# A Logistic Trigonometric Generalized Class of Distribution Characteristics, Applications, and Simulations

Authors: Zafar Mahmood; Hazar A. Khogeer; Eslam Hussam Hafez; Md. Moyazzem Hossain
Journal: Journal of Mathematics
 (2022)

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## Abstract

We propose a trigonometric generalizer/generator of distributions utilizing the quantile function of modified standard Cauchy distribution and construct a logistic-based new G-class disbursing cotangent function. Significant mathematical characteristics and special models are derived. New mathematical transformations and extended models are also proposed. A two-parameter model logistic cotangent Weibull (LCW) is developed and discussed in detail. The beauty and importance of the proposed model are that its hazard rate exhibits all monotone and non-monotone shapes while the density exhibits unimodal and bimodal (symmetrical, right-skewed, and decreasing) shapes. For parametric estimation, the maximum likelihood approach is used, and simulation analysis is performed to ensure that the estimates are asymptotic. The importance of the proposed trigonometric generalizer, G class, and model is proved via two applications focused on survival and failure datasets whose results attested the distinct better fit, wider flexibility, and greater capability than existing and well-known competing models. The authors thought that the suggested class and models would appeal to a broader audience of professionals working in reliability analysis, actuarial and financial sciences, and lifetime data and analysis.


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## Body

## 1. Introduction and Motivations

Generalizing a classical distribution is an old practice in distribution theory. The earliest work regarding generalizing a distribution was conducted by Pearson [1] using differential equations. Since 1985, the families (or classes) of distributions have been derived adopting the following famous methodologies: differential equation method, transformation techniques, compounding methodology, skewed distributions generation, parametric induction, quantile based approach, transformed (T)-transformer (X) mechanism, exponentiated T-X system, T-R\{Y\} approach, etc., It is possible to more readily manage data that is highly skewed with each new model that is developed because of its enhanced heavy-tailed, tractable core functions, and simplified simulation technique.The literature review explores the following four critical points which prove the basis for this study. (i) In the statistical literature, a slew of new families and models for algebraically generalizers/distribution’s generatorsWGx have been introduced in contrast of neglecting trigonometric equations (trigonometric generalizers). (ii) The interest in modelling directional and proportional data led applied researchers to develop and employ trigonometry function-based models capable of handling these datasets more smoothly and economically. (iii) The use of algebraic and trigonometric functions in mixture generalizers is still to be researched and investigated. (iv) Using the stated transformation, any current G class or model may be simply reversed in a subsequent version. The major motives are considered to be influenced and inspired by the generated results in terms of accuracy, adaptability, and goodness of fit (gof) which are included below:(i)
To introduce a generator of distributions based on cotangent function (that is a combination of algebraic and trigonometric functions and generators concurrently).(ii)
To introduce a new G-class called a new logistic-G class of distributions (LCG for short) in trigonometric scenario.(iii)
There are several advantages to the suggested class, including its simplification, lack of non-identifiability, and lack of over-parametrization.(iv)
Because of the injection of the cotangent function, the new CDF can increase flexibility, giving rise to new efficient and flexible models.(v)
Non-generalized and non-exponentiated models, according to the literature, give insufficient gof.The famous generalizers and corresponding G-families are presented in Table1.Table 1
The real line supporting generalizers and corresponding G-families.
Range of TGeneratorWGxModels of the T-X familyInventor(s)−∞,∞logGx/G¯xLog odd logistic familyTorabi and Montazeri [2]−∞,∞log−logG¯xLogistic-X familyTahir et al. [3]−∞,∞1−1+−logGx−λ−1Logistic type 2-G familyHassan et al. [4]−∞,∞log−logGxLogistic-G familyMansoor et al. [5]Moreover, the new generators and G-classes are introduced regarding several purposes. Among them, the recent and remarkable include the following. (A) To present a new family utilizing the additive model structure (see [6]). (B) New generator WGx is defined with quantile function (see [7]). (C) To achieve better gof and more flexibility than existing classical models (see [8]). (D) Type I half logistic Burr XG family has been constructed by Algarni et al. [9]. (E) The bivariate Weibull-G family based on copula function using odd classes has been introduced by El-Sherpieny et al. [10]. (F) A significant amount of distribution families was proposed using parameter induction (inserting one or maybe more new parameters (s) to the baseline), for example, a new one was put forward by Cordeiro et al. [11]. (G) Adopting the T-X family methodology only (see [12]). (H) Presenting a flexible family which deals with both monotone and non-monotone hazard rate function (see [13]). (I) Using a flexible family to introduced flexible generalized Pareto distribution (see [14]). The whole real line interval −∞,∞ distributions naturally come up when random variables should vary in the infinite real interval and several distributions such as logistic, normal, Laplace, t, Chen, and Gumbel distributions are supported on this interval. It has been used to describe the distribution of income and wealth in a fairly basic way. The logistic distribution has numerous uses in statistical analysis and has a form that resembles the normal distribution.A previous study [15] introduced logistic distribution whose main functions (cdf and pdf) in a new format are given below:(1)Fx=1+e−x−μ/σ−1,fx=e−x−μ/σσ1+e−x−μ/σ2,x∈ℝ.Alzaatreh et al. [16] contributed transformed (T)-transformer (X) family of distributions (for short, T-X family) which expanded vision about generators of distributions WGx, and in this article, via the T-X method, a new logistic-G class of statistical distributions is proposed adopting a cotangent-based trigonometric generator −cotπGx.This paper is outlined as follows. Section1 is about introduction and motivations while in Section 2, cotangent generator and LCG class are developed. In Section 3, some special models are presented while in Section 4, the characteristics of LCG class are deduced. In Section 5, a new model LCW along with significant properties is discussed and a simulation work is performed in Section 6. The importance of the new class and model is confirmed by two real-life applications using failure and survival datasets in Section 7. Finally, in Section 8, the conclusions are presented.


## 2. Development of Cotangent Generator

In this part, let us assume thatX is a random variable (r.v.) that follows the famous distribution, the standard Cauchy distribution, and its location parameter has value equal to μ=0 while the other parameter which is called the scale parameter has value equal to σ=1; then, its cdf and qf, respectively, are(2)Fx=12+tan−1xπ,x>0,(3)Qu=tanπu−0.5=tanπu−π2=−cotπu.Replacingu by Gx, WGx=−cotπGx, a new trigonometric generator based on cotangent function is achieved having support −∞,∞.


### 2.1. Genesis of LCG Class

By assuming thatrt is considered as the pdf of a r.v. T∈a,b for −∞≤a<b<∞ and WGx=−cotπGx, it fulfills the T-X family’s following requirements. (i) Since πGx∈0,π, then −cotπGx∈−∞,∞. (ii)  −cotπGx is differentiable and monotonically non-decreasing because cotangent function is differentiable and monotonically non-decreasing on 0,π. (iii) Since Gx⟶0   such that  x⟶−∞ , then  πGx⟶−∞  and  −cotπGx⟶−∞. Similarly, Gx⟶1   as  x⟶∞ ; then,  πGx⟶∞  and  −cotπGx⟶∞.Now, the main functions of LCG class in T-X format can be written as(4)Fx=∫−∞−cotπGxrtdt.=R−cotπGx,(5)fx=πgxcsc2πGxr−cotπGx,(6)hx=πgxcsc2πGxr−cotπGx1−R−cotπGx.


#### 2.1.1. LCG Class in Cotangent Scenario

LetT be a logistic r.v. with cdf Rt=1+e−t−1 and pdf rt=e−t1+e−t−2. Putting WGx=−cotπGx in (4), the cdf of new class is obtained and presented as(7)Fx=1+ecotπGx−1.The pdf corresponding to (7) reduces to(8)fx=πgxcsc2πGxecotπGx1+ecotπGx−2.


#### 2.1.2. LCG Class in Tangent Scenario

PuttingWGx=tanπGx−0.5 in (4), the cdf of new class may be expressed as below:(9)Fx=1+e−tanπGx−0.5−1.The pdf corresponding to (9) reduces to(10)fx=πgxsec2πGx−0.5e−tanπGx−0.51+e−tanπGx−0.5−2.


