On The Modified Burr XII-Power Distribution: Development, Properties, Characterizations and Applications

In this paper, a flexible lifetime distribution with increasing, decreasing and bathtub hazard rate called the Modified Burr XII-Power (MBXII-Power) is developed on the basis of the T-X family technique. The density function of the MBXII-Power is arc, exponential, left-skewed, right-skewed, J, reverse-J and symmetrical shaped.  Descriptive measures such as moments, moments of order statistics, incomplete moments, inequality measures, residual life functions and reliability measures are theoretically established. The MBXII-Power distribution is characterized via different techniques. Parameters of the MBXII-Power distribution are estimated using maximum likelihood method. The simulation study is performed on the basis of graphical results to see the performance of maximum likelihood estimates (MLEs) of the MBXII-Power distribution. The potentiality of the MBXII-Power distribution is demonstrated by its application to real data sets: survival times of pigs, survival times of patients and quarterly earnings


Introduction
A flexible model for the analysis of lifetime data sets is often always attractive to the researchers. Pareto, power function, exponential and Weibull distribution are of interest due to the attractive flexibility and simplicity.
In recent decades, many continuous univariate distributions have been developed but various data sets from reliability, insurance, finance, climatology, biomedical sciences and other areas do not follow these distributions. Therefore, modified, extended and generalized distributions and their applications to problems in these areas is a clear need of day.
The modified, extended and generalized distributions are obtained by the introduction of some transformation or addition of one or more parameters to the baseline distribution.
The odds ratio for the power random variable X is Gurvich et al. (1997) replaced "x" with odds ratio in the Weibull distribution for the development of a class of extended Weibull distributions. Alzaatreh et al. (2013) developed the cdf of the T-X family of distributions as

Transformations and Compounding
The MBXII-Power distribution is derived through (i) ratio of exponential and gamma random variables and (ii) compounding generalized Weibull Power (GW-P) and gamma distributions.
where W(G(x)) is function of G(x) and r(t) is the pdf of a non-negative random variable.

Basic Structural Properties
The survival, hazard, cumulative hazard, reverse hazard, elasticity functions and the Mills ratio of a random variable X with the MBXII-Power distribution are given, respectively, by The quantile function of the MBXII-Power distribution is ( ) The MBXII-Power random number generator is ( ) where the random variable Z has the uniform distribution on ( ) 0,1 .

Sub-Models
The MBXII-Power distribution has the following sub models.

Shapes of the MBXII-Power Density and Hazard Rate Functions
The following graphs show that shapes of the MBXII-Power density are arc, left-skewed, right-skewed, J, reverse-J, exponential and symmetrical (Fig. 1). The MBXII-Power distribution has increasing, decreasing, decreasing-increasing, inverted bathtub and bathtub hazard rate function ( Fig. 2 and Fig. 3).

Moments about Origin
The th ordinary moment for the MBXII-Power distribution is, The factorial moments of the MBXII-Power distribution are given by   ( ) The r th moment about means, Pearson's measures for skewness 1  and kurtosis 2  , moment generating function and cumulants of X for the MBXII-Power distribution are achieved from the relations ( ) ( ) 1 1,

Moments of Order Statistics
Moments of order statistics have applications in reliability and life testing. Moments of order statistics are also designed for replacement policy to predict of failure of future items obtained from few initial failures.
The pdf of the mth order statistic m:n X is The pdf of mth order statistic m:n X for the MBXII-Power distribution is Moments about the origin of mth order statistic m:n X for the MBXII-Power distribution are

Incomplete Moments
Incomplete moments are used in mean inactivity life, mean residual life function, and other inequality measures. The th incomplete moment for the MBXII-Power distribution is where ( ) The mean deviation about mean is ( ) ( ) and mean deviation about median is

Residual Life Functions
The residual life says ( ) n mz of X for the MBXII-Power distribution has the following n th moment The average remaining lifetime of a component at time, z say ( )

RELIABILITY MEASURES
In this section, different reliability measures for the MBXII-Power distribution are studied.

Stress-Strength Reliability of the MBXII-Power Distribution Let
The probability , sm R in the (26) is called reliability in a multicomponent stress-strength model.

