# Sine Inverse Lomax Generated Family of Distributions with Applications

Authors: Aisha Fayomi; Ali Algarni; Abdullah M. Almarashi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper introduces a new family of distributions by combining the sine produced family and the inverse Lomax generated family. The new proposed family is very interested and flexible more than some old and current families. It has many new models which have many applications in physics, engineering, and medicine. Some fundamental statistical properties of the sine inverse Lomax generated family of distributions as moments, generating function, and quantile function are calculated. Four special models as sine inverse Lomax-exponential, sine inverse Lomax-Rayleigh, sine inverse Lomax-Frèchet and sine inverse Lomax-Lomax models are proposed. Maximum likelihood estimation of model parameters is proposed in this paper. For the purpose of evaluating the performance of maximum likelihood estimates, a simulation study is conducted. Two real life datasets are analyzed by the sine inverse Lomax-Lomax model, and we show that providing flexibility and more fitting than known nine models derived from other generated families.


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## Body

## 1. Introduction

Authors have recently been interested in produced families of distributions. Among them are the beta-G [1], Kumaraswamy (Ku)-G [2],T-X [3], Type-II half-logistic-G [4], [0, 1] truncated Fréchet-G [5], extended odd Fréchet-G [6], truncated Lomax-G [7], Ku Type-I half-logistic-G [8], transmuted (Tr) odd Fréchet-G [9], truncated Burr X-G [10], and exponentiated transmuted G [11].According to [12], the inverse Lomax (IL)-G has the distribution function (CDFu) and density function (PDFu) shown below (PDFu):(1)Rx;ξ=1+βG¯x;ξGx;ξ−α(2)rx;ξ=αβgx;ξGx;ξ21+βG¯x;ξGx;ξ−α−1,x∈R,α,β>0.By entering the scale parameterβ=1, we are interested. Then, the PDFu and CDFu of IL-G have the following:(3)rx;ξ=αgx;ξGx;ξ21+G¯x;ξGx;ξ−α−1,x∈R,α>0,and(4)Rx;ξ=1+G¯x;ξGx;ξ−α.The sine generated (S-G) family of distributions studied by [13], and it has the following CDFu and PDFu:(5)Fx;ξ=sinπ/2Rx;ξ,x∈R,and(6)fx;ξ=π2rx;ξcosπ2Rx;ξ,x∈R.Indeed, the characteristics of the models generated from the S-G family have inspired numerous comprehensive families of continuous distributions that are likewise centered on trigonometric functions, such as Beyond the S-G [14] and sine TL-G [15]. The bulk of these families are focused on the S-G structure and do not include any extra tuning parameters or modifications.The primary goal of this work is to introduce and explore a novel family of probability distributions, based on the S-G and IL-G families. The sine IL family is the name given to the new family (SIL-G). Thus, the SIL-G family differs from other modified S-G families in terms of general simplicity and the advantage of adjustable skewness. As a result, the SIL-G can give intriguing models for a variety of fitting purposes. This work develops this practical element, as well as key theoretical conclusions.This work is divided into seven sections as follows. We will look at the new family and some of the statistical features of the sine inverse Lomax produced family of distributions in Section2. Section 3 examines four new models that are unique to the SIL-G family. Maximum likelihood (ML) estimation of the family’s parameters is described in Section 4. SIL-Lomax simulation results are shown in Section 5. Two real-world data sets are utilized in Section 6 to test the SIL-Lomax model. Section 7 concludes with a few words.


