Transmuted Weibull distribution and its applications

In this paper we study new distribution called transmuted Weibull distribution. Some properties of this distribution are described. The usefulness of the distribution for modelling data is illustrated using real data set.


Introduction
The Weibull distribution was originally introduced by Swedish physicist Waloddi Weibull [1]. He applied it on modelling the distribution of the yield strength of materials. The Weibull distribution is popular and widely used, with many applications in areas such as engineering (reliability, failure analysis, lifetime analysis …), material science, quality control, physics, medicine, meteorology, hydrology and others [2].
However, there are cases when standard Weibull distribution fails to model data suitably enough. This is where it is necessary to apply generalized distribution because of its flexibility and better fit of the data than standard Weibull distribution. The importance of such generalization has been proved in recent years on various problems and many standard distributions have been generalized. Some recent generalizations which are based on the Weibull distribution, are discussed in [3].
An interesting idea of generalization where the distribution is derived using the quadratic rank transmutation map, was studied first by Shaw and Buckley in [4]. Recently, many transmuted distributions have been proposed in literature. See, for example, transmuted Gumbel distribution [5], transmuted Rayleigh distribution [6], transmuted modified Weibull distribution [7], transmuted inverse Weibull distribution [8], transmuted Pareto distribution [9] and others.
Aryal and Tsokos [10] used the quadratic rank transmutation map to develop new generalization of the Weibull distribution, called the transmuted Weibull distribution. They studied mathematical properties of this distribution and the maximum likelihood estimates of the parameters.
In this paper, we focus on demonstrating the usefulness of the transmuted Weibull distribution for modelling lifetime, illustrated on three real data sets. All three data sets are modelled with the transmuted Weibull distribution, the 2-parameter and 3-parameter Weibull distribution. For comparison of the distributions the following criteria are used: the Akaike's information criterion, the corrected Akaike's information criterion, the Bayesian information criterion, the coefficient of determination and the root mean square error.
The parameters of the probability distributions are estimated using the maximum likelihood method. The calculations are performed using statistical software STATISTICA and software MATLAB.
The rest of the paper is organized as follows: In Section 2 we introduce the transmuted Weibull distribution. In Section 3 we define basic statistical properties including moments, quantiles, reliability and hazard rate functions. The maximum likelihood estimates of the transmuted Weibull parameters are presented in Section 4. Finally, in Section 5 we provide the real data applications of the transmuted Weibull distribution.

Transmuted Weibull distribution
A random variable is said to have the 3-parameter Weibull distribution (WD) W(a, b, c) with parameters a > 0, b > 0, c ≥ 0 if its cumulative distribution function (CDF) for x ≥ c is given by and probability density function (PDF) is given by where a is the dimensionless shape parameter, b is the scale parameter and c is the location parameter.
The transmuted distribution can be obtained by adding real number , |≤ into CDF that provides more flexibility in the form of new distribution [7].  denotes the transmuting parameter.
A random variable X is said to have the transmuted distribution if its CDF F(x) and PDF f(x) are given by where || ≤ 1, G(x) and ( ) are CDF and PDF of the base distribution, respectively.
Observe that for we have the base distribution of the random variable X.

Statistical properties
In this section we define statistical properties including moments, quantiles, reliability and hazard rate functions. The origin and other aspects of this distribution can be found in [10].

Moments and quantiles
The r-th moment, , of the TWD is defined as Consequently, the mean E(X) and the variance Var(X) are given by respectively, where ( ) is gamma function defined by ( ) ∫ , .
The -th quantile, ( ), of the TWD is given by

Reliability analysis
The TWD can be useful in characterization of the lifetime data analysis. The reliability function if the TWD is given by This function can be interpreted as probability that an item does not fail prior to some time x.
Another important characteristics is the hazard rate function that can be interpreted as conditional probability of failure when the item has already survived to time x.
The hazard rate function of the TWD is given by

Random number generation
The random numbers from the TWD can be generated using the inverse function of CDF (5) where U has uniform distribution over the interval ( ).

Maximum likelihood estimates
As a method for estimation of parameters of the TWD we choose the maximum likelihood method (MLM). Let be a random sample of size from the TWD with PDF (6) and let be a realization of the random sample. The likelihood function ( ) of this random sample is given by and the log-likelihood function ( ) of this random sample is given by Differentiating (15) with respect to a, b and , respectively, and equating each derivative to zero we obtain the equations The MLM estimates ̂, ̂, ̂ of the parameters a, b,  are obtained iteratively from the nonlinear system of equations (16) -(18).

Application
In this section we present the analysis of the real data sets using the TWD and compare it with the WD. In order to compare the distributions we consider the criteria:  the Akaike's information criterion (AIC) -calculated using [11] ( ̂) where ( ̂) ( ̂) is the maximized value of the likelihood function for the estimated model, ̂ is the MLM estimate of the parameter , m is number of parameters to be estimated, n is number of observed data;  the corrected Akaike's information criterion (AICC) -calculated using [12] ( )  the Bayesian information criterion (BIC) -calculated using [13] ( ̂) ( )  the coefficient of determination (R 2 ) -calculated using where ̂( ) is estimated cumulative distribution function and Function ( ) is the empirical distribution function, defined as follows The distribution better fitting the data corresponds to smaller values of AIC, BIC, AICC and RMSE and larger value of R 2 .

Data set 1
The first data set represents lifetimes of Kevlar 49/epoxy strands subjected to constant sustained pressure at 90% stress level until the strand failure. For previous studies with the data see [14 -17]. The data are as follows:  Table 1 gives the descriptive statistics for the data set. We can see that the empirical distribution is skewed to the right. Table 2 presents the MLM estimates of the parameters together with the log-likelihood ln L, AIC, AICC and BIC values. Results are rounded to four decimal places. In Table 2 we see that values of AIC, AICC and RMSE are smallest ones and value of ln L and R 2 are highest ones for TWD, compared to values for 2-parameter and 3parameter WDs. Hence, we can conclude that the TWD provides better fit to the data than the other two distributions.

Conclusion
In this paper we studied modification of the Weibull distribution, called the transmuted Weibull distribution. The usefulness of this distribution for modelling lifetime was illustrated using three real data sets. We compared the performance of the transmuted Weibull distribution with performances of the 2-parameter and the 3-parameter Weibull distributions. Comparison of distributions was based on criteria as the Akaike's information criterion, the corrected Akaike's information criterion, the Bayesian information criterion, the coefficient of determination and the root mean square error. We demonstrated that the transmuted Weibull distribution is more flexible and it better models lifetime data of animate (breast tumour patients, infected guinea pigs) and inanimate (Kevlar/epoxy strands) objects than the 2-parameter or the 3-parameter Weibull distribution. This paper was supported by the Slovak Grant Agency VEGA through the project No. 1/0812/17.