A New Lifetime Distribution: Properties, Copulas, Applications, and Different Classical Estimation Methods

A new continuous version of the inverse flexible Weibull model is proposed and studied. Some of its properties such as quantile function, moments and generating functions, incomplete moments, mean deviation, Lorenz and Bonferroni curves, the mean residual life function, the mean inactivity time, and the strong mean inactivity time are derived. 'e failure rate of the new model can be “increasing-constant,” “bathtub-constant,” “bathtub,” “constant,” “J-HRF,” “upside down bathtub,” “increasing,” “upside down-increasing-constant,” and “upside down.” Different copulas are used for deriving many bivariate and multivariate type extensions. Different non-Bayesian well-known estimation methods under uncensored scheme are considered and discussed such as the maximum likelihood estimation, Anderson Darling estimation, ordinary least square estimation, Cramér-von-Mises estimation, weighted least square estimation, and right tail Anderson Darling estimation methods. Simulation studies are performed for comparing these estimation methods. Finally, two real datasets are analyzed to illustrate the importance of the new model.


Introduction
e Weibull model [1] is a very useful distribution in modeling real data exhibiting monotonic hazard rate function (HRF). But it cannot be used in modeling and studying data which have nonmonotonic HRF such as the "bathtub shape (U-HRF)." For avoiding this drawback, Bebbington et al. [2] have defined a new two-parameter distribution which is an extension of the Weibull distribution referred to as a flexible Weibull (FW) extension distribution; it has a failure function that can be "decreasing," "increasing," or "bathtub-shaped." Analogously, El-Goharyet et al. [3] derived the two-parameter inverse flexible Weibull (IFW) model which is the reciprocal of a random variable (RV) which has FW model. Several mathematical properties of this distribution such as the mode, moments, and moment generating function (MGF) have been discussed. El-Gohary et al. [3] proved that the hazard rate function (RRF) of the IFW model can be "upside down constant," the cumulative distribution function (CDF) of IFW distribution is given by where the two parameters α > 0 and θ > 0 control the shape of the distribution. e corresponding probability density function (PDF) is (TIITL-G) family of distributions. Due to [11], the CDF of the TIITL-G family is given by e PDF is defined by where λ > 0 is a shape parameter, g ψ (z) � dG ψ (z)/dz is the baseline PDF, and G ψ (z) is a baseline CDF. e remainder of the paper is organized as follows: in Section 2, we define the CDF, PDF, and HRF of TIITLIFW model and provide a simple expansion of the PDF. Simple type copula is derived in Section 3. Various mathematical properties are discussed in Section 4. Non-Bayesian estimation methods under uncensored schemes are given in Section 5. Section 6 presents a comparison under the non-Bayesian estimation methods using uncensored schemes via a simulation study. Section 7 presents a comparison under uncensorship with some competitive models. Concluding remarks are contained in Section 8.

The New Model
Using (1) in (3), the CDF of the TIITLIFW distribution can be written as e corresponding PDF is given by (7) Figure 1 shows some plots of the PDF of the TIITLIFW distribution for some different values of the parameters. Figure 2 shows some plots of the HRF of the TIITLIFW distribution for some different parameter values. Based on Figure 1, we conclude that the new PDF can have many right skewed heavy tail shapes. Based on Figure 2, it is observed that the new HRF can be "increasing-constant," "bathtubconstant," "bathtub," "constant," "J-HRF," "upside down bathtub," "increasing," "upside down-increasing-constant," and "upside down" e PDF of the TIITLIFW distribution can be written as Proof. Supposing |w 1 /w 2 | < 1, w 3 > 0 is a real noninteger, we have the power series expansion using the power series (9) in equation (8), and the fact that expanding exp[− 2(i + 1)exp((α/z) − θz)] using Taylor series again, using a series expansion of exp[(j + 1)(α/z)], and after some algebras, the PDF can be written as where

Bivariate TIITLIFW (BTIITLIFW) via Morgenstern
Family. e CDF of the Morgenstern family of two random RVs (X 1 , X 2 ) can be derived as where setting 2 Complexity then we have

