Estimation of Sine Inverse Exponential Model under Censored Schemes

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics and Statistics, College of Science Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia Department of Mathematics, College of Science and Arts in Ar Rass, Qassim University, Ar Rass, Saudi Arabia ,e Higher Institute of Commercial Sciences, Al Mahalla Al Kubra 31951, Algharbia, Egypt


Introduction
In the recent years, inverse and half-inverse problems are studied in general operator theory [1][2][3], numerous authors have attracted the attention of generated families of distributions such as Kumaraswamy-G by [4], sine generated (S-G) by [5], Kumaraswamy Type-I half-logistic-G [6], Weibull-G [7], odd Fréchet -G by [8], the Burr type X-G by [9], Kumaraswamy Kumaraswamy-G [10], truncated Cauchy power-G by [11], generalized odd half-Cauchy-G by [12], and among others. e cumulative distribution function (CDFu) and PDFu of S-G are Letting g(x; ξ) and G(x; ξ), the PDFu and CDFu of IE distribution, it has the following form: e main idea for this paper was to introduce a new oneparameter model that is more flexible than the IE model by using the S-G family. e new model is called the SIE model. e SIE model is more flexible than the IE model and it has many applications in physics, medicine, nanophysics, and nanoscience [13][14][15][16]. is manuscript is arranged as follows. Section 2 presents materials and methods. In Section 3, statistical inference of the SIE model under the censored scheme is studied. Section 4 presents results and discussion. At the end of article, conclusions are discussed.

e New SIE Model.
Letting random variable X to have SIE distribution, then the CDFu, PDFu, survival function (SFu), and HRFu of X are where θ is a scale parameter. Figures 1-3 show the plots of PDFu, CDFu, and HRFu of the SIE model. e PDFu of the SIE model can be right skewed and unimodal shaped, while the HRFu of the SIE model can be increasing or J-shaped.

Quantile and Median.
If X∼ SIE, then the QFu of SIE is as follows: and by taking u � 0.5, we get the median (M) as MacGillivray's skewness function is defined in [17] as Proof. Let X be a r. v. with pdf equation (6). e r th Mo of SIE distribution is computed as follows: By inserting the expansion cos[G( e last equation can be rewritten as follows: where en, e MGFu of X is e ICMo, denoted by φ s (t), of the SIE distribution is By using equation (18), φ s (t) will be as where c(s, t) � t 0 y s− 1 e − y dy is the lower ICGFu where ICGFu is incomplete gamma function. e CMo, denoted by τ s (t), of the SIE distribution is By using equation (20), τ s (t) will be as given where Γ(s, t) � e pdf of X (m) can be expressed as  Specially, the pdfs of the lowest and greatest OS can be computed as

Statistical Inference under Censored Samples
For different reasons, such as time constraints, money, or other resources, reliability or lifespan testing trials are typically censored. Generally speaking, there are two types of censorship schemes: Type-I and Type-II CS. Estimation using these two censoring techniques will be discussed in this section of the paper. If we use type-I censoring, we have a set time, say X, but the amount of things that fail during the trial is completely random. Type-II censoring, on the contrary, is a process that continues until the stated number of failures is reached.

ML Estimation under Type-I Censor.
Assume that X 1 , X 2 , . . . , X r be a type-I CS of size r obtained from lifetime testing experiment on k items whose lifetime follows the PDFu for SIE. e likelihood function (LLFu) of type-I CS is given as (26) e log-LLFu corresponding to equation (26) is given by e ML equations for the SIE distribution are as follows: en, the ML estimators for the parameter θ are computed by putting (z ln L/zθ) � 0 and solving.

ML Estimation under Type-II Censor.
Let X 1 , X 2 , . . . , X r be a type-II CS of size r observed from lifetime testing experiment on k items whose lifetime has the PDFu for SIE. e LLFu of type-II CS is e log-LLFu corresponding to equation (29) is given by e ML equations for the SIE distribution are as follows: en, the ML estimators for the parameter θ is calculated by putting (z ln L/zθ) � 0 and solving. e first data are known as ball bearing data, and it represents the number of rotations before ball bearing failure obtained [18]. e second data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by

Conclusion
In

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.