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New class of Lindley distributions: properties and applications
Journal of Statistical Distributions and Applications volume 8, Article number: 11 (2021)
Abstract
A new generalized class of Lindley distribution is introduced in this paper. This new class is called the TLindley{Y} class of distributions, and it is generated by using the quantile functions of uniform, exponential, Weibull, loglogistic, logistic and Cauchy distributions. The statistical properties including the modes, moments and Shannon’s entropy are discussed. Three new generalized Lindley distributions are investigated in more details. For estimating the unknown parameters, the maximum likelihood estimation has been used and a simulation study was carried out. Lastly, the usefulness of this new proposed class in fitting lifetime data is illustrated using four different data sets. In the application section, the strength of members of the TLindley{Y} class in modeling both unimodal as well as bimodal data sets is presented. A member of the TLindley{Y} class of distributions outperformed other known distributions in modeling unimodal and bimodal lifetime data sets.
Introduction and motivation
The Lindley distribution was first introduced as a one scale parameter distribution by Lindley (1958). In the recent years, researchers have given Lindley distribution a special attention for its importance in modelling complex real lifetime data. Some researchers went in the track of studying the Lindley distribution and its properties in more details. Ghitany et al. (2008) studied some properties of the one parameter Lindley distribution, and in the application part, they showed that it is more flexible and works better in modelling lifetime data than the known exponential distribution. Other researchers have introduced more flexible generalizations of Lindley by compounding Lindley with other wellknown distributions. A two parameters extension of Lindley distribution was investigated by Ghitany et al. (2011), Nadarajah et al. (2011), Shanker et al. (2013), and Shanker et al. (2017). More recently, another twoparameter Lindley distribution was introduced by Dey et al. (2019), which provides a better fit to skewed real data than the inverse Lindley distribution introduced by Sharma et al. (2015). With comparison to Weibull distribution, Arslan et al. (2017) proposed the use of Generalized Lindley distribution introduced by Nadarajah et al. (2011) as an alternative to the Weibull distribution when modeling wind speed data. A notable amount of attention in the literature is given to the threeparameter Lindley distribution generalization. Many threeparameter generalizations have been defined, analyzed and presented as a competitive models to wellknown distributions (From these threeparameter generalization; the one proposed by Zakerzadeh and Dolati (2009), Elbatal et al. (2013), and another threeparameter Lindley was introduced by Ashour and Eltehiwy (2015), which was extended by the exponentiation of Lindley distribution. The various threeparameter Lindley generalizations defined over the past decade assembled strength and flexibility in modelling the different shapes of lifetime data. As a result, less interest was given to studying Lindley generalization with more than three parameters. One of the few fourparameter generalizations of Lindley distribution is named the betageneralized Lindley distribution, and it was proposed by Oluyede and Yang (2015).
Generalizing distributions mainly depends on adding more flexibility to known distributions which result from implanting a basic distribution into more capable structure. The literature of Distribution Theory is full of different techniques to generalize continuous distributions to enhance their abilities in modeling real world data. Lee et al. (2013) discussed the different methods for generating distributions with more details.
In this paper, we use the transformtransformer framework (TX class) introduced by Alzaatreh et al. (2013) to generalize the one parameter Lindley distribution, and named it the TLindley{Y} class of distributions. Alzaatreh et al. (2014) refined the TX class by defining the TR{Y} framework. The TR{Y} method can be briefly defined as follows. Let T, R and Y be random variables with the respective CDFs F_{T}(x)=P(T≤x),F_{R}(x)=P(R≤x), and F_{Y}(x)=P(Y≤x). The PDFs of T, R and Y are f_{T}(x),f_{R}(x), and f_{Y}(x), respectively. Define the quantile function of the random variable Y as Q_{Y}(p)= inf{y:F_{Y}(y)≥p}, 0<p<1. The CDF and the PDF of the random variable X, following a TR{Y} family of distributions, are respectively defined as
Generalizing distributions using the TR{Y} framework involves adding more parameters to the generalized distribution. Hence there is more flexibility in modeling lifetime data. In the recent years, many new classes of distributions using the TR{Y} framework were introduced as a generalization to known distributions; Alzaatreh et al. (2014) used this technique to define the Tnormal{Y} family of distributions as a generalization to the normal distribution. Hamed et al. (2018) introduced a generalization for Pareto distribution using the transformtransformer framework. Alzaghal et al. (2013) proposed an exponentiation to the TR{Y} family of distributions by adding an extra parameter to the random variable T.
In this paper, the TR{Y} framework was used to generalize the Lindley distribution. The main motivation for using this frame, is to extend the characteristics of the baseline Lindley model to fit different shapes of data including left skewed, symmetric and bimodal. Moreover, to provide better fit than other distributions with the same or more numbers of parameters when modeling real world data sets.
The rest of the paper is structured as follows: In Section 2, the definition of the TLindley{Y} class of distributions, and six different subclasses of theTLindley{Y} are proposed. Some statistical properties of this new class of distributions such as modes, moments, and Shannon’s entropies are investigated in Section 3. In Section 4, some new members of this new class are introduced and studied in more details. The maximum likelihood estimation method is used to estimate the parameters of the normalLindley{Cauchy} distribution and a simulation study is performed in Section 5. The flexibility of this new class of distributions in fitting four different shapes of data is illustrated in Section 6. Finally, a brief conclusion of this paper is given in Section 7.
The TLindley{Y} class of distributions
The cumulative distribution function (CDF) and probability density function (PDF) of the one parameter Lindley distribution are, respectively, given by
Using Eq. (1) with F_{R}(x) to be the CDF defined in Eq. (3), the CDF and PDF of the random variable X following the general TLindley{Y} class of distributions are, respectively, given as
and
Table 1 provides the six different quantile functions that are used in generating six different subclasses of the TLindley{Y} class of distributions. The different subclasses of TLindley{Y} class introduced in this paper are different generalized classes of Lindley distribution with a maximum of three parameters.
2.1 New TLindley{Y} subclasses of distributions
Using the different quantile functions listed in Table 1, six new subclasses of the TLindley{Y} are defined in this subsection.
2.1.1 TLindley{uniform} class of distributions
By using the quantile function of the uniform distribution, Q_{Y}(p)=p, the corresponding CDF to (4) is
and the corresponding PDF to (5) is
2.1.2 TLindley{exponential} class of distributions
By using the quantile function of the exponential distribution, Q_{Y}(p)=−b log(1−p), the corresponding CDF to (4) is
and the corresponding PDF to (5) is
where \(\overline {F}{(x)}=1F(x).\) Using the hazard, h_{R}(x), and cumulative hazard, H_{R}(x), functions for the Lindley distribution, the CDF and PDF of the TLindley{exponential} class can be written as F_{X}(x)=F_{T}{−bH_{R}(x)} and f_{X}(x) = bh_{R}(x)f_{T}{−bH_{R}(x)}, respectively. Therefore, the TLindley{exponential} class of distributions arises from the hazard function of the Lindley distribution.
2.1.3 TLindley{Weibull} class of distributions
By using the quantile function of the Weibull distribution, Q_{Y}(p)=β(− log(1−p))^{1/α}, the corresponding CDF to (4) is
and the corresponding PDF to (5) is
Note that, if α=1 in Eq. (8), then the PDF of the TLindley{Weibull} class of distributions reduces to the PDF of the TLindley{exponential} class of distributions.
2.1.4 TLindley{Loglogistic} class of distributions
By using the quantile function of the loglogistic distribution, Q_{Y}(p)=a(p/(1−p))^{1/b}, the corresponding CDF to (4) is
and the corresponding PDF to (5) is
Note that, if a=b=1, then the family of distributions in Eq. (10) arising from the odds of the Lindley distribution and it is given by
2.1.5 TLindley{Logistic} class of distributions
By using the quantile function of the logistic distribution, Q_{Y}(p)=a+b log(p/(1−p)), the corresponding CDF to (4) is
and the corresponding PDF to (5) is
If a=0 and b=1, then the family of distribution in Eq. (11) arising from the logit function of the Lindley distribution and it is given by
2.1.6 TLindley{Cauchy} class of distributions
By using the quantile function of the Cauchy distribution, Q_{Y}(p)= tan(π(p −0.5)), the corresponding CDF to (4) is
and the corresponding PDF to (5) is
SOME structural properties of the TLindley{Y} class of distributions
In this section, some structural properties of the new proposed class of distributions is discussed in details. Proofs are not provided for obvious results.
Lemma 1
Let T be a random variable with PDF f_{T}(x), then the random variable X=Q_{R}(F_{Y}(T)) follows the T Lindley{Y} class of distributions, where Q_{R}(·) is the quantile function of Lindley distribution. As a result, X can be simplified to
where \(K_{W_{1}}(Z)= \frac {{1 + \theta }}{\theta }  \frac {1}{\theta }{W_{ 1}}\left (Z (\theta + 1){e^{ (\theta + 1)}}\right)\) and W_{−1} denotes the negative branch of the Lambert W function. For more details about the negative branch of the Lambert function; see Lazri and Zeghdoudi (2016).
Corollary 1
Based on Lemma 1, we have

