A new modified Lehmann type – II G class of distributions: exponential distribution with theory, simulation, and applications to engineering sector [version 1; peer review: awaiting peer review]

Background: Modeling against non-normal data a challenge for theoretical and applied scientists to choose a lifetime model and expect to perform optimally against experimental, reliability engineering, hydrology, ecology, and agriculture sciences, phenomena. Method: We have introduced a new G class that generates relatively more flexible models to its baseline and we refer to it as the new modified Lehmann Type – II (ML–II) G class of distributions. A list of new members of ML–II-G class is developed and as a sub-model the exponential distribution, known as the ML-II-Exp distribution is considered for further discussion. Several mathematical and reliability characters along with explicit expressions for moments, quantile function, and order statistics are derived and discussed in detail. Furthermore, plots of density and hazard rate functions are sketched out over the certain choices of the parametric values. For the estimation of the model parameters, we utilized the method of maximum likelihood estimation. Results: The applicability of the ML–II–G class is evaluated via ML–II–Exp distribution. ML–II–Exp distribution is modeled to four suitable lifetime datasets and the results are compared with the wellknown competing models. Some well recognized goodness–of–fit including -Log-likelihood (-LL), Anderson-Darling (A*), Cramer-Von Mises (W*), and Kolmogorov-Smirnov (K-S) test statistics are considered for the selection of a better fit model. Open Peer Review Reviewer Status AWAITING PEER REVIEW Any reports and responses or comments on the article can be found at the end of the article. Page 1 of 22 F1000Research 2021, 10:483 Last updated: 15 JUL 2021


Introduction
Over the past two decades, the growing attention of researchers towards the development of new G families has explored the remarkable characteristics of baseline models. New models open new horizons for theoretical and applied researchers to address real-world problems, to proficiently and adequately fit them to asymmetric and complex random phenomena. Accordingly, several classes of distributions have been developed and discussed in the literature. For more details, we encourage the reader to see the credible work of some notable scientists including Marshall  ; the exponentiated generalized (EG) class proposed by Cordeiro et al. 6 the T-X family proposed by Alzaatreh et al. 7 ; the Weibull-G family proposed by Bourguignon et al. 8  The method of maximum likelihood estimation is used to estimate the unknown model parameters and develop some simulation results to assess the performance of maximum likelihood estimations (MLEs) in Inference. Applications are discussed in four real data applications and finally, the conclusion is reported in Conclusions.
A new modified Lehmann Type -II-G Class of distributions Lehmann 15 proposed the Lehmann Type-I (L -I) distribution, which was the simple exponentiated version of any arbitrary baseline model. Accordingly, the first credit goes to Gupta et al. 16 because they applied L -I on exponential distribution. The associated cumulative distribution function (CDF) is given by Cordeiro et al. 4 deserves to be acknowledged as they use dual transformation that yielded the Lehmann Type -II (L-II) G class of distributions. The associated CDF is given by The closed-form feature of the L-II distribution assists one to derive and study its numerous properties and in literature, both the approaches (L-I and L-II) have been extensively utilized to study the unexplored characteristics of the classical and modified models.
We develop a new G class, known as a modified Lehmann Type-II (ML-II) G class of distributions. The corresponding CDF is given by where G x; ξ ð Þϵ 0,1 ð Þis a CDF of any arbitrary baseline model based on the parametric vector ξ, dependent on (r x 1) with À∞ < α < 1, and α, β > 0 are the scale and shape parameters, respectively. Let g x;ξ ð Þ¼dG x;ξ ð Þ=dx be the density function of any baseline model. The associated probability density function (PDF), hazard rate function (HRF), and quantile function of ML-II-G class of distributions are given by, respectively Now and onward, the modified Lehmann Type-II-G random variable X corresponding to f ML-IIÀG x; α,β,ξ ð Þwill be denoted by X~ML-II-G x; α,β,ξ ð Þ : This study aimed to propose a new G class of distributions that generates more flexible alternative continuous models relative to its parent distribution. From a computational point of view, new models are very simple to interpret. New models offer greater distributional flexibility and can provide a better fit over the complex random phenomena that exclusively arise in engineering sciences. However, to the best of our knowledge, no study has been conducted previously that discusses our new G class, deliberates the unexplained complex random phenomena so well and advances the fit to a diverse range of sophisticated lifetime data.
Linear combination provides a much more informal approach to discuss the CDF and PDF than the conventional integral computation when determining the mathematical properties. For this, binomial expansion is given as follows: Infinite linear combinations of CDF (1) and PDF (2) of ML-II-G class are given by, respectively The r-th ordinary moment (say μ = r ) of X is given by Further, by (1), Lehmann Type-II-G class and baseline G x ð Þ model can be traced back at α ¼ 0, and α ¼ 0, β ¼ 1, respectively. The expansions of CDF (5) and PDF (6) provide us with the Exp-G class, which is quite useful for the generalization of models.

