Abstract
This paper proposes the new three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution which generalizes the inverse Weibull model. The 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. The importance of the TIHLIW model is studied via a real data application.
1. Introduction
The 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]).
The cumulative distribution function (CDF) of the IW model is
Its associated probability density function (PDF) has the following form:
The statistical literature contains several extensions of the IW model, see, for example, beta IW by Khan [5], generalized IW by de Gusmão et al. [6], modified IW by Khan and King [4], gamma IW by Pararai et al. [7], Kumaraswamy generalized IW by Oluyede and Yang [8], Kumaraswamy modified IW by Aryal and Elbatal [9], Marshall-Olkin IW by Okasha et al. [10], and alpha power IW by Basheer [11].
Many generalized classes of distributions have been proposed for modeling real-life data in several applied fields such as reliability, engineering, biological studies, economics, medical sciences, environmental sciences, and finance. For example, Marshall and Olkin [12] proposed Marshall-Olkin-G, Shaw and Buckley [13] defined transmuted-G, Cordeiro and de Castro [14] pioneered Kumaraswamy-G, Alizadeh et al. [15] proposed the extended odd Weibull-G, and Cordeiro et al. [16] studied the odd Lomax-G family.
Cordeiro et al. [17] proposed the type I half-logistic-G (TIHL-G) class. The CDF of the TIHL-G family of distributions is given (for ) bywhere refers to the baseline CDF with a parameter vector . The CDF in (3) is a wider class which can be used to generate more flexible extended distributions.
The associated PDF of (3) has the formwhere is the baseline PDF. The 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. The proposed model can be used, as a good alternative to some existing distributions, in modeling several real data.
The paper is outlined as follows. In Section 2, we define the TIHLIW distribution and derive a useful representation for its PDF. The 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. These 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.
2. The TIHLIW Distribution
The CDF of the three-parameter TIHLIW distribution follows, by replacing equation (1) in (3), as
The corresponding PDF of (4) reduces to
The 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.
The TIHLIW distribution is a very flexible model that approaches to different distributions as special submodels:(1)If , then the TIHLIW distribution reduces to the type I half-logistic Fréchet (TIHLFr) distribution(2)If , TIHLIW distribution reduces to the type I half-logistic inverse exponential (TIHLIE) distribution(3)If , we have the type I half-logistic inverse Rayleigh (TIHLIR) distribution
2.1. Linear Representation
In this section, we express the TIHLIW PDF as a mixture linear representation of IW densities.
Consider the power series:
Expanding using (9), we can write (7) as
Consider the power series:
Using the power series (11) and after some algebra, the TIHLIW PDF reduces towhere refers to the IW PDF with parameters and , and
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 in (1). For , the ordinary and incomplete moments of are given byrespectively, where is the complete gamma function and is the lower incomplete gamma function.
3. 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.
3.1. Quantile and Moment-Generating Functions
As has CDF of TIHLIW distribution, the quantile function (QF) is defined bywhere . This relation is used to find the QF of TIHLIW distribution as follows:
The above equation can be used to generate TIHLIW random variates. Here, we obtain the MGF of the IW distribution (1) by setting :
By expanding and calculating the integral, we can write
Using the Wright generalized hypergeometric function [18],
The MGF of the IW distribution has the form
Using equations (12) and (20), the MGF of the TIHLIW distribution reduces to
3.2. Moments
The ordinary moment of is
For , we obtain
Setting in (23), we obtain the mean of .
The incomplete moment of the TIHLIW distribution is defined by .
Using equation (12), we can write
Hence, we obtain the incomplete moment of the TIHLIW distribution:
The first incomplete moment which follows by setting in the above equation is
3.3. Mean Residual Life and Mean Waiting Time
The mean residual life (MRL) has useful applications in economics, life insurance, biomedical sciences, demography, product quality control, and product technology (see [19]). The MRL refers to the expected additional life length for a unit that is alive at age , and it is defined by , .
The MRL of can be calculated by the formula:where is the survival function of .
