A New Inverse Weibull Distribution: Properties and Applications

Corresponding Author: Mona Mustafa Elbiely Department of Statistics, Mathematics and Insurance, Damanhur University, Egypt Email: elbielym@gmail.com Abstract: In this work, we introduce a new flexible extension of the Inverse Weibull distribution. Some important mathematical properties of the proposed model are derived along with a numerical analysis of mean, variance, skewness and kurtosis measures of the proposed model. The performance of the maximum likelihood method is assesses via a comprehensive simulation studies in terms of mean squared errors. The new model is better than some other important competitive extensions of the Inverse Weibull in modeling the breaking stress data, the glass fibers data and the relief time data. Some plots are given to illustrate the suitability of the new model to fit the used data sets.


Introduction
A Random Variable (RV) T is said to have the Inverse Weibull (IW) distribution (Gusmao et al., 2011) if its Probability Density Function (PDF) and Cumulative Distribution Function (CDF) are given by: (2) respectively, where a > 0 is a scale parameter and c, b > 0 are a shape parameters, respectively. For c = 1, we get the standard IW model. For b = 2 in (2), we get the Generalized Inverse Rayleigh model (GIR). For a = 1 in (2), we get the generalized Inverse Exponential model (GIEx). For c = 1 and b = 2 in (2), we get the IR model. For c = a = 1 we get the IEx model.
Recently, Cordeiro et al. (2016) proposed a new class of distributions called the generalized odd log-logistic-G (GOLL-G) family with two extra shape parameters. For an arbitrary baseline CDF ( ) G x Φ and Φ the parameters vector, the CDF of the GOLL-G family is given by: The PDF corresponding to (3) is given by: Here, we define the generalized odd log-logistic Inverse Weibull (GOLLIW) model and provide some plots of its PDF and hazard rate function (HRF) [h (α,θ,c,a,b) (x)]. The GOLLIW CDF is given by: The PDF corresponding to (5) is given by: The Hazard Rate Function (HRF) for the new model can be get from f (α,θ,c,a,b) Table 1 provide the sub-models of the GOLLIW model. As illustrated in Table 1, the new model generalizes nineteen sub-model, ten of them are quite new.
The density of the proposed model can be right skewed and symmetric (see left panel in Fig. 1). The HRF of the proposed model can be only upside down (see right panel in Fig. 1 and 3 to 5).
For simulation of this new model, we obtain the Quantile Function (QF) of X (by inverting (5)), say x u = Q(u) = F -1 (u), as:  IEx Keller and Kamath (1982) We used (7) for simulating the new model (see Section 4).

Useful Representations
Following Cordeiro et al. (2016) and after some algebra, the PDF (6) of X can be expressed as:    and shape parameter b. So, the new density (6) can be expressed as a double linear mixture of the IW density. Then, several of its structural properties can be obtained from Equation (8) and those properties of the IW model. By integrating Equation (8), the CDF of X becomes:

Moments and Incomplete Moments
The r th ordinary moment of X is given by: where: Setting r = 1, 2, 3 and 4 in (10), we have: ( ) We can obtain variance Var(X) skewness (Ske(X)) and kurtosis (Ku(X)) measures using the well known relationships.
Numerical Analysis for the E(X), Var(X), Ske(X) and Ku(X) Measures Numerical analysis for the E(X), Var(X), Ske(X) and Ku(X) are calculated in Table 2 using (10) and wellknown relationships for some selected values of parameter α, θ, c, a and b using the R software. Based on Table 2 we note that: 1-The skewness of the proposed model is always positive. 2-The kurtosis of the proposed model can be only more than 3. 3-The parameter θ has no effect on the Ske(X) and Ku(X). As illustated in Table 2, skewness = 3.3797 and kurtosis = 74.56 for all values of parameters. 4-The parameter c has no effect on the Ske(X) and Ku(X). As illustated in Table 2, skewness = 2.5307 and kurtosis = 31.1798 for all values of parameters. 5-The parameter a has no effect on the Ske(X) and Ku(X). As illustated in Table 2, skewness = 1.555626 and kurtosis = 9.54016 for all values of parameters. 6-The mean of the proposed model increases as α and b decreases. 7-The mean of the proposed model decreases as θ and a decreases.
The rth incomplete moment, say ϕ r (t), of X can be expressed, from (8), as: where, γ(κ, q) is the incomplete gamma function, where: is a confluent hypergeometric function. The first incomplete moment given by (11) with r = 1 is: The MGF M X (t) = E (e tX ) of X can be derived from Equation (8) We also can determine the generating function of π (c = 1,a,b) (t) in (1) by setting x −1 = y, we can write this MGF as: By expanding the first exponential and calculating the integral, we have: Consider the Wright generalized hypergeometric function (Wright (1935)) defined by: Then, we can write M(t, α, β) as: Combining expressions (10) and (12), we obtain the MGF of X, say M(t), as:

Residual Life and Reversed Residual Life Functions
The n th moment of the residual life is given by: the n th moment of the residual life of X is given by: Let x 1 ,…, x n be a random sample from the GOLLIW distribution with parameters α, θ, c, a and b. Let Θ = (α, θ, c, a, b) T be the 5×1 parameter vector. For determining the Maximum Likelihood Estimations (MLEs) of Θ, we have the log-likelihood function: The components of the score vector, U(Θ) = To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ. For interval estimation of the parameters, we obtain the 5×5 observed information matrix: Whose elements can be computed numerically.