(i)
\(X = K_{W_{1}} \left ({1T} \right)\) follows the TLindley{uniform} class,

(ii)
\(X = K_{W_{1}} \left ({ {e^{ \left ({T/b} \right)}}} \right)\) follows the TLindley{exponential} class,

(iii)
\(X = K_{W_{1}} \left ({{e^{ {{{\left ({T/\beta } \right)}^{\alpha } }}}}} \right)\) follows TLindley{Weibull} class,

(iv)
\(X = K_{W_{1}}\left ({ {{\left ({1 + {{\left ({T/a} \right)}^{b}}} \right)}^{ 1}}} \right)\) follows the TLindley{loglogistic} class,

(v)
\(X = K_{W_{1}}\left ({{{\left ({1 + {e^{\left ({T  a} \right)/b}}} \right)}^{ 1}}} \right)\) follows the TLindley{logistic} class,

(vi)
\(X = K_{W_{1}}\left ({(0.5  (\arctan T)/\pi)} \right)\) follows the TLindley{Cauchy} class.
The importance of Lemma 1 is that it shows the relationship between the random variable X and the random varaiable T. As an example, we can generate the random variable X that follows the TLindley{Cauchy} distribution in Eq. (12) by first simulating the random variable T from the PDF f_{T}(x) and then computing \(X = K_{W_{1}}\left ({(0.5  (\arctan T)/\pi)} \right)\), which has the CDF F_{X}(x)
Lemma 2
If Q_{X}(p),0<p<1 denote a quantile function of the random variable X. Then, the quantile function for TLindley{Y} class is given by Q_{X}(p)=Q_{R}{F_{Y}(Q_{T}(p)}, which can be reduced to
Corollary 2
Based on Lemma 2, the quantile functions for the (i) T Lindley{uniform}, (ii) T Lindley{exponential}, (iii) T Lindley{Weibull}, (iv) T Lindley{loglogistic}, (v) T Lindley{logistic}, and (vi) T Lindley{Cauchy} classes of distributions, are respectively

(i)
\(X = K_{W_{1}} \left ({1{Q_{T}}(p)} \right)\),

(ii)
\(X = K_{W_{1}} \left ({ {e^{ \left ({{Q_{T}}(p)/b} \right)}}} \right)\),

(iii)
\(X = K_{W_{1}} \left ({{e^{ {{{\left ({{Q_{T}}(p)/\beta } \right)}^{\alpha } }}}}} \right)\),