## 2.1. Genesis of LCG Class

By assuming thatrt is considered as the pdf of a r.v. T∈a,b for −∞≤a<b<∞ and WGx=−cotπGx, it fulfills the T-X family’s following requirements. (i) Since πGx∈0,π, then −cotπGx∈−∞,∞. (ii)  −cotπGx is differentiable and monotonically non-decreasing because cotangent function is differentiable and monotonically non-decreasing on 0,π. (iii) Since Gx⟶0   such that  x⟶−∞ , then  πGx⟶−∞  and  −cotπGx⟶−∞. Similarly, Gx⟶1   as  x⟶∞ ; then,  πGx⟶∞  and  −cotπGx⟶∞.Now, the main functions of LCG class in T-X format can be written as(4)Fx=∫−∞−cotπGxrtdt.=R−cotπGx,(5)fx=πgxcsc2πGxr−cotπGx,(6)hx=πgxcsc2πGxr−cotπGx1−R−cotπGx.


### 2.1.1. LCG Class in Cotangent Scenario

LetT be a logistic r.v. with cdf Rt=1+e−t−1 and pdf rt=e−t1+e−t−2. Putting WGx=−cotπGx in (4), the cdf of new class is obtained and presented as(7)Fx=1+ecotπGx−1.The pdf corresponding to (7) reduces to(8)fx=πgxcsc2πGxecotπGx1+ecotπGx−2.


### 2.1.2. LCG Class in Tangent Scenario

PuttingWGx=tanπGx−0.5 in (4), the cdf of new class may be expressed as below:(9)Fx=1+e−tanπGx−0.5−1.The pdf corresponding to (9) reduces to(10)fx=πgxsec2πGx−0.5e−tanπGx−0.51+e−tanπGx−0.5−2.


## 2.1.1. LCG Class in Cotangent Scenario

LetT be a logistic r.v. with cdf Rt=1+e−t−1 and pdf rt=e−t1+e−t−2. Putting WGx=−cotπGx in (4), the cdf of new class is obtained and presented as(7)Fx=1+ecotπGx−1.The pdf corresponding to (7) reduces to(8)fx=πgxcsc2πGxecotπGx1+ecotπGx−2.


## 2.1.2. LCG Class in Tangent Scenario

PuttingWGx=tanπGx−0.5 in (4), the cdf of new class may be expressed as below:(9)Fx=1+e−tanπGx−0.5−1.The pdf corresponding to (9) reduces to(10)fx=πgxsec2πGx−0.5e−tanπGx−0.51+e−tanπGx−0.5−2.


## 3. Special Models

Some special models of LCG class with their main functions and corresponding graphs are presented subsequently.


### 3.1. The Logistic Cot Exponential (LCE) Distribution

By assuming that the variableX follows an exponential distribution, then we may be able to express the central functions for the LCE distribution in the form as follows:(11)Fx=1+ecotπ1−e−αx−1,x>0,fx=παe−αxcsc2π1−e−αxecotπ1−e−αx1+ecotπ1−e−αx−2,hx=παe−αxcsc2π1−e−αx1+ecotπ1−e−αx−1.


### 3.2. The Logistic Cot Lindley (LCLi) Distribution

AssumingX is a Lindley random variable, we can write the new one-parameter LCLi model that has the following cdf, pdf, and hazard function. Figure 1 demonstrates the graph plots of the LCE, and Figure 2 demonstrates the the graph plots of the LCLi.(12)Fx=1+ecotπ1−e−αx1+αx/1+α−1,x,α>0,fx=π1+xα2e−αx/1+αcsc2π1−e−αx1+αx/1+αecotπ1−e−αx1+αx/1+α1+ecotπ1−e−αx1+αx/1+α2,hx=π1+xα2e−αx1+αcsc2π1−e−αx1+αx1+α1+ecotπ1−e−αx1+αx/1+α−1.Figure 1
Graphical representation for the plots of the LCE: (a) density and (b) hazard rate function using certain values of the parameters.
(a)(b)Figure 2
Graphical representation for the plots of the LCLi: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


### 3.3. The Logistic Cot Gamma (LCGa) Distribution

LetX be a gamma random variable; then, the LCGa model has the following main functions:(13)Fx=1+ecotπγα,βx/Γα−1,x>0,fx=πβαΓαxα−1e−βxcsc2πγα,βxΓαecotππγα,βx/Γα1+ecotππγα,βx/Γα−2,hx=πβαΓαxα−1e−βxcsc2πγα,βxΓα1+ecotππγα,βx/Γα−1.The plots of the LCGa are shown as graphs in Figure3.Figure 3
Graphical representation for the plots of the LCGa: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


### 3.4. The Logistic Cot Dagum (LCD) Distribution

By assuming that for the possibility thatX is a Dagum random variable, then the LCD distribution has the following primary functions:(14)Fx=1+ecotπ1+αx−β−λ−1,x>0,fx=παβλx−β+11+αx−β−λ+1csc2π1+αx−β−λecotπ1+αx−β−λ1+ecotπ1+αx−β−λ2,hx=παβλx−β+11+αx−β−λ+1csc2π1+αx−β−λ1+ecotπ1+αx−β−λ−1.The graphs in Figure4 show the plots of the LCD.Figure 4
Graphical representation for the plots of the LCD: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


## 3.1. The Logistic Cot Exponential (LCE) Distribution

By assuming that the variableX follows an exponential distribution, then we may be able to express the central functions for the LCE distribution in the form as follows:(11)Fx=1+ecotπ1−e−αx−1,x>0,fx=παe−αxcsc2π1−e−αxecotπ1−e−αx1+ecotπ1−e−αx−2,hx=παe−αxcsc2π1−e−αx1+ecotπ1−e−αx−1.


## 3.2. The Logistic Cot Lindley (LCLi) Distribution

AssumingX is a Lindley random variable, we can write the new one-parameter LCLi model that has the following cdf, pdf, and hazard function. Figure 1 demonstrates the graph plots of the LCE, and Figure 2 demonstrates the the graph plots of the LCLi.(12)Fx=1+ecotπ1−e−αx1+αx/1+α−1,x,α>0,fx=π1+xα2e−αx/1+αcsc2π1−e−αx1+αx/1+αecotπ1−e−αx1+αx/1+α1+ecotπ1−e−αx1+αx/1+α2,hx=π1+xα2e−αx1+αcsc2π1−e−αx1+αx1+α1+ecotπ1−e−αx1+αx/1+α−1.Figure 1
Graphical representation for the plots of the LCE: (a) density and (b) hazard rate function using certain values of the parameters.
(a)(b)Figure 2
Graphical representation for the plots of the LCLi: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


## 3.3. The Logistic Cot Gamma (LCGa) Distribution

LetX be a gamma random variable; then, the LCGa model has the following main functions:(13)Fx=1+ecotπγα,βx/Γα−1,x>0,fx=πβαΓαxα−1e−βxcsc2πγα,βxΓαecotππγα,βx/Γα1+ecotππγα,βx/Γα−2,hx=πβαΓαxα−1e−βxcsc2πγα,βxΓα1+ecotππγα,βx/Γα−1.The plots of the LCGa are shown as graphs in Figure3.Figure 3
Graphical representation for the plots of the LCGa: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


## 3.4. The Logistic Cot Dagum (LCD) Distribution

By assuming that for the possibility thatX is a Dagum random variable, then the LCD distribution has the following primary functions:(14)Fx=1+ecotπ1+αx−β−λ−1,x>0,fx=παβλx−β+11+αx−β−λ+1csc2π1+αx−β−λecotπ1+αx−β−λ1+ecotπ1+αx−β−λ2,hx=παβλx−β+11+αx−β−λ+1csc2π1+αx−β−λ1+ecotπ1+αx−β−λ−1.The graphs in Figure4 show the plots of the LCD.Figure 4
Graphical representation for the plots of the LCD: (a) density and (b) hazard rate using certain values of the parameters.
(a)(b)


## 4. Mathematical Properties of LCG Class

Here, important properties of the new class are presented.


### 4.1. The Inverse Function for Both pdf and cdf (Quantile Function)

We present an additional property ofX which is qf:(15)Qu=QGv.Here,QGv=G−1v is the parental qf, whereas v=1/πcot−1ln1−u/u. So, we can write the quantile density function Q′u as follows:(16)Q′u=π−1cot−1ln1−u/u−2πu1−u1+ln1−u/u2.


### 4.2. Useful Reliability Functions

In this part of the paper, we will concentrate our efforts to introduce the most important reliability functions. First we will define the survival function (sf)Sx, after that we write the equation of the hazard rate function (hrf) hx, and the reversed hazard rate function rx is as below, and the cumulative hazard rate function (chrf) Hx and mills’ ratio mx are, respectively, given below.(17)Sx=ecotπGx1+ecotπGx,hx=πgxcsc2πGx1+ecotπGx−1,rx=πgxcsc2πGxecotπGx1+ecotπGx−1,Hx=lnecotπGx1+ecotπGx−1,mx=1+ecotπGxπgxcsc2πGx2.