By inserting (26) in (27), the MRL of the TIHLIW distribution follows as
The mean inactivity time (MIT) (mean waiting time) is defined by , , and it can be calculated by the formula:
By substituting (26) in (29), the MIT of the TIHLIW distribution follows as
3.4. Order Statistics
Consider the random sample from the TIHLIW denoted by and its associated order statistics denoted by . The PDF of the order statistic is denoted by and can be expressed as
The PDF of is defined also by the formula:
Using the PDF and CDF of the TIHLIW distribution, equation (32) reduces to
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 writewhere denotes the IW PDF with parameters and , and
Equation (38) reveals that the PDF of the TIHLIW order statistics is a mixture of IW densities. Hence, the moment of has the form
4. Estimation Methods
The estimation of the TIHLIW parameters is investigated using four methods of estimation, namely, the maximum likelihood estimators (MLEs), least squares estimators (LSEs), weighted least squares estimators (WLSEs), and Cramér–von Mises estimators (CVMEs).
4.1. Maximum Likelihood
We determine the MLEs of the TIHLIW parameters. Let be a random sample of size from TIHLIW where . The log-likelihood function for has the form
We can maximize the above log-likelihood equation by solving the nonlinear likelihood equations which follow by differentiating it. The associated components of the score vectorare given by
And,
The MLEs of can be constructed by solving the nonlinear system that cannot be analytically solved; hence, statistical programs are utilized to solve them numerically via iterative techniques such as a Newton–Raphson algorithm.
4.2. Ordinary and Weighted Least Squares
The least square and weighted least square methods are used to estimate the parameters of beta distribution [20]. Let be the order statistics of a sample from the TIHLIW distribution, and then the LSEs and WLSEs of , and can be obtained by minimizing the following function with respect to , and :where in the case of LSEs and 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 , and :where
4.3. Cramér–von Mises
Based on the Cramér–von Mises (CVM) method [21, 22], the CVMEs of the TIHLIW parameters can be constructed by minimizingwith respect to , and . Further, the CVMEs can also be obtained by solving the following nonlinear equations simultaneously with respect to , and :where , , and are, respectively, defined in equations (47)–(49).
5. 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. Mathematica 9 is utilized to obtain simulation results from replications. The mean values of the MLEs, LSEs, WLSEs, and CVMEs and their MSEs of the TIHLIW parameters by considering sample sizes , 100, and 200, are given for different values of parameters , , and . The mean values of the MLEs, LSEs, WLSEs, and CVMEs and MSEs of the TIHLIW parameters are listed in Tables 1–3.
From Tables 1–3, we conclude the following:(1)The MSE of , , and for all estimation methods decreases as increases.(2)Table 1 shows that the MLEs have the lowest MSE in most cases of and . Also, the WLSEs have the least MSE in most cases of at , , and .(3)Table 2 shows that CVMEs get the least MSE in most situations of and . Also, the WLSEs have the lowest MSE in most cases of at , , and .(4)Table 3 shows that CVMEs have the lowest MSE in most cases of , , and at , , and .
6. Data Analysis
In this section, we present an application to a real data set to illustrate the performance and flexibility of the TIHLIW distribution. The data refer to relief times of a sample of 20 patients who receive an analgesic [23]. These data have been analyzed by Afify et al. [24] and Cordeiro et al. [25].
For these data, we compare the TIHLIW model with some rival models, namely, the beta generalized inverse Weibull geometric (BGIWGc) by Elbatal et al. [26], transmuted complementary Weibull geometric (TCWGc) by Afify et al. [27], beta transmuted Weibull (BTW) by Afify et al. [28], McDonald log-logistic (McLL) by Tahir et al. [29], beta Weibull (BW) by Lee et al. [30], McDonald Weibull (McW) by Cordeiro et al. [31], exponentiated transmuted generalized Rayleigh (ETGR) by Afify et al. [32], and new modified Weibull (NMW) [33] distributions. The PDFs of these distributions are given (for ) by BGIWGc: , where , and . BTW: where and . TCWGc: , where and . McLL: , where and . McW: , where and . BW: , where and . ETGR: , where and . NMW: , where and .
We consider some criteria including (where 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 , AIC, CAIC, HQIC, , and 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.
The fitted PDF, estimated CDF, estimated survival function (SF), and PP plots of the TIHLIW distribution are shown in Figures 2 and 3.
7. Conclusions
We proposed a three-parameter type I half-logistic inverse Weibull (TIHLIW) distribution as a new extension of the inverse Weibull model. The 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. The 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
This 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
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful to the Deanship of Scientific Research at King Saud University represented by the Research Center at the College of Business for financially supporting this research.