A New Cubic Transmuted Inverse Weibull Distribution: Theory and Applications

This paper introduces a new cubic transmutation of the inverse Weibull distribution, known as a cubic transmuted inverse Weibull distribution. The model is thought to be useful for the analysis of complex life data, modeling failure times, accessing product reliability, and many other fields like economics, hydrology, biology


Introduction
The probability distributions have been in use for modeling several real-world phenomena.However, it frequently occurs that some real-world issues cannot be adequately modeled by the standard probability models and hence some extension or generalization is required.The generalization of the probability distribution is usually done to gain more flexibility in modeling of the data and to use them in more challenging real-world issues.In this paper, we have expanded the inverse Weibull (IW) distribution so that the resulting distribution will enable us to handle more complicated real-world problems.The inverse Weibull distribution has been considered as a useful probability distribution for modeling dependability data and in characterizing degradation phenomena of mechanical components.The distribution was initially discussed by Keller et al. (1982).The cumulative distribution function (cdf) of the IW distribution is where 0   is the shape parameter and 0   is the rate parameter.
The IW distribution has both increasing and decreasing hazard rates depending on the shape parameter.The inverse exponential distribution appears as a special case of the IW distribution for 1   .Calabria and Pulcini (1990) have estimated the parameters of the IW distribution using maximum likelihood and least square estimation techniques.Mahmoud et al. (2003) have obtained order statistics from the inverse Weibull distribution.Sultan (2008) has derived

Pakistan Journal of Statistics and Operation Research
Bayes estimates of the parameters by using different loss functions.Kundu and Howaldar (2010) have conducted the Bayesian inference and prediction for the IW based on Type-II censored data.De Gusmao (2011) has introduced a generalized inverse Weibull distribution.Aryal and Tsokos (2011) have introduced transmuted Weibull distribution and have also discussed its distributional properties.Khan and King (2012) have introduced a five-parameter generalized IW distribution.Khan and King (2013) have proposed a transmuted modified Weibull distribution.Khan and King (2014) have also proposed a three-parameter transmuted IW distribution.Recently some researchers have also conducted some other research using the IW distribution.These include Alslman and Helu (2022) where where This study aims to use the family of distributions given in (3) and to generate a new cubic transmuted inverse Weibull (CTIW) distribution.The paper also aims to see the effect of the model parameters on the shape and hazard rate function of the distribution.The structure of the paper follows.A new CTIW distribution is proposed in Section 2. Statistical properties including the moments, generating functions, quantile functions, random number generation, and reliability function for the proposed CTIW distribution are given in Section 3. The distribution of order statistics from the proposed CTIW distribution are given in Section 4. Section 5 contains parameter estimation for the CTIW distribution.Section 6 is based upon the consistency of the estimation method on the basis of extensive simulation study.Section 7 is based upon some real data application of the proposed CTIW distribution.Some concluding remarks are given in Section 8.

A New Cubic Transmuted Inverse Weibull Distribution
A random variable X is said to have an inverse Weibull (IW) distribution if it has cdf as given in (1).The density function corresponding to (1) is given as where 0   is the shape parameter and 0   is the rate parameter.Khan and King (2014) have introduced a transmuted inverse Weibull (TIW) distribution by using (1) in (2).The cdf of TIW distribution is    is the transmutation parameter.A new cubic transmuted inverse Weibull (CTIW) is obtained by using (1) in (3).The cdf of the proposed CTIW distribution is ; , , e 1 e 3 2e ; where 0   is the shape parameter, 0   is the rate parameter, and is the transmutation parameter.The probability density function (pdf) of CTIW distribution is readily written, from (5), as , , e 1 6 e 1 e ; It is easy to see that the CTIW distribution reduces to the inverse Weibull distribution for 0  The plot of the density function shows that the distribution is unimodal.We will, now, discuss some useful distributional properties of the proposed CTIW distribution.

Distributional Properties
In this section, we have discussed some useful properties of CTIW distribution.These properties are discussed in the following sub-sections.