## 2. The New SIL-G Family

A new family of continuous probability distributions known as the sine inverse Lomax generated (SIL-G) family will be introduced in this section. It is possible to obtain the new family’s CDFu by substituting (3) into (5):(7)Fx;ξ=sinπ/21+G¯x;ξGx;ξ−α,x∈R,α>0.The PDFu of the SIL-G family is(8)fx;ξ=παgx;ξ2Gx;ξ21+G¯x;ξGx;ξ−α−1cosπ/21+G¯x;ξGx;ξ−α,α>0,x∈R.The survival function (SFu), hazard rate function (HRFu), reversed HRFu, and cumulative HRFu are(9)F¯x;ξ=1−sinπ/21+G¯x;ξGx;ξ−α,hx;ξ=παgx;ξ1+G¯x;ξ/Gx;ξ−α−1cosπ/21+G¯x;ξ/Gx;ξ−α2Gx;ξ21−sinπ/21+G¯x;ξ/Gx;ξ−α,τx;ξ=παgx;ξ2Gx;ξ21+G¯x;ξGx;ξ−α−1cotπ/21+G¯x;ξGx;ξ−α,and(10)Hx;ξ=−ln1−sinπ/21+G¯x;ξGx;ξ−α,respectively. The quantile function of SIL-G, say Qu of X, is obtained by inverting (7) as(11)Qu=G−112/πArcsinu−1/α−1.The following expansion is used to get PDFu (8) expansion:(12)cosGx=∑i=0∞−1i2i!Gx2i.The PDFu of SIL-G family may be written by inserting (9) into (8):(13)fx;ξ=παgx;ξ2Gx;ξ2∑i=0∞−1i2i!π/2i1+G¯x;ξGx;ξ−α2i+1−1.By using the next binomial series,(14)1+Z−α=∑j=0∞−1jα+j−1jZj,α>0,Z<1.We may write the PDFu of the SIL-G family by inserting (11) into (10):(15)fx;ξ=παgx;ξ2Gx;ξ2∑i,j=0∞−1i+j2i!α2i+1+jjπ/2iG¯x;ξGx;ξj.Alternatively, we may express the last equation as follows:(16)fx;ξ=παgx;ξ2Gx;ξ2∑i,j=0∞−1i+j2i!α2i+1+jjπ/2iG¯x;ξ1−1−Gx;ξj.Again, using the following binomial theory,(17)1−Z−α=∑k=0∞α+k−1kZk,α>0,Z<1.By inserting (13) in (12) then, the PDFu of SIL-G is given by(18)fx;ξ=παgx;ξ2Gx;ξ2∑i,j,k=0∞−1i+j2i!α2i+1+jjj+k−1kπ/2i1−Gx;ξj+k.Again, using the following binomial theory,(19)1−Zα=∑m=0∞−1mαmZm,α>0,Z<1.By inserting (15) in (14), we obtain(20)fx;ξ=∑m=0∞ηmgx;ξGx;ξm−2,where ηm=πα/2∑i,j,k=0∞−1i+j+m/2i!α2i+1+jjj+k−1kj+kmπ/2i.Therth moment of the SIL−G family is calculated as(21)μ˙r=EXr=∫−∞∞xrfx;ξdx=∑m=0∞ηm∫−∞∞xrgx;ξGx;ξm−2dx.Then,(22)μ˙r=∑m=0∞ηmτr,m−2.The moment generating function of SIL-G is(23)MXt=∑r=0∞trr!μ˙r=∑m,r=0∞trr!ηmτr,m−2.


## 3. New Submodels

We will be discussing four new versions of the SIL-G in this section: SIL-exponential, SIL-Rayleigh, SIL- Frèchet, and SIL-Lomax.


### 3.1. SIL-Exponential Distribution

Forgx;θ=θe−θx,x>0, and Gx;θ=1−e−θx, the PDFu of SIL-exponential (SILE) is(24)fx=παθe−θx21−e−θx21+eθx−1−1−α−1cosπ/21+eθx−1−1−α.The corresponding CDFu is(25)Fx=sinπ/21+eθx−1−1−α.


### 3.2. SIL-Rayleigh Distribution

Let us consider the Rayleigh distribution hasgx;θ=2θxe−θx2,x,θ>0, and Gx;θ=1−e−θx2, we obtain the SIL-Rayleigh (SILR) PDFu as(26)fx=παθxe−θx21−e−θx221+eθx2−1−1−α−1cosπ/21+eθx2−1−1−α.The corresponding CDFu is(27)Fx=sinπ/21+eθx2−1−1−α.


### 3.3. SIL-Frèchet Distribution

The Frèchet (F) model with PDFu;gx;μ,δ=δμδx−δ−1e−μ/xδ,x,μ,δ>0, and CDFu Gx;μ,δ=e−μ/xδ; hence, the SIL-Frèchet (SILF) PDFu is as follows:(28)fx=παδμδx−δ−12e−αμxδcosπ2e−αμxδ.The CDFu of the SILF distribution is(29)Fx;α,μ,δ=sinπ2e−αμxδ.