CHARACTERIZATIONS
In this section, the MBXII-Power distribution is characterized via: (i) conditional expectation; (ii) truncated moment; (iii) hazard function; (iv) Mills ratio; (v) certain functions of the random variable and (vi) conditional expectation of record values.
We present our characterizations in six subsections.

Characterization Via Conditional Expectation
The MBXII-Power distribution is characterized via conditional expectation..   F.A., Hamedani, G. G. Korkmaz, M. C. and Proof. If X has cdf (4), then ( ) ( ) Differentiating (28) with respect to t, we obtain  After simplification and integration we arrive at

Characterizations via Truncated Moment of a Function of the Random Variable
Here we characterize the MBXII-Power distribution via relationship between truncated moment of a function of X and another function. This characterization is stable in the sense of weak convergence (Glänzel;1990).   F.A., Hamedani, G. G. Korkmaz, M. C. and where D is a constant.

Characterization via Hazard Function
In this sub-section, the MBXII-Power distribution is characterized via hazard function.
be continuous random variable .The pdf of X is (5) if and only if its hazard function, ( ) F hx , satisfies the first order differential equation Proof. If X has pdf (5), then the above differential equation holds. Now if the differential equation holds, then ( ) which is the hazard function of the MBXII-Power distribution.

Characterization via Mills Ratio
In this sub-section, the MBXII-Power distribution is characterized via Mills ratio. ( ) Let X: 0, → be continuous random variable .The pdf of X is (5) if and only if the Mills ratio satisfies the first order differential equation (5), then the above differential equation surely holds. Now if the differential equation holds, then which is Mills ratio of the MBXII-Power distribution.

Characterization via Certain Function of the Random Variable
The MBXII-Power distribution is characterized through certain function of the continuous random variable X. Hamedani (2013) used this technique for characterization.
After differentiation the above equation with respect to x, and then reorganizing the terms, we obtain F.A., Hamedani, G. G. Korkmaz, M. C. and Integrating the last equation from 0 to x, we have , Proposition 5.5.1 provides a characterization of (4). Clearly there are other choices of these functions.

Characterization via Conditional Expectation of the Record Values
Nagaraja (1988)

MAXIMUM LIKELIHOOD ESTIMATION
In this section, parameters estimates are derived using maximum likelihood method.  Where  is assumed to known because of its maximum likelihood is equal to maximum order statistics. In order to compute the estimates of the parameters , , ,     of the MBXII-Power distribution, the following nonlinear equations must be solved simultaneously: The above equations 30-33 can be solved either directly or using the R (optim and maxLik functions), SAS (PROC NLMIXED) and Ox program (sub-routine Max BFGS), or employing non-linear optimization methods such as the quasi-Newton algorithm.

Fig. 5: Fitted pdf, cdf, survival and pp plots of the MBXII-Power distribution for Survival Times of Pigs data
We can observe that the MBXII-Power distribution is close to empirical data (Fig. 5).

Application II: Survival Times of Patients:
The data are collected (Feigl and Zelen;1965)     The MBXII-Power distribution is best fitted model than the other sub-models because the values of all criteria of goodness of fit are significantly smaller for the MBXII-Power distribution.

Fig. 6: Fitted pdf, cdf, survival and pp plots of the MBXII-Power distribution for Survival Times of Patients
We can observe that the MBXII-Power distribution is close to empirical data (Fig.6).    We can observe that the MBXII-Power distribution is close to empirical data (Fig.7).

CONCLUDING REMARKS
We have developed the MBXII-Power distribution along with properties such as submodels, moments, inequality measures, residual and reverse residual life function, stressstrength reliability and multicomponent stress-strength reliability model. The MBXII-Power distribution is characterized via different techniques. Maximum Likelihood estimates are computed. The simulation study is performed on the basis of graphical results by using the MBXII-Power distribution to see the performance of MLEs. Goodness of fit show that the MBXII-Power distribution is a better fit. Applications of the MBXII-Power model to survival times of pigs, survival times of patients and quarterly earnings are presented to show its significance and flexibility. We have proved that the MBXII-Power distribution is empirically better for survival times of pigs, survival times of patients and quarterly earnings.

PP Plot for MBXII-Power distribution for Quartly Earning Data
Observed Probabilites Expected Probabilites