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
x=\mathit{\mu}+{\sigma}^{2}\varphi \left(x\right)\left\{\frac{{{{Q}^{\prime}}^{\prime}}_{Y}(\Phi (x\left)\right)}{{{Q}^{\prime}}_{Y}(\Phi (x\left)\right)}+\frac{{{f}^{\prime}}_{T}\left({Q}_{Y}\right(\Phi \left(x\right)\left)\right)}{{f}_{T}\left({Q}_{Y}\right(\Phi \left(x\right)\left)\right)}{{Q}^{\prime}}_{Y}(\Phi (x\left)\right)\right\}.
(3.1)
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
\begin{array}{ll}\text{(i)}\phantom{\rule{0.6em}{0ex}}x& =\mathit{\mu}+{\sigma}^{2}{h}_{\varphi}\left(x\right)\left\{1+\frac{{{f}^{\prime}}_{T}\left({H}_{\varphi}\left(x\right)\right)}{{f}_{T}\left({H}_{\varphi}\left(x\right)\right)}\right\},\phantom{\rule{2em}{0ex}}\\ \text{(ii)}\phantom{\rule{0.6em}{0ex}}x& =\mathit{\mu}+{\sigma}^{2}{h}_{\varphi}\left(x\right)\left\{2+\frac{{{f}^{\prime}}_{T}\left(\Phi \left(x\right)/(1\Phi (x\left)\right)\right)}{(1\Phi (x\left)\right){f}_{T}\left(\Phi \left(x\right)/(1\Phi (x\left)\right)\right)}\right\},\phantom{\rule{2em}{0ex}}\\ \text{(iii)}\phantom{\rule{0.6em}{0ex}}x& =\mathit{\mu}+{\sigma}^{2}\frac{{h}_{\varphi}\left(x\right)}{\Phi \left(x\right)}\left\{\frac{{{f}^{\prime}}_{T}\left(\text{log}\left\{\Phi \left(x\right)/(1\Phi (x\left)\right)\right\}\right)}{{f}_{T}\left(\text{log}\left\{\Phi \left(x\right)/(1\Phi (x\left)\right)\right\}\right)}+2\Phi \left(x\right)1\right\},\phantom{\rule{2em}{0ex}}\\ \text{(iv)}\phantom{\rule{0.6em}{0ex}}x& =\mathit{\mu}+{\sigma}^{2}\frac{{h}_{\varphi}\left(x\right)}{{H}_{\varphi}\left(x\right)}\left\{\frac{{{f}^{\prime}}_{T}\left\{\text{log}\right({H}_{\varphi}\left(x\right)\left)\right\}}{{f}_{T}\left\{\text{log}\right({H}_{\varphi}\left(x\right)\left)\right\}}+{H}_{\varphi}\left(x\right)1\right\}.\phantom{\rule{2em}{0ex}}\end{array}
(3.2)
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
x=\mathit{\mu}+{\sigma}^{2}{h}_{\varphi}\left(x\right)\left\{1+\frac{{{f}^{\prime}}_{T}\left({H}_{\varphi}\left(x\right)\right)}{{f}_{T}\left({H}_{\varphi}\left(x\right)\right)}\right\}=\mathit{\mu}+{\sigma}^{2}{h}_{\varphi}\left(x\right)\left[(\mathit{\alpha}1)/{H}_{\varphi}\left(x\right){\mathit{\beta}}^{1}+1\right].
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
{\sigma}_{X}={\sigma}_{T}+E\left(\text{log}\phantom{\rule{1em}{0ex}}{f}_{Y}\left(T\right)\right)+\text{log}(\sigma \sqrt{2\pi})+\frac{1}{2{\sigma}^{2}}E{(X\mathit{\mu})}^{2}.
(3.3)
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
{\sigma}_{X}={\sigma}_{T}+E\left(\text{log}{f}_{Y}\left(T\right)\right)E\left(\text{log}{f}_{R}\left(X\right)\right).
