On Burr III Marshal Olkin Family: Development, Properties, Characterizations and Applications

In this paper, a flexible family of distributions with unimodel, bimodal, increasing, increasing and decreasing, inverted bathtub and modified bathtub hazard rate called Burr III-Marshal Olkin-G (BIIIMO-G) family is developed on the basis of the T-X family technique. The density function of the BIIIMO-G family is arc, exponential, left- skewed, right-skewed and symmetrical shaped. Descriptive measures such as quantiles, moments, incomplete moments, inequality measures and reliability measures are theoretically established. The BIIIMO-G family is characterized via different techniques. Parameters of the BIIIMO-G family are estimated using maximum likelihood method. A simulation study is performed to illustrate the performance of the maximum likelihood estimates (MLEs). The potentiality of BIIIMO-G family is demonstrated by its application to real data sets.


Introduction
developed a new family of distributions with an additional shape parameter called Marshall and Olkin-G (MO-G) family. The survival function of MO-G family is where G(x, ψ) is the baseline cumulative distribution function(cdf) which may depend on the vector parameter ψ. Many famous MO-G families and its special distributions are available in literature such as Marshall-Olkin-G (Marshall and Olkin;1997), the MO extended Lomax (Ghitany et al.;2007), MO semi-Burr and MO Burr (Jayakumar and Mathew;2008), MO q-Weibull 2010), MO extended Lindley (Ghitany et al.;2012), the generalized MO-G (Nadarajah et al. (2013), the MO Fréchet (Krishna et al;, the MO family