Beta Kumaraswamy Burr Type X Distribution and Its Properties

We proposed a so-called Beta Kumaraswamy Burr Type X distribution which gives the extension of the Kumaraswamy-G class of family distribution. Some properties of this proposed model were provided, like: the expansion of densities and quantile function. We considered the Bayes and maximum likelihood methods to estimate the parameters and also simulate the model parameters to validate the methods based on different set of true values. Some real data sets were employed to show the usefulness and flexibility of the model which serves as generalization to many sub-models in the fields of engineering, medical, survival and reliability analysis.


Introduction
Here, we provide a brief explanation on the Beta distribution which has been receiving more attention in the area of statistical modeling over a century [1].The beta distribution has PDF and CDF defined as: F Beta (t; ν, κ) = I t (ν, κ) t (0, 1). ( where the shape parameters, ν and κ, are positive.Its functions are unimodal, uni-anti modal, bathtub or constant depending on the parameter values.Beta models are adaptable in-terms of versatility and this flexibility motivates its usage in many applications.On the other side, we have Kumaraswamy (Kum) distribution [2], which fails to accept the fact that Beta model does not fit hydrological data.The PDF and CDF of the Kum Model, Kum(ϕ, ψ), is given as: F Kum (t; ϕ, ψ) = 1 − (1 − t ϕ ) (ψ) t (0, 1).( 4) 1.1.Beta-Generated (Beta-G) Family of Distribution Beta-G family was proposed by [3]and [31] is a very versatile and rich class of generalized distributions.It recieves a quite well response over the recent years.It has a PDF and CDF as follows: (6) where S(t) is the survival function of Beta-G distribution.Some of the well known works related to Beta-G family of distributions received an impact for a given distribution with PDF and CDF of Beta-G proposed by [3], [4] and [5] family of distribution.

Kumaraswamy-G (Kum-G) Family of Distribution:
Kumaraswamy-G family received a quite well attention in the recent years in the area of applied statistics proposed by [6], this family called Kum-G family motivate us in which we applied the method which proposed by [7] in developing our model having PDF and CDF define respectively as: where, t > 0, g(t) = d dt G(t) and ϕ, ψ > 0, denotes the shape parameters on the notion for other parameters in the baseline distribution.Kumaraswamy-G class by [6], received a quite well attention to researchers in the last few decades in the literature.

Beta Generated Kumaraswamy-G (Beta Kum-G) Family of Distribution:
Recently [8]- [32], proposed this new family of distribution called Beta Generated Kumaraswamy-G (Beta Kum-G) distribution having θ = p, q ϕ, ψ, ϑ and τ where they adopt the method of Beta-Generator proposed by [3] and reviewed by [4]- [5].This new family called beta Kum-G family motivate us in which we applied the method by [7], in developing our model.It has a PDF and CDF given as: This new family can also be referred to as the new generalized Kumaraswamy-G (GKum-G) family of distribution.

Burr Type X (BX)
This model was proposed by [9] and revisited by [10]- [11], whom contribute a lot to family of continuous distributions and played a vital role in medical, survival and reliability analysis.Our first motivation arises to the failure or hazard rate function having some important features including the Burr-Type X (BX) property which exhibits the fitting of engineering and medical data sets, more especially big data, as it was recently observed by [12], that BX model is very effective and versatile in modeling reliability strength and lifetime data sets.They also thought BX model can be exponentiated Rayleigh (ER) or generalized Rayleigh (GR), but they prefer calling it BX model due to its suitability to Burr family of distribution and properties.They observed that the GR or BX model with two-parameter also has a quite common properties with gamma, generalized exponential and Wei-bull distributions respectively.Although, the BX model density and cumulative distribution functions have a simple close form and its also has a convenient and flexible feature in modeling censored (incomplete) data, unlike gamma, GE and Weibull distributions.The twoparameter BX has a monotonically increasing and decreasing hazard function features, which can be used for practical aspects in statistical distribution and modeling of applications.Recently, authors have been studying BX model due to its ability and flexibility in modeling reliability data sets such as, [13] whom proposed the two parameter Burr type X distribution which authors like; [14]and [15] made some extension.The Burr type X distribution can be used in modeling general lifetime data.Based on this model distribution in [10], whose PDF and CDF are given by: These two Figure (1) and Figure (2), were originally plotted by [11], showing the flexibility (left-skewed, right-skewed and symmetry), shapes (bathtub and increasing) respectively.The two parameters BX or GR model in equation (11) above, implies BX(ϑ, τ ), if ϑ = 1, BX distribution reduces to a well-known one parameter Rayleigh distribution.Where ϑ and τ are the shape and scale parameters respectively.The two parameters BX or GR model in equation 11  above, implies BX(ϑ, τ ), if ϑ = 1, BX distribution reduces to a well-known one parameter Rayleigh distribution.If ϑ = 1, this plays the role of τ , which is the scale parameter.On the other hand, if ϑ ≤ 1 2 , therefore the PDF of BX model will be decreasing function of the model while for ϑ > 1  2 , it proves that it is a unimodal right skewed function.If it has a mode written as to τ , where t o is called the non-linear systems of equation given as: The above mode of BX model shows clearly that it is decreasing function of the scale parameter τ and also the increasing function of ϑ respectively.The PDF figures at different forms resemble that of Weibull and gamma functions.Moreover, the median of the BX model occurs when the given quantile at: This also shows that the non-increasing form of τ , and the non-decreasing form of ϑ.The remaining part of the section were organized as follows.In section 2, we provide the distributional properties, the expansions for the PDF and CDF.In section 3, the expansion of densities, sub-models.In Section 4, we obtain the moments, moment generating function and the order statistics, Re nyi entropy, the quantile function, Skewness and Kurtosis and maximum likelihood estimation.In section 5, application of real data and comparison of the new model with some sub-models was done and follow by the conclusion in section ?? respectively.