(3.4)
For the T  N {Y} family we have f_{
R
}(x) = ϕ (x), so
\text{log}(\varphi (x\left)\right)=\text{log}(\sigma \sqrt{2\pi}){\left[\right(x\mathit{\mu})/\sigma ]}^{2}/2.
(3.5)
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
\begin{array}{ll}\text{(i)}\phantom{\rule{1em}{0ex}}{\sigma}_{X}& =\text{log}(\sigma \sqrt{2\pi}){\mathit{\mu}}_{T}+{\sigma}_{T}+E{(X\mathit{\mu})}^{2}/\left(2{\sigma}^{2}\right),\phantom{\rule{2em}{0ex}}\\ \text{(ii)}\phantom{\rule{1em}{0ex}}{\sigma}_{X}& =\text{log}(\sigma \sqrt{2\pi})2E\left(\text{log}\right(1+T\left)\right)+{\sigma}_{T}+E{(X\mathit{\mu})}^{2}/\left(2{\sigma}^{2}\right),\phantom{\rule{2em}{0ex}}\\ \text{(iii)}\phantom{\rule{1em}{0ex}}{\sigma}_{X}& =\text{log}(\sigma \sqrt{2\pi})2E\left(\text{log}\right(1+{e}^{T}\left)\right)+{\mathit{\mu}}_{T}+{\sigma}_{T}+E{(X\mathit{\mu})}^{2}/\left(2{\sigma}^{2}\right),\phantom{\rule{2em}{0ex}}\\ \text{(iv)}\phantom{\rule{1em}{0ex}}{\sigma}_{X}& =\text{log}(\sigma \sqrt{2\pi})E\left({e}^{T}\right)+{\mathit{\mu}}_{T}+{\sigma}_{T}+E{(X\mathit{\mu})}^{2}/\left(2{\sigma}^{2}\right).\phantom{\rule{2em}{0ex}}\end{array}
(3.6)
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
\text{(i)}\phantom{\rule{1em}{0ex}}E\left({X}^{r}\right)=\sum _{j=0}^{r}\sum _{{k}_{1},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}\sum _{i=0}^{2{s}_{j}+j}{2}^{j/2}{\sigma}^{j}{\mathit{\mu}}^{rj}A\left(\underline{k}\right)\left(\genfrac{}{}{0ex}{}{r}{j}\right)\left(\genfrac{}{}{0ex}{}{2{s}_{j}+j}{i}\right){(2)}^{i}{M}_{T}(i),
(3.7)
\text{(ii)}\phantom{\rule{1em}{0ex}}E\left({X}^{r}\right)=\sum _{j=0}^{r}\sum _{{k}_{1},\phantom{\rule{0.3em}{0ex}}{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}{2}^{j/2}{\sigma}^{j}{\mathit{\mu}}^{rj}A\left(\underline{k}\right)\left(\genfrac{}{}{0ex}{}{r}{j}\right)E\left\{{\left(\frac{T1}{T+1}\right)}^{2{s}_{j}+j}\right\},
(3.8)
\text{(iii)}\phantom{\rule{1em}{0ex}}E\left({X}^{r}\right)=\sum _{j=0}^{r}\sum _{{k}_{1},\phantom{\rule{0.3em}{0ex}}{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}{2}^{j/2}{\sigma}^{j}{\mathit{\mu}}^{rj}A\left(\underline{k}\right)\left(\genfrac{}{}{0ex}{}{r}{j}\right)E\left\{{\left(\frac{{e}^{T}}{1+{e}^{T}}\right)}^{2{s}_{j}+j}\right\},
(3.9)
\text{(iv)}\phantom{\rule{1em}{0ex}}E\left({X}^{r}\right)=\sum _{j=0}^{r}\sum _{{k}_{1},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}\sum _{i=0}^{2{s}_{j}+j}{2}^{j/2}{\sigma}^{j}{\mathit{\mu}}^{rj}A\left(\underline{k}\right)\left(\genfrac{}{}{0ex}{}{r}{j}\right)\left(\genfrac{}{}{0ex}{}{2{s}_{j}+j}{i}\right){(2)}^{i}{M}_{{e}^{T}}(i),
(3.10)
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
{\Phi}^{1}(1{e}^{T})=\sqrt{2}\phantom{\rule{0.3em}{0ex}}\sigma \text{er}{\text{f}}^{1}(12{e}^{T})+\mathit{\mu},
the r^{th} moments can be written as
E\left({X}^{r}\right)=E{\left(\sqrt{2}\phantom{\rule{0.3em}{0ex}}\sigma \text{er}{\text{f}}^{1}(12{e}^{T})+\mathit{\mu}\right)}^{r}=\sum _{j=0}^{r}\left(\genfrac{}{}{0ex}{}{r}{j}\right){2}^{j/2}{\sigma}^{j}\phantom{\rule{0.3em}{0ex}}E\left\{{\text{(er}{\text{f}}^{1}(1{e}^{T}))}^{j}\right\}{\mathit{\mu}}^{rj}.
