Some Theoretical and Computational Aspects of the Inverse Generalized Power Weibull Distribution

This paper introduces a new three-parameter distribution called inverse generalized power Weibull distribution. This distribution can be regarded as a reciprocal of the generalized power Weibull distribution. The new distribution is characterized by being a general formula for some well-known distributions, namely inverse Weibull, inverse exponential, inverse Rayleigh and inverse Nadarajah-Haghighi distributions. Some of the mathematical properties of the new distribution including the quantile, density, cumulative distribution functions, moments, moments generating function and order statistics are derived. The model parameters are estimated using the maximum likelihood method. The Monte Carlo simulation study is used to assess the performance of the maximum likelihood estimators in terms of mean squared errors. Two real datasets are used to demonstrate the flexibility of the new distribution as well as to demonstrate its applicability.


Introduction
Weibull distribution is one of the most important distributions used in reliability engineering and other disciplines. Also, it adequately describes the observed failure times of many different types of components and phenomena. Therefore, the Weibull distribution was more widely used as a basis for several generalizations. See, for example, the exponentiated Weibull distribution by Mudholkar and Srivastava (1993), beta-Weibull distribution by Lee et al. (2007) and Kumaraswamy Weibull distribution by Cordeiro et al. (2010). For more detail on the generalizations of Weibull distribution, refer to the books by Murthy et al. (2004), Rinne (2008) and Lai (2014). Also, Haghighi and Nikulin (2006)  (2) where α and θ are two shape parameters and λ is a scale parameter. The Weibull distribution is a special case of (1) when α = 1. Its hazard rate function according to Nikulin and Haghighi (2009) can be constant, monotone, unimodal, bathtub-shaped. In the literature, some extensions of the generalized power Weibull distribution proposed by many authors, such as Selim and Badr (2016) proposed the Kumaraswamy generalized power Weibull distribution, Selim (2018) proposed the generalized power generalized Weibull distribution, Khan (2018) proposed the transmuted generalized power Weibull distribution and Pena-Ramirez et al. This paper aims to introduce a reciprocal of the generalized power Weibull distribution named inverse generalized power Weibull (IGPW) distribution and studies its mathematical properties. The motivations for deriving the inverse generalized power Weibull distribution are to provide more usefulness and flexibility of the ordinary distribution and to improve its goodness-of-fit in comparison with the well-known distributions in lifetime data analysis.
The rest of this paper is organized as follows. The inverse generalized power Weibull distribution and the special cases thereof are introduced in Section 2. Some of the mathematical properties of IGPW distribution are derived in Section 3, including the quantile function, skewness, kurtosis, ordinary moments, moment generating function and order statistics. The maximum likelihood estimation of the model parameters is introduced in Section 4. In Section 5, the Monte Carlo simulation study is used to assess the performance of the maximum likelihood estimators in terms of mean squared errors. Two real data sets are used to illustrate the usefulness of the IGPW distribution in Section 6. The final Section is devoted to the conclusion.

Inverse Generalized Power Weibull Distribution
The inverse generalized power Weibull distribution can be derived using the transformation = 1⁄ , whereupon if the random variable follows the GPW distribution, the random variable follows the IGPW distribution. The cdf and pdf of IGPW distribution are given, respectively, by where is scale parameter and α, are shape parameters. This model has inverse Weibull (IW) distribution as a special case when = 1. Hence, it can also be considered as an extension of the inverse exponential distribution which is developed by  when = θ = 1. The graphs of the pdf and cdf for selected values of the model parameters are plotted in Fig.  1.
The survival ( ) and the hazard rate ℎ( ) functions of the distribution are given, respectively by

Special cases of the distribution
A number of the important distributions can be obtained as special cases of the distribution, are specifically inverse Weibull ( ) , inverse exponential ( ) , inverse Nadarajah-Haghighi ( ) and inverse Rayleigh ( ) distributions. The special cases of distribution for selected values of the parameters ( , ) are listed in Table 1.

The Statistical Properties
In this section, some of the statistical properties of IGPW distribution including the quantile function, random variables generation function, moments, moment generating function, skewness, kurtosis and order statistics are derived.

Quantile function and simulation
The quantile function has a number of important applications, for example, it can be used to obtain the median, skewness, kurtosis and can be also used to generate random variables. The q-th quantile is a solution of the following equation ( ) = , 0 ≤ ≤ 1. The random variables of IGPW distribution can be simulated using equation (9) as following where u ~ the uniform (0, 1) distribution and ~( , , ).

Skewness and kurtosis
The shortcomings of the classical skewness and kurtosis measures can be avoided by using the skewness and kurtosis measures based on quantiles like Bowley's skewness and Moors' kurtosis. The Bowley's skewness measure based on quartiles ((Kenney and Keeping 1962)) is given by and the Moors' kurtosis measure based on octiles (Moors (1988)) is given by The Fig. 3, shows the behaviors of median, skewness and kurtosis of the IGPW distribution as a function of the parameters α and θ.

