Modified Extended Inverted Weibull Distribution with Application to Neck Cancer Data

This work introduces a new three-parameter modified extended inverted Weibull (MEIW) distribution which is a hybrid of the one-parameter inverted Weibull distribution. The density function of the MEIW can be expressed as a linear combination of the inverted Weibull densities. Some mathematical properties of the proposed MEIW model such as ordinary and incomplete moments, mean residual life


Introduction
The Inverted Weibull distribution (IW) distribution is also known as one-parameter inverse Weibull distribution, which can be applied in biological and reliability studies with some monotone failure rates. Several modifications of the IW distribution have been studied, which includes: Flaih et al. (2012) studied the properties of Exponentiated Inverted Weibull distribution and applied it to model the breaking strength of carbon fibers; Ogunde et al. (2017aOgunde et al. ( , 2017b investigated the properties of the transmuted inverted Weibull distribution and Exponentiated transmuted inverted Weibull respectively; Ogunde et al. (2020) proposed and studied the properties of Alpha Power Extended Inverted Weibull Distribution.
The cumulative distribution function (CDF) of the IW distribution is given by Its corresponding probability density function (PDF) has the following form: The main motivation of this paper is to extend the standard inverted Weibull distribution to the standard modified extended inverted Weibull distribution by adding two extra shape parameters; the shape parameter are to be address the lack of fit of the inverted Weibull distribution for modeling lifetime data which exhibited non-monotone failure rates.

Modified Extended Inverted Weibull Model
The inverted Weibull (IW) distribution is extended using the Exponentiated-G family of distributions given by Cordeiro et al. (1998), to obtain Where and are positive shape parameters. The associated PDF is given by Suppose that the random variable X has the Extended Inverted Weibull distribution where its CDF and PDF are given in equations (3) and (4). Given N, let 1 , … , be independent and identically distributed random variables with Extended Inverted Weibull distribution. Let be distributed according to the zero truncated Poisson distribution by Cohen (1960) PDF which is the modified exponentiated Inverted Weibull distribution denoted by MEIW( , , ) is defined by the marginal cdf of X, that is, (1 − − ) , > 0; , , > 0 (6) The density function is given by The expression for the survival and the hazard function is, respectively, given by and   The MEIW distribution is a flexible model that contains some distributions as special models: (i) If = 1, then the MEIW distribution reduces to the Extended Inverted Weibull (EIW) distribution.

Linear Representation
Here, we present the MEIW PDF as a mixture linear representation of IW density. Consider the following power series: Using equation (11) to simplify [− (1−(1− − − ) )] , the resulting equation can be written as It should be noted that (12) is an Exponentiated Generalized Inverted Weibull distribution with parameters and ( + 1) Further, applying (10) in (12), after some algebra, the MEIW distribution PDF reduces to Where ( ; ( + 1), ) is the IW PDF with parameters ( + 1) and . This implies that (13) is a linear combination of the IW densities and some mathematical properties related to MEIW distribution can be obtained from those of the IW distribution.
Consider a random variable (RV) ~( ) in (1). For < , the ℎ ordinary and incomplete moments of are given by ,

Ordinary Moments of MEIW Distribution
The ℎ ordinary moment of RV is The coefficient of variation ( )), Skewness ( ), and kurtosis ( ) can easily be obtained. The variance ( 2 ), coefficient of variation ( ), coefficient of skewness ( ) and coefficient of kurtosis ( ) are given by respectively. Table 1 present the moments of MEIW distribution with some selected values of the parameters and for a fixed value of = 6.5.  (16), we obtain an expression for the mean of .
Thus, the ℎ incomplete moment of RV is Additionally, we obtain an expression for the ℎ incomplete moment of MEIW distribution given by An expression for the incomplete moment of MEIW distribution given in (19) can be used to obtain an expression for Bonferroni and Lorenz curves which have applications in medicine, insurance, reliability, demography, and insurance, which is also useful in economics for the study of income and poverty analysis

Mean Residual Life and Mean Waiting Time
The mean residual life (MRL) can be applied in economics, biomedical sciences, product quality control, insurance, and demography. It can be define as the expected additional life length for a unit that is alive at age , and it is represented mathematically by The MRL of can be obtained by using the formula: Where ( ) is the survival function of . By inserting (19) in (20), the value of the MEIW distribution is given as The mean inactivity time (MIT) (mean waiting time) is defined by ( ) = ( / ≤ ), > 0, and it can be obtained by the formula: By substituting (19) in (22), the MIT of the MEIW distribution is given as

Order Statistics
Consider a random sample from the MEIW ( , , ) denoted by of size and have the following order statistics denoted by 1: < 2: <. . . < : . Then, the PDF of the ℎ order statistics is given by Considering + −1 ( ) ( ) and further applying Taylor series given in (10), we have Combining (24) and (25), we have an expression for the ℎ order statistics of MEIW distribution as

Entropy
Measure of entropy plays a vital role in reliability analysis. It is used to determine the amount of variation and uncertainty in the dataset. If the value of entropy is large, it indicates large uncertainty in the data. Hence, for measuring the amount of uncertainty of a random variable x following the MEIW distribution, Renyi (1960) and Havarda and Charvat (1967) entropies are considered. The entropies are as follows: where ( ) = ∑(−1) + + ( ) ( Where ( ) = ∑(−1) + + ( ) ( Proof. By definition, Renyi and q-entropy are: Then By applying equations (10) and (11) in the equation above, we have The q-entropy, introduced by Havarda and Charvat (1967), is defined as Moreover, by applying equations (10) and (11) in the equations above, we have

Stochastic Ordering
The MEIW distribution is ordered with respect to the strongest "likelihood ratio" ordering as demonstrated in the following theorem.