(3.11)
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
{\left({\text{erf}}^{1}(12{e}^{T})\right)}^{j}=\sum _{{k}_{1},\phantom{\rule{0.3em}{0ex}}{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}A({k}_{1},{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{k}_{j}){(12{e}^{T})}^{2{s}_{j}+j},
(3.12)
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
{\left(\text{er}{\text{f}}^{1}(12{e}^{T})\right)}^{j}=\sum _{{k}_{1},\phantom{\rule{0.3em}{0ex}}{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{k}_{j}=0}^{\infty}\sum _{i=0}^{2{s}_{j}+j}A({k}_{1},{k}_{2},\cdots \phantom{\rule{0.3em}{0ex}},{k}_{j})\left(\genfrac{}{}{0ex}{}{2{s}_{j}+j}{i}\right){(2)}^{i}{e}^{\mathit{\text{iT}}}.
(3.13)
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
\begin{array}{ccc}\text{(i)}\phantom{\rule{1em}{0ex}}& D(\mathit{\mu})& =\sqrt{2}\sigma \sum _{k=0}^{\infty}\sum _{i=0}^{2k+1}A\left(k\right)\left(\genfrac{}{}{0ex}{}{2k+1}{i}\right){(2)}^{i+1}{S}_{{e}^{u}}(\mathit{\mu},0,i),\end{array}
(3.14)
\begin{array}{cc}D\left(M\right)& =\sqrt{2}\sigma \sum _{k=0}^{\infty}\sum _{i=0}^{2k+1}A\left(k\right)\left(\genfrac{}{}{0ex}{}{2k+1}{i}\right){(2)}^{i+1}{S}_{{e}^{u}}(M,0,i),\end{array}
(3.15)
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)).
\begin{array}{ccc}\text{(ii)}\phantom{\rule{1em}{0ex}}& D(\mathit{\mu})& =\sqrt{8}\sigma \sum _{k=0}^{\infty}A\left(k\right){S}_{\frac{u1}{u+1}}(\mathit{\mu},0,2k+1),\end{array}
(3.16)
\begin{array}{cc}D\left(M\right)& =\sqrt{8}\sigma \sum _{k=0}^{\infty}A\left(k\right){S}_{\frac{u1}{u+1}}(M,0,2k+1),\end{array}
(3.17)
where Q_{
Y
}(Φ (c)) = Φ (c)/(1  Φ (c)).
\begin{array}{ccc}\text{(iii)}\phantom{\rule{1em}{0ex}}& D(\mathit{\mu})& =\sqrt{8}\sigma \sum _{k=0}^{\infty}A\left(k\right){S}_{\frac{{e}^{u}}{1+{e}^{u}}}(\mathit{\mu},\infty ,2k+1),\end{array}
(3.18)
\begin{array}{cc}D\left(M\right)& =\sqrt{8}\sigma \sum _{k=0}^{\infty}A\left(k\right){S}_{\frac{{e}^{u}}{1+{e}^{u}}}(M,\infty ,2k+1),\end{array}
(3.19)
where Q_{
Y
}(Φ (c)) = log {Φ (c)/(1  Φ (c))}.