### 3.4. TIL-Lomax Distribution

Forgx;a,b=b/a1+x/a−b+1 and Gx;a,b=1−1+x/a−b, we obtain the SIL-Lomax (SILL) PDFu as(30)fx=παb1+x/a−b+12a1−1+x/a−b21+1+xab−1−1−α−1cosπ/21+1+xab−1−1−α.The CDFu of the SILL distribution is(31)Fx=sinπ/21+1+xab−1−1−α,x,α,a,b>0.Plots of PDFu for the SILE, SILR, SILF, and SILL are displayed in Figure1.Figure 1
Plots of PDFu of the SILE, SILR, SILF, and SILL distributions.We can note from Figure1. The pdf can be right skewness, unimodal, symmetric, and decreasing.


## 3.1. SIL-Exponential Distribution

Forgx;θ=θe−θx,x>0, and Gx;θ=1−e−θx, the PDFu of SIL-exponential (SILE) is(24)fx=παθe−θx21−e−θx21+eθx−1−1−α−1cosπ/21+eθx−1−1−α.The corresponding CDFu is(25)Fx=sinπ/21+eθx−1−1−α.


## 3.2. SIL-Rayleigh Distribution

Let us consider the Rayleigh distribution hasgx;θ=2θxe−θx2,x,θ>0, and Gx;θ=1−e−θx2, we obtain the SIL-Rayleigh (SILR) PDFu as(26)fx=παθxe−θx21−e−θx221+eθx2−1−1−α−1cosπ/21+eθx2−1−1−α.The corresponding CDFu is(27)Fx=sinπ/21+eθx2−1−1−α.


## 3.3. SIL-Frèchet Distribution

The Frèchet (F) model with PDFu;gx;μ,δ=δμδx−δ−1e−μ/xδ,x,μ,δ>0, and CDFu Gx;μ,δ=e−μ/xδ; hence, the SIL-Frèchet (SILF) PDFu is as follows:(28)fx=παδμδx−δ−12e−αμxδcosπ2e−αμxδ.The CDFu of the SILF distribution is(29)Fx;α,μ,δ=sinπ2e−αμxδ.


## 3.4. TIL-Lomax Distribution

Forgx;a,b=b/a1+x/a−b+1 and Gx;a,b=1−1+x/a−b, we obtain the SIL-Lomax (SILL) PDFu as(30)fx=παb1+x/a−b+12a1−1+x/a−b21+1+xab−1−1−α−1cosπ/21+1+xab−1−1−α.The CDFu of the SILL distribution is(31)Fx=sinπ/21+1+xab−1−1−α,x,α,a,b>0.Plots of PDFu for the SILE, SILR, SILF, and SILL are displayed in Figure1.Figure 1
Plots of PDFu of the SILE, SILR, SILF, and SILL distributions.We can note from Figure1. The pdf can be right skewness, unimodal, symmetric, and decreasing.


## 4. ML Method of Parameter Estimation

SupposeX1,X2,…,Xn be a series of observed values from the SIL−G family with the set of parameter Φ=α,ξT. According to the following formula, the log-likelihood function (LLF) is(32)lnL,Φ=nlnπ2+nlnα+∑i=1nlngxi;ξ−2∑i=1nlnGxi;ξ−α+1∑i=1nlnZi+∑i=1nlncosπ/2Zi−α,where Zi=1+G¯xi;ξ/Gxi;ξ. Score vector components UΦ=Uα,UξkT are used to calculate the first partial derivatives of the LLF with respect to α and ξ as(33)Uα=nln2+nα−∑i=1nlnZi+π/2∑i=1nZi−αlnZitanπ/2Zi−αand(34)Uξk=∑i=1ngk′xi;ξgxi;ξ−2∑i=1nGk′xi;ξGxi;ξ−α+1∑i=1n∂Zi/∂ξkZi−πα2∑i=1nZi−α−1∂Zi∂ξktanπ/2Zi−α.where gk′xi;ξ=∂gxi;ξ/∂ξk and Gk′xi,ξ=∂Gxi;ξ/∂ξk.