Real Data Modeling
In this section, we provide three applications to real data sets to illustrate the importance of the GOLLIW distribution. To evaluate performance of considered model, the MLEs of the parameters for the considered models are calculated and three goodness-of-fit statistics are used to compare the new distribution. The measures of goodness of fit including anderson-Darling (A ★ ), Cramérvon Mises (W ★ ) and Kolmogrov-Smirnov (K-S) statistics are computed to compare the fitted models. The statistics A ★ and W ★ are described in details in Chen and Balakrishnan (1995). In general, the smaller are the values of these statistics, the better the fit to the data. The required computations are carried out in the R-language for the first three application. The numerical values of the model selection statistics A ★ , W ★ and K-S are listed in Tables 4, 6 and 8. Tables 5, 7 and 9 list the MLEs and their corresponding standard errors (in parentheses) of the model parameters. The Total Time Test (TTT) plots for the three data set indicates that the HRFs are increasing, increacing and increasing.
The statistics of the fitted models for the 1st data set are presented in Table 4 and the MLEs and corresponding standard errors are given in Table 5. We note from the figures in Table 4 that the GOLLIW model has the lowest values of the A ★ , W ★ and K-S statistics (for the 1st data set) as compared to their submodels, suggesting that the GOLLIW model provide the best fit.
The histogram of the 1st data and other important plots are displayed in Fig. 2.
We compare the fits of the GOLLIW distribution with other models such as Inverse Weibull (  The parameters of the above densities are all positive real numbers except for the TIW distribution for which |α|≤1.

Breaking Stress Data
The 1st data set is an uncensored data set consisting of 100 observations on breaking stress of carbon fibers (in Gba) given by Nichols and Padgett (2006) and these data are used by Mahmoud and Mandouh (2013) to fit the transmuted IW distribution. The statistics of the fitted models are presented in Table 4 and the MLEs and corresponding standard errors are given in Table 5. We note from Table 4  K-S statistics (for the 1st data set) as compared to further models and therefore the new one can be chosen as the best one. The histogram of the 1st data and other important plots are displayed in Fig. 3.

Glass Fibers Data
The 2nd data set is generated data to simulate the strengths of glass fibers which was given by Smith and Naylor (1987).
The statistics of the fitted models are presented in Table 6 and the MLEs and corresponding standard errors are given in Table 7. It is clear from Table 6 that the GOLLIW gives the lowest values the A ★ , W ★ and K-S statistics (for the 2nd data set) as compared to other models and therefore our new model can be chosen as the best one. The histogram of the 2nd data are displayed in Fig. 3. We compare the fits of the GOLLIW distribution with other IW models such as KIW, BIW, EIW, IW, TIW, MOIW and McIW.

Relief Time Data
The 3rd data set (Wingo data) represents a complete sample from a clinical trial describe a relief time (in hours) for 50 arthritic patients (Wingo (1983)).
We will compare the fits of the GOLLIW distribution with other models such as IW, EIW and TIW. The statistics of the fitted models are presented in Table 8 and the MLEs and corresponding standard errors are given in Table 9. It is clear from Table 8 that the GOLLIW gives the lowest values of the A ★ , W ★ and K-S statistics (for the 3rd data set) as compared to other models and therefore the new model can be chosen as the best one. The histogram of the 3rd data ar displayed in Fig. 4.

Concluding Remarks
We introduce a new flexible extension of the Inverse Weibull distribution. Some important mathematical properties of the proposed model are derived along with a numerical analysis of mean, variance, skewness and kurtosis measures of the proposed model. The performance of the maximum likelihood method is assesses via a comprehensive simulation studies in terms of mean squared errors. The new model is better than some other important competitive extensions of the Inverse Weibull in modeling the breaking stress data, the glass fibers data and the relief time data. Some plots such as estimated PDF, estimated CDF, P-P plot, estimated HRF and TTT are given to illustrate the suitability of the new model to fit the used data sets.
Based the numerical analysis of mean, variance, skewness and kurtosis measures we summarize the following concluding remakes: 1-The skewness of the proposed model is always positive. 2-The kurtosis of the proposed can be less than 3 and more than 3 3-The parameter θ has no effect on the skewness and kurtosis: As illustated in Table 2, skewness = 3.3797 and kurtosis = 74.56 for all values of parameters 4-The parameter c has no effect on the Ske(X) and kurtosis: As illustated in Table 2, skewness = 2.5307 and kurtosis = 31.1798 for all values of parameters 5-The mean of the proposed model increases as α and b decreases 6-The mean of the proposed model decreases as θ and a decreases