Order statistics
In reliability analysis and life testing of a component in quality control, order statistics (OS) and its moments are considered as a noteworthy measure. Let X 1 ,X 2 ,…, X n be a random sample of size n that follows the Lehmann Type-II-G class and X 1:n ð Þ < X 2:n ð Þ < … < X n:n ð Þ be the corresponding OS. The random variables X i ð Þ ,X 1 ð Þ , and X n ð Þ be the i-th, minimum, and maximum OS of X.
The PDF of X i ð Þ is given by where ¼ 1,2, 3,…,n,F x; ξ ð Þ, and f x;ξ ð Þare the associated CDF (5) with corresponding PDF (6) of the Lehmann Type-II-G family. Using the fact that By placing the last expression in f i:n ð Þ x; ξ ð Þ, we get the most refined form of OS PDF and one may determine it by integrating (5)(6) and the expression may be given as follows A modified Lehmann Type -II Exponential (Ml-II-Exp) distribution (sub-model) In this section, we introduce a sub-model of ML-II-G class of distributions, known as a ML-II-Exp distribution.
For this, we have the CDF and PDF of the exponential distribution G Exp x;θ ð Þ¼1 À e Àθx , and g Exp x; θ ð Þ¼θe Àθx , for θ > 0,0 < x < ∞, respectively. The associated CDF and corresponding PDF of the ML-II-Exp distribution are obtained by following (1-2) and its analytical expressions are given by respectively: where α is a scale and β,θ > 0 are the shape parameters, respectively. By following (1-2), linear representation of CDF and PDF are given as follows, respectively (11) is expected to be quite supportive in the forthcoming computations of various mathematical properties of the ML-II-Exp distribution.

Reliability characteristics
One of the imperative roles of probability distribution in reliability engineering is to analyze and predict the life of a component. One may define the reliability function as the probability that a component survives until the time x and analytically it can be written as R The reliability function of X is given by In reliability theory, the significant contribution of a function that measures the failure rate of a component in a particular time t is sometimes referred to as the HRF, failure rate function, or the force of mortality, and mathematically it can be written as h The HRF of X is given by Numerous notable reliability measures for the ML-II-Exp distribution can be discussed and derived, such as reverse HRF by

and Odd function by
Shapes Different plots of PDF and HRFs of the ML-II-Exp distribution are sketched over the selected and fixed combinations of the model parameters, respectively. Figure 1 (a, b, c) presents the reversed-J, constant, unimodal, and right-skewed shapes of the PDF and Figure 2 (a, b, c) illustrates the decreasing and increasing HRF. However, an increasing HRF with some interesting facts are identified when suddenly spikes arise at the tail end of HRF is unexpectedly detected. Such kinds of trends are often observed in non-stationary time series lifetime phenomena.

Limiting behavior
Here we study the limiting behavior of CDF, PDF, reliability, and HRFs of the ML-II-Exp distribution present in (8), (9), (13), and (14) at x! 0 and x! ∞.  Proposition-1: Limiting behavior of CDF, PDF, reliability, and HRFs of the ML-II-Exp distribution at x! 0 is followed by Proposition-2: Limiting behavior of CDF, PDF, reliability, and HRFs of the ML-II-Exp distribution at x ! ∞ is followed by Limiting behaviors developed in the above expressions may illustrate the effect of parameters on the tail of the ML-II-Exp distribution.

Moments and associated measures
Moments have a remarkable role in the discussion of the distribution theory, to study the significant characteristics of a probability distribution such as mean; variance; skewness, and kurtosis.
limits: as x⟶0 ) x⟶∞; y⟶0 ) y⟶∞: By placing the above information in (15), we get by making simple computation on the last expression leads us to the r-th ordinary moment, in terms of the gamma function and it is given by where Γ : The derived expression in (16) may serve a supportive and useful role in the development of several statistical measures.