(iv)
\(X = K_{W_{1}}\left ({ {{\left ({1 + {{\left ({{Q_{T}}(p)/a} \right)}^{b}}} \right)}^{ 1}}} \right)\),

(v)
\(X = K_{W_{1}}\left ({{{\left ({1 + {e^{\left ({{Q_{T}}(p)  a} \right)/b}}} \right)}^{ 1}}} \right)\),

(vi)
\(X = K_{W_{1}}\left ({(0.5  (\arctan {Q_{T}}(p))/\pi)} \right)\).
Theorem 1
The mode(s) of the T Lindley{Y} class are the solutions of the equation
where M=θ−Ψ{f_{T}(Q_{y}(F_{R}(x)))} and Ψ(f)=f^{′}/f.
Proof
Using the fact that fR′(x)=(1/(1+x)−θ)f_{R}(x), the derivative of f_{X}(x) can be written as fX′(x)=f_{X}(x)R(x), where R(x)=1/(1+x)−θ+Ψ{f_{T}(Q_{y}(F_{R}(x)))}−Ψ{f_{y}(Q_{y}(F_{R}(x)))}. The equation to be solved to find the mode(s) of f_{X}(x) can be obtained by solving the equation R(x)=0. □
Corollary 3
Based on Theorem 1, the mode(s) of the (i) T Lindley{uniform}, (ii) T Lindley{exponential}, (iii) T Lindley{Weibull}, (iv) T Lindley{loglogistic}, (v) T Lindley{logistic}, and (vi) T Lindley{Cauchy} distributions are solutions of the following equations, respectively,

(i)
x+1=M^{−1},

(ii)
x+1=(M−θ^{2}(1+x)/(1+θ(1+x)))^{−1},

(iii)
\(x + 1 = {\left ({M+ \frac {{\left ({1 + x} \right){\theta ^{2}}}}{{\left ({1 + \theta + x\theta } \right)}}\left ({\frac {{\left ({1  \alpha } \right)}}{{\alpha \log \left [ {{\mbox{{e}}^{ x\theta }}\left ({1 + x\theta /(1 + \theta)} \right)} \right ]}}  1} \right)} \right)^{ 1}},\)

(iv)
\(x + 1 = {\left ({M  \frac {{\left ({1 + x} \right){\theta ^{2}}\left ({  2b\left ({1 + \theta + x\theta } \right) + \left ({b + 1} \right){\mbox{{e}}^{x\theta }}\left ({1 + \theta } \right)} \right)}}{{b\left ({  1  \left ({1 + x} \right)\theta + {\mbox{{e}}^{x\theta }}\left ({1 + \theta } \right)} \right)\left ({1 + \theta + x\theta } \right)}}} \right)^{ 1}},\)

(v)
\(x + 1 = {\left ({M  \frac {{\left ({1 + x} \right){\theta ^{2}}\left ({  {\mbox{{e}}^{x\theta }}\left ({1 + \theta } \right) + 2\left ({1 + \theta + x\theta } \right)} \right)}}{{\left ({1 + \theta + x\theta } \right)\left ({1 + \theta + x\theta  {\mbox{{e}}^{x\theta }}\left ({1 + \theta } \right)} \right)}}} \right)^{ 1}},\)

(vi)
\(x + 1 = ({M} {\left. {  2\pi {\mbox{{e}}^{ x\theta }} \frac {(1 + x)} {{(1 + \theta)}}{\theta ^{2}}\tan \left [ {\pi \left ({0.5  {\mbox{{e}}^{ x\theta }}\left ({1 + \frac {{x\theta }}{{1 + \theta }}} \right)} \right)} \right ]} \right)^{ 1}}.\)
In Section 4, the normalLindley {Cauchy } distribution is an example of a bimodal distribution, which means that Corollary 3 (vi) could have more than one solution to represent a bimodal distribution.
The entropy of a random variable X is a measure of variation of uncertainty. Entropy has several applications in information theory, physics, chemistry and engineering. The Shannon’s entropy for a continuous random variable X with PDF f(x) is defined as η_{X}=E[− logf(x)] (Shannon 1948).
Theorem 2
The Shannon’s entropy for the TLindley{Y} class is given by
where, η_{T} is the Shannon’s entropy for the random variable T and μ_{X} is the mean of the random variable X.
Proof
By the definition of the Shannon entropy,
Using the fact that the random variable T=Q_{y}{F_{R}(X)} for the T Lindley{Y} class, the η_{X} can be written as
Now, log(f_{R}(x))= log(θ^{2}/(1+θ))+ log(1+x)−θx, which implies
Hence, η_{X}=η_{T}+E(logf_{Y}(T))− log(θ^{2}/(1+θ))−E(log(1+X))−θμ_{X}. □
Corollary 4
Based on Theorem 2, the Shannon’s entropies of the (i) T Lindley{uniform}, (ii) T Lindley{exponential}, (iii) T Lindley{Weibull}, (iv) T Lindley{loglogistic}, (v) T Lindley{logistic}, and (vi) T Lindley{Cauch} classes of distributions, respectively, are given by

(i)
\({\eta _{X}} = E_{W_{1}}((1  T))  \log \left ({\theta /(1 + \theta)} \right),\)

(ii)
\({\eta _{X}} = E_{W_{1}}\left ({e^{ (T/b)}}\right) {\mu _{T}}/b  \log \left ({\theta /(1 + \theta)} \right),\)

(iii)
\({\eta _{X}} = E_{W_{1}}\left ({e^{ {{\left ({T/\beta } \right)}^{\alpha } }}}\right) + (\alpha  1)E(\log T)  E\left ({{T^{\alpha } }} \right)/{\beta ^{\alpha }}  \log \left ({\alpha \theta /{\beta ^{\alpha } }(1 + \theta)} \right)\),