## 5. Simulation Study

The ML estimators’ performance is measured according to the number of sample sizen. Evaluation based on numerical information of the performance of ML estimations for the SILL model is performed. The mean square errors (A1), lower bound (A2), upper bound (A3), average length (A4) of confidence interval, and ML estimations are assessed. The simulation method is carried out with the help of the MATHEMATICA (9) package:(1)
The SILL distribution is used to create a random sampleX1,X2,…Xn of sizes n=30,50, and 100.(2)
Nine sets of parameter values evaluated.(3)
The SILL model’s ML estimate is assessed using parameter values and sample sizes.(4)
Repeat this procedure 5000 times to obtain the means and A1 of the ML estimations for model parameter values. Table1 summarizes empirical findings. Furthermore, the following observations have been made.(5)
From Tables1–9, we can note that when n increases the values of ML and A4 are decreased.Table 1
MLE, M1, M2, M3, and M4 of the SILL model atα=0.3,b=0.5,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.321410.038380.004570.638250.63368−0.056100.698920.755030.476890.015570.179140.774630.595500.122120.831650.709530.493120.008300.344150.642090.297950.315620.670620.35500500.299650.021330.035890.563410.52753−0.014620.613920.628540.526280.011700.295450.757110.461660.251250.801310.550060.493160.003230.371820.614500.242680.348580.637730.289151000.334230.011810.172210.496240.324040.141180.527270.386090.461150.008690.324010.598290.274290.297740.624550.326810.516920.002570.443840.590010.146160.429850.604000.17415Table 2
MLE, M1, M2, M3, and M4 of the SILL model atα=0.5,b=0.5,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.465010.025280.055600.874420.81882−0.022800.952820.975620.550440.031680.136610.964270.827660.057371.043520.986150.484260.005860.349470.619050.269580.323660.644860.32121500.575740.099490.284380.867100.582720.228590.922890.694300.381500.103190.118500.644490.525990.068140.694860.626710.524860.010030.426570.623140.196570.407750.641960.234211000.512530.016090.314060.711000.396940.276060.749010.472950.531400.016670.332610.730180.397570.294550.768240.473700.503500.002190.440830.566160.125330.428830.578160.14933Table 3
MLE, M1, M2, M3, and M4 of the SILL model atα=0.7,b=0.5,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.668140.051670.141731.194551.052820.040931.295351.254430.670150.093110.052921.287371.23445−0.065271.405561.470830.489730.004780.357630.621830.264200.332330.647120.31479500.672320.016750.302031.042600.740570.231131.113500.882380.621890.040050.189561.054230.864670.106771.137011.030240.486100.001490.397140.575060.177920.380100.592090.211991000.628880.008280.407690.850060.442370.365340.892410.527070.598030.024620.349710.846350.496640.302160.893900.591740.491570.000340.434520.548630.114110.423590.559550.13596Table 4
MLE, M1, M2, M3, and M4 of the SILL model atα=0.3,b=0.3,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.491960.023350.119030.864880.745850.047620.936290.888670.349410.029580.032050.666770.63472−0.028720.727540.756260.507740.005070.373430.642050.268630.347710.667770.32007500.568730.021940.288920.848530.559610.235340.902110.666770.288370.012280.056890.519850.462970.012560.564180.551620.516750.002470.424170.609330.185160.406440.627060.220621000.481520.004910.314660.648380.333720.282710.680330.397620.308830.003220.171730.445930.274200.145470.472180.326710.484480.000650.428650.540310.111670.417950.551000.13305Table 5