\begin{array}{ccc}\text{(iv)}\phantom{\rule{1em}{0ex}}& D(\mathit{\mu})& =\sqrt{2}\sigma \sum _{k=0}^{\infty}\sum _{i=0}^{2k+1}A\left(k\right)\left(\genfrac{}{}{0ex}{}{2k+1}{i}\right){(2)}^{i}{S}_{{e}^{{e}^{u}}}(\mathit{\mu},\infty ,i),\end{array}
(3.20)
\begin{array}{cc}D\left(M\right)& =\sqrt{2}\sigma \sum _{k=0}^{\infty}\sum _{i=0}^{2k+1}A\left(k\right)\left(\genfrac{}{}{0ex}{}{2k+1}{i}\right){(2)}^{i}{S}_{{e}^{{e}^{u}}}(M,\infty ,i),\end{array}
(3.21)
where Q_{
Y
}(Φ (c)) = log { log(1  Φ (c))}.
Proof.
By definitions of D (μ) and D (M),
\begin{array}{ll}D(\mathit{\mu})& ={\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}\mathit{\mu}}(\mathit{\mu}x)\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}+{\int}_{\phantom{\rule{0.3em}{0ex}}\mathit{\mu}}^{\phantom{\rule{0.3em}{0ex}}\infty}(x\mathit{\mu})\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}=2{\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}\mathit{\mu}}(\mathit{\mu}x)\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}\phantom{\rule{2em}{0ex}}\\ =2\mathit{\mu}{F}_{X}(\mathit{\mu})2{\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}\mathit{\mu}}x\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathrm{dx.}\phantom{\rule{2em}{0ex}}\end{array}
(3.22)
\begin{array}{ll}D\left(M\right)& ={\int}_{\infty}^{M}(Mx)\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}+{\int}_{M}^{\infty}(xM)\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}\phantom{\rule{2em}{0ex}}\\ =2{\int}_{\infty}^{M}(Mx)\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}+E\left(X\right)M\phantom{\rule{2em}{0ex}}\\ =\mathit{\mu}2{\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}M}x\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathrm{dx.}\phantom{\rule{2em}{0ex}}\end{array}
(3.23)
We first proof the results (3.14) and (3.15) for the T  N{exponential} family. Defining the integral
{I}_{c}={\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}c}x\phantom{\rule{0.3em}{0ex}}{f}_{X}\left(x\right)\mathit{\text{dx}}={\int}_{\phantom{\rule{0.3em}{0ex}}\infty}^{\phantom{\rule{0.3em}{0ex}}c}\frac{\mathrm{x\varphi}\left(x\right)}{1\Phi \left(x\right)}{f}_{T}\left\{\text{log}(1\Phi (x)\right\}\mathit{\text{dx}},
(3.24)
and using the substitution u =  log (1  Φ (x)), (3.24) can be written as
{I}_{c}={\int}_{\phantom{\rule{0.3em}{0ex}}0}^{\phantom{\rule{0.3em}{0ex}}\text{log}(1\Phi (c\left)\right)}{\Phi}^{1}(1{e}^{u}){f}_{T}\left(u\right)\mathrm{du.}
(3.25)
By using similar approach as in Theorem 3, the equation (3.25) can be written as
{I}_{c}=\mathit{\mu}{F}_{X}\left(c\right)+\sqrt{2}\sigma \sum _{k=0}^{\infty}\sum _{i=0}^{2k+1}A\left(k\right)\left(\genfrac{}{}{0ex}{}{2k+1}{i}\right){(2)}^{i}{S}_{{e}^{u}}(c,0,i),
(3.26)
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).