### 4.3. The Density Function Analytical Formulas

By solving (18), we can get the roots of the equation, and we can call it the solutions of the density function of the new class; it is possible to provide an analytical description of the forms of the new density:(18)g′xgx+πgxcsc2πGx2ecotπGx1+ecotπGx−2cotπGx−csc2πGx=0.(18) can be multi-rooted equation. Let λx=d2logfx/dx2; then,(19)λx=g′′xgx−g′x2gx2+πg′xcsc2πGx2ecotπGx1+ecotπGx−2cotπGx−csc2πGx+πgx2πgxcsc2πGx1+cotπGx+πgx1+cotπGx221+ecotπGx−2πgxcsc2πGxecotπGxcotπGx+csc2πGx+2πgxcsc2πGx2ecotπGx2.


### 4.4. The Hazard Function Analytical Formulas

In order to find the roots of (20), we must obtain the crucial points of the hrf hx.(20)g′xgx−2πgxcotπGx−πgxcsc2πGxecotπGx1+ecotπGx=0.As we can see, this equation has many solutions or we can say many roots are available for this equation. Suppose thatλ′x=d2loghx/dx2. We have(21)λ′x=g″xgx−g′x2gx2−2πg′xcotπGx−πgx2csc2πGx+π1+ecotπGx3g′xcsc2πGxecotπGx−πgx2csc2πGx2ecotπGx−2πgx2csc2πGxcotπGxecotπGx−π1+ecotπGx4×πgx2csc2πGx2ecotπGx2.Using any numerical software and (18) and (20), we can investigate local maximums and minimums, as well as inflexion points.


### 4.5. Linear Representation

The cdf of LCG class presented in (7) can be written as follows:(22)Fx=1+ecotπGx−1.Using the relation1+et−1=1−1+et−1,Fx may be easily formulated as(23)Fx=1−1+e−cotπGx−1.Through WolframAlpha,1+x−1=∑i=0∞−1ixi,  forx<1. We can demonstrate the cdf (2.6) in this form after applying this series and exponent series ex=∑j=0∞xj/j!, respectively.(24)Fx=∑i=1∞∑j=0∞ij−1i+j+1j!cotπGxj.Now, for the termcotπGxj, we will use the power series to expand it cotxs=∑i=0∞aisx2i−s, such that a0s=1 , a1s=−s/3 , a2s=s5s−7/90, etc,. are obtained by the aid of the highly speed MATHEMATICA software see Tahir [17]. Hence, cotπGxj=∑k=0∞akjπ2k−jGx2k−j, and(25)Fx=∑m=1∞Wi,j,k,lHmx.We know thatHmx is considered as the exponentiated distribution function parameter m in the power and(26)Wi,j,k,l=∑i,l=1∞∑j,k=0∞ij−1i+j+l+m+1j!2k−jlmlakjπ2k−j,Hmx=Gxm.With the expansion of (8), the following formula may be obtained from the previously mentioned idea of exponentiated distributions:(27)fx=∑m=1∞vi,j,k,lhmx,such that hmx can be noted as the exponentiated density having a power parameter m and(28)vi,j,k,l=∑i,l=1∞∑j,k=0∞ij−1i+j+l+m+1j!2k−jlmlakjπ2k−j,a0s=1,a1s=−s/3,a2s=s5s−7/90,and(29)hmx=mgxGxm−1.


### 4.6. Moments, Incomplete Moments, Mean Deviation, Skewness, and Kurtosis

Suppose that we have a random variable calledYm that follows the the exp-G distribution that has a power parameter m, i.e., having density hmx. Now we gave two formulas for the nth moment of X which follows (27), and the first one can expressed as follows:(30)EXn==∑i,l=1∞∑j,k=0∞vi,j,k,lEYmn.Thenth moment of X can be written in a second form as it may be deduced from (30) in terms of the G qf like the following equations:(31)EXn=∑i,l=1∞∑j,k=0∞vi,j,k,lm∫−∞∞xnGxm−1gxdx,EXn=∑m=0∞dm−1τn,m−1,where dm−1=∑i,l=1∞∑j,k=0∞m and τn,m−1=∫−∞∞xnGxm−1gxdx=∫01QGunum−1du. These integrals can be calculated numerically.


### 4.7. Weighted Moments

In this section, we introduce the equation of the weighted moments. So, we can write ther,sth probability weighted moment (PWM) of X as(32)ρr,s=EXrFXs=∫0∞xrFxsfxdx.Then,(33)Fxs=1+ecotπGx−s.Putting the pdf of LCG class (given below)(34)fx=πgxcsc2πGxecotπGx1+ecotπGx−2,in (32), after applying the binomial expansion and exponential series, we get(35)ρr,q=∑i,j=0∞π−s+2ii+1jj!∫−∞∞xrgxcsc2πGxcotπGxjdx.RegardingcotπGxj, we can use power series expansion cotxs=∑k=0∞aksx2k−s, such that a0s=1, a1s=−s/3, a2s=s5s−7/90, etc. Similarly, for csc2πGx=∑l=0∞cl2x2l−2, we get(36)ρr,q=∑i,j,k,l∞−s+2ii+1jj!π2k+l−j−1akjcl2∫−∞∞xrgxGx2k+l−j−2dx,ρr,q=∑i,j,k,l∞Ui,j,k,l∫−∞∞xr2k+l−j−1gxGx2k+l−j−2dx,where Ui,j,k,l=−s+2ii+1j/j!π2k+l−j−1akjcl2/2k+l−j−1.


### 4.8. Generating Function

This section is devoted to introduce the moment generating function (mgf), and it can be written as follows:(37)MXt=Eetx=∫−∞∞etxfxdx=∫−∞∞etx∑i,l=1∞∑J,k=0∞vi,J,k,lmGxm−1gxdx.


### 4.9. Order Statistics

In this section, we will focus our attention on one of the most important properties which is the order statistics. We can easily write the form ofIth order statistic density function as follows:(38)fI:Nx=D∑J=0N−I−1JN−IJfxFxJ+I−1,such that D=N!/I−1!N−I!.Considering the mathematical advancements described in PWM, fI:Nx becomes(39)fI:Nx=∑m,N,p=0∞W∗J,l,m,N,ph2p+N−m−1x,such that h2p+N−m−1x is exp-G density possessing power parameter 2p+N−m−1 for J,l,m,N,p≥0.(40)W∗J,l,m,N,p=∑J,l=0∞D−1JN−IJ−J+I+1ll+1mπ2p+N−m+1ap2dnmm!2p+N−m+1,and(41)h2p+N−m−1x=2p+N−m−1gxGx2p+N−m−1−1.It should come as no surprise that the density of the LCG order statistics is a linear combination of exp-G densities; this is extremely evident, as revealed by (39), which is the main result to be demonstrated.


### 4.10. Entropy Measures

The Shannon entropy is defined asηX=E−logfX, and for the LCG class, it may be formulated as the following equation:(42)ηX=E−logfx=−logπ−Eloggx−Elogcsc2πGx−EcotπGx+2Elog1+ecotπGx.Proof.
Alzaatreh et al. [16] deduced Shannon entropy of T-X family. Since here WGx=−cotπGx, adopting the same methodology, we get ηX as(43)ηX=EloggG−1e−eT−EeT+μT+ηT,where μT is the mean of r.v. T. Using (43), we can easily prove the Shannon entropy of the LCG class given in (42) where T follows logistic distribution.
Rényi entropy is(44)Iδf=11−δlogIδ,such that Iδf=∫ℜfδxdx,δ>0 and δ≠1.(45)fδx=πgxcsc2πGxecotπGx1+ecotπGx−2δ.
We get the following result after performing the expansion by the aid of power series:(46)fδx=πδ∑i=0∞∑j=0∞−2δiδ+ijj!gxδcsc2πGxδcotπGxj.
After incorporating the result, the Rényi entropy will reduce to(47)Iδf=11−δlog∫−∞∞πδ∑i=0∞∑j=0∞−2δiδ+ijj!gxδcsc2πGxδcotπGxj.
Now, we can write the final equation of the Rényi entropy as follows:(48)Iδf=11−δlog∑i,j=0∞Di,j,δ∫−∞∞gxδcsc2πGxδcotπGxj,where Di,j,δ=πδ−2δiδ+ij/j!.