MLE, M1, M2, M3, and M4 of the SILL model atα=0.5,b=0.3,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.261730.03201−8.333028.8564817.18950−9.9788310.5023020.481100.283930.01557−3.753974.321828.07578−4.527185.095039.622210.491420.01072−4.799005.7818310.58080−5.812056.7948912.60690500.259860.02093−2.389022.908745.29776−2.896253.415976.312230.316080.01174−0.964911.597062.56196−1.210201.842353.052550.492170.00563−0.981571.965922.94749−1.263782.248123.511901000.268820.010360.129470.408160.278690.102790.434850.332060.328410.005790.231610.425210.193600.213080.443750.230670.483440.002010.413320.553560.140240.399890.566990.16709Table 6
MLE, M1, M2, M3, and M4 of the SILL model atα=0.8,b=0.8,a=0.8.
nMLA190%95%A2A3A4A2A3A4300.823180.102580.254661.391701.137050.145791.500571.354780.683120.153130.060631.305601.24497−0.058571.424801.483370.829930.027850.566951.092910.525970.516591.143270.62669500.670740.109700.251611.089870.838270.171351.170130.998780.838880.050220.376381.301390.925010.287811.389951.102140.752910.024330.565110.940720.375610.529140.976680.447541000.684950.050280.390610.979280.588680.334251.035650.701400.962910.076340.623231.302580.679350.558191.367630.809440.759370.008200.637630.881110.243470.614320.904420.29010Table 7
MLE, M1, M2, M3, and M4 of the SILL model atα=0.5,b=0.8,a=0.8.
nMLA190%95%A2A3A4A2A3A4300.534990.06024−0.195201.265181.46039−0.335031.405011.740030.877040.168880.210471.543611.333140.082831.671251.588420.802820.018510.395041.210590.815550.316961.288670.97172500.428410.079710.012490.844320.83183−0.067150.923970.991110.924220.101190.522731.325720.802990.445841.402600.956760.740100.029930.491410.988790.497380.443791.036420.592631000.474370.026380.220610.728130.507520.172020.776730.604710.802630.031950.565031.040240.475210.519531.085730.566210.787950.008760.643830.932080.288250.616230.959680.34345Table 8
MLE, M1, M2, M3, and M4 of the SILL model atα=0.5,b=1.2,a=0.5.
nMLA190%95%A2A3A4A2A3A4300.629920.125150.049281.210571.16129−0.061911.321761.383671.131950.137800.358061.905831.547770.209872.054031.844160.544600.010460.370120.719080.348950.336710.752490.41578500.439540.073930.073810.805270.731460.003780.875300.871521.243130.134480.731921.754341.022410.634031.852231.218190.481870.007690.370390.593350.222970.349040.614700.265661000.598280.043650.313950.882620.568670.259500.937060.677571.045730.060540.684481.406990.722510.615301.476160.860860.533930.003170.454980.612890.157920.439860.628010.18816Table 9
MLE, M1, M2, M3, and M4 of the SILL model atα=0.5,a=1.5,b=0.5.
nMLA190%95%A2A3A4A2A3A4300.676100.316490.069431.282771.21334−0.046741.398941.445681.405320.547890.546462.264171.717710.382002.428632.046630.558150.025850.381670.734630.352960.347870.768420.42055500.481800.038070.080260.883330.803070.003370.960220.956851.466300.075180.866462.066141.199680.751602.181001.429400.515680.003140.397520.633840.236320.374890.656460.281571000.511120.020730.229840.792410.562560.175980.846270.670291.452740.048061.045171.860310.815140.967131.938350.971230.494660.001350.420690.568630.147940.406520.582800.17627