Residual and reversed residual life functions
The residual life function/conditional survivor function of random variable R t ð Þ ¼ X þ t=X > t X is the probability that a component whose life says x, survives in an additional interval at t≥0. Analytically it can be written as The residual life function of X is given by with associated CDF PDF and HRF corresponding to (18) are given as follows, respectively Mean residual life function is given by Moreover, the reverse residual life can be defined as: R t ð Þ ¼ t À X=X≤t.
The reverse residual life function of X is given by with associated CDF PDF and HRF corresponding to (22) are given as follows, respectively Mean reversed residual life function/mean waiting time is given by One may derive the strong mean inactivity time of X by following are termed as the first and second lower incomplete moments of X, respectively, with η i ¼ α i 1 À α ð Þ ÀβÀi Àβ À 1 i , and À∞ < α < 1:

Entropy
When a system is quantified by disorderedness, randomness, diversity, or uncertainty, in general, it is known as entropy.
Rényi 17 entropy of X is given by By following (9), we simplify f x ð Þ in terms of f ζ x ð Þ, we get ð Þ by placing the above expression in (25), we get Quantile function, and mode The q th quantile function of ML-II-Exp distribution is obtained by inverting the CDF. Quantile function is defined as The quantile function of X is given by To obtain the 1 st quartile, median and 3 rd quartile of X, place q = 0.25, 0.5, and 0.75 respectively in (26). Henceforth, to generate random numbers, one may assume that the CDF in (8) follows the uniform distribution u = U (0, 1).
The modal value of X is calculated by following the constraint f = x; α, β, θ ð Þ¼ df x; α, β, θ ð Þ dx ¼ 0: For convenience, f x;α,β,θ ð Þ can be rewritten as The simplified form of f = x;α,β,θ ð Þis given by Hence, solving simple algebra on the previous equation may provide us with the most suitable form of the mode of X in support of f = x; α, β, θ ð Þ¼0 and it is given bŷ Stressstrength reliability Let X 1 and X 2 be defined to discuss the strength and stress of a component, respectively, followed by the same uni-variate family of distributions, which will work in order if X 2 < X 1 . To discuss the reliability (say R) of X, it is given by R ¼ P X 2 < X 1 ð Þ : Theorem 2: Let X 1 $ ML-II-Exp (x;α,β 1 ,θ) and X 2 $ ML-II-Exp (x; α,β 2 ,θ) be independent random variables following the ML-II-Exp distribution; then the reliability is given by Proof: Reliability (R) is defined as R of X can be written by following (9), as limits: as x⟶0 ) t⟶1; x⟶∞ ) t⟶0: By placing the above information in (27), we have Hence, the simple computation of the above expression provides us with the reduced form of R in terms of β 1 and β 2 , as we presume that the R is a function of β 1 with increasing behavior and it is given by

Order statistics
In reliability analysis and life testing of a component in quality control, order statistics OS and its moments are considered as a noteworthy measure. Let X 1 , X 2 ,…,X n be a random sample of size n following the ML-II-Exp distribution and X 1:n ð Þ < X 2:n ð Þ < … < X n:n ð Þ be the corresponding OS. The random variables X i ð Þ , X 1 ð Þ , and X n ð Þ be the i-th, minimum, and maximum OS of X.
The PDF of X i ð Þ is given by By following (8) and (9), the PDF of X i ð Þ takes the form by utilizing some the techniques of binomial expansion (mentioned in the new modified Lehmann Type -II-G Class of distributions) to simplify (28), we get the reduced form of f i:n ð Þ x; α,β,θ ð Þand it is given as follows and we determine the linear representation of (29) and it can be written as Indeed, (29) has a supportive role in the calculation of r-th moment OS and hereafter, straightforward computation of (29) leads us to the r-th moment OS of X and it is given as follows The CDF of X i ð Þ is given by Furthermore, the minimum and maximum OS of X follows directly from (28) with i = 1 and i = n, respectively.

