The Extended Inverse Weibull Distribution: Properties and Applications

'is paper proposes the new three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution which generalizes the inverse Weibull model. 'e density function of the TIHLIW can be expressed as a linear combination of the inverse Weibull densities. Some mathematical quantities of the proposed TIHLIW model are derived. Four estimation methods, namely, the maximum likelihood, least squares, weighted least squares, and Cramér–von Mises methods, are utilized to estimate the TIHLIW parameters. Simulation results are presented to assess the performance of the proposed estimation methods. 'e importance of the TIHLIW model is studied via a real data application.


Introduction
e inverse Weibull (IW) distribution is also known as reciprocal Weibull distribution (see [1,2]). Keller et al. [3] used the IW distribution to describe the degradation phenomena of mechanical components such as crankshaft and pistons of diesel engines. Further, the IW model has many important applications in reliability engineering, infant mortality, useful life, wear-out periods, life testing, and service records (see [4]). e cumulative distribution function (CDF) of the IW model is Its associated probability density function (PDF) has the following form: where G(x; δ) refers to the baseline CDF with a parameter vector δ. e CDF in (3) is a wider class which can be used to generate more flexible extended distributions. e associated PDF of (3) has the form where g(x; δ) is the baseline PDF. e hazard rate function (HRF) of the TIHL-G family is In this paper, we propose a new lifetime model called the type I half-logistic inverse Weibull (TIHLIW) model. e proposed model can be used, as a good alternative to some existing distributions, in modeling several real data. e paper is outlined as follows. In Section 2, we define the TIHLIW distribution and derive a useful representation for its PDF. e mathematical properties of the TIHLIW distribution are derived in Section 3. In Section 4, the TIHLIW parameters are estimated via four methods, namely, the maximum likelihood, least squares, weighted least squares, and Cramér-von Mises estimators. ese estimators are compared via some simulations in Section 5. In Section 6, we illustrate the flexibility and potentiality of the TIHLIW model using a real data set. Finally, some concluding remarks are offered in Section 7.

The TIHLIW Distribution
e CDF of the three-parameter TIHLIW distribution follows, by replacing equation (1) in (3), as e corresponding PDF of (4) reduces to e HRF of the TIHLIW distribution takes the form Figure 1 provides some shapes of the PDF and HRF of the TIHLIW distribution for some different values of the parameters.
e TIHLIW distribution is a very flexible model that approaches to different distributions as special submodels: Expanding [1 + (1 − e − αx − β ) λ ] − 2 using (9), we can write (7) as Consider the power series: Using the power series (11) and after some algebra, the TIHLIW PDF reduces to where g(x; α(j + 1), β) refers to the IW PDF with parameters α(j + 1) and β, and 2 Complexity Equation (12) means that the PDF of the TIHLIW is a linear combination of the IW densities and can be used to calculate some mathematical quantities of the TIHLIW model from those of the IW distribution.
Consider the random variable (RV) Y ∼ IW(α, β) in (1). For n < β, the nth ordinary and incomplete moments of Y are given by

Some Properties
In this section, we studied some statistical properties of the TIHLIW distribution, such as quantile function, ordinary moments, moment generating function, incomplete moment, and mean deviation.

Quantile and Moment-Generating Functions.
As RV X has CDF of TIHLIW distribution, the quantile function (QF) is defined by where (0 < u < 1). is relation is used to find the QF of TIHLIW distribution as follows: , u ∈ (0, 1). (16) e above equation can be used to generate TIHLIW random variates. Here, we obtain the MGF of the IW distribution (1) by setting w � x − 1 : By expanding exp(t/w) and calculating the integral, we can write Using the Wright generalized hypergeometric function [18], e MGF of the IW distribution has the form Using equations (12) and (20), the MGF of the TIHLIW distribution reduces to  Complexity 3

Moments.
e sth ordinary moment of RV X is For (s < β), we obtain Setting (s � 1) in (23), we obtain the mean of X.
Using equation (12), we can write Hence, we obtain the sth incomplete moment of the TIHLIW distribution: e first incomplete moment which follows by setting (s � 1) in the above equation is

Mean Residual Life and Mean Waiting Time.
e mean residual life (MRL) has useful applications in economics, life insurance, biomedical sciences, demography, product quality control, and product technology (see [19]). e MRL refers to the expected additional life length for a unit that is alive at age t, and it is defined by m e MRL of X can be calculated by the formula: where S(t) is the survival function of X. By inserting (26) in (27), the MRL of the TIHLIW distribution follows as , t > 0, and it can be calculated by the formula: By substituting (26) in (29), the MIT of the TIHLIW distribution follows as