Distributional Properties
2.1.Beta Kumaraswamy Burr Type X (Beta Kum-BX) Distribution: In this section, we present our new model Beta Kum-BX with six parameters as the properties.Our second motivation, lies within the wide usage of Kum-G family define in equation (7) and the baseline distribution known as the BX define in equation (11) together with the property of beta-G family define in equation ( 5) and ( 6) above due the relative flexibility and capability in modeling agriculture, engineering and medical datasets in respect to the model suitability at different tractable or complex situations.However, we were motivated to introduced this new model called Beta Kumaraswamy Burr Type X (Beta Kum-BX) with six parameters θ = ν, κ, ϕ, ψ, ϑ and τ by confounding equation ( 9) and ( 11) and also equation ( 10) and ( 12) by the methods of beta-G generator proposed by [8] and thereby Burr Type X as the baseline distribution which we obtain the probability density function PDF given as: We now show that the t 0 f (t)dt = 1.Therefore from equation ( 13), above we simplify the model and proof the theorem.
Therefore, equation (13), becomes more like a complete beta function given as; The corresponding cumulative distribution function, survival function S(t), hazard or failure rate function h(t) and cumulative hazard function H(t) respec-tively.
Based on the relative usage of Kumaraswamy and Burr Type X baseline distributions we were motivated to propose this six parameters Beta Kum-BX due to the wideness of the BX with two parameters which generalizes (Rayleigh) and it provides a continuous function among the twelve Burr family of distribution which makes it suitable in modeling several complex situations.Also, the model transformation in equation ( 14) are not submissive meaning not mostly tractable just like the Weibull distribution where the skewness and kurtosis at different shapes were flexible with monotonically non-decreasing and non-increasing hazard functions.
Our last motivation was noted from the above CDF of the new model equation (14), F BetaKum−BX (t, θ) m 1 Y as t → 0 also 1 − F BetaKum−BX (t, θ) ∼ m 2 exp −(τ t) 2 as t → ∞ where m 1 and m 2 denotes the model constraints.This new model BKBX encloses a lower tail performance for the exponentiated Rayleigh model and the upper tail acts like the Burr-Type X model.Kumaraswamy Burr-Type X which are the (Kumaraswamy and Burr-Type X) motivates us with its great varieties of fitting failure, engineering (reliability), medicine and agriculture etc.This six parameter model encompasses many sub-models with great flexibility which in their bathtub shapes, monotonically non-increasing and decreasing failure function and also has some common properties with Kumaraswamy-Weibull by [18] which fit more perfectly well than the reduce modified Kumaraswamy Burr type X model relating to flexibility and suitability towards multivariate and censored data respectively.
Figure (3) and Figure (4), describes the shapes of the probability density function and cumulative distribution function functions for the given parameter values.These function represent different kind of forms depending on choosing values of BKBX model parameters.We noticed that the additional shape parameters allows for high level of flexibility.

