Modelling to Engineering Data: Using a New Two-Parameter Lifetime Model

A novel two-parameter continuous lifespan model is developed, based on a truncated Fréchet produced family of distributions known as the truncated Fréchet inverted Lindley distribution. It includes a thorough discussion of statistical features such as the quantile function, moments, order statistics, incomplete moments, and Lorenz and Bonferroni curves. .e greatest likelihood approach for estimating population parameters is described. Finally, one real-world data set to application is utilized to demonstrate the new distribution’s utility. .e data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm.


Introduction
Adding parameter(s) to baseline distributions is a traditional approach for generating families of probability distributions. ese families have the capacity to increase the desirable aspects of probability distributions as well as extract additional information from a variety of data sets, which may be used in a variety of fields such as engineering, economics, biology, and environmental sciences. Another generator utilizes the shortened random variable. In this context, significant research on the truncated (T)-G families is the T Fréchet-G [1], T Weibull-G [2], Type II T Fréchet-G (TIITFG) [3],T Burr X-G [4], T Lomax-G [5], T power Lo-G (TPLoG) [6], TX family of distributions [7], T log-logistic-G [8], generalized odd Weibull-G [9], Topp-Leone-G [10], transmuted odd Fréchet-G [11] and truncated Cauchy power [12].
Aldahlan [3] proposed the TIITFG family with the following cumulative distribution function (cdf ): and the density function (pdf ) e following exponential series is used to generate the expansion of pdf (2): Regarding the existing binomial series can be used, Employing (3) and (4) in (2), then the pdf of TIITFG, where b is real, is where e quantile function Q (u) of X is given by Sharma et al. [13] investigated the inverted Lindley (ILi) distribution, which has the following pdf and cdf, respectively Many statisticians have generalized the ILi distribution in recent years, such as [14] who studied the extended ILi distribution, [15] who proposed the generalized ILi distribution, [16] who proposed the power ILi distribution, a new extension of Ili proposed by [17,18] who studied weighted Ili distribution, [19]who studied alpha power transformed ILi (APTILi) distribution, and [20] who studied extended exponentiated ILi distribution, and logarithmic ILi model was studied by [21] and generalized Marshall Olkin ILi by [22]. e main goal of this article is to present the Type II truncated Fréchet inverted Lindley (TIIFILi) distribution, a novel two-parameter life-time model. e new model is very flexible. e pdf can be symmetric, right skewness, and unimodal. Also, the hrf can be unimodal, increasing, and J-shaped. Investigate some of its statistical features to discuss the statistical inference of the TIIFILi model and to give leading fits than some known models with favourable results for the TIIFILi model. e new model is extremely adaptable, and we may obtain the cdf and pdf by adding (8) and (9), as shown in (1) and (2).
Using the three equations (3)-(5), we can rewrite (11) as follows: We may rewrite the above equation using the binomial expansion as where e TIIFILi distribution function, the hazard rate function (hrf ), the inverted hazard rate function, and the cumulative hazard rating function are given when a random variable Z follows the TIIFILi model, and Figures 1 and 2 show the pdf and hrf plots of the TIIFILi distribution. Figure 1 demonstrates how the pdf might be unimodal and tilted to the right. Figure 2 depicts various potential hrf forms, including monotone increasing, up-side-down, and J-shaped. e remainder of this article is arranged as follows: Section 2 investigates distribution's mathematical characteristics of the proposed model. Section 3 covers the estimate of distribution parameters using the maximum likelihood method of estimation. Section 4 provides actual data applications to illustrate the potential of the new distribution, and Section 5 ends with remarks.

Mathematical Properties
In this section, we will study some statistical properties such as the quantile function, median, order statistics, moments, moment generating function, incomplete moments, and Lorenz and Bonferroni curves. e quantile function By multiplying (9) both sides by − (1 + θ)e − (1+θ) , en,

Corollary 1. If Z ∼ TIITFILi, the median M of Z is given by
Assuming Z 1 < Z 2 < · · · < Z n is an order sample from TIITFILi population, the pdf of the i th ordered statistics is given as Substituting (10) and (11), and applying general binomial series expansion, (13) becomes We can get the first and last order statistics at i � 1 and i � n, respectively, as follows: e r th moments of TIITFILi distribution are defined as follows: Letting y � (θ(j + 1)/z), z � (θ(j + 1)/y), dz � (− θ(j + 1)/y 2 )dy, and simplifying further, then e moment generating function of TIITFILi model can be calculated by e incomplete moments, for example, ϖ s (t) are provided by Using (10), then, ϖ s (t) can be taken, the next formula e lower incomplete gamma function is then used to produce (32)

Journal of Function Spaces 5
In this case, ](s, t) is the lower incomplete gamma function.
e Lorenz and Bonferroni curves are given by . (33)

Maximum Likelihood Estimation
Assume that Z 1 , Z 2 , ..., Z n is a random sample of size n from a population with TIITFILi pdf and that the log-likelihood function is provided by e score functions which correspond to equating the first-order partial derivative of the last equation to zero is given by where N i � 1 − (1 + (θ/(1 + θ)z i ))e (− θ/z i ) , M i � (zN i /zθ) are the solutions, say b and θ. e maximum likelihood estimators of the TIITFILi distribution correspond to the scoring functions. e score functions, on the other hand, are nonlinear functions; the numerical values of the maximum likelihood estimates may be derived using the Newton Raphson iterative optimisation technique.

Modelling to Data Sets
To describe the performance of the TIITFILi model in reality, actual data sets are explored. e whole first set of data comes from [23]. e outcomes of the fits are compared in data set to the power Li (PLi) by [24], alpha power transformed Li (APTLi) by [25], and APT extended exponential (APTEE) by [26]  e maximum likelihood estimates (MLEs), standard errors (SErs) of parameters, and the above statistics measures for the both data sets are given in Table 1. Figures 3-6 provide further information.   Table 1 reveals that the TIITFILi model is best suitable than the APTLi, APTEE, and PLi models. Figures 3 to 6 attempt to estimate pdfs, cdfs, sfs, and pp plots for the fitted models. We infer that now the TIITFILi model fits the data set better.

Conclusion
is research suggested a novel two-parameter truncated Fréchet inverted Lindley model (TIITFILi) distribution for modelling engineering data and other applications. e TIITFILi model generalizes and extends the inverted Lindley distribution. e TIITFILi distribution's hazard rate might be increasing, unimodal, and J-shaped. Mathematical properties of the new model such as ordinary moments, incomplete moments, and the quantile function, order statistics, are discussed. e maximum likelihood approach is used to estimate the parameters of the new distribution. e novel distribution's value and potential are proven by comparing its fit to a real-world data set to those of existing distributions. According to the goodness-of-fit statistics, the new distribution fits better than the other competing distributions.

Data Availability
Please contact the relevant author if you would like to acquire the numerical data set used to perform the study presented in the paper.

Conflicts of Interest
e author declares no conflicts of interest.