### 4.11. Some Transformations to Develop New G-Classes

(i)
If the r.vY∼ standard logistic distribution, then X =  cot−1Y/π∼ logistic cotangent-G class of distributions.(ii)
If the r.vY∼ standard logistic distribution, then X =  cot−1Y/π1/α∼ logistic cotangent exponentiated-G class of distributions.(iii)
If the r.vY∼ standard logistic distribution, then X =  cot−1logYα/π∼ generalized logistic cotangent-G class of distributions.(iv)
If the r.vY=u1/α−1∼ standard logistic distribution, then X =  cot−1logY/π∼ generalized logistic cotangent exponentiated-G class of distributions.


## 4.1. The Inverse Function for Both pdf and cdf (Quantile Function)

We present an additional property ofX which is qf:(15)Qu=QGv.Here,QGv=G−1v is the parental qf, whereas v=1/πcot−1ln1−u/u. So, we can write the quantile density function Q′u as follows:(16)Q′u=π−1cot−1ln1−u/u−2πu1−u1+ln1−u/u2.


## 4.2. Useful Reliability Functions

In this part of the paper, we will concentrate our efforts to introduce the most important reliability functions. First we will define the survival function (sf)Sx, after that we write the equation of the hazard rate function (hrf) hx, and the reversed hazard rate function rx is as below, and the cumulative hazard rate function (chrf) Hx and mills’ ratio mx are, respectively, given below.(17)Sx=ecotπGx1+ecotπGx,hx=πgxcsc2πGx1+ecotπGx−1,rx=πgxcsc2πGxecotπGx1+ecotπGx−1,Hx=lnecotπGx1+ecotπGx−1,mx=1+ecotπGxπgxcsc2πGx2.


## 4.3. The Density Function Analytical Formulas

By solving (18), we can get the roots of the equation, and we can call it the solutions of the density function of the new class; it is possible to provide an analytical description of the forms of the new density:(18)g′xgx+πgxcsc2πGx2ecotπGx1+ecotπGx−2cotπGx−csc2πGx=0.(18) can be multi-rooted equation. Let λx=d2logfx/dx2; then,(19)λx=g′′xgx−g′x2gx2+πg′xcsc2πGx2ecotπGx1+ecotπGx−2cotπGx−csc2πGx+πgx2πgxcsc2πGx1+cotπGx+πgx1+cotπGx221+ecotπGx−2πgxcsc2πGxecotπGxcotπGx+csc2πGx+2πgxcsc2πGx2ecotπGx2.


## 4.4. The Hazard Function Analytical Formulas

In order to find the roots of (20), we must obtain the crucial points of the hrf hx.(20)g′xgx−2πgxcotπGx−πgxcsc2πGxecotπGx1+ecotπGx=0.As we can see, this equation has many solutions or we can say many roots are available for this equation. Suppose thatλ′x=d2loghx/dx2. We have(21)λ′x=g″xgx−g′x2gx2−2πg′xcotπGx−πgx2csc2πGx+π1+ecotπGx3g′xcsc2πGxecotπGx−πgx2csc2πGx2ecotπGx−2πgx2csc2πGxcotπGxecotπGx−π1+ecotπGx4×πgx2csc2πGx2ecotπGx2.Using any numerical software and (18) and (20), we can investigate local maximums and minimums, as well as inflexion points.


## 4.5. Linear Representation

The cdf of LCG class presented in (7) can be written as follows:(22)Fx=1+ecotπGx−1.Using the relation1+et−1=1−1+et−1,Fx may be easily formulated as(23)Fx=1−1+e−cotπGx−1.Through WolframAlpha,1+x−1=∑i=0∞−1ixi,  forx<1. We can demonstrate the cdf (2.6) in this form after applying this series and exponent series ex=∑j=0∞xj/j!, respectively.(24)Fx=∑i=1∞∑j=0∞ij−1i+j+1j!cotπGxj.Now, for the termcotπGxj, we will use the power series to expand it cotxs=∑i=0∞aisx2i−s, such that a0s=1 , a1s=−s/3 , a2s=s5s−7/90, etc,. are obtained by the aid of the highly speed MATHEMATICA software see Tahir [17]. Hence, cotπGxj=∑k=0∞akjπ2k−jGx2k−j, and(25)Fx=∑m=1∞Wi,j,k,lHmx.We know thatHmx is considered as the exponentiated distribution function parameter m in the power and(26)Wi,j,k,l=∑i,l=1∞∑j,k=0∞ij−1i+j+l+m+1j!2k−jlmlakjπ2k−j,Hmx=Gxm.With the expansion of (8), the following formula may be obtained from the previously mentioned idea of exponentiated distributions:(27)fx=∑m=1∞vi,j,k,lhmx,such that hmx can be noted as the exponentiated density having a power parameter m and(28)vi,j,k,l=∑i,l=1∞∑j,k=0∞ij−1i+j+l+m+1j!2k−jlmlakjπ2k−j,a0s=1,a1s=−s/3,a2s=s5s−7/90,and(29)hmx=mgxGxm−1.


## 4.6. Moments, Incomplete Moments, Mean Deviation, Skewness, and Kurtosis

Suppose that we have a random variable calledYm that follows the the exp-G distribution that has a power parameter m, i.e., having density hmx. Now we gave two formulas for the nth moment of X which follows (27), and the first one can expressed as follows:(30)EXn==∑i,l=1∞∑j,k=0∞vi,j,k,lEYmn.Thenth moment of X can be written in a second form as it may be deduced from (30) in terms of the G qf like the following equations:(31)EXn=∑i,l=1∞∑j,k=0∞vi,j,k,lm∫−∞∞xnGxm−1gxdx,EXn=∑m=0∞dm−1τn,m−1,where dm−1=∑i,l=1∞∑j,k=0∞m and τn,m−1=∫−∞∞xnGxm−1gxdx=∫01QGunum−1du. These integrals can be calculated numerically.


## 4.7. Weighted Moments

In this section, we introduce the equation of the weighted moments. So, we can write ther,sth probability weighted moment (PWM) of X as(32)ρr,s=EXrFXs=∫0∞xrFxsfxdx.Then,(33)Fxs=1+ecotπGx−s.Putting the pdf of LCG class (given below)(34)fx=πgxcsc2πGxecotπGx1+ecotπGx−2,in (32), after applying the binomial expansion and exponential series, we get(35)ρr,q=∑i,j=0∞π−s+2ii+1jj!∫−∞∞xrgxcsc2πGxcotπGxjdx.RegardingcotπGxj, we can use power series expansion cotxs=∑k=0∞aksx2k−s, such that a0s=1, a1s=−s/3, a2s=s5s−7/90, etc. Similarly, for csc2πGx=∑l=0∞cl2x2l−2, we get(36)ρr,q=∑i,j,k,l∞−s+2ii+1jj!π2k+l−j−1akjcl2∫−∞∞xrgxGx2k+l−j−2dx,ρr,q=∑i,j,k,l∞Ui,j,k,l∫−∞∞xr2k+l−j−1gxGx2k+l−j−2dx,where Ui,j,k,l=−s+2ii+1j/j!π2k+l−j−1akjcl2/2k+l−j−1.


## 4.8. Generating Function

This section is devoted to introduce the moment generating function (mgf), and it can be written as follows:(37)MXt=Eetx=∫−∞∞etxfxdx=∫−∞∞etx∑i,l=1∞∑J,k=0∞vi,J,k,lmGxm−1gxdx.


## 4.9. Order Statistics

In this section, we will focus our attention on one of the most important properties which is the order statistics. We can easily write the form ofIth order statistic density function as follows:(38)fI:Nx=D∑J=0N−I−1JN−IJfxFxJ+I−1,such that D=N!/I−1!N−I!.Considering the mathematical advancements described in PWM, fI:Nx becomes(39)fI:Nx=∑m,N,p=0∞W∗J,l,m,N,ph2p+N−m−1x,such that h2p+N−m−1x is exp-G density possessing power parameter 2p+N−m−1 for J,l,m,N,p≥0.(40)W∗J,l,m,N,p=∑J,l=0∞D−1JN−IJ−J+I+1ll+1mπ2p+N−m+1ap2dnmm!2p+N−m+1,and(41)h2p+N−m−1x=2p+N−m−1gxGx2p+N−m−1−1.It should come as no surprise that the density of the LCG order statistics is a linear combination of exp-G densities; this is extremely evident, as revealed by (39), which is the main result to be demonstrated.


