Transmuted Sushila Distribution and its Application to Lifetime Data

The Sushila distribution is generalized in this article using the quadratic rank transmutation map as developed by Shaw and Buckley (2007). The newly developed distribution is called the Transmuted Sushila distribution (TSD). Various mathematical properties of the distribution are obtained. Real lifetime data is used to compare the performance of the new distribution with other related distributions. The results shown by the new distribution perform creditably well.


Introduction
Last few years have witnessed generalizations of various lifetime distributions. This is achieved by compounding the distribution with any of new families of distributions. The process involves introduction of new shape parameter(s) to improve flexibility of the baseline distribution. Among well-known generalized families of distributions are: Marshall-Olkin family of distributions [1]; Beta G distributions [2]; Quadratic Transmuted family of distributions [3]; Kumaraswamy G distributions [4]; Gamma G distributions [5]; Exponentiated generalized G distributions [6]; Weibull G distributions [7]; and Alpha Power Transformation [8]. Researchers in sciences and engineering have applied these families of distribution to improve modelling of various lifetime data.
In this article, we generalize the Sushila distribution [9] using the Quadratic Transmuted family of distributions [3] and the new generalization is called the Transmuted Sushila Distribution (TSD). A random variable X is said to have the Sushila distribution if its distribution function (CDF) is given as: It is observed that is a special case of the Lindley distribution [10] when λ = 1.
Given a baseline distribution with the CDF G(x), the Quadratic Transmuted (QT) family of distributions has the cdf : (1.2) QT was applied to some probability distributions by [11,12] with the resulting distributions offering more flexibility. [13] also discussed some mathematical properties for the QT family of distributions. With generally acceptability, researchers have applied (1.2) and introduced different new members of the QT family for diverse lifetime distributions. List of some QT distributions was provided by [14].

Transmuted Sushila Distribution
The Transmuted Sushila Distribution (TSD) is obtained by putting (1.1) into (1.2). Hence, a random variable X is said to have the TSD, i.e. X ∼ T SD(θ, λ, a) if its CDF is given as: (2.1) Figures 1 illustrates the cdf of the TSD for some selected values of scale and shapes parameters. The probability distribution function (PDF) of TSD is obtained by differentiating (2.1) once. Hence, the PDF of X ∼ T SD(θ, λ, a) is: Note: (i) The TSD becomes the Sushila Distribution due to [9] if a = 0.
(ii) The TSD becomes the Lindley Distribution due to [10] if a = 0 and λ = 1. The pdf of the TSD for some selected values of scale and shapes parameters is shown in figure 2.

Reliability Analysis
Survival Function: The probability of an item not failing prior to a particular time is defined by its reliability or survival function S(x). This is defined by S(x) = 1 − F(x). Therefore, if a random variable X ∼ T SD(θ, λ, a), then its survival function is given by:

Hazard Rate Function (HRF):
This is the risk a system has in experiencing and event in an instantaneous time provided it has not experienced it at present time. It is a measure of proneness to an event and it is obtained as: . For a random variable that has the TSD, the hrf is given as: .
(3.2) Figure 4 shows the HRF of the TSD for some values of scale and shapes parameters.
The trend depicting the cumulative HRF of the TSD for some values of scale and shapes parameters is shown in figure 5.

Order Statistics
Given that X 1,n < X 2,n < ... < X n,n is a set of ordered random variable of size n, if X ∼ T SD(θ, λ, a), then, the PDF of the r th order statistics is given as: Therefore, the rth order statistics of X is given as: If r = 1 and r = n, the 1st and nth order statistic for X are respectively given in (4.2) and (4.3).

Quantiles Function
The quantile function of a random variable X ∼ T SD(θ, λ, a), is given as: Therefore, the first, the second, and the third quartiles for the random variable are obtained by respectively setting u to 0.25, 0.50, and 0.75. These are given by: where the Lambert function W is a complex function with multiple values which is defined as the solution for the equation W (u) e W (u) = u.

Skewness and Kurtosis
Variability in a data set can be investigated using skewness and kurtosis, and classical measures can be susceptible to outliers. Given the quantile function as in (4.4), the Moor's Kurtosis [16] based on octiles is given as: (4.5) Also, the Bowley's measure of skewness [17] based on quartiles is given as: (4.6)

Moments
Proposition 4.1. If a random variable X follows TSD with pdf as given in (2.2), then the kth raw moment is given by: Proof. Therefore, Hence,

Mean and Variance of TSD:
The mean of a random variable X follows TSD is obtained by putting k = 1 in (4.7) above. Hence the mean is given as: Also, the variance of X is obtained as var(x) = E(x 2 ) − (E(x)) 2 , where E(x 2 ) is obtained by equating k = 2 in (4.7).
Hence, the variance of X is given by:

Moment Generating Function
Proposition 4.2. The moment generating function of a random variable X that follows the TSD is given as: (4.8)

Parameter Estimation
Proposition 5.1. Given that X i , i = 1, 2, ..., n are iid random variables from TSD, then the log-likelihood function of X is defined as: Proof. The likelihood function of a random variable X that follows TSD is: Hence, the log-likelihood function of a random variable X that follows TSD is: The MLE of (θ, λ, a) can be obtained by maximizing (5.1). This gives the set of normal equations below: Obtaining solutions for the set of normal equations analytically is tedious. Using the MaxLik function in R language [18], the solutions are obtained numerically using algorithms like Newton-Raphson.

Asymptotic Confidence Bounds of TSD
Using the variance-covariance matrix I −1 o , a (100 − α)% confidence intervals of the parameters θ, λ, a can be obtained.

Application
The QT technique has been applied to obtain new set of distributions by compounding the Lindley distribution. The Transmuted Lindley distribution (TLD) was introduced by [20]. [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] introduced the Transmuted Two-Parameter Lindley distribution (TTPLD). The Transmuted Generalized Quasi Lindley distribution (TGQLD) was introduced by [23]. All these newly transmuted distributions have the Lindley distribution has special case. This attribute is also shared by the TSD newly introduced in this research. The probability distribution functions of all the models compared for data application are presented in table 1 below.
The data set used to observe the performance of TSD is the remission times (months) of a sample of 128 bladder cancer patients. This data has been applied in various survival analysis [19,20].