(iv)
\({\eta _{X}} = E_{W_{1}}\left ({e^{\frac {T}{b}}}\right)  \log \left (\frac {\beta \theta }{\alpha (1 + \theta)} \right) + (\beta  1)E\left ({\log \frac {T}{\alpha }} \right)  2E\left ({\log \left ({1 + {{(\frac {T}{\alpha })}^{\beta } }} \right)} \right)\),

(v)
\({\eta _{X}} = E_{W_{1}}\left ({{\left ({1 + {e^{\frac {T  a}{b}}}} \right)}^{ 1}}\right)  \left (\frac {{\mu _{T}}  a}{b} \right) 2E\left ({\log \left ({1 + {e^{\frac {T + a}{b}}}} \right)} \right)  \log \left ({\frac {\theta }{b(1 + \theta)}} \right)\),

(vi)
\({\eta _{X}} = E_{W_{1}}\left (0.5  (\arctan T)/\pi \right)  E\left ({\log ({T^{2}} + 1)} \right)  \log \left ({\theta /\pi (1 + \theta)} \right).\)
Where, \(E_{W_{1}}(Z)={\eta _{T}}  \theta {\mu _{X}}  E\left ({\log \left ({  1  {W_{ 1}}\left \{ {  Z(\theta + 1){e^{ (\theta + 1)}}} \right \}} \right)} \right)\) and μ_{T} is the mean for the random variable T.
Theorem 3
The r^{th} noncentral moments for the TLindley{Y} class of distributions are given by
where\({b_{n,r}} = {(n{a_{0}})^{ 1}}\sum \limits _{l = 1}^{n} {\left [ {r(l + 1)  n} \right ]{a_{l}}{b_{n  l,r}}},{b_{0,r}} = a_{0}^{r},{a_{n}} =\frac {{{{(n + 1)}^{n  1}}}}{{(n)!}}{(\theta + }\) 1)^{n+1}e^{−(n+1)(θ+1)}, and\({c_{n,r}}= \binom {r}{j} {\left ((\theta +1)\right)^{rj}}\).
Proof
From lemma 1,
By using the binomial expansion and the series expansion for the Lambert W_{−1} function,
whenever z<1/e, the X^{r} can be written as
where c_{n} and a_{n} defined in the statement of Theorem 3. Therefore, \(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n = 0}^{\infty } {{c_{n,r}}}} \,{b_{n,r}}\,E{\left ({{\overline {F}}_{Y}(T)} \right)^{n + 1}}.\) See Gradshteyn and Ryzhik (2007), where b_{n,r} can be obtained from the recurrence relation defined in Theorem 3. □
Corollary 5
Based on Theorem 3, the r^{th} noncentral moments for the (i) T Lindley{uniform}, (ii) T Lindley{exponential}, (iii) T Lindley{Weibull}, (iv) T Lindley{loglogistic}, (v) T Lindley{logistic}, and (vi) T Lindley{Cauchy} classes of distributions, respectively, are given by

(i)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n = 0}^{\infty } {{c_{n,r}}}} {b_{n,}}_{r} E{\left ({1  T} \right)^{n + 1}}\), exists if E(1−T)^{n+1} exist.

(ii)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n = 0}^{\infty } {{c_{n,r}}} }{b_{n,r}}{M_{T}}\left (\frac {(n + 1)}{b}\right)\), exists if \({M_{T}}\left (\frac {(n + 1)}{b}\right) < \infty \).

(iii)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n = 0}^{\infty } {{c_{n,r}}} }{b_{n,r}}{M_{{T^{\alpha } }}}\left (\frac {(n + 1)}{{\beta ^{\alpha } }}\right)\), exists if \({M_{{T^{\alpha } }}}\left (\frac {(n + 1)}{{\beta ^{\alpha } }}\right)<\infty.\)

(iv)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n = 0}^{\infty } {{c_{n,r}}}} {b_{n,r}} E{\left ({1 + {{(\frac {T}{a})}^{ b}}} \right)^{ (n + 1)}}\), exists if \(E{\left ({1 + {{(\frac {T}{a})}^{ b}}} \right)^{ (n + 1)}}\) exist.

(v)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n,i = 0}^{\infty } {\binom { (n + 1)}{i} {{c_{n,r}}}} }{b_{n,r}}{M_{T  a}}(\frac {i}{b})\), exists if \({M_{T  a }}(\frac {i}{b})<\infty.\)

(vi)
\(E({X^{r}}) = {\theta ^{ r}}\sum \limits _{j = 0}^{r} {\sum \limits _{n,i = 0}^{\infty } {\binom {n+1}{i} {{{(0.5)}^{n + 1  i}}{{( \pi)}^{ i}}{c_{n,r}}}} } {b_{n,r}} E{\left ({\arctan T} \right)^{i}}\), exists if E(arctanT)^{j} exist.
Where M_{X}(t)=E(e^{tX}).
The next theorem is about the mean deviation from the mean, D(μ), and the mean deviation from the median, D(M), for the TLindley{Y} class of distributions.
Theorem 4
The D(μ) and D(M) for the TLindley{Y} class of distributions, respectively, are given by
where μ and M are the mean and median for X, and \({I_{q}} = \left ({  (1 + \theta)/\theta } \right){F_{X}}(q) + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {{a_{n}}} \int _{ \infty }^{{Q_{Y}}({F_{R}}(q))} {{f_{T}}(u)\,{{\left ({{\overline {F}_{y}}(u)} \right)}^{n}}\,du} \), where \({a_{n}} = \frac {{{n^{n  2}}}}{{(n  1)!}}{(\theta + 1)^{n}}{e^{ n(\theta + 1)}}.\)
Proof
For a nonnegative random variable X, we have D_{μ}=E(X−μ) =2μ F_{X}(μ)−2I_{μ}, and D_{M}=E(X−M)=μ−2I_{M}, where \({I_{q}} = \int _{0}^{q} {x\,{f_{X}}(x)\,dx}\). From Eq. (2) and Lemma 1, we have \({I_{q}} = \int _{ \infty }^{{Q_{Y}}({F_{R}}(q))} {{f_{T}}(u)\,{Q_{R}}({F_{Y}}(u))\,du.}\) By using the series expansion of Lambart W function given in Eq. (14), Q_{R}(·) can be written as \({Q_{R}}(p) =  \frac {{1 + \theta }}{\theta } + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {{a_{n}}} {\left ({1  p} \right)^{n}}\), where \({a_{n}} = \frac {{{n^{n  2}}}}{{(n  1)!}}{(\theta + 1)^{n}}\) e^{−n(θ+1)}. In turn, implies the result in Theorem 4. □
Corollary 6
Based on Theorem 4, the I(q)’s for (i) TLindley{uniform}, (ii) T Lindley{exponential}, (iii) T Lindley{Weibull}, (iv) T Lindley{loglogistic}, (v) T Lindley{logistic}, and (vi) T Lindley{Cauchy} classes of distributions, respectively, are given by