## 4.10. Entropy Measures

The Shannon entropy is defined asηX=E−logfX, and for the LCG class, it may be formulated as the following equation:(42)ηX=E−logfx=−logπ−Eloggx−Elogcsc2πGx−EcotπGx+2Elog1+ecotπGx.Proof.
Alzaatreh et al. [16] deduced Shannon entropy of T-X family. Since here WGx=−cotπGx, adopting the same methodology, we get ηX as(43)ηX=EloggG−1e−eT−EeT+μT+ηT,where μT is the mean of r.v. T. Using (43), we can easily prove the Shannon entropy of the LCG class given in (42) where T follows logistic distribution.
Rényi entropy is(44)Iδf=11−δlogIδ,such that Iδf=∫ℜfδxdx,δ>0 and δ≠1.(45)fδx=πgxcsc2πGxecotπGx1+ecotπGx−2δ.
We get the following result after performing the expansion by the aid of power series:(46)fδx=πδ∑i=0∞∑j=0∞−2δiδ+ijj!gxδcsc2πGxδcotπGxj.
After incorporating the result, the Rényi entropy will reduce to(47)Iδf=11−δlog∫−∞∞πδ∑i=0∞∑j=0∞−2δiδ+ijj!gxδcsc2πGxδcotπGxj.
Now, we can write the final equation of the Rényi entropy as follows:(48)Iδf=11−δlog∑i,j=0∞Di,j,δ∫−∞∞gxδcsc2πGxδcotπGxj,where Di,j,δ=πδ−2δiδ+ij/j!.


## 4.11. Some Transformations to Develop New G-Classes

(i)
If the r.vY∼ standard logistic distribution, then X =  cot−1Y/π∼ logistic cotangent-G class of distributions.(ii)
If the r.vY∼ standard logistic distribution, then X =  cot−1Y/π1/α∼ logistic cotangent exponentiated-G class of distributions.(iii)
If the r.vY∼ standard logistic distribution, then X =  cot−1logYα/π∼ generalized logistic cotangent-G class of distributions.(iv)
If the r.vY=u1/α−1∼ standard logistic distribution, then X =  cot−1logY/π∼ generalized logistic cotangent exponentiated-G class of distributions.


## 5. LCW Distribution

In this section, a two-parameter special model logistic cot Weibull (LCW) with its properties is presented.


### 5.1. Methodology

TakingGx, as the cdf of Weibull distribution while gx as the corresponding density, it follows the two-parameter LCWm we can write three of cdf, and its corresponding pdf and associated with its hazard function, as formulated below respectively, are:(49)Fx=1+ecotπ1−e−λxα−1,x>0,fx=πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−2,hx=πλαxα−1e−λxαcsc2π1−e−λxα1+ecotπ1−e−λxα−1.The graphs represented in Figure5 demonstrate the plots of the LCW.Figure 5
Graphical representation for the plots of the LCW: (a) density and (b) hazard rate for some parametric values.
(a)(b)


### 5.2. Reliability Functions

We have(50)Sx=1−Fx=1−1+ecotπ1−e−λxα−1,x>0.hx=πλαxα−1e−λxαcsc2π1−e−λxα1+ecotπ1−e−λxα−1,rx=πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−21+ecotπ1−e−λxα−1,Hx=−lnSx=ln1−1+ecotπ1−e−λxα−1−1,mx=1−1+ecotπ1−e−λxα−1πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−2.


### 5.3. Quantile Function

This section is devoted to demonstrate the quantile function equation of LCW which is(51)Qu=QGv,where QGv=G−1v is the parent qf and v=−1/λln1+1/πcot−1ln1−u/u−α.


### 5.4. Residual and Reverse Residual Life

The residual life has several uses in probability and statistics and risk assessment. Suppose thatX represents a unit’s lifespan and X≥0 with PX=1; then, the r.v. Xt=t−X|X≤t, for a fixed t>0, is known as time since failure. The residual lifetime of LCW r.v. X is denoted by Rtx and is defined as(52)Rtx=ecotπ1−e−λx+tα1+ecotπ1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λx+tα.Additionally, the reversed hazard rate functionR¯tx is written as(53)R¯tx=ecotπ1−e−λx−tα1+ecotπ1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λx−tα.


### 5.5. Stochastic Ordering

Stochastic ordering is a tool for analysing the constitutional properties of stochastic structures that are intricate. There are several forms of stochastic orderings that can be used to sort random variables according to their distinguished properties. SupposeS and K are independent random variables with cdfs FS and FK, respectively; then, S is said to be smaller than K iff it satisfies the following.(i)
Stochastic order (S≤st (K)) if FSS≥FKS for all S.(ii)
Hazard rate order (S≤hr (K)) if hsS≥hKS for all S.(iii)
Mean residual life order (S≤mrl (K)) if msS≥mkS for all S.(iv)
Likelihood ratio order (S≤lr (K)) if fXS/fyS decrease in S.The LCW distributionλ,α is ordered according to the strongest “likelihood ratio” ordering, as demonstrated in the following theorem. The versatility of two-parameter LCW distribution λ,α is demonstrated. Let S follow LCW λ1,α1 and K follow LCW λ2,α2. Then, the likelihood ratio is(54)fSSfKS=λ1α1π1−1/λ1log1−w1/π11−1/α1λ2α2π2−1/λ2log1−w2/π21−1/α2,where w1=π11−e−λ1Sα1 and w2=π21−e−λ2Sα2. Again,(55)ddxlogfSSfKS=−α1−α2logπ1e−λ1Sα1/π1−1logπ2e−λ2Sα2/π2−1α1α2π1e−λ1Sα1π2e−λ2Sα2.Ifλ1=λ2=λ and α1≥α2, then d/dslogfsS/fKS<0, and hence S≤lrK, S≤hrS,S≤mrlY and S≤stK.


### 5.6. Stress-Strength Reliability

In this section, we will introduce and define one of the most important properties of any distribution which is the reliability functionR; this function may be represented as(56)R=PX>K=∫−∞∞F2xf1xdx.Suppose bothX and K are LCW independent random variables with parameters α1,λ1 and α2,λ2 and fixed scale parameter σ. Then,(57)R=∫0∞π1λ1α1xα1−1e−λ1xα1csc2π1λ1α1xα1−1e−λ1xα1ecotπ1λ1α1xα1−1e−λ1xα11+ecotπ11−e−λ1xα1−21+ecotπ21−e−λ2xα2−1dx.After applying the binomial and exponent series expansion, then substitutingu1=π1λ1α1xα1−1e−λ1xα1 and u2=π2λ2α2xα2−1e−λ2xα2, the above equation reduces to(58)R=∑i=0∞∑j=0−2∫0∞π1λ1i+1α1−1ii!−1−2j−1λ1log1−u1π1icsc2u1ej+1cotu11+ej+1cotu1−1du.Solving complicated integration is very hard but with the aid of super computers and advanced mathematical software, we easily find the value of the hard integral introduced above (Table2).Table 2
Submodels of LCW.
Serial No.αSubmodel of LCWRemarks about model1.1Logistic cot exponential (LCE) distributionNew in literature2.2Logistic cot Rayleigh (LCR) distributionNew in literature


### 5.7. Submodels of LCW

### 5.8. Estimation

One of the most famous estimators is the maximum likelihood equation as we can easily obtain and write the log-likelihood function regarding the distribution’s parameters of the distribution under consideration by getting the logarithms function for the likelihood functionΘ=α,λ⊤. So as a result, we may formulate the log-likelihood function as follows:(59)ℓ=nlogαλπ−α−1∑i=1nlogx−λ∑i=1nxα+∑i=1nlogcsc2π1−e−λxα+∑i=1ncotπ1−e−λxα−2∑i=1nlog1+ecotπ1−e−λxα.Consider the following formulae to be the score vector's componentsUΘ, or in other words the derivative of the vector with respect to the two parameters:(60)Uα=nα+∑i=1nlogx−λ∑i=1nxαlogα−2πλα∑i=1nxα−1e−λxαcotπ1−e−λxα+πλα∑i=1nxα−1e−λxαcsc2π1−e−λxα−2πλα∑i=1nxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−1,(61)Uλ=nλ−∑i=1nxα−2πxαe−λxα+πxαe−λxαcsc2π1−e−λxα−2πxαe−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−1.The MLEs may be derived by putting the equations in the previous sentence to zero and solving them concurrently (see, for instance, [18]).