Moments and moment generating function
The moments and moment generating function of the IGPW distribution are given by the following theorems: Theorem 1. If has the IGPW distribution, then the th moments of for integer value of −1 is where Γ(a, b) denotes the upper incomplete gamma function and is Euler's number.
Proof. The ℎ moment of is defined as follows Let = (1 + ) , the above expression reduce to Then, by applying the binomial expansion of By integrating the incomplete gamma function in (18) we get the r th moment of as follows If α = θ = 1, we get the moments of inverse exponential distribution as follows And if θ = 1, we get the moments of inverse Weibull distribution as follows

1
Theorem 2. If ~ distribution, then for any integer value of −1 , the moment generating function is Proof. The moment generating function is defined as follows , we get By inserting Eq. (14) in Eq. (20), yields the moment generating function of IGPW distribution as in (19).

Order statistics
Assuming that (1) , (2) , … , ( ) are the order statistics of a random sample follows a continuous distribution with cdf ( ) and pdf ( ), then the pdf of ( ) is given by Let is a random variable of IGPW distribution, then the density function of the k-th order statistics of the IGPW distribution is If = 1, the pdf of order statistics is and if = n, the pdf of order statistics is

Maximum Likelihood Estimation
This section is devoted to discussing the maximum likelihood estimation ( ) and the approximate confidence intervals for the unknown parameters of IGPW distribution. Let These nonlinear equations cannot be analytically solved, but the statistical software like R program (Team (2015)) can be used to solve them numerically using iterative techniques.
The asymptotic variance-covariance matrix of the MLEs for the three parameters , and is the inverse of the observed Fisher information matrix as follows The elements of the sample Fisher information matrix can be obtained by deriving the second derivatives of the log-likelihood function (26) and evaluating them at the MLEs ( (Cohen 1965)). These elements can be derived as follow The asymptotic normality of the MLE can be used to compute the approximate confidence intervals for the parameters α, and θ as follow where z τ/2 is an upper (τ /2)100% of the standard normal distribution.

Simulation Study
In this section, the simulation study is executed to assess the performance of the proposed MLE method for estimating the parameters of IGPW distribution. Monte Carlo experiments were carried out based on generated data from IGPW distribution. By using the inversion method in Section 3.1, We generated 1000 samples of size n = 20, 50, 100 from IGPW distribution for different combinations of parameters α, λ and θ. The mean square errors (MSE) of the MLEs were computed using the "CG" optimization' method in R program. The simulation results were displayed in Table 2. The main conclusion from the figures in Table 2, is that the mean square errors of MLEs decrease with increasing the sample size. This indicates that the MLE method is suitable for estimating the unknown parameters of IGPW distribution.

Real Data Illustration
This section illustrates the usefulness of the IGPW distribution using two real datasets. These datasets are described as follows: The

The data set (II): Remission times data
The second data set represents the remission times (in months) of a random sample of 128 bladder cancer patients (see Lee and Wang (2003) Table 3. The fitted models were compared by using Cramér-von Mises (W * ), Anderson Darling (A * ), Kolmogorov-Smirnov ( − ), -Log-likelihood (−lnL) , Akaike Information Criterion (AIC) , Consistent Akaike Information Criterion ( ) , Bayesian Information Criterion (BIC) and Hannan-Quinn Information Criterion (HQIC). Based on these criteria, the best model is the one that achieves the lowest values for the information criteria and goodness-of-fit statistics. Hence, it is clear from the numerical results in Table 4, that the IGPW model provides a better fit than the other competing models. The Figures 4 and 5 display the graphical comparison of the fitted models for datasets I and II, respectively. Also, these figures graphically illustrate that IGPW distribution provides the best fit to our data sets, as compared to the other considered models. Therefore, the IGPW model can be used as a possible alternative to the well-known models like inverse exponential and inverse Weibull models.

Conclusion
This paper introduces a new three-parameter distribution, called the inverse generalized power Weibull distribution. This distribution is considered as a reciprocal of the generalized power Weibull distribution and a generalization of inverse Weibull distribution. Some of the statistical properties of the inverted generalized power Weibull distribution, including the moments, hazard rate function, quantile function and order statistics are derived. The maximum likelihood method is used to estimate the model parameters. The performances of the maximum likelihood estimators are assessed in terms of mean squared errors using Monte Carlo simulation. The practical applications have established that the proposed distribution is quite useful for dealing with reliability data and behaves better than its four special cases (inverse Weibull, inverse exponential, inverse Rayleigh and inverse Nadarajah-Haghighi distributions).