function of the T  normal {Y} family is given by ${h}_{X}\left(x\right)={h}_{\varphi}\left(x\right)\times \frac{{h}_{T}\left({Q}_{Y}\right(\Phi \left(x\right)\left)\right)}{{h}_{Y}\left({Q}_{Y}\right(\Phi \left(x\right)\left)\right)}$, where h_{ ϕ }(x) = ϕ (x)/(1  Φ (x)).
The Tnormal {Y} family is a general framework for generating many different generalizations of the normal distribution. Various existing generalizations of normal distributions can be obtained based on this framework. The beta normal (Eugene et al. 2002), Kumaraswamy normal (Cordeiro and de Castro 2011), and generalized betagenerated normal (Alexander et al. 2012) belong to the T  normal{standard uniform} families. The gammanormal distribution studied by Alzaatreh, et al. (2014) is a member of T  normal{standard exponential} family. For distribution “parsimony”, we will focus on the quantile functions of standard distributions in order to limit the number of parameters. Generalizations from using nonstandard quantile functions or adding exponentiated and/or scale parameters can be derived in a straightforward manner. In the following, we define the families of generalized normal (GN) distributions, Tnormal{Y}, using the standard quantile functions (b)(e) defined in Table 1.
2.1 Family of GN distributions from the quantile function of exponential distribution (T N{exponential})
By using the quantile function (b) in Table 1: Q_{ Y }(Φ (x)) =  log(1  Φ (x)), the corresponding CDF to (2.5) is
and the corresponding PDF is
where h_{ ϕ }(x) and H_{ ϕ }(x) =  log[ 1  Φ (x)] are the hazard and cumulative hazard functions for the normal distribution, respectively. Thus, this family of GN distributions is denoted as T  N{exponential}, which arises from the “hazard function of the normal distribution”.
2.2 Family of GN distributions from the quantile function of loglogistic distribution (T N{loglogistic})
By using the quantile function (c) in Table 1: Q_{ Y }(Φ (x)) = Φ (x)/(1  Φ (x)), the corresponding CDF to (2.5) is
and the corresponding PDF is
The family of GN distributions in (2.9) is denoted as T  N{loglogistic}, which arises from the “odds of the normal distribution”.
2.3 Family of GN distributions from the quantile function of logistic distribution (T N{logistic})
By using the quantile function (d) in Table 1: Q_{ Y }(Φ (x)) = log (Φ (x)/(1  Φ (x))), the corresponding CDF to (2.5) is
and the corresponding PDF is
The family of GN distributions in (2.11) is denoted as TN{logistic}, which arises from the “logit function of the normal distribution”.
2.4 Family of GN distributions from the quantile function of extreme value distribution (T N{extreme value})
By using the quantile function (e) in Table 1: Q_{ Y }(Φ (x)) = log( log(1  Φ (x)), the corresponding CDF to (2.5) is
and the corresponding PDF is
The family of GN distributions in (2.13) is denoted as T  N{extreme value}, which arises from the “extreme value function of the normal distribution”.
3 Some properties of the T normal family of distributions
In this section, some of the general properties of the Tnormal family will be discussed.
Lemma 1
(Transformation). For any random variable T with PDF f_{ T }(x), then the random variable

(i)
X = Φ ^{ 1}(1  e ^{ T}) follows the distribution of T  N{exponential} family in (2.7).

(ii)
X = Φ ^{ 1} (T/(1 + T)) follows the distribution of T  N{loglogistic} family in (2.9).

(iii)
X = Φ ^{ 1}(e ^{T}/(1 + e ^{T})) follows the distribution of T  N{logistic} family in (2.11).