Shapes of the BKBX Distribution
On the other hand, Figure (5) and Figure (6) below, shows the failure rate or hazard function of BKBX model increasing and decreasing or bathtub shapes.This new model with six parameters is more flexible than Kumaraswamy Burr Type X and Burr Type X distribution with four and two parameters due to flexibility of Kumaraswamy distribution with two parameters, that leads more smooth and vital.This new model will be useful in modeling and analyzing real life, censored and uncensored data in medical, engineering, pure science and agricultural areas.

Expansion of PDF and CDF
We provide expansion of the PDF of BKBX distribution in terms of infinite weighted sum.We obtain some structural properties based on the pdf expansions such as moment, moment generating function, mean division and others.The expansion is being done by using the equation of the PDF from equation (13).According to [6], if |W | < 1 and ω > 0 is a real non-integer, we have the series representation : If ω is an integer number, the series representation for finding CDF expansion is: By using the expansion of equation ( 18) and ω > 0 real non-integer, By using the binomial expansion in equation ( 13), above which is the density function of BKBX distribution with six parameters, θ = (ν, κ, ϕ, ψ, ϑ, τ ) and θ
=f KBX (t; θ 2 ) Where, ψj = ψ j (j + n) Alternatively, we can expand the PDF also used by [6]: Where, η l = ψj Also we can expand the CDF with the following result using an incomplete beta function also provided by [5].
Also, on the other hand, we have the CDF of BKBX as: Using Equation ( 24), above we obtained: By exchanging the indices j and k in the symbol, we get: Where, On the other hand, the expansion of the CDF of BKBX can be derived similarly as provided by [33], Where,
Show that equation ( 13) reduces to Beta Burr Type X with four parameters Let ϕ = ψ = 1 in equation ( 13), given as; We can as well reduce to BBX model as follows: Hence,

Preprints
2. When ν = κ = 1, the BKBX in Equation ( 13) above reduces to Kumaraswamy Burr Type X (KBX) with four parameters from the family of Kumaraswamy-G proposed by [6] also a special case to Kumaraswamy-Weibull distribution proposed by [14] respectively.
Show that equation ( 13) reduces to Kumaraswamy Burr Type X with four parameters Let ν = κ = 1 in equation ( 13), given as;

Hence, by law of indices let
We can as well reduces to BX model as follows: 4. When ν = κ = ϕ = ϑ = τ = 1, the BKBX in Equation ( 13) above reduces to Rayleigh distribution with one scale parameter by [19].
Show that equation ( 13) reduces to Rayleigh with with one scale parameter.Let ν = κ = ϕ = ϑ = τ = 1 in equation ( 13), given as; We can as well reduces to R model from BX model substituting ϑ = 1, by the law of indices let C = {1 − e −(τ t) 2 } and C 0 = 1,: Figure (7) below shows the flow chart of all the sub-models of BKBX distribution at each level, where it provides how the six parameters reduces to its last submodel with one scale parameter Rayleigh (R) by [19] and one shape parameter Burr Type X (BX1) [9] respectively.Probability weighted r th moments (PWMs), It was initially introduced by [20], defined to be the expectation of some functions of a random variable x and y defined.The (m, n, t)) th PWM of T is defined by: These follows an incomplete gamma function provided by [14]. Where,

Moment Generating Function (MGF)
The MGF of BKBX distribution can be obtained and expressed in form of exponential Kum-G family of distribution from the results we obtained in the moments above using Equation 20 above, Where M X (s) is the MGF of a BKBX distribution.

Order Statistics
These two notable statisticians [10]and [11], proposed the procedure of deriving the order statistics of Burr Type X (BX) distribution.We adopted this method by assuming a random sample from a population T 1 , T 2 , . . ., T n , from BKBX model with six parameters θ = (ν, κ, ϕ, ψ, ϑ, τ ) and KBX model parameters, θ 2 = (ϕ, ψ, ϑ, τ ).We denote T r:n as the r th order statistics, given the PDF of T r:n can be express as: Therefore, equation (30) reduces to this form by law of indices, By adopting the binomial expansion by [16], the PDF and CDF of the BKBX distribution.We obtained the PDF of the r th order statistics of is as follows: Hence, we obtained the r th order statistics of the BKBX distribution as: ξ y = ξy (y + 1),

Rényi Entropy
An entropy for a given random variable is called the measure of uncertainty used in many areas of research like: applied statistics and reliability in engineering.In the recent years, [33], modified and derived the general formula which was originally proposed by [34], over 5 decades ago.Moreover, [24], provides the expression of this entropies for models with more than one variates.It is given by definition:

Rényi Entropy
An entropy for a given random variable is called the measure of uncertainty used in many areas of research like: applied statistics and reliability in engineering.In the recent years, [24], modified and derived the general formula which was originally proposed by [17], over 5 decades ago.Moreover, [16], provides the expression of this entropies for models with more than one variates.It is given by definition: Entropies are measures which quantified the randomness or diversity of a random variable X .They express the expected information uncertainty or content of a PDF.Entropy is a measure that provide important tools to analyze evolutionary processes over time (technical change) and to indicate the variety in the distributions at a particular moment of time.The large value of the Rényi entropy indicates that there is greater uncertainty in the data.Where > 0 and = 1.Using binomial expansion also provided by [16], we can expressed it by denoting: The Rényi entropy for BKBX distribution can be found from equation ( 13), above as follows: Therefore, we obtained the Rényi entropy by substituting equation (34), in equation ( 33) as follows: i,j=0 (ν − 1) j Thus, the Rényi entropy is, (35)

Quantile Function
Let Q ν, κ (u) be the beta quantile function with parameters ν and κ.The quantile function of the BKBX model with θ = (ν, κ, ϕ, ψ, ϑ, τ ) parameters, let t = Q(u), using the cumulative distribution function given as: multiply both powers by "1/ψ" to deduce the equation: Similarly, we again multiply both powers by "1/ϑϕ" to deduce the equation: Take a logarithm and multiply for both sides by -1 we obtained: by taking the square root of both sides: Therefore, the quantile function of BKBX distribution is given below by taking the inverse making "t" the subject of the formula.

Skewness and Kurtosis
Skewness: is a measure of symmetry, or more precisely, the lack of symmetry.A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.The Bowley skewness by [25], is actually a statistical procedure to find the positive or negative skewed distribution on based on different kinds of data depending on area and the suitability of the functions.It is the most popular tool procedure in finding a skewness fit given by.
On the other hand we have, Kurtosis: is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution.That is, data sets with high kurtosis tend to have heavy tails, or outliers.Data sets with low kurtosis tend to have light tails, or lack of outliers.A uniform distribution would be the extreme case.The Moors Kurtosis by [26], can be calculated by using the formula given below:    Figure (9), shows the measure of the relative peakedness of a distribution.For a normal bellshaped distribution (mesokurtic) and a platykurtic distribution (flatter than a normal distribution with shorter tails).Also a leptokurtic distribution (more peaked than a normal distribution with longer tails) respectively.

Simulation Studies
We conduct a Monte Carlo simulation using the inverse CDF method considering the quantile function of the new BKBX model from Equation (43), used to generate the random numbers for the simulation results and also compare with any existing model for validating the efficiency of developing the new model.
A simulation studies was done to access and validate the efficiency of the new model as we adopt the Monte Carlo simulation using the inverse CDF method of sampling considering the quantile function above of the new model which we study to access the performance of the MLE's of θ = (ν, κ, ϕ, ψ, ϑ, τ ) parameters.
We also generate different "n" sample observation from Equation (43).The parameter estimation was done by maximum likelihood method.We considered different sample size n =(50,150,300,500) and the number of repetition is 1000 respectively.The true parameters value at different sets for each model were used respectively.
Table (1) and Table (2), shows that the mean, bias and MSE of the estimate parameters.We observed that, when we increase sample size "n", the mean is converging towards the true-value while, the bias and MSE are becoming very smaller close to zero for both BKBX and BWB models.Likewise, the blue colour estimated values indicates how the bias and MSE are decreasing with increase of the sample sizes n=(50,150,300,500) respectively."The following data is about 346 nicotine measurements made from several brands of cigarettes in 1998.The data have been collected by the Federal Trade Commission which is an independent agency of the US government, whose main mission is the promotion of consumer protection by [29].

Criterion:
The criterion like: log-likelihood, Akaike information criterion, consistent Akaike information criterion and Bayesian information criterion for the data set above so as to compare the models and to check which have least or smaller -2 , A k IC , A k ICC, B ayes IC, K-SM and p − V alue values.The distribution of the datasets was skewed to the left, right and symmetric respectively.