(i)
\({I_{q}} =H_{q} + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {\sum \limits _{j = 1}^{n} \binom {n}{j} {{{( 1)}^{j}}{a_{n}}}} {S_{u}}(q,0,j),\) where \(H_{q}=\left ({  (1 + \theta)/\theta } \right){F_{X}}(q),{S_{\xi } }(q,z,r) = \int _{z}^{{Q_{Y}}({F_{R}}(q))} {{\xi ^{r}}} {f_{T}}(u)du,\) and Q_{Y} for uniform distribution.

(ii)
\({I_{q}} =H_{q} + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {{a_{n}}} {S_{{e^{u/b}}}}(q,0,  n),\) where Q_{Y} for exponential distribution.

(iii)
\({I_{q}} = H_{q}+ \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {{a_{n}}} {S_{{e^{{{\left ({u/\beta } \right)}^{\alpha } }}}}}(q,0,  n),\) where Q_{Y} for Weibull distribution.

(iv)
\({I_{q}} = H_{q} + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {\sum \limits _{j = 0}^{\infty } {a_{n}}\binom {n}{j}} {S_{{{(u/a)}^{b}}}}(q,0,j),\) where Q_{Y} for loglogistic distribution.

(v)
\({I_{q}} = H_{q}+ \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {{a_{n}}} {S_{{e^{(u  a)/b}}}}(q,  \infty,j),\) where Q_{Y} for logistic distribution.

(vi)
\({I_{q}} = H_{q} + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {\sum \limits _{j = 0}^{n} \binom {n}{j}{{\left ({\frac {1}{2}} \right)}^{n  j}}{{( 1)}^{j}}\frac {{{a_{n}}}}{{{\pi ^{j}}}}{S_{\arctan (u)}}\left ({q,  \infty,j} \right)},\) where Q_{Y} for Cauchy distribution.
Theorem 4 and Corollary 6 can be used to obtain the mean deviations for T Lindley{uniform}, T Lindley {exponential}, T Lindley{Weibull}, T Lindley{loglogistic}, T Lindley{logistic}, and T Lindley{Cauchy} distributions.
Some members of the TLindley{Y} class of distributions
In this section, three new distributions of the class of TLindley{Y} are studied. The first is a member of the TL{E} subclass, the second is a member of the TL{LL} subclass, and the last one is a member of the TL{C} subclass.
4.1 The WeibullLindley{Exponential} distribution
Let the random variable T follows the Weibull distribution with parameters γ and α, the CDF of T is then \(\phantom {\dot {i}\!}{F_{T}}(x) = 1 e^{{{(x/\gamma)}^{\alpha } }}\), where x≥0,γ,α>0. Using Eq. (6), the CDF of the WeibullLindley{exponential} (WL{E}) distribution is defined as
With β=b/γ and using Eq. (7), the corresponding PDF of WL{E} distribution is given by
When α=1, Eq. (15) reduces to the exponentialLindley{exponential} distribution, when α=β=1, Eq. (15) is reduced to exponentiatedLindley distribution, and when α=β=θ=1, Eq. (15) is simply the Lindley distribution.
In Fig. 1, various plots of the WL{E} are provided for different values of the parameters θ,α, and β. The graphs show that the WL{E} can be unimodal with monotonically decreasing (reversed Jshape), skewed to the right, symmetric, or skewed to the left.
Using the general properties of the TLindley{Y} class of distributions derived in Section 2, the following properties of the WL{E} distribution are obtained:

(i)
The Quantile function: By using Corollary 2 part (ii), the quantile function of the WL{E} is given by
$${Q_{X}}(p) =  \frac{{\theta + 1}}{\theta}  \frac{1}{\theta }{W_{ 1}}\left\{ {  (\theta + 1){e^{ \left({(\theta + 1) + {{\left({  \log (1  p)} \right)}^{1/\alpha }}/\beta} \right)}}} \right\}. $$ 
(ii)
Mode: By using corollary 3 part (ii), the mode of WL{E} distribution is the solution of the following equation which can be evaluated numerically
$$x + 1 = {\left({\theta  \Psi \left\{ {\alpha \,{z^{\alpha  1}}{e^{z}}} \right\}  {\theta^{2}}(1 + x)/\left({1 + \theta (1 + x)} \right)} \right)^{ 1}}, $$where \(\phantom {\dot {i}\!}z = {\left ({  \beta \log \left \{ {\left ({1 + \theta x/(\theta + 1)} \right){e^{ \theta x}}} \right \}} \right)^{\alpha  1}}{e^{ {{\left ({\beta \log \left \{ {\left ({1 + \theta x/(\theta + 1)} \right){e^{ \theta x}}} \right \}} \right)}^{\alpha } }}}.\)