## 5.1. Methodology

TakingGx, as the cdf of Weibull distribution while gx as the corresponding density, it follows the two-parameter LCWm we can write three of cdf, and its corresponding pdf and associated with its hazard function, as formulated below respectively, are:(49)Fx=1+ecotπ1−e−λxα−1,x>0,fx=πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−2,hx=πλαxα−1e−λxαcsc2π1−e−λxα1+ecotπ1−e−λxα−1.The graphs represented in Figure5 demonstrate the plots of the LCW.Figure 5
Graphical representation for the plots of the LCW: (a) density and (b) hazard rate for some parametric values.
(a)(b)


## 5.2. Reliability Functions

We have(50)Sx=1−Fx=1−1+ecotπ1−e−λxα−1,x>0.hx=πλαxα−1e−λxαcsc2π1−e−λxα1+ecotπ1−e−λxα−1,rx=πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−21+ecotπ1−e−λxα−1,Hx=−lnSx=ln1−1+ecotπ1−e−λxα−1−1,mx=1−1+ecotπ1−e−λxα−1πλαxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−2.


## 5.3. Quantile Function

This section is devoted to demonstrate the quantile function equation of LCW which is(51)Qu=QGv,where QGv=G−1v is the parent qf and v=−1/λln1+1/πcot−1ln1−u/u−α.


## 5.4. Residual and Reverse Residual Life

The residual life has several uses in probability and statistics and risk assessment. Suppose thatX represents a unit’s lifespan and X≥0 with PX=1; then, the r.v. Xt=t−X|X≤t, for a fixed t>0, is known as time since failure. The residual lifetime of LCW r.v. X is denoted by Rtx and is defined as(52)Rtx=ecotπ1−e−λx+tα1+ecotπ1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λx+tα.Additionally, the reversed hazard rate functionR¯tx is written as(53)R¯tx=ecotπ1−e−λx−tα1+ecotπ1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λx−tα.


## 5.5. Stochastic Ordering

Stochastic ordering is a tool for analysing the constitutional properties of stochastic structures that are intricate. There are several forms of stochastic orderings that can be used to sort random variables according to their distinguished properties. SupposeS and K are independent random variables with cdfs FS and FK, respectively; then, S is said to be smaller than K iff it satisfies the following.(i)
Stochastic order (S≤st (K)) if FSS≥FKS for all S.(ii)
Hazard rate order (S≤hr (K)) if hsS≥hKS for all S.(iii)
Mean residual life order (S≤mrl (K)) if msS≥mkS for all S.(iv)
Likelihood ratio order (S≤lr (K)) if fXS/fyS decrease in S.The LCW distributionλ,α is ordered according to the strongest “likelihood ratio” ordering, as demonstrated in the following theorem. The versatility of two-parameter LCW distribution λ,α is demonstrated. Let S follow LCW λ1,α1 and K follow LCW λ2,α2. Then, the likelihood ratio is(54)fSSfKS=λ1α1π1−1/λ1log1−w1/π11−1/α1λ2α2π2−1/λ2log1−w2/π21−1/α2,where w1=π11−e−λ1Sα1 and w2=π21−e−λ2Sα2. Again,(55)ddxlogfSSfKS=−α1−α2logπ1e−λ1Sα1/π1−1logπ2e−λ2Sα2/π2−1α1α2π1e−λ1Sα1π2e−λ2Sα2.Ifλ1=λ2=λ and α1≥α2, then d/dslogfsS/fKS<0, and hence S≤lrK, S≤hrS,S≤mrlY and S≤stK.


## 5.6. Stress-Strength Reliability

In this section, we will introduce and define one of the most important properties of any distribution which is the reliability functionR; this function may be represented as(56)R=PX>K=∫−∞∞F2xf1xdx.Suppose bothX and K are LCW independent random variables with parameters α1,λ1 and α2,λ2 and fixed scale parameter σ. Then,(57)R=∫0∞π1λ1α1xα1−1e−λ1xα1csc2π1λ1α1xα1−1e−λ1xα1ecotπ1λ1α1xα1−1e−λ1xα11+ecotπ11−e−λ1xα1−21+ecotπ21−e−λ2xα2−1dx.After applying the binomial and exponent series expansion, then substitutingu1=π1λ1α1xα1−1e−λ1xα1 and u2=π2λ2α2xα2−1e−λ2xα2, the above equation reduces to(58)R=∑i=0∞∑j=0−2∫0∞π1λ1i+1α1−1ii!−1−2j−1λ1log1−u1π1icsc2u1ej+1cotu11+ej+1cotu1−1du.Solving complicated integration is very hard but with the aid of super computers and advanced mathematical software, we easily find the value of the hard integral introduced above (Table2).Table 2
Submodels of LCW.
Serial No.αSubmodel of LCWRemarks about model1.1Logistic cot exponential (LCE) distributionNew in literature2.2Logistic cot Rayleigh (LCR) distributionNew in literature


## 5.7. Submodels of LCW

## 5.8. Estimation

One of the most famous estimators is the maximum likelihood equation as we can easily obtain and write the log-likelihood function regarding the distribution’s parameters of the distribution under consideration by getting the logarithms function for the likelihood functionΘ=α,λ⊤. So as a result, we may formulate the log-likelihood function as follows:(59)ℓ=nlogαλπ−α−1∑i=1nlogx−λ∑i=1nxα+∑i=1nlogcsc2π1−e−λxα+∑i=1ncotπ1−e−λxα−2∑i=1nlog1+ecotπ1−e−λxα.Consider the following formulae to be the score vector's componentsUΘ, or in other words the derivative of the vector with respect to the two parameters:(60)Uα=nα+∑i=1nlogx−λ∑i=1nxαlogα−2πλα∑i=1nxα−1e−λxαcotπ1−e−λxα+πλα∑i=1nxα−1e−λxαcsc2π1−e−λxα−2πλα∑i=1nxα−1e−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−1,(61)Uλ=nλ−∑i=1nxα−2πxαe−λxα+πxαe−λxαcsc2π1−e−λxα−2πxαe−λxαcsc2π1−e−λxαecotπ1−e−λxα1+ecotπ1−e−λxα−1.The MLEs may be derived by putting the equations in the previous sentence to zero and solving them concurrently (see, for instance, [18]).


## 6. Results Deduced from the Simulation Work

In this phase of the study, we employed Monte Carlo simulation to assess the distribution’s effectiveness all across the estimation procedure. The MLEs of the model parameters of the LCW distribution are evaluated. For each sample sizen=30, 50,100,200,500,1000, this simulation study is repeated 1,000 times. The parametric values are (I) α=0.1, λ=0.5, (II) α=0.5, λ=0.5,  (III): α=0.05, λ=1.0, and (IV) α=0.01, λ=1.2. The outputs related to the biases of the MLEs, mean square errors (MSEs), lower bounds, and upper bounds are displayed in Tables 3 and 4. As the sample size n increases, in general, the biases, MSEs, L.bounds, and U.bounds of X decrease while the CPs of the confidence intervals are quite close to the 95% nominal levels which indicate that the MLEs have good performance for estimating the parameters of the LCW distribution; also, we will use the conducted results to find the upper and lower bounds for the estimates for the parameters of the model. Tables 3 and 4 contain a summary of all simulation outcomes.Table 3
MSEs, biases, lower bounds, and upper bounds.
ParameternBiasesMSEsLower boundsUpper boundsIα300.0630.0040.1380.189500.0590.0040.1390.1791000.0570.0030.1430.1712000.0560.0030.1460.1655000.0550.0030.1490.16110000.0550.0030.1510.159λ301.4482.2941.5462.350501.3922.0241.5902.1951001.3421.8241.6332.0512001.3231.7591.6761.9705001.3181.7401.7251.91010001.3161.7331.7501.881IIα300.3200.1110.6900.950500.2960.0920.6980.8951000.2820.0810.7140.8502000.2800.0790.7320.8285000.2760.0760.7460.80610000.2760.0760.7540.797λ301.4672.3641.5592.374501.3942.0211.5902.1981001.3351.8041.6272.0432001.3271.7711.6801.9745001.3191.7431.7261.91110001.3151.7311.7501.881Table 4
Monte Carlo simulation results: biases, MSEs, lower bounds, and upper bounds.
ParameternBiasesMSEsLower boundsUpper boundsIIIα300.0320.0010.0690.095500.0300.0010.0700.0891000.0290.0010.0720.0852000.0280.0010.0730.0835000.0280.0010.0750.08110000.0270.0010.0750.080λ305.32334.2344.3918.256504.73823.9964.4567.0211004.47020.3664.6266.3152004.37019.2564.7895.9525004.33018.7964.9655.69410004.31618.6545.0605.573IVα300.0060.0000.0140.019500.0060.0000.0140.0181000.0060.0000.0140.0172000.0050.0000.0150.0165000.0050.0000.0150.01610000.0050.0000.0150.016λ307.28363.8685.58511.380506.40444.1895.7089.5001006.00636.7915.9728.4412005.91335.2696.2557.9715005.85534.3966.5187.59210005.84734.2456.6687.426