(iv)
$$X={\Phi}^{1}\left(1{e}^{{e}^{T}}\right)$$
follows the distribution of T  N {extreme value} family in (2.13).
Proof.
The result follows immediately from Remark 1(i).
Lemma 1 gives the relationships between the random variable X and the random variable T. These relationships can be used to generate random samples from X by using T. For example, one can simulate the random variable X which follows the distribution of T  N{exponential} family in (2.7) by first simulating random variable T from the PDF f_{ T }(x) and then computing X = Φ^{ 1} (1  e^{ T}), which has the CDF F_{ X }(x).
Lemma 2.
The quantile functions for the (i) T  N{exponential}, (ii) T  N{loglogistic}, (iii) T  N{logistic}, and (iv) T  N{extreme value} distributions, are respectively,

(i)
$${Q}_{X}\left(\phantom{\rule{0.3em}{0ex}}p\right)={\Phi}^{1}\left\{1{e}^{{Q}_{T}\left(\phantom{\rule{0.3em}{0ex}}p\right)}\right\}$$
,

(ii)
Q _{ X }(p) = Φ ^{ 1} {Q _{ T }(p)/(1 + Q _{ T }(p))},

(iii)
$${Q}_{X}\left(\phantom{\rule{0.3em}{0ex}}p\right)={\Phi}^{1}\left\{{e}^{{Q}_{T}\left(\phantom{\rule{0.3em}{0ex}}p\right)}/\left(1+{e}^{{Q}_{T}\left(\phantom{\rule{0.3em}{0ex}}p\right)}\right)\right\}$$
,