Summary Of Comments On The Simulation Results
In summary we proposed a new model called Beta Kumaraswamy Burr-Type X (BKBX) with six parameters θ = (ν, κ, ϕ, ψ, ϑ, τ ) that extends and generalized the Kum-G family, Burr type X distributions and many sub-models by the family of BK-G family which was proposed by [32].
We obtained the distributional properties like: probability density function, cumulative distribution function, hazard function and their expansions.Also, the statistical properties like: quantile function, Browley skewness, Moors Kurtosis were also obtained.The parameters were estimated by using maximum likelihood estimation (MLE) methods.A simulation study for the model parameters was conducted by given different true values as we can see the higher you increase the "n" bias the smaller the mean square error becomes at each level of the simulation process.
Based on the simulation results for the comparison using the Akaike information criterion (AIC) as to check the best model in-terms of providing a meaningful and exact true value and also how the mean, bias and root mean square errors were obtained.
From Table (1) and Table (2), it was shown that as the sample sizes at each level of the process the bias and the mean square error (MSE) decreases with increasing of the sample sizes, where it proven that the BKBX is the best compare to Beta-Weibull model in-terms of comparison but on the other hand the AIC value of BKBX gives the smallest value of the context that not always the best but rather based on the true values used in the simulation study respectively.

Summary Of Comments On The Real Data
Table 3, 4 and 5 above and below shows MLEs for the individual fitted model for the adopted datasets and the estimated (-2 , A k IC , A k ICC and B ayes IC, K-SM and p−V alue), where the results obtained indicates clearly that the Beta Kumaraswamy-Burr Type X is a stronger with highest peak, flexible and vital to its sub-models and other existing models like: Beta-Weibull, generalized Gompertz and generalized exponential were used here for fitting the data set respectively.
These model as fitted based on the above datasets proves that it is a good example for modeling left, right skewed and symmetric datasets.Also, the  Based on the data above ω = 20.567> 5.991 = χ 4;0.05 , we therefore reject the null hypothesis.Figure 10, 11 and 12 above and below for the plot of densities which compares the models with the histogram empirical graphs proves that Beta Kum-BX is closer to the histogram peak point than the other models which shows the flexibility of the new model with strong properties.On contrast it shows that based on this three applicable datasets from different areas which fits Beta kumaraswamy Burr type X (Beta Kum-BX) distribution having the smallest (-2 , A k IC , A k ICC, B ayes IC, K-SM and p − V alue ) among all.This suggest that the Beta Kum-BX distribution is very good in modeling left and right skewed and also some symmetric datasets respectively.In other to validate the goodness-of-fit for the new model based on the datasets at different level applied like; laboratory dataset, Hiv/Aids (censored) and nicotine measurement for smokers with all comes from science and medical respectively.The used of the test statistics for each model both the new and existing are: Cramer-von Mises (CvM) and the Watson (tests) by [30].In general view, the suitability of a model based on any datasets apply depends on how smaller the values obtained at each test result for a given datasets where it clearly proves that the Kolmogorov-Smirnov test, p − V alue, Cramer-von Mises (CvM) and the Watson (tests) were obtained in all the three provided tables showing the flexibility and efficiency of the new models Beta Kum-BX with six parameters having a better fitness as well as the histogram and empirical graphs respectively.Lastly we hope that the proposed model and its generated models will attract wider applications in several areas among others.

Figure 1 :
Figure 1: Plot of the Burr Type X probability density function for the shape parameter ϑ.

Figure 2 :
Figure 2: Plot of the Burr Type X hazard function for the shape parameter ϑ.

Figure ( 8 )
Figure(8), shows measure of relative symmetry as normally zero indicates symmetry.The larger its absolute value the more asymmetric the distribution.The positive parameters values indicates a long right tail skeweness.Figure(9), shows the measure of the relative peakedness of a distribution.For a normal bellshaped distribution (mesokurtic) and a platykurtic distribution (flatter than a normal distribution with shorter tails).Also a leptokurtic distribution (more peaked than a normal distribution with longer tails) respectively.

I
νν I νκ I νϕ I νψ I νϑ I ντ I κν I κκ I κϕ I κψ I κϑ I κτ I ϕν I ϕκ I ϕϕ I ϕψ I ϕϑ I ϕτ I ψν I ψκ I ψϕ I ψψ I ψϑ I ψτ I ϑν I ϑκ I ϑϕ I ϑψ I ϑϑ I ϑτ

Figure 12 :
Figure 12: Graph of the histogram and empirical distribution for Beta Kum-BX with existing different models for the Nicotine Measurements dataset.

Table 3 :
The ML estimates, -2 log-likelihood, A k IC , A k ICC and Bayes IC, K-SM and p − V alue values.for the life of fatigue fracture of Kevlar dataset.

Table 4 :
The ML estimates, log-likelihood,A k IC, A k ICC, Bayes IC for the measurements of the AIDS clinical trials group study 320 dataset.

Table 5 :
The ML estimates, log-likelihood, A k IC, A k ICC, Bayes IC for Nicotine measurements dataset.