Via Clayton Copula.
e BTIITLIFW type extension: the weighted version of the Clayton copula can be expressed as Let us assume that X ∼ TIITLIFW (V 1 ) and Y ∼ TII-TLIFW (V 2 ). en, setting the associated CDF of the BTIITLIFW type distribution can be written as A straightforward M-dimensional extension from the above will be

Via Modified Farlie-Gumbel-Morgenstern (FGM)
Copula. e joint CDF of the modified FGM copula is given as Type I: Recalling the following functional form for both A(u) and V(w). en, the BTIITLIFW-FGM (Type-I) can be derived from where erefore, Type II: Let A(u) * and V(w) * be two functional forms, satisfying all the conditions mentioned above, where en, the corresponding BTIITLIFW-FGM (Type-II) can be derived from en, Type-III: In this case, one can also derive a closed form expression for the associated CDF of the BTIITLIFW-FGM (Type-III) from as follows . en, the Renyi's entropy copula can be expressed as en, the associated BTIITLIFW can be directly derived

Quantile Function.
For RV Z has CDF of the TIITLIFW distribution, the quantile function Z q of the TIITLIFW distribution is given by the following equation: where

Moments and Generating
Functions. e r th moment of the TIITLIFW distribution is obtained using the formula hence using equation (7) we obtain Setting y � (j + 1)θz, it follows that where is a gamma function. In particular, if r � 1 and r � 2, we obtain the mean and variance of the TIITLIFW distribution. e MGF of the TIITLIFW is given by e mathematical form of the Galton skewness and Moors kurtosis of TIITLIFW distribution can be computed using the quantile function and well-known relationships. e first four moments, skewness, and kurtosis of the TIITLIFW distribution for different values of parameters are represented in Table 1. Table 1 shows that the skewness is always positive, and kurtosis is always greater than three. Figure 3 shows two plots of the skewness of the TII-TLIFW distribution. Figure 4 presents two plots of the kurtosis of the TIITLIFW distribution. Based on Figures 3 and 4, we conclude that the TIITLIFW model can have many useful skewness and kurtosis shapes.

Incomplete Moments.
e s th lower and upper incomplete moments of Z are defined by respectively, for any real � s > 0. e s th lower incomplete moment of the TIITLIFW distribution is where is the lower incomplete gamma function. Similarly, the s th upper incomplete moment of the TIITLIFW distribution is where is the upper incomplete gamma function.

Mean Deviation, Lorenz, and Bonferroni Curves. For RV
, the mean deviation about the mean and median, respectively, is given by where Complexity e Lorenz curve for a positive RV Z is defined as en, we have where q � G − 1 (p). Also, Bonferroni curve is defined by  8 Complexity en, en, the MIT can be derived as e strong mean inactivity time (SMIT) is a new reliability function given by erefore, the SMIT can be expressed as

e MLE Method.
Let Z 1 , Z 2 , . . . , Z n be a random sample of size n from TIITLIFW. e log-likelihood function for the vector of parameters V can be written as e associated score function is given by e log(L) can be maximized by solving the following nonlinear likelihood equations U n (V |α) � z log(L)/zα, U n (V |θ) � z log(L)/zθ and U n (V |λ) � z log(L)/zλ. en, e maximum likelihood estimation (MLE) of V, say V ⌢ , is obtained by solving the system of nonlinear equations

e CVME Method.
e CVMEs of the parameters α, θ, and λ [12] are obtained by minimizing the following expression with respect to the parameters α, θ, and λ, respectively, where Complexity 9 CVME V ( ) � where refers to the empirical estimate of the CDF at z [i: n] computed from a certain sample and (63) en, CVMEs of the parameters α, θ, and λ are obtained by solving the following nonlinear system: where zβ , zb , za . (65)

e OLSE and WLSE Method. Let F V (z [i: n]
) denote the CDF of the TIITLIFW model and z 1: n < z 2: n < · · · < z n: n be the n ordered RS. e OLSEs [13] are obtained by minimizing where (68) e OLSEs are obtained by solving where D (α) (z [i: n] ; V), D (θ) (z [i: n] ; V), and D (λ) (z [i: n] ; V) are as defined above. e WLSE is obtained by minimizing the function WLSE (V) with respect to α, θ, and λ. en, where 10 Complexity (71) e WLSEs are obtained by solving

e ADE Method. e ADEs are obtained by minimizing the function
where en, the parameter estimates are derived by solving the nonlinear equations (75)

e ADE (R-T) Method. e ADEs (R-T) are obtained by minimizing
Complexity en, the estimates follow by solving the nonlinear equations (77)