(iv)
$${Q}_{X}\left(\phantom{\rule{0.3em}{0ex}}p\right)={\Phi}^{1}\left\{1{e}^{{e}^{{Q}_{T}\left(\phantom{\rule{0.3em}{0ex}}p\right)}}\right\}$$
.
Proof.
The result follows directly from Remark 1(ii).
Theorem 1.
The mode(s) of the T  N {Y} family are the solutions of the equation
Proof.
One can show the result in (3.1) by setting the derivative of the equation (2.6) to zero and then using the fact that ϕ^{′} (x) =  σ^{ 2} (x  μ) ϕ (x).
Corollary 1.
The mode(s) of the (i) T  N {exponential}, (ii) T  N {loglogistic}, (iii) T  N {logistic}, and (iv) T  N {extreme value} distributions, respectively, are the solutions of the equations
Note that the results in Theorem 1 do not imply that the mode is unique. It is possible that there is more than one mode for some of these GN distributions. For example, the logistic  N {logistic} distribution given in section 4 is a bimodal distribution. If T follows the gamma distribution with parameters α and β, equation (3.2) can be simplified to
This agrees with the result obtained by Alzaatreh et al. (2014) for the gammanormal distribution.
The entropy of a random variable X is a measure of variation of uncertainty (Rényi 1961). Shannon’s entropy for a random variable X with PDF g(x) is defined as E{ log(g(X))}.
Theorem 2.
The Shannon’s entropies for the T  N {Y} family is given by
Proof.
Since $X\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}{Q}_{R}\left({F}_{Y}\right(T\left)\right)$, this implies that $T\stackrel{d}{=}\phantom{\rule{0.3em}{0ex}}{Q}_{Y}\left({F}_{R}\right(X\left)\right)$. Hence, from (2.3) we have ${f}_{X}\left(x\right)=\frac{{f}_{T}\left(t\right)}{{f}_{Y}\left(t\right)}\times {f}_{R}\left(x\right)$. This implies
For the T  N {Y} family we have f_{ R }(x) = ϕ (x), so
The result in (3.3) follows from (3.4) and (3.5).
Corollary 2.
The Shannon’s entropies for the (i) T  N {exponential}, (ii) T  N {loglogistic}, (iii) T  N {logistic}, and (iv) T  N {extreme value} distributions, respectively, are given by
Proof.
The results in (i)(iv) can be easily shown using (3.3) and the fact that f_{ Y }(T) = e^{ T}, (1 + T)^{ 2}, e^{T}(1 + e^{T})^{ 2} and ${e}^{T}{e}^{{e}^{T}}$ for exponential, loglogistic, logistic and extreme value, respectively.
Theorem 3.
The r^{th} noncentral moments of the (i) T  N{exponential}, (ii) T  N {loglogistic}, (iii) T  N{logistic}, and (iv) T  N {extreme value} distributions, respectively, can be expressed as
where $A\left(\underline{k}\right)=A({k}_{1},{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{k}_{j})={(\sqrt{\pi}/2)}^{2{s}_{j}+j}{a}_{{k}_{1}}{a}_{{k}_{2}}\cdots {a}_{{k}_{j}}$, s_{ j }= k_{1} + k_{2} + ⋯ + k_{ j }, M_{ T }(  i) =E (e^{ iT}), ${a}_{k}=\frac{{c}_{k}}{2k+1}$, ${c}_{k}=\sum _{j=0}^{k1}\frac{{c}_{j}{c}_{k1j}}{(\phantom{\rule{0.3em}{0ex}}j+1)(2j+1)}$, and c_{0} = 1.
Proof.
We first show (3.7). By using Lemma 1, the r^{th} moments for the T  N{exponential} distribution can be written as E (X^{r}) = E (Φ^{ 1}(1  e^{ T}))^{r}. Since
the r^{th} moments can be written as
On using the series representation for erf^{ 1} (1  2e^{ T}) (Wolfram.com, 2014), we get ${\text{erf}}^{1}(12{e}^{T})=\sum _{k=0}^{\infty}{a}_{k}{(\sqrt{\pi}/2)}^{2k+1}{(12{e}^{T})}^{2k+1}$, where ${a}_{k}=\frac{{c}_{k}}{2k+1}$, ${c}_{k}=\sum _{j=0}^{k1}\frac{{c}_{j}{c}_{k1j}}{(\phantom{\rule{0.3em}{0ex}}j+1)(2j+1)}$, and c_{0} = 1. This implies