## 6. Real Data Analysis

SILL distribution as an example for the family is exemplified in this section based on two datasets. A number of authors have used this data to illustrate the usefulness of competing models. Also, the models’ fit is evaluated formatively and compared to that of other distributions. The Akaike IC (E1), Consistent Akaike IC (E2), Hannan–Quinn IC (E3), Anderson–Darling (E4), and Cramér-von Mises IC (E5) are used to evaluate the fitted models. This is because, in general, smaller values of these statistics indicate a better fit to the data.Tahir et al. [16] investigated the first dataset, which represented the 84 failure times for a certain windshield device. For these data, we will compare the SILL distribution’s fits to the exponentiated Lomax (ExL) [17], Lomax (L), inverse Lomax (IL), Ku Weibull (KuW) [18], McDonald Weibull (MW) [19], beta Weibull (BeW) [20], Tr Marshall–Olkin F (TrMOF) [21], Tr log-logistic (TrLL) [13], and log-logistic (LL) models.The second dataset (in months): a random sample of 128 bladder cancer patients was investigated by [22]. The SILL is compared to other distributions such as the type-II TL power Lomax (TIITLPoL) [23], Tr complementary Weibull geometric (TrCoWG) [24], Ku exponentiated BurrXII (KuExBXII) [25], beta exponentiated Burr XII (BeExBXII) [26], beta F (BeF) [27], exponentiated Tr generalized Rayleigh (ExTrGeR) [28], Tr modiﬁed Weibull (TrMW) [29], and generalized Tr Weibull (GeTrW) [30] models.Tables10 and 11 show estimated parameters and associated standard errors (SErs). Tables 12 and 13 provide the statistics for the fitted models. E1, E2, E3, E4, and E5 values for SILL are the lowest among the competitor models, as shown in Tables 12 and 13. Both datasets are best fit by SILL distribution. See Figures 2 and 3, for further information on this topic.Table 10
Estimates and SErs for the first data.
Fitted modelsEstimates and SErsSILL (α, a, b)3.132 (0.489)797.744 (2694)416.678 (1404)ExL (α, a, b)3.630 (0.624)20074.510 (2041.826)26257.680 (99.742)TrMOF (α, σ, a, b)200.747 (87.275)1.952 (0.125)0.102 (0.017)−0.869 (0.101)L (a, b)51425.350 (5933.490)131789.7800 (296.120)TrLL (α, a, b)2.060 (0.280)3.103 (0.341)−0.490 (0.385)IL (a, b)4.002 (1.290)1.981 (0.750)MW (c, b, a, β, α)16.721 (9.622)33.640 (19.994)17.690 (6.222)0.310 (0.050)1.940 (1.011)LL (a, b)2.391 (0.140)3.224 (0.297)BeW (β, α, b, a)0.298 (0.060)1.360 (1.002)11.496 (6.730)34.180 (14.840)KuW (a, b, α, β)34.660 (17.530)81.850 (52.014)14.433 (27.100)0.204 (0.042)Table 11
Estimates and SErs for the second data.
Fitted modelsMLEs and SErsSILL (α, a, b)1.517 (0.279)15.419 (9.989)1.933 (0.743)TIITLPoL (θ, α, a, b)8.430 (42.442)0.785 (3.350)0.814 (0.128)8.557 (15.373)GeTrW (λ, b, a, β, α)0.002 (1.770)0.013 (7.214)2.800 (1.117)0.654 (0.121)0.299 (0.151)TrMW (α, a, λ, β)0.1208 (0.024)0.0002 (0.011)0.407 (0.410)0.8955 (0.626)KuExBXII (b, c, a, k, β)67.636 (104.728)0.340 (0.390)2.780 (44.510)0.840 (1.723)3.083 (49.353)BeExBXII (b, a, k, β, c)20.278 (17.300)22.190 (21.960)1.310 (1.080)1.780 (1.076)0.224 (0.144)ExTrGeR (α, β, λ, δ)7.376 (5.390)0.050 (0.004)0.120 (0.260)0.050 (0.036)BeF (β, α, b, a)0.169 (0.104)27.753 (71.510)33.342 (36.350)12.526 (24.470)TrCoWG (α, β, a, b)106.070 (124.800)1.712 (0.100)0.217 (0.610)0.010 (0.007)Table 12
The E1, E2, E3, E4, and E5 statistics for the first data.
Fitted modelsE1E2E3E4E5SILL277.235277.535280.1661.1280.125ExL288.800289.100291.3281.7440.220TrMOF309.472309.980319.2002.4040.320L333.977334.123335.6631.3980.167TrLL284.772285.0720287.7031.4920.184IL360.950361.100362.6344.3840.538MW283.900284.670296.0531.5910.199LL283.163283.311285.1171.5200.187BeW305.028305.534314.7513.2200.465KuW281.434281.941291.1601.5060.185Table 13
The E1, E2, E3, E4, and E5 statistics for the second data.
Fitted modelsE1E2E3E4E5SILL825.820826.014829.2970.1760.027TIITLPoL828.200828.523832.8330.2220.034GeTrW831.350831.840837.1420.3060.047TrMW836.450836.780847.8600.1250.760KuExBXII831.651832.143837.4500.3200.048BeExBXII841.270841.760855.5300.9000.134ExTrGeR866.350866.680877.7602.3610.398BeF842.970843.290854.3701.1210.168TrCoWG829.996830.320834.630.3060.043Figure 2
Fitted cdf, pdf, sf, and pp plots for the first data.Figure 3
Fitted cdf, pdf, sf, and pp plots for the second data.


## 7. Concluding Remarks

In this manuscript, we propose the SIL-G family, a novel one parameter produced family of distributions. There are four new unique models specified. We investigate many essential characteristics of the SIL-G family, such as PDFu expansions, explicit formulations for moments, generating function, and quantile. We estimate the parameters using ML estimation techniques. To examine the limited sample behavior of the ML estimates, we undertake a Monte Carlo simulation analysis on one SILL model as an example of the new family. The new family’s relevance is demonstrated through applications to two real-world datasets.In our future works, we plan to study some new generating families of distributions depending on trigonometric functions as sine function. Also, some new models depending on the sine family and statistical inference by using different methods of parameter and various algorithms and comparing each other will be studied. Applications for many fields as physics, economics, engineering, and medicine will be studied.



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