Comparing the Non-Bayesian Estimation Methods under Uncensored Schemes via a Simulation Study
A numerical simulation study is conducted to compare the non-Bayesian estimation methods. e simulation study is based on N � 1000 which generated datasets from the TIITLIFW distribution where n � 20, 60, 100, 200, and 500 and α � 2, θ � 3, and λ � 1.1. e comparison is performed based on the bias (BIAS (V) ) and root mean -standard error (RMSE (V) ).
From Table 2, we note the following: 1) e BIAS (V) tends to 0 as n increases and tends to ∞ which means that all estimators are nonbiased.
2) e RMSE (V) tends to 0 as n increases and tends to ∞ which means incidence of consistency property.

Comparing Models under Uncensorship
We illustrate the flexibility and the performance of the TIITLIFW distribution as compared to some alternative models using two real data applications. e goodness-of-fit statistics for this distribution are compared with other competitive distributions. e MLEs of the distribution parameters are determined numerically. To compare the distributions, we consider the measures of goodness-of-fit, such as Akaike information criterion (C 1 ), consistent Akaike information criterion (C 2 ), and Bayesian information criterion (C 3 ) statistic. e better distribution to fit the data corresponds to smaller values of these statistics.
We consider two uncensored datasets for comparing competitive models. e first data present the remission times (in months) of a random sample of 128 bladder cancer patients [14] [15], the total time in test (TTT) plots, box plots, quantilequantile (QQ) plots, and kernel density estimation (KDE) plots are shown in Figure 5 and Figures 6(a) and 6(c) for bladder cancer data and lifetimes data, respectively. Based on Figures 6(a) and 6(c), the HRF of the bladder cancer data is "upside down" or "reversed U-shape." Based on Figures 6(a) and 6(c), the HRF of the lifetimes data is "U-shape." e box plots (middle panels) are presented along with its corresponding normal quantile-quantile plot (right panels) in Figures 5 and 6 for discovering the outliers and normality. e following competitive models are considered in the comparison: the exponentiated IFW (Exp-IFW) [12], IFW [3], exponentiated generalized IW (ExpG-IFW) [16], generalized IW (G-IW) [17], IW [18], and [2]. Tables 3 and 4 present the MLEs for the bladder cancer data and lifetimes data. Tables 5 and 6 show the statistics criteria for the bladder cancer data and lifetime data. From Tables 5 and 6, it is clear that the TIITLIFW distribution provides the best fits for the two datasets. Figures 7 and 8 show the estimated PDFs (EPDFs) (left panel) and the estimated HRF (EHRFs) (right panel) for bladder cancer data and lifetimes data, respectively. Figures 9 and 10 show the profile of the log-likelihood function for bladder cancer data and lifetimes data, respectively. From

Concluding Remarks
A three-parameter lifetime distribution, so-called the TII-TLIFW distribution, is introduced as an extension of the inverse flexible Weibull distribution. Some explicit expressions for mathematical quantities of the TIITLIFW distribution are derived. e hazard rate function allows constant, decreasing, increasing, upside down bathtub, or bathtubshaped forms. We consider six different estimation methods to estimate the parameters of the TIITLIFW distribution. e performance of these proposed estimation methods is conducted via some simulations. A real data application proves that the TIITLIFW model provides consistently better fits compared to some other well-known competitive models.

Data Availability
e data used to support the findings in this study are included within the paper.

Conflicts of Interest
e author declares no conflicts of interest regarding the publication of this paper.