The Moments
The moments are very important in studying certain properties of a distribution.The rth moment of a random variable X is obtained as The rth moment for CTIW distribution is given in the following Theorem.

Theorem 1:
The rth moment of a random variable X having a CTIW distribution is given as Proof: The rth moment of CTIW distribution is given as Solving the integral we have (7) and this completes the proof.
The mean of the distribution is easily obtained by using r = 1 in ( 7) and is   The variance of the CTIW distribution can be easily obtained by using  is obtained by using r = 2 in (7).The coefficients of skewness and kurtosis can also be obtained by computing higher order moments from (7).

The Moment Generating Function
The moment generating function of a random variable is very useful in computing its moments.The moment generating function is defined as The moment generating function of the proposed CTIW distribution is given in the following theorem.

Theorem 2:
The moment generating function of CTIW distribution is Proof: The proof is simple.
A more useful function is the characteristic function and is defined as The characteristic function of CTIW distribution is readily written from (8) as

Quantile Function and Median
In this section, we will derive the quantile function of CTIW distribution.The quantile function of a random variable X, having cdf   F x , is obtained as a solution of   F x p  for x.Now, for CTIW distribution the quantile function is obtained as a solution of for x.Writing   e x w     , the above equation can be written as 0 where 2 ; 3 and The real solution of the above cubic equation is (10), and solving for x, the quantile function of CTIW distribution is where and The quantiles of CTIW can be obtained by using 0 1 p   in (11).Specifically, the median can be obtained by using p = 0.5 in (11) and then solving for x.It is to be noted that for p = 0.5, the quantity 2 0   and 3  reduces to

Random Number Generation
The random data can be generated from CTIW distribution on the lines of the quantiles.Specifically, a random observation can be generated from any distribution, with cdf   F x , by solving   . Now, the random observation from CTIW distribution can be generated by solving for x where u is a random observation from   0,1 U .Specifically, a random observation can be generated by using (11) where p is replaced by u, a random observation from   0,1 U .

Reliability Analysis
The reliability function is a function of time that gives the probability of an item operating for a certain period without failure.It is the complement of a distribution function.The reliability function for the proposed CTIW distribution is given as ; The hazard rate function   h t , also known as the force of mortality rate or failure rate.The hazard rate is the ratio of the density function to the reliability function.The hazard rate function for the CTIW distribution is obtained as the ratio of ( 6) to (12) and is The plots of reliability and hazard rate functions, for different values of the parameters, are given in Figure 2 below

Order Statistics
Order statistics is a useful branch of statistics and helps study the behavior of extremes in random sampling from some probability distribution.In this section we will discuss the distribution of order statistics when a random sample is available from CTIW distribution.Suppose that 1 2 , , , n X X X … is a random sample of size n from some distribution, then 1: are the order statistics.The distribution of rth order statistics is Now, using the density and distribution functions of CTIW distribution, the distribution of rth order statistics is for x > 0 and 1, 2, , r n  … .The distribution of the smallest and largest observations can be easily obtained by using r = 1 and r = n, respectively in (14).It is to be noted that for 0   , in (14), we can obtain the distribution of the rth order statistics from the inverse Weibull distribution.
The moments of order statistics from CTIW distribution can be numerically computed from (14) 1 above contains the mean of order statistics for different values of n, r, and λ.From this table we can see that for fixed n and λ, the mean of order statistics increases with an increase in r.Further, for fixed r and λ, the mean of order statistics decreases with an increase in n.We can also see, from Table 1, that for fixed n and r, the mean increases with an increase in λ when 2 r n  and decreases with an increase in λ for 2 r n  .The variance also shows a trend similar to the mean except for r = 1 when n and λ are fixed.