Inference
In this section, we estimate the parameters of the ML-II-Exp distribution by following the method of MLE, as this method provides the maximum information about the unknown model parameter. Let X 1 ,X 2 ,X 3 ,…,X n be a random sample of size n from the ML-II-Exp distribution, then the likelihood function L MLÀIIÀExp ψ ¼ α,β,θ ð Þ¼ Q n i¼1 f MLÀII-Exp x i ;ψ ð Þof X is given by The log-likelihood function, l ψ ð Þ is given by Partial derivatives of (33) w.r.t. α, θ, and β yield, respectively The maximum likelihood estimates (ψ i ¼α,β,θ) of the ML-II-Exp distribution can be obtained by maximizing (33) or by solving the above non-linear equations simultaneously. These non-linear equations, however, do not provide an analytical solution for the MLEs and the optimum value of α, β and θ. Consequently, iterative techniques such as the Newton-Raphson type algorithm are an appropriate choice in the support of MLEs.

Simulation study
In this sub-section, we discuss the performance of MLEs using the following algorithm.
Step 4: Estimated bias, root mean square error (RMSE), variance, and mean values are presented in Table 5.
Step 5: Each sample is replicated N = 500 times.
Step 6: A gradual decrease in SEs, biases, RMSE, variances, means, and MLEs pretty close to the true parameters are observed with increase in the sample sizes.
Step 7: Finally, the estimates present in Tables 4 and 5 help us to specify that the method of maximum likelihood works consistently for the ML-II-Exp distribution.
The in-practice measures for the development of average estimate (AE), SE, bias, and RMSE are given as follows: Four real data applications In this section, we explore four suitable lifetime datasets to model the ML-II-Exp distribution. These datasets are associated with the engineering sector. The first dataset relates to the study of failure times of 84 windshields for a particular model of aircraft (the unit for measurement is 1000 hours) that was first discussed by Ramos et al. 18 The second dataset relates to the study of service times of 63 aircraft windshields (the unit for measurement is 1000 hours) that was discussed by Tahir et al. 19 The third dataset follows the discussion of the breaking stress of carbon fibers (in Gba) that was initially developed by Nicholas and Padgett 20 and finally the fourth one relates to the study of fatigue life of 6061 -T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle of 31,000 psi. This dataset was pioneered by Birnbaum and Saunders. 21 Subsequently, this data was discussed by Shanker et al., 22 after subtracting 65 from each observation. The datasets are given in the underlying data.      The ML-II-Exp distribution is compared with the well-known models (CDF list is mentioned in Table 6). We follow some recognized selection criteria including -Log-likelihood (-LL), Anderson-Darling (A*), Cramer-Von Mises (W*), and Kolmogorov-Smirnov (K-S) test statistics. Some common results of descriptive statistics such as minimum value, 1 st quartile, means, 3 rd quartile, 95% confidence interval, and the maximum value, are tabulated in Table 3. The parameter estimates, standard errors (in parenthesis), and the goodness-of-fit are confirmed in Tables 7-10, respectively. The minimum value of the goodness-of-fit is the criteria of the better fit model that the ML-II-Exp distribution perfectly satisfies. Hence, we affirm that the ML-II-Exp distribution is a better fit than its competitors.
Furthermore, for a visual comparison the fitted density and distribution functions, Kaplan-Meier survival and probabilityprobability (PP) plots, total time on test transform (TTT), and box plots, are presented in Figures 3-6 (a, b, c, d, e, f, g, h), respectively. These plots provide sufficient information about the closest fit to the data. All the numerical results in the subsequent tables are calculated with the assistance of statistical software RStudio-1.2.5033. with its package Adequa-cyModel. The explored datasets are given in the underlying data.

Conclusion
In this article, we introduced and studied a more flexible G class, called the modified Lehmann Type-II (ML-II) G class of distributions along with explicit expressions for the moments, quantile function, and OS. The exponential distribution was used as the baseline distribution for ML-II-G class, known as ML-II-Exp distribution. It was discussed comprehensively, which demonstrated the reversed-J, constant, unimodal, and right-skewed shapes of a density function. The method of MLE along with the simulation was carried out to investigate the performance of the proposed method. The efficiency of the ML-II-G class was evaluated when the most efficient and consistent results of ML-II-Exp distribution competed the well-known models and explored the dominance along with a better fit in four real-life datasets.
We hope that in the future, the proposed class and its sub-models will explore the wider range of applications in diverse areas of applied research and will be considered as a choice against the baseline models.   This project contains the following extended data:

Data availability
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