Sine Inverse Lomax Generated Family of Distributions with Applications

)is paper introduces a new family of distributions by combining the sine produced family and the inverse Lomax generated family. )e 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 LomaxLomax 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.

(4) e sine generated (S-G) family of distributions studied by [13], and it has the following CDFu and PDFu: and f(x; ξ) � π 2 r(x; ξ)cos π 2 R(x; ξ) , x ∈ R. (6) 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]. e bulk of these families are focused on the S-G structure and do not include any extra tuning parameters or modifications. e 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. e sine IL family is the name given to the new family (SIL-G).
us, 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. is work develops this practical element, as well as key theoretical conclusions.
is 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 Section 2. 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.

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): e PDFu of the SIL-G family is e survival function (SFu), hazard rate function (HRFu), reversed HRFu, and cumulative HRFu are and respectively. e quantile function of SIL-G, say Q (u) of X, is obtained by inverting (7) as e following expansion is used to get PDFu (8) expansion: e PDFu of SIL-G family may be written by inserting (9) into (8): By using the next binomial series, We may write the PDFu of the SIL-G family by inserting (11) into (10): Alternatively, we may express the last equation as follows: Again, using the following binomial theory, By inserting (13) in (12) then, the PDFu of SIL-G is given by Again, using the following binomial theory, By inserting (15) in (14), we obtain where e r th moment of the SIL − G family is calculated as en, e moment generating function of SIL-G is

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.

SIL-Exponential Distribution
e corresponding CDFu is

TIL-Lomax
e CDFu of the SILL distribution is Plots of PDFu for the SILE, SILR, SILF, and SILL are displayed in Figure 1.
We can note from Figure 1. e pdf can be right skewness, unimodal, symmetric, and decreasing.

ML Method of Parameter Estimation
Suppose X 1 , X 2 , . . . , X n be a series of observed values from the SIL − G family with the set of parameter Φ � (α, ξ) T . According to the following formula, the log-likelihood function (LLF) is where T are used to calculate the first partial derivatives of the LLF with respect to α and ξ as and where

Simulation Study
e ML estimators' performance is measured according to the number of sample size n. Evaluation based on numerical information of the performance of ML estimations for the SILL model is performed. e mean square errors (A1), lower bound (A2), upper bound (A3), average length (A4) of confidence interval, and ML estimations are assessed. e simulation method is carried out with the help of the MATHEMATICA (9) package: (1) e SILL distribution is used to create a random sample X 1 , X 2 , . . . X n of sizes n � 30, 50, and 100.

Concluding Remarks
In this manuscript, we propose the SIL-G family, a novel one parameter produced family of distributions. ere 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. e 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.

Data Availability
e data used to support the findings of the study can be obtained from corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.