Order Statistics.
Consider the random sample from the TIHLIW (λ, α, β) denoted by (X 1 , . . . , X n ) and its associated order statistics denoted by (X (1) , . . . , X (n) ). e PDF of the rth order statistic is denoted by (X (r) , 1 ≤ r ≤ n) and can be expressed as e PDF of X (r) is defined also by the formula: Using the PDF and CDF of the TIHLIW distribution, equation (32) reduces to 4 Complexity Using expansion (9) and the power series, We can write Applying the power series (11), equation (35) reduces to By replacing (36) in equation (32), we obtain Hence, we can write where g(x; α(k + 1), β) denotes the IW PDF with parameters β and α(k + 1), and Equation (38) reveals that the PDF of the TIHLIW order statistics is a mixture of IW densities. Hence, the qth moment of X (r) has the form

Maximum Likelihood.
We determine the MLEs of the TIHLIW parameters. Let (x 1 , x 2 , . . . , x n ) be a random sample of size n from TIHLIW (ϕ) where ϕ � (λ, α, β) T . e log-likelihood function for ϕ has the form ℓ � n log(2λαβ) We can maximize the above log-likelihood equation by solving the nonlinear likelihood equations which follow by differentiating it. e associated components of the score vector are given by And, e MLEs of ϕ can be constructed by solving the nonlinear system U n (ϕ) � 0 that cannot be analytically solved; hence, statistical programs are utilized to solve them numerically via iterative techniques such as a Newton-Raphson algorithm.

Ordinary and Weighted Least Squares.
e least square and weighted least square methods are used to estimate the parameters of beta distribution [20]. Let x (1) < x (2) < · · · < x (n) be the order statistics of a sample from the TIHLIW distribution, and then the LSEs and WLSEs of α, λ, and β can be obtained by minimizing the following function with respect to α, λ, and β: where (A i � 1) in the case of LSEs and (A i � ((n + 1) 2 (n + 2))/(i(n − i + 1))) in the case of WLSEs. Further, the LSEs and WLSEs of the TIHLIW parameters are also obtained by solving the following nonlinear equations simultaneously with respect to α, λ, and β: where

Simulation Study
In this section, we conduct a simulation study to compare the performance of the different estimators based on the mean square error criterion. We compare the performances of the MLEs, LSEs, WLSEs, and CVMEs based on the mean square errors (MSEs) for different sample sizes.  Tables 1-3.  From Tables 1-3, we conclude the following: (1) e MSE of α, λ, and β for all estimation methods decreases as n increases.
(2) Table 1 shows that the MLEs have the lowest MSE in most cases of α and λ. Also, the WLSEs have the least MSE in most cases of β at α � 0.5, λ � 0.5, and β � 0.5.

Data Analysis
In this section, we present an application to a real data set to illustrate the performance and flexibility of the TIH-LIW distribution. e data refer to relief times of a sample of 20 patients who receive an analgesic [23]. ese data have been analyzed by Afify et al. [24] and Cordeiro et al. [25].
We consider some criteria including − 2ℓ (where ℓ is the maximized log-likelihood), AIC (Akaike information criterion), HQIC (Hannan-Quinn information criterion), CAIC (corrected Akaike information Criterion), AD (Anderson-Darling statistic), and CVM (Cramér-von Mises statistic), where Table 4 lists the numerical values of − 2ℓ, AIC, CAIC, HQIC, A * , and W * for all fitted models, whereas MLEs and their standard errors (SEs) (in parentheses) are given in Table 5. From Table 4, the TIHLIW has the lowest values for all goodness-of-fit measures, and hence it provides close fits to relief times data than other fitted models. e fitted PDF, estimated CDF, estimated survival function (SF), and PP plots of the TIHLIW distribution are shown in Figures 2 and 3.

Conclusions
We proposed a three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution as a new extension of the inverse Weibull model. e TIHLIW density is a linear combination of the inverse Weibull densities. Some explicit expressions for mathematical quantities of the TIHLIW distribution are derived. We consider four methods of estimation, namely, the maximum likelihood, least squares, weighted least squares, and Cramér-von Mises methods, to estimate the TIHLIW parameters. e performance of these proposed estimation methods is conducted via some simulations. A real data application proves that the TIHLIW model provides consistently better fits compared to some other rival models.

Data Availability
is work is mainly a methodological development and has been applied on secondary data related to the relief times of 20 patients who received an analgesic, but if required, data will be provided.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.