where $A({k}_{1},{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{k}_{j})={(\sqrt{\pi}/2)}^{2{s}_{j}+j}{a}_{{k}_{1}}{a}_{{k}_{2}}\cdots {a}_{{k}_{j}}$ and s_{ j }= k_{1} + k_{2} + ⋯ + k_{ j }. By using the binomial expansion on ${(12{e}^{T})}^{2{s}_{j}+j}$, (3.12) can be written as
The result of (3.7) follows by using equation (3.13) in equation (3.11). The results of (3.8)(3.10) can be obtained by applying the same technique for (3.7).
If T follows the gamma distribution with parameters α and β for the T  N {exponential}, we obtain the term M_{ T }(  i)=(1 + β i)^{ α} in (3.7). Thus, (3.7) gives the r^{th} noncentral moment of gammaN{exponential} distribution as shown in Alzaatreh et al. (2014).
The deviation from the mean and the deviation from the median are used to measure the dispersion and the spread in a population from the center. The mean deviation from the mean is denoted by D (μ), and the mean deviation from the median M is denoted by D (M).
Theorem 4.
D (μ) and D (M) for each of (i) T  N{exponential}, (ii) T  N {log  logistic}, (iii) T  N{logistic}, and (iv) T  N{extreme value} distributions, respectively, are
where ${S}_{\xi}(c,a,\mathit{\alpha})={\int}_{a}^{{Q}_{Y}(\Phi (c\left)\right)}{\xi}^{\mathit{\alpha}}{f}_{T}\left(u\right)\mathit{\text{du}}$ and Q_{ Y }(Φ (c)) =  log(1  Φ (c)).
where Q_{ Y }(Φ (c)) = Φ (c)/(1  Φ (c)).
where Q_{ Y }(Φ (c)) = log {Φ (c)/(1  Φ (c))}.
where Q_{ Y }(Φ (c)) = log { log(1  Φ (c))}.
Proof.
By definitions of D (μ) and D (M),
We first proof the results (3.14) and (3.15) for the T  N{exponential} family. Defining the integral
and using the substitution u =  log (1  Φ (x)), (3.24) can be written as
By using similar approach as in Theorem 3, the equation (3.25) can be written as
where A (k) is defined in the proof of Theorem 3, ${S}_{\xi}(c,a,\mathit{\alpha})={\int}_{a}^{{Q}_{Y}(\Phi (c\left)\right)}{\xi}^{\mathit{\alpha}}{f}_{T}\left(u\right)\mathit{\text{du}}$ and Q_{ Y }(Φ (c)) =  log (1  Φ (c)). The results in (3.14) and (3.15) follow by using (3.26) in (3.22) and (3.23). Applying the same techniques of showing (3.14) and (3.15), one can show the results of (3.16) and (3.17) for (ii), (3.18) and (3.19) for (iii), and (3.20) and (3.21) for (iv).
4 Some examples of GN families of distributions with different T distributions
In this section different T distributions are used to generate different GN distributions. In the following subsections, we present four new GN distributions namely, Weibull  N{exponential}, exponential  N{loglogistic}, logistic  N {logistic} and logistic  N{extreme value}. For illustrative purposes, we study some properties of the Weibull  N{logistic} distribution. To conserve space, properties of other GN distributions are not given. One can follow the same method to study the properties of other GN distributions.
4.1 The Weibull  N{exponential} distribution
If a random variable T follows the Weibull distribution with parameters c and γ, then ${f}_{T}\left(x\right)=c{\gamma}^{1}{\left(\frac{x}{\gamma}\right)}^{c1}{e}^{{\left(\frac{x}{\gamma}\right)}^{c}},\phantom{\rule{2.77626pt}{0ex}}c,\phantom{\rule{0.3em}{0ex}}\gamma >0$. From (2.8), the PDF of the Weibull  N {exponential} is defined as
Remark 2