(iii)
The r^{th} noncentral moments: By using Corollary 5 part (ii), the r^{th} noncentral moments of WL{E} distribution are given by
$$E({X^{r}}) = {\theta^{ r}}\sum\limits_{j = 0}^{r} {\sum\limits_{n, i = 0}^{\infty} {{{c_{n,r}}{b_{n,r}}\frac{{{{\left({  1} \right)}^{i}}}}{{i!}}\frac{{{{(n + 1)}^{i}}}}{{{\beta^{i}}}}\Gamma (1 + i/\alpha)}} }. $$ 
(iv)
The Mean deviations: By using Theorem 4 and Corollary 6 part (ii), the mean deviation from the mean and the mean deviation from the median of WL{E} are given by
$${D_{\mu}} = 2\mu \,{F_{T}}({Q_{Y}}({F_{R}}(\mu)))  2{I_{\mu}} \, \text{and} \, {D_{M}} = \mu  2{I_{M}}, $$where I_{q} is given by \({I_{q}} =H_{q} + \frac {1}{\theta }\sum \limits _{n = 1}^{\infty } {\sum \limits _{i = 0}^{\infty } {\frac {{{{( 1)}^{i}}{n^{i}}}}{{i!}}}} \,{a_{n}}\,\Gamma \left [ {1 + \frac {i}{\alpha },{{\left ({  \log {{{\overline {F}}_{R}}(q)}} \right)}^{\alpha } }} \right ],\) and \(\Gamma (\alpha,x) = \int _{0}^{x} {{u^{\alpha  1}}{e^{ u}}du} \) is the incomplete gamma function.
4.2 The exponentialLindley{Loglogistic} distribution
Let the random variable T follows the exponential distribution with parameter γ and with the CDF F_{T}(x)=1−e^{−x/γ}. Using Eq. (9), the CDF of the exponentialLindley{LogLogistic} (EL{LL}) distribution is defined as
With α=a/γ and using Eq. (10), the PDF of the EL{LL} distribution is given by
Figures 2 and 3 provide different graphs of the EL{LL} distribution for various values of θ,α and β. The plots show that the EL{LL} distribution can be unimodal with either a monotonically decreasing behavior or skewed.
4.3 The NormalLindley{Cauchy} distribution
Let the random variable T follows the normal distribution with parameters μ and σ, then the CDF and the PDF of T are \(F_{T}(x)=\Phi \left (\frac {x\mu }{\sigma }\right)\) and \({f_{T}}(x)={\sigma ^{1}}\phi \left (\frac {x\mu }{\sigma }\right)=\frac {1}{{\sqrt {2\pi }\sigma }} e^{\left ({  {{(x  \mu)}^{2}}/2{\sigma ^{2}}} \right)},\) where ϕ(x) is N(μ,σ), Φ(x) is the CDF of ϕ(x),σ>0, and −∞<x,μ<∞.
Using Eq. (12), the CDF of the normalLindley{Cauchy}(NL{C}) distribution is defined as
Using Eq. (13), the PDF of the NL{C} distribution is given by
Figures 4 and 5 provide different graphs of the NL{C} distribution for various values of θ,μ and σ. Figure 4 shows that this new class of Lindley distribution can be unimodal with either skewed right, left, or symmetric curves. While Fig. 5 is showing that the NL{C} distribution, with only three parameters, is flexible to assemble bimodality behavior.
Estimation and simulation for the parameters of the NL{C} distribution
In this section, the unknown parameters of the NL{C} distribution are estimated using the maximum likelihood (ML) estimation method. Then, a simulation study to assist the performance of the maximum likelihood estimates (MLEs) is presented.
5.1 Estimation
Let X_{1},X_{2},....,X_{n} be a random sample of size n from NL{C} and Θ=(θ,μ,σ)^{T} be vector of parameters of dimension 3.
By setting \({z_{i}} = \frac {{1 + \theta + \theta {x_{i}}}}{{1 + \theta }}e^{\theta {x_{i}}}\), the loglikelihood function for Θ is given by
where \(C_{1}=0.5n\log (\frac {\pi }{2}) + 2n\log \theta + n\log (1 + \theta).\)
By setting \({t_{i}} = \frac {{\left ({1 + {x_{i}}} \right)}}{{\left ({1 + \theta + \theta {x_{i}}} \right)}}  \frac {1}{{\left ({1 + \theta } \right)}}  {x_{i}},\) the score vector
for the parameters θ,μ, and σ are derived analytically as
respectively. By setting the equations U_{μ}=0 and U_{σ}=0, the MLEs of \(\hat \mu \) and \(\hat \sigma \) are given by
Hence, we first maximize the loglikelihood function
with respect to θ, which gives the MLE of \(\hat \theta \), then substitute \(\hat \theta \) into Eq. (16) to find the MLE \(\hat \mu \) for the parameter μ, and substitute \(\hat \theta \) and \(\hat \mu \) into Eq. (17) to find the MLE \(\hat \sigma \) for the parameter σ. The SAS software was used to run all the needed analysis. The initial value for the parameter θ is obtained by assuming the random sample x_{i},i=1,2…,n is from Lindley distribution with parameter θ.
5.2 Simulation
A simulation is used to investigate the performance of the MLEs for the parameters’ of the NL{C} distribution. To generate a random sample form NL{C}, we first generate a random sample (t_{i},i=1,2…,n) from the normal distribution with parameters μ and σ, then apply the transformation in Corollary 1 part (vi); \(X = K_{W_{1}}\left ({(0.5  (\arctan T)/\pi)} \right)\), the resulting random sample will follow the NL{C} class.
Five different sample sizes are considered (n=25,50,100,200,500) with six different combinations of parameters (θ=0.5,1.5,2,μ=0,0.5,1,2,σ=0.5,1.5,2). For each parameters’ combination and each sample size, the simulation process is repeated 500 times. Table 2 gives the average biases (actualestimated) and the standard deviations of \(\hat \theta, \hat \mu \) and \(\hat \sigma \). It can be concluded from the table that the efficiency of the ML estimation method increased with the increase of the sample size where the bias and the standard deviation got smaller.
Table 2 shows that the ML estimation method is an appropriate technique for estimating the parameters of the NL{C} distribution. Similar estimation analysis was conducted for WL{E} and EL{LL} distributions. The results show that the ML method is an appropriate method for estimating the parameters of TLindley{Y} class of distributions.
Applications of some TLindley{Y} distributions
In this section, the applicability of WL{E}, EL{LL} and NL{C} as members of the TLindley{Y} class of distributions in modeling real data set is presented. Four different data sets including right skewed, left skewed, symmetric and bimodal shapes are considered. The flexibility of the TL{Y} members are compared with other wellknown distributions The computations and the statistical analysis for the different applications were done using the SAS software. For each one of the application, the ML estimation method is used to estimate the parameters of the fitted distributions. The initial value for the parameter θ for the WL{E} and EL{LL} distributions is obtained in a similar manner to the one used for the NL{C} and the initial value for the rest of their parameters is set to 1.
To compare the different fitted models, the following goodness of fit tests were carried: the value of two times the minus loglikelihood function −2 logl, Akaike information criterion (AIC), Bayesian information criterion (BIC), KolmogorovSmirnov (KS) and its corresponding pvalue. We have also considered the Anderson–Darling (A^{∗}) and Cramér–von Mises (W^{∗}), see Chen and Balakrishnan (1995) for details regarding these statistics. In general, the smaller the value of any of the goodness fit test correspond to a better fit for the data. Except for the pvalue of the KS test, the higher the pvalue the better the fit. The first three applications illustrate the different shapes of a unimodal data sets; right skewed, symmetric and left skewed. The fourth and last application represents the modeling of a bimodal data set.
6.1 The United Kingdom quarterly gas consumption between the years 19601986