Stress-Strength Reliability
In engineering practice, it is of common interest that the life of a component is subjected to a random stress. The random strength can be modeled by a random variable 1 and the random stress can be modeled by a random variable 2 . The probability that the component functions satisfactorily is given by = ( 1 < 2 ) and is regarded as a measure of component reliability in several applications. Let 1 be a random variable following the MEIW distribution with parameters , 1 and 1 with CDF 1 ( ) and 2 be a random variable following the MEIW distribution with parameters , 2 and 2 with PDF 1 ( ). Then an expression for Stress-strength reliability of MEIW distribution can be obtained using the relation given by = ( 1 < 2 ) = ∫ 1 ( ; , 1 , 1 ) 2 ( ; , 2 , 2 ) ∞ −∞ (35) Using (10) and (11) in (36) and subsequently carrying out a simple algebraic manipulation, we obtain an expression for stress-strength reliability for MEIW distribution as

Maximum Likelihood
This method is the most widely used method for parameter estimation (Casella and Berger, 1990), because it has several flexible properties including asymptotic efficiency, consistency, and invariance property, and many others. Suppose ( 1 , 2 … , ) is a random sample of size n from MEIW( ), where = ( , , ). Then, the likelihood function for is of the form

=1
Correspondingly, the log-likelihood function, [ ( )] = is given as The maximum likelihood estimators of ̂, ̂ and ̂ of the parameters , , and can be obtained numerically by optimizing the log-likelihood function in (39). Alternatively, the log-likelihood equation in (39) For a given set of observations, the matrix given in equation (44) is obtained after convergence of the Newton-Raphson procedure in R or MATLAB software.

Asymptotic Confidence Intervals
The asymptotic confidence intervals for the parameters of the MEIW distribution are presented. Similar results can be obtained for any other models under the same class of distributions. Numerically, expectations in the Fisher Information Matrix (FIM) can be obtained. Let ∆= (̂,̂,̂) be the maximum likelihood estimate of ∆ = ( , , ). Under satisfying conditions for parameters in the interior parameter space, but not on the boundary, we have: √ ∆ − ∆ → 3 (0, −1 (∆)) with (∆) as the expected Fisher information matrix. Replacing the expected Fisher information matrix with the observed information matrix, the asymptotic behavior remains valid. The multivariate normal distribution 3 (0, (∆) −1 ) with mean vector 0 = (0, 0, 0) , can be used to construct the confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions. That is, the approximate 100(1 − )% two-sided confidence intervals for , , and are given by: respectively, where −1 (∆), −1 (∆) and −1 are the diagonal elements of −1 (∆) and 2 ⁄ is the upper 2 ⁄ ℎ percentile of the distribution of the standard normal.

The Likelihood Ratio Test
The likelihood ratio (LR) test statistic can be used to compare the MEIW distribution with its sub models. The unrestricted estimates, ̂,̂, ̂ and restricted estimates ̆,̆ and ̆ can be computed to construct the LR statistics for testing hypotheses concerning the sub models of the MEIW distribution. For example, to test = 1, (MEIW against EIW) the LR statistic reduces to = 2[ ( ( ̂,̂,̂ )) − ( ( ̆,̆, 1))], and the LR test rejects the null hypothesis, when > χ 2 , where χ 2 denote the upper 100 % point of the χ 2 distribution with 1 degrees of freedom.

Applications
In this section, we present an application of the Modified Exponentiated Inverted Weibull distribution on real data set. We shall compare the fit of the MEIW with the Exponentiated Inverted Weibull (EIW), Exponentiated Frechet ( The real data represents the survival times of patients suffering from neck cancer disease. The patients in this group were treated with combined radiotherapy and chemotherapy (CT & RT  Kumar et al. (2015) fitted this data to the inverse Lindley distribution, Saeed and Ali (2020) fitted the data set to the Sinh Inverted Exponential distribution. The Exploratory data analysis for the neck cancer data is given in Table 3 which shows that the data is over-dispersed with an excess kurtosis of 1.87 (leptokurtic). The results of the estimated values of the parameters (2*Log-likelihood, AIC, BIC, CAIC, and HQIC) are listed in Table 4. The Total time on Test (TTT) which shows that the neck cancer data exhibits non-monotone failure rate and the empirical density plot which clearly indicates that the data is moderately positively skewed is given in Figure 5 , the fitted PDF and estimated CDF of the MEIW curve to this data are given in Figures 6. The selection criterion is that the lowest 2*Log-likelihood and AIC correspond to the best model fitted. The MLEs, AIC, BIC, CAIC and HQIC are shown in Table 4. From the Table, we can observe that the MEIW model shows the smallest 2*Log-likelihood, AIC, BIC, CAIC and HQIC than other competing distributions.
Diagram I Diagram II  The LR statistic was obtained for testing the hypotheses 0 : =1 1 = 0 e, that is to compare the MEIW model with the EIW model. The LR statistic = 2{655.717-523.512} = 132.205( − < 0.01), sufficing to show that the model is a better model that can be used in fitting the data. Figure 5. Plots of estimated CDF and histogram fitted PDF of the fitted models for the neck cancer disease data from left to right

Conclusion
In this study, a three-parameter model called Modified Exponentiated Inverted Weibull distribution is proposed and studied. Various properties of MEIW model are derived. Maximum likelihood estimation method is used to estimate the parameters of the model. The application of MEIW model is demonstrated by using a neck cancer data and it provides a reasonable good parametric fit to the data set than some other competitive models considered in this work because the MEIW model contains the minimum information loss as a result of its smallest AIC, BIC CAIC and HQIC values.