(i)
When c = 1, the Weibull  N {exponential} reduces to the exponentialnormal distribution with θ = 1/γ.

(ii)
When c = γ = 1, the Weibull  N {exponential} reduces to the normal distribution.

(iii)
When c = 1 and γ ^{1} = n ∈ N, the PDF in (4.1) reduces to the distribution of the minimum order statistics, x _{(1)}, from a normal random sample of size n.
By using (2.7), the CDF of the Weibull  N{exponential} is given by
In Figures 1 and 2, various graphs of f_{ X }(x) when μ = 0, σ = 1 and for various values of c and γ are provided. These Figures indicate that the Weibull  N{exponential} PDF can be left skewed, right skewed, or symmetric. Also, the WeibullN{exponential} is left skewed whenever γ > 1 and right skewed whenever γ < 1. For fixed γ, the peak increases as c increases.
Some properties of the Weibull  N {exponential} are obtained in the following by using the general properties discussed in section 3.

(1)
Quantile function: By using Lemma 2, the quantile function of the Weibull  N {exponential} distribution is given by Q _{ X }(p) = Φ ^{ 1}{1  exp (  γ ( log (1  p))^{1/c})}.

(2)
Mode: By using Corollary 1, the mode of WeibullN{exponential} distribution is the solution of the following equation
$$x=\mathit{\mu}+{\sigma}^{2}{h}_{\varphi}\left(x\right)\left\{\frac{c1}{{H}_{\varphi}\left(x\right)}c{\gamma}^{c}{\left({H}_{\varphi}\right(x\left)\right)}^{c1}+1\right\}.$$ 
(3)
Shannon entropy: By using Corollary 2 and the fact that μ _{ T } = γ Γ (1 + 1/c) and σ _{ T }=1 + ξ (1  1/c) + log (γ/c) (see Song 2001), one can easily obtain the Shannon entropy of Weibull  N{exponential} distribution as
$${\mathit{\eta}}_{X}=\text{log}(\sigma \sqrt{2\pi})\gamma \Gamma (1+1/c)+\xi (11/c)+\text{log}(\gamma /c)+E{(X\mathit{\mu})}^{2}/\left(2{\sigma}^{2}\right)+1.$$ 
(4)
Moments: By using Theorem 3, a series representation of the r ^{th} moments of the WeibullN{exponential} distribution can be obtained by replacing M _{ T }( i) with
$$\sum _{k=0}^{\infty}\frac{{(1)}^{k}{\gamma}^{k}}{k!}\Gamma \left(1+\frac{k}{c}\right)$$
in equation (3.7).