In this first application, the quarterly logged demand for gas in the United Kingdom between the years 1960 1986 is used. The gas consumption data set is heavily skewed to the right. Tahir et al. (2016) fitted this data using the WeibullDagum distribution (WD) defined based on the WeibullG class that is proposed by Bourguignon et al. (2014). The WD flexibility in modeling this data set was compared to the one of BetaDagum distribution (BD) introduced by Domma and Condino (2013). The BD distribution is a subdistribution of the betaG class presented by Eugene et al. (2002).The five parameter DB with the additional two positive shape parameters, placed second in fitting this data set compared to the WD. Tahir et al. (2016) tests’ results for DW and BD are included in Tables 3 and 4. To examine the flexibility of the three members of TL{Y} class of distributions with three parameters in fitting this data set, another two competitive Lindley generalized distributions with three parameters were used in this comparison. The first one is the Beta Lindley distribution (BL) proposed by Merovci and Sharma (2014), and the second one is the Generalized Lindley distribution (GL) due to Zakerzadeh and Dolati (2009). On examining the results in Tables 3 and 4, we observe that all the distributions specifically the TL{Y} members give an adequate fit to the data. However, the threeparameter NL{C} distribution provides the smallest −2 logl, AIC, BIC, KS, A^{∗} and W^{∗} values and the highest KS pvalue compared to the other competing six distributions. This put the NL{C} in top at fitting this skewed right data among all the considered models including the fourparameter WD and the fiveparameter BD distributions.
Figure 6 displays the histogram and the fitted density functions for the UK gas consumption data set.
6.2 The annual maximum temperatures at England cities
The second data set is an approximately symmetric data set with 80 observations and is about the annual maximum temperatures recorded in Oxford and Worthing at England between the years 19011980. The data was first analyzed by Chandler and Bate (2007) and recently, Alzaatreh et al. (2015) used members of the Weibullgamma{Y} family to fit this data set. The results of the four parameter Weibullgamma{exponential} (WG{E}) and Weibullgamma{loglogistic} (WG{LL}) were included in this study. In addition to that, the flexibility of WL{E}, EL{LL} and NL{C} distributions in fitting this data set were compared to the performance of the BL and GL distributions (see application one).
With the lowest AIC, BIC and A^{∗} values, the threeparameter distributions NL{C} provides a good fit to this data set compared to the other competing distributions. With the lowest W^{∗} and KS test values and the highest pvalue for the KS test, the fourparameter WG{LL} is also providing a good fit for this data set. But, with one less parameter and very similar values of the KS and W^{∗} tests, the NL{C} (once again) is considered the best in fitting this data set. See Tables 5 and 6.
The histogram of the annual maximum temperature and the fitted PDFs of WL{E}, NL{C}, WG{E} and WG{L} distributions are presented in Fig. 7.
6.3 Time to AIDS
The third data set is about the times in years to infection with AIDS for 295 patients. Those patients were infected with AIDS virus from a contaminated blood transfusion, and the time in years it took each one of them to develop the AIDS was measured from the date of infection. This data is taken from Klein and Moeschberger (1997). Recently, Weibull Lindley distribution (WL) due to Asgharzadeh et al. (2018) and the extended Lindley distribution (EL) due to Bakouch et al. (2015) were used to fit this left skewed data set by Asgharzadeh et al. (2018). Both of these Lindley generalizations provided an adequate fit to this data set using multiple measures. In this compression, The same TL{Y} members used in the previous applications; WL{E},EL{LL} and NL{C} were fitted to this data in addition to the BL,GL, WL and the EL distributions defined earlier. The parameter estimates, and the various goodness of fit measures for these seven distributions in fitting this data set are recorded in Tables 7 and 8. It is obvious from the goodness of fit measures that the threeparameter TL{Y} members compete well with the other distributions. But, based on all of the used goodness of fit measures the EL{LL} rank first in fitting this data set by providing the lowest test values provided in Tables 7 and 8. While the WL and EL distributions provided an adequate fit, they still ranked second and third in fitting this data sets with the second and third lowest goodness of tests’ values. This application illustrates the flexibility of the EL{LL} distribution in fitting a left skewed data set compared to other wellknown Lindley generalizations. In Fig. 8, the histogram of the time to AIDS data set and the fitted PDFs are presented.
The three previous applications show the flexibility of WL{E}, EL{LL} and NL{C} members of the TL{Y} class of distributions in fitting the different shapes assembly by unimodal data set well including skewed left, skewed right as well as symmetric. The fourth and last application illustrates the flexibility of the NL{C} distribution, a member of the TL{Y} class of distributions, in fitting a bimodal data set.
6.4 Times to death of psychiatric patients
The bimodal data set is about the times to death of twenty six psychiatric patients admitted to the University of Iowa hospital during the period 19351948. This data set is taken from Klein and Moeschberger (1997). Recently, Alzaghal and Hamed (2019) analyzed this data using the bimodal normalLomax{Cauchy} distribution (NLo{C}). Tables 9 and 10 provide the parameter estimates, and the different goodness of fit tests’ results for the different distributions included in this comparison. The performance of the threeparameter NL{C} distribution in modeling this data set was compared to the following bimodal competitive distributions: the fourparameter betanormal distribution (BN) defined by Famoye et al. (2004), the fourparameter Weibullgamma{log logistic} distribution (WG{LL}) introduced by Alzaatreh et al. (2015), the threeparameter logisticnormal{logistic} distribution (LN{L}), a member of the Tnormal{Y} introduced by Alzaatreh et al. (2014), and the fourparameter NLo{C}. Finally, the threeparameter WL distribution was also fitted to this data set. The NL{C} distribution with only threeparameter fitted this data well with the smallest AIC, BIC, KS statistics and with the highest KS pvalue. The fourparameter NLo{C} distribution got the smallest A^{∗} and W^{∗} tests’ values, and the same −2 logl test value as the NL{C} making it a strong competitive to the NL{C} distribution in fitting this data set. But, with one less parameter the NL{C} is providing a superior fit to this data set. Comparing only the threeparameter distributions applicability in fitting this data set, the WL distribution rank second with the second smallest AIC, BIC, KS, A^{∗},W^{∗} tests’ values and with the second largest pvalue based on the KS statistic after the NL{C} distributions. In Fig. 9, the bimodality of the data set is clearly captured by the NL{C} distribution showing the superiority of the NL{C} in fitting this data set followed by the NLo{C} fit.
Conclusion
In this paper, we proposed a new class of distributions, socalled the TLindley{Y} class of distributions. This new Lindley generalization is based on the TR{Y} methodology. The TLindley{Y} class of distributions have a variety of shapes varying between unimodal and bimodal. Therefore, members of this class can effectively be used in analyzing unimodal as well as bimodal realworld data as presented in the application section. Different statistical properties of the new proposed class of distributions are investigated. Six new subclasses based on the quantile functions of uniform, exponential, Weibull, loglogistic, logistic and Cauchy are introduced. Three members from three different subclasses are studied in more details. A simulation analysis is carried to study the performance of the maximum likelihood estimation method in estimating the unknown parameters of the threeparameter NL{C} distribution. In the application section, the NL{C} distribution shows a superiority in fitting three out of the four data fitted in comparison to other known distributions.
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Abbreviations
 TX class:

The transformtransformer framework
 CDF:

The Cumulative Distribution Function
 PDF:

The Probability Density Function
 Q _{Y}(p):

The Quantile Function
 h _{R}(x):

The Hazard Function
 H _{R}(x):

The Cumulative Hazard
 W _{−1} :

The negative branch of the Lambert W function
 D(μ):

The mean deviation from the mean
 D(M):

The mean deviation from the median
 WL{E}:

The WeibullLindley{exponential} distribution
 EL{LL}:

The exponentialLindley{LogLogistic} distribution
 NL{C}:

The normalLindley{Cauchy} distribution
 ML:

The maximum likelihood
 MLEs:

The maximum likelihood estimates
 −2 logl :

Two times the minus loglikelihood function
 AIC:

The Akaike information criterion
 BIC:

The Bayesian information criterion
 KS:

The KolmogorovSmirnov criterion
 A ^{∗} :

The Anderson– Darling test
 W ^{∗} :

The Cramér–von Mises test
 WD :

The WeibullDagum distribution
 BD :

The BetaDagum distribution
 BL :

The Beta Lindley distribution
 GL :

The Generalized Lindley distribution
 WG{E}:

The Weibullgamma exponential
 WG{LL}:

The Weibullgamma{loglogistic}
 WL :

The Weibull Lindley distribution
 EL :

The extended Lindley distribution
 NLo{C}:

The normalLomax{Cauchy} distribution
 BN :

The betanormal distribution
 WG{LL}:

The Weibullgamma{log logistic} distribution
 LN{L}:

The logisticnormal{logistic} distribution
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Acknowledgements
The authors wish to thank the Organisers of the ICOSDA 2019 Conference for the opportunity to present in this great and well organized conference.The authors extend their sincere gratitude to the Editors of the Journal of Statistical Distributions and Applications for waving of the APC payment. Also, the authors would like to thank the editor and anonymous referees for their valuable comments and suggestions that improved the quality of the paper.
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Hamed, D., Alzaghal, A. New class of Lindley distributions: properties and applications. J Stat Distrib App 8, 11 (2021). https://doi.org/10.1186/s4048802100127y
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Keywords
 Two parameters Lindley distribution
 Moments
 Maximum likelihood estimation
 The TR{Y} framework