## 7. Applications and Data Analysis

Two different applications to actual datasets were shown by us in order to demonstrate the utility of the distribution that was suggested (LCW). The criteria of goodness of fit proved that it can be used in place of famous two, three, and four-parameter models and many others. All calculations are performed using theR script Adequacy Model. Moreover, the proposed LCW is compared with McDonald Weibull (McW), Exponentiated Kumaraswamy Weibull (EKumW), Burr Weibull (BurrW), Beta Weibull (BW), Gompertz Weibull (GoW), Exponentiated Odd Weibull (EOW), Topp-Leone Weibull (TLW), Odd Log Logistic Weibull (OLLW), Marshall Olkin Weibull (MOW), and many other Weibull based models.


### 7.1. Application 1: Failure Time Data

The first data correspond to the failure time of 50 components (per 1000 h) which were collected from Murthy et al. [19]. The dataset can be found easily in [19]. We avoid adding the data in the paper as they can be easily accessed. We have provided some statistics on the data used to make the reader comfortable in reading the paper. The summary statistics for this dataset are as follows: n = 50, median = 1.4140, x¯ = 3.3422, s = 4.18, Q1 = 0.2075, Q3 = 4.4988, skewness = 1.38, and kurtosis = 0.92.We can easily recognize that Figure6 represents the graphical representation of the histogram, TTT plot, box plot, and kernel density for failure time data. Figure 7 represents the comparative cdf and pdf of LCW and other models using failure time data.Figure 6
Histogram, TTT plot, box plot, and kernel density for failure time data.Figure 7
The comparative cdf and pdf of LCW and other models using failure time data.Table5 provides the MLEs of the parameters while Table 6 provides the values of AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for each model. On the basis of the statistics given in these tables, the best fit model is LCW and has the potential to fit right-skewed data with the increasing failure rate.Table 5
MLEs and their standard errors (in parentheses) for failure time data.
DistributionδαβλθLCW0.5515—0.5669——(0.0510)—(0.0342)——McW1.70770.10981.13210.08295.2260(0.0269)(0.0304)(0.0029)(0.0144)(1.6100)EKumW4.81847.94890.73650.06240.3578(0.2392)(8.3781)(0.0001)(0.0279)(0.2505)BurrW0.256456.40860.19543.4080—(3.5884)(161.976)(2.0084)(35.0251)—BW7.84680.09960.51283.6685—(0.0298)(0.0158)(0.0249)(1.6635)—GoW0.5845—0.44391.06180.3756(0.1166)—(17.7316)(42.4056)(15.0100)EOW1.31430.82600.0920—0.4360(1.1942)(0.6978)(0.2778)—(0.2273)TLW0.84270.07070.4825—22.5648(1.2441)(0.0979)(0.4253)—(35.2538)OLLW0.33000.52871.1378——(0.1721)(0.1784)(0.3284)——MOW0.38050.60380.7274——(0.2614)(0.5527)(0.1395)——LW0.8904—0.93360.9755—(1.5009)—(13.3937)(13.9952)—W0.5412—0.6612——(0.0994)—(0.0747)——Table 6
ℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for failure time data.
Distributionℓ^AICCAICBICHQICA∗W∗K-SP valueLCW94.8185193.6372193.8925197.4612195.09340.24330.03290.08510.8316McW99.1402208.2805209.6441217.8406211.92100.59400.08210.11740.4604EKumW100.6444211.2888212.6525220.8489214.92940.90320.14580.16160.1312BurrW102.4136212.8272213.7161220.4753215.73960.96040.15330.12630.3706BW100.9767209.9534210.8423217.6015212.86590.88710.14330.12660.3684GoW101.9336211.8671212.7560219.5152214.77960.90460.14150.13740.2754EOW101.6597211.3194212.2083218.9675214.23180.89830.12780.14940.1934TLW101.7022211.4044212.2933219.0525214.31680.89630.14290.11870.4473OLLW101.0825208.1650208.6867213.9010210.34930.81840.12040.14910.1953MOW102.2050210.4099210.9317216.1460212.59420.95190.15260.11460.4917LW105.4720216.9440217.4658222.6801219.12841.32740.21370.13540.2907W102.3533208.7066208.9619212.5306210.16280.95370.15210.12690.3649


### 7.2. Application 2: Survival Time Data

The upcoming data are for the mortality periods, measured in weeks, of 33 individuals with acute myelogenous leukemia which were used by Feigl and Zelen [20]. The dataset is given below. 65, 156, 100, 134, 16, 108, 121, 4, 39, 143, 56, 26, 22, 1, 1, 5, 65, 56, 65, 17, 7, 16, 22, 3, 4, 2, 8, 4, 3, 30, 4, 43.The summary statistics for this data set are as follows:n = 33, median = 22.00, x¯ = 42.06, s = 46.94, Q1 = 4.0, Q3 = 65.00, skewness = 1.07, and kurtosis = -0.16. Figure 8 represents the histogram, TTT plot, box plot, and kernel density for survival time data. Figure 9 represents the comparative cdf and pdf of LCW and other models using survival time data.Figure 8
Histogram, TTT plot, box plot, and kernel density for survival time data.Figure 9
The comparative cdf and pdf of LCW and other models using survival time data.Table7 provides the MLEs of the parameters, and Table 8 contains the values of AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for each model. On the basis of the statistics presented in these tables, the best fitted model is LCW and has the potential to fit right-skewed data with increasing failure rate.Table 7
MLEs and their standard errors (in parentheses) for survival time data.
DistributionδαβλθLCW0.0845—0.6684——(0.0201)—(0.0513)——McW0.86560.33340.74010.084010.8047(0.0025)(0.1019)(0.0024)(0.0160)(0.1870)EKumW1.36572.54050.65000.08270.9868(0.0057)(0.1880)(0.0027)(0.0188)(0.2236)BurrW0.230326.59260.19174.1946—(5.4000)(66.0689)(3.0635)(67.0145)—BW1.51070.08660.63034.0026—(0.0058)(0.5643)(0.0022)(2.2471)—GoW0.7010—0.11050.58510.2048(0.1759)—(3.2008)(16.9612)(5.9728)EOW0.46540.09166.6955—5.2082(0.01367)(0.0164)(0.0037)—(0.0037)TLW0.00723.17170.9891—0.3050(0.0083)(7.1664)(0.2090)—(0.4779)OLLW0.00900.56401.2916——(0.0113)(0.1806)(0.3135)——MOW0.03190.60250.8728——(0.0439)(0.6717)(0.2050)——LW0.7210—0.11099.8869—(1.7845)—(0.8397)(74.8148)—W0.0845—0.6684——(0.0201)—(0.0513)——Table 8
ℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for survival time data.
Distributionℓ^AICCAICBICHQICA∗W∗K-SP valueLCW146.0607296.1215296.5353299.5300297.09320.22280.03040.09100.9534McW148.3388306.6776308.9853314.0063309.10690.41050.06000.11650.7776EKumW149.4336308.8673311.1750316.1959311.29650.49940.07360.11630.7790BurrW150.2305308.4611309.9425314.3240310.40450.55700.07960.12750.6752BW149.0827306.1654307.6469312.0284308.10880.48600.07360.12830.6676GoW149.8920307.7839309.2654313.6468309.72730.55040.07750.12550.6943EOW147.0092302.0183303.4998307.8813303.96170.30880.04360.09830.9163TLW149.9365307.8730309.3545313.7360309.81640.53420.07650.12420.7064OLLW149.4551304.9103305.7674309.3075306.36780.53930.07530.12510.6983MOW150.0520306.1041306.9612310.5013307.56160.54460.07850.12520.6974LW152.1233310.2465311.1037314.6437311.70410.75930.11830.12840.6667W150.1510304.3020304.7158307.2335305.27370.55220.07870.12720.6784