(5)
Mean deviations: By using Theorem 4, the mean deviation from the mean and the mean deviation from the median of Weibull  N{exponential} distribution can be obtained by replacing S (μ,0,i) and S (M,0,i) with $\frac{c}{{i}^{c}{\gamma}^{c}}\sum _{k=0}^{\infty}\frac{{(1)}^{k}}{{\gamma}^{\mathit{\text{ck}}}{i}^{\mathit{\text{ck}}}k!}\Gamma \left[c\right(k+1),i\text{log}(1\Phi (\mathit{\mu})\left)\right]$ and $\frac{c}{{i}^{c}{\gamma}^{c}}\sum _{k=0}^{\infty}\frac{{(1)}^{k}}{{\gamma}^{\mathit{\text{ck}}}{i}^{\mathit{\text{ck}}}k!}\Gamma \left[c\right(k+1),i\text{log}(1\Phi \left(M\right)\left)\right]$ in equations (3.14) and (3.15) respectively, where $\Gamma (\mathit{\alpha},x)={\int}_{0}^{x}{u}^{\mathit{\alpha}1}{e}^{u}\mathit{\text{du}}$ is the incomplete gamma function.
4.2 The exponential  N{loglogistic} distribution
If a random variable T follows the exponential distribution with parameter λ, then f_{ T }(x) = λ e^{ λx}, λ > 0. From (2.10), the PDF of the exponentialN{loglogistic} is defined as
From (2.9), the CDF of (4.2) is given by ${F}_{X}\left(x\right)=1\text{exp}\left[\frac{\mathrm{\lambda \Phi}\left(x\right)}{1\Phi \left(x\right)}\right].$
In Figure 3, various graphs of f_{ X }(x) when μ = 0, σ = 1 and for various values of λ are provided. These graphs indicate that the exponential  N {log  logistic} distribution is always left skewed. Also, the skewness increases as λ decreases.
4.3 The logistic  N{logistic} distribution
If a random variable T follows the logistic distribution with parameter λ, then f_{ T }(x) = λ e^{ λx}(1 + e^{ λx})^{2}, λ > 0. From (2.12), the PDF of logisticN{logistic} distribution is defined as
From (2.11), the CDF of (4.3) is given by ${F}_{X}\left(x\right)=\frac{{\Phi}^{\mathit{\lambda}}\left(x\right)}{{\Phi}^{\mathit{\lambda}}\left(x\right)+{(1\Phi (x\left)\right)}^{\mathit{\lambda}}}.$
When λ = 1, (4.3) reduces to the normal distribution. In Figure 4, various graphs of f_{ X }(x) when μ = 0, σ = 1 and for various values of λ are provided. These graphs indicate that the PDF of logistic  N {logistic} can be bimodal and the bimodality occurs for small values of λ. Also, it is easy to see from the PDFs in (4.3) that the distribution is symmetric for all values of λ.
4.4 The logistic  N{extreme value} distribution
If a random variable T follows the logistic distribution with parameter λ, then f_{ T }(x) = λ e^{ λx}(1 + e^{ λx})^{ 2}, λ > 0. From (2.14), the PDF of the logistic  N {extreme value} distribution is defined as
From (2.13), the CDF of (4.4) is given by ${F}_{X}\left(x\right)=\frac{{{H}_{\varphi}}^{\mathit{\lambda}}\left(x\right)}{1+{{H}_{\varphi}}^{\mathit{\lambda}}\left(x\right)}.$
In Figure 5, various graphs of f_{ X }(x) when μ = 0, σ = 1 and for various values of λ are provided. These graphs indicate that the distribution is always right skewed. Also, the skewness increases as λ decreases.
5 Applications
To illustrate the flexibility of the GN distributions, we fit some GN distributions to a unimodal data set and a bimodal data set. The unimodal data with n = 66 in Table 2 is obtained from Nichols and Padgett (2006) on the breaking stress of carbon fibers of 50 mm in length. Alzaatreh at el. (2013) fitted the data set to the gammanormal distribution. They showed that the standard gammanormal distribution with μ = 0 and σ = 1 provides a good fit to the data set. The standard form of exponential  N {exponential}, exponentiated exponential  N {exponential} and Weibull  N {exponential} distributions with μ = 0 and σ = 1 are applied to fit the data set and the results compared with the results from standard gammanormal distribution. The maximum likelihood estimates, the loglikelihood value, the AIC (Akaike Information Criterion), the KolmogorovSmirnov (KS) test statistic, and the pvalue for the KS statistic for the fitted distributions are reported in Table 3. The results in Table 3 show that all the generalized normal distributions give an adequate fit to the data. However, the KS values indicate that the gamma  N {exponential} distribution provides the best fit among the distributions. Figure 6 displays the histogram and the fitted density functions for the data.
The second application is on a bimodal data set obtained from Emlet et al. (1987) on the asteroid and echinoid egg size. The data is available from the first author. The data consists of 88 asteroid species divided into three types; 35 planktotrophic larvae, 36 lecithotrophic larvae, and 17 brooding larvae. Since the logarithm of the egg diameters of the asteroids data has a bimodal shape, Famoye et al. (2004) applied the betanormal distribution to the logarithm of the data set. We apply the logisticN{logistic} distribution, which can be bimodal, to fit the same data. The results of the maximum likelihood estimates, the loglikelihood value, the AIC (Akaike Information Criterion), the KolmogorovSmirnov (KS) test statistic, and the pvalue for the KS statistic for the fitted distributions are reported in Table 4. The results in Table 4 show that both the betanormal and logisticN{logistic} distributions give an adequate fit to the data. However, the KS values indicate that the logisticN{logistic} distribution provides a better fit. Figure 7 displays the histogram and the fitted density functions for the data.
6 Summary and conclusions
The normal distribution is the most commonly used distribution in both statistical theory and applications. The generalization of the normal distribution is studied using the T  X framework proposed by Alzaatreh et al. (2013). Four types of generalized normal families from the quantile functions of the (i) exponential, (ii) loglogistic, (iii) logistic, and (iv) extreme value distributions are proposed. Some general properties are studied. Four generalized normal distributions are described and some of their properties investigated. It is noticed that the shapes of GN distributions can be symmetric, skewed to the right, skewed to the left or bimodal. This gives the families some flexibility in fitting real world data. Because the GN distributions include the normal distribution as a special case, using the GN distributions to fit data enables one to check if the additional parameters characterize the deviation from the normal distribution. Many types of generalizations of the normal distribution can be derived using the methodology described in this paper. Due to the fact that GN distributions are natural extensions from the normal distribution, statistical modeling by assuming the error term follows some form of GN distribution will be an interesting topic for future research.
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Acknowledgments
The authors are very grateful to the Associate Editor and the three anonymous reviewers for various constructive comments and suggestions that have greatly improved the presentation of the paper. The authors particularly thank one of the reviewers for suggesting the unified notation to define the TR{Y} family of distributions.
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The authors, viz AA, CL and FF with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Ayman Alzaatreh, Carl Lee and Felix Famoye contributed equally to this work.
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Alzaatreh, A., Lee, C. & Famoye, F. Tnormal family of distributions: a new approach to generalize the normal distribution. J Stat Distrib App 1, 16 (2014). https://doi.org/10.1186/21955832116
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Keywords
 TX distributions
 Hazard function
 Quantile function
 Shannon entropy