Estimation of the Model Parameters
In this section, we have discussed the maximum likelihood estimation of the parameters of CTIW distribution.Suppose that a random sample of n observations is available from the CTIW distribution.The likelihood function is then The log-likelihood function is , , ; ln ln 1 ln ln 1 6 e 1 e .
The maximum likelihood estimates of the parameters are obtained by differentiating (15) with respect to the unknown parameters, equating the derivatives to zero, and simultaneously solving the resulting equations.Now, the derivatives of the log-likelihood function with respect to the unknown parameters are The maximum likelihood estimators of ,   and  can be obtained by equating the derivatives in ( 16)-( 18 , where entries of the variance-covariance matrix can be obtained by inverting the Fisher information matrix; whose entries are given as

Simulation Study
In this section, we have presented a simulation study to see the performance of the estimates.The simulation study is conducted by drawing random samples of sizes 50, 100, 200, 500, and 1000 from CTIW distribution with 1   ,

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  and 1    .For each generated sample of a specific size, the maximum likelihood estimates of the unknown parameters are computed; by using the "bbmle" package of R, see Bolker and Bolker (2017); and then the average estimate is computed by using The results of the simulation study are given in Table 3 below The results of Table 3 show that the estimates converges to the population parameters as the sample size increases.We can also see, from the above table, that the mean square error of the estimates decreases with an increase in the sample size.

Real Data Applications
In this section, we have used some real data sets to see the suitability of the proposed CTIW distribution.We have used the following five data sets for the analysis  6.The results of this table also indicate that the CTIW is the best fit for all five data sets.We have, also, constructed the plots of empirical distribution functions for the five data sets alongside the fitted distribution functions for different distributions.These plots are given in Figure 3 below.The figures also indicate that the CTIW distribution is the best fit for all of the data sets.real data sets that the proposed model is a better fit than other competitive models.We hope that the proposed CTIW distribution may be highly competitive in real-life data sets and will be extensively used in different areas of statistics.
is the transmutation parameter and   G x is the distribution function of any standard probability model.Rahman et al. (2019) have proposed a cubic-transmuted family of distributions and have used the proposed family to introduce a cubic-transmuted uniform (CTU) distribution.The CTU distribution can be used as an alternative to the Beta and Kumaraswamy distributions.Many distributional models have already been generated from this family of distribution and have shown better performance than other competing distributions in handling complex datasets.The cdf of the cubic transmuted family of distributions, proposed by Rahman et al. (2019), is cubic transmuted inverse exponential distribution appears as a special case of CTIW distribution for 1   , Also, for 2   , a cubic transmuted inverse Rayleigh distribution appears as a special case of CTIW distribution.Some plots of the pdf and cdf of CTIW distribution for different values of the parameters are given in Figure-1 below.

Figure 1 :
Figure 1: The Density and Distribution Functions of Cubic Transmuted Inverse Weibull Distribution

Figure 2 :
Figure 2: The Reliability and Hazard Rate Functions of Cubic Transmuted Inverse Weibull Distribution ) to zero and numerically solving the resulting equations.The asymptotic distribution of maximum likelihood estimators is given as; see for example Rehman et al. (2018a,2018b) and Sarhan and Zaindin (2009);

Figure 3 :
Figure 3: Empirical and Fitted Distribution Functions Carbon Fiber Data Breast Cancer Data Repairable Items Data , Haj Ahmed et al. (2023), and Tashkandy et al. (2023), among others.Shahbaz et al. (2012) have introduced and studied a Kumaraswamy inverse Weibull distribution.Basheer (2019) has introduced a new generalized alpha power IW distribution.Kumar and Nair (2021) have introduced a generalization of the log-transformed version of the IW distribution and have also applied the proposed distribution in cancer research.Hassan and Nassr (2018) have introduced a new family of univariate distribution called the IW generated family.A lot of new models can be generated from this new family.The transmuted family of distributions has been introduced by Shaw and Buckley (2007) and has since been used by several authors to propose new probability distribution.The cdf of this family of distributions is given as

Table 4 : Summary Measures of Various Data Sets
for these criteria.The maximum likelihood estimates of the model parameters of CTIW, TIW, and IW distributions for different data sets are given in Table5below.From this table, we can see that the CTIW distribution seems a reasonable fit for all the data sets as it has the smallest value of the log-likelihood function in comparison with the other two distributions.The values of various fitted criteria for different distributions and for different data sets are given in Table