## 7.1. Application 1: Failure Time Data

The first data correspond to the failure time of 50 components (per 1000 h) which were collected from Murthy et al. [19]. The dataset can be found easily in [19]. We avoid adding the data in the paper as they can be easily accessed. We have provided some statistics on the data used to make the reader comfortable in reading the paper. The summary statistics for this dataset are as follows: n = 50, median = 1.4140, x¯ = 3.3422, s = 4.18, Q1 = 0.2075, Q3 = 4.4988, skewness = 1.38, and kurtosis = 0.92.We can easily recognize that Figure6 represents the graphical representation of the histogram, TTT plot, box plot, and kernel density for failure time data. Figure 7 represents the comparative cdf and pdf of LCW and other models using failure time data.Figure 6
Histogram, TTT plot, box plot, and kernel density for failure time data.Figure 7
The comparative cdf and pdf of LCW and other models using failure time data.Table5 provides the MLEs of the parameters while Table 6 provides the values of AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for each model. On the basis of the statistics given in these tables, the best fit model is LCW and has the potential to fit right-skewed data with the increasing failure rate.Table 5
MLEs and their standard errors (in parentheses) for failure time data.
DistributionδαβλθLCW0.5515—0.5669——(0.0510)—(0.0342)——McW1.70770.10981.13210.08295.2260(0.0269)(0.0304)(0.0029)(0.0144)(1.6100)EKumW4.81847.94890.73650.06240.3578(0.2392)(8.3781)(0.0001)(0.0279)(0.2505)BurrW0.256456.40860.19543.4080—(3.5884)(161.976)(2.0084)(35.0251)—BW7.84680.09960.51283.6685—(0.0298)(0.0158)(0.0249)(1.6635)—GoW0.5845—0.44391.06180.3756(0.1166)—(17.7316)(42.4056)(15.0100)EOW1.31430.82600.0920—0.4360(1.1942)(0.6978)(0.2778)—(0.2273)TLW0.84270.07070.4825—22.5648(1.2441)(0.0979)(0.4253)—(35.2538)OLLW0.33000.52871.1378——(0.1721)(0.1784)(0.3284)——MOW0.38050.60380.7274——(0.2614)(0.5527)(0.1395)——LW0.8904—0.93360.9755—(1.5009)—(13.3937)(13.9952)—W0.5412—0.6612——(0.0994)—(0.0747)——Table 6
ℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for failure time data.
Distributionℓ^AICCAICBICHQICA∗W∗K-SP valueLCW94.8185193.6372193.8925197.4612195.09340.24330.03290.08510.8316McW99.1402208.2805209.6441217.8406211.92100.59400.08210.11740.4604EKumW100.6444211.2888212.6525220.8489214.92940.90320.14580.16160.1312BurrW102.4136212.8272213.7161220.4753215.73960.96040.15330.12630.3706BW100.9767209.9534210.8423217.6015212.86590.88710.14330.12660.3684GoW101.9336211.8671212.7560219.5152214.77960.90460.14150.13740.2754EOW101.6597211.3194212.2083218.9675214.23180.89830.12780.14940.1934TLW101.7022211.4044212.2933219.0525214.31680.89630.14290.11870.4473OLLW101.0825208.1650208.6867213.9010210.34930.81840.12040.14910.1953MOW102.2050210.4099210.9317216.1460212.59420.95190.15260.11460.4917LW105.4720216.9440217.4658222.6801219.12841.32740.21370.13540.2907W102.3533208.7066208.9619212.5306210.16280.95370.15210.12690.3649


## 7.2. Application 2: Survival Time Data

The upcoming data are for the mortality periods, measured in weeks, of 33 individuals with acute myelogenous leukemia which were used by Feigl and Zelen [20]. The dataset is given below. 65, 156, 100, 134, 16, 108, 121, 4, 39, 143, 56, 26, 22, 1, 1, 5, 65, 56, 65, 17, 7, 16, 22, 3, 4, 2, 8, 4, 3, 30, 4, 43.The summary statistics for this data set are as follows:n = 33, median = 22.00, x¯ = 42.06, s = 46.94, Q1 = 4.0, Q3 = 65.00, skewness = 1.07, and kurtosis = -0.16. Figure 8 represents the histogram, TTT plot, box plot, and kernel density for survival time data. Figure 9 represents the comparative cdf and pdf of LCW and other models using survival time data.Figure 8
Histogram, TTT plot, box plot, and kernel density for survival time data.Figure 9
The comparative cdf and pdf of LCW and other models using survival time data.Table7 provides the MLEs of the parameters, and Table 8 contains the values of AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for each model. On the basis of the statistics presented in these tables, the best fitted model is LCW and has the potential to fit right-skewed data with increasing failure rate.Table 7
MLEs and their standard errors (in parentheses) for survival time data.
DistributionδαβλθLCW0.0845—0.6684——(0.0201)—(0.0513)——McW0.86560.33340.74010.084010.8047(0.0025)(0.1019)(0.0024)(0.0160)(0.1870)EKumW1.36572.54050.65000.08270.9868(0.0057)(0.1880)(0.0027)(0.0188)(0.2236)BurrW0.230326.59260.19174.1946—(5.4000)(66.0689)(3.0635)(67.0145)—BW1.51070.08660.63034.0026—(0.0058)(0.5643)(0.0022)(2.2471)—GoW0.7010—0.11050.58510.2048(0.1759)—(3.2008)(16.9612)(5.9728)EOW0.46540.09166.6955—5.2082(0.01367)(0.0164)(0.0037)—(0.0037)TLW0.00723.17170.9891—0.3050(0.0083)(7.1664)(0.2090)—(0.4779)OLLW0.00900.56401.2916——(0.0113)(0.1806)(0.3135)——MOW0.03190.60250.8728——(0.0439)(0.6717)(0.2050)——LW0.7210—0.11099.8869—(1.7845)—(0.8397)(74.8148)—W0.0845—0.6684——(0.0201)—(0.0513)——Table 8
ℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and P values for survival time data.
Distributionℓ^AICCAICBICHQICA∗W∗K-SP valueLCW146.0607296.1215296.5353299.5300297.09320.22280.03040.09100.9534McW148.3388306.6776308.9853314.0063309.10690.41050.06000.11650.7776EKumW149.4336308.8673311.1750316.1959311.29650.49940.07360.11630.7790BurrW150.2305308.4611309.9425314.3240310.40450.55700.07960.12750.6752BW149.0827306.1654307.6469312.0284308.10880.48600.07360.12830.6676GoW149.8920307.7839309.2654313.6468309.72730.55040.07750.12550.6943EOW147.0092302.0183303.4998307.8813303.96170.30880.04360.09830.9163TLW149.9365307.8730309.3545313.7360309.81640.53420.07650.12420.7064OLLW149.4551304.9103305.7674309.3075306.36780.53930.07530.12510.6983MOW150.0520306.1041306.9612310.5013307.56160.54460.07850.12520.6974LW152.1233310.2465311.1037314.6437311.70410.75930.11830.12840.6667W150.1510304.3020304.7158307.2335305.27370.55220.07870.12720.6784


## 8. Concluding Remarks

A novel logistic-G family of distributions is developed, which employs trigonometric and algebraic generalizers based on cotangent functions. This class has been shown to be more adaptable and useful in a variety of practical applications, particularly survival, dependability, and failure modelling. Furthermore, a two-parameter model (LCW) with various density shapes, as well as hazard rate different shapes, is developed. This work also derives and presents many statistical and mathematical properties of the proposed family.In parametric estimating, the maximum likelihood method is used, and a Monte Carlo simulation analysis is used to determine whether or not the estimates are suitable. To ascertain which distribution is most suitable for modelling the real datasets, we employ a number of goodness-of-fit measures that decide which one is the superior one among all its competitors. We show that, even with a higher number of parameters, this suggested distribution consistently delivers superior fits than other existing and competing Weibull models. We believe that the suggested class and related models would find wider applicability in sectors such as dependability and survival studies, hydrology, geology, and others.


## 9. Future Work

In the upcoming work, we will apply the proposed distribution and the new family of distribution to censored sample scheme. We will try different kinds of censoring schemes like type-I and type-II censored sample and we will generate random censored samples from the new distribution. We can extend our work to apply the proposed model to accelerated life test with different types such as constant and partially constant and maybe progressive stress accelerated life tests. At last, we will use different optimality criteria to the censored samples generated from the proposed model.



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*Source: 7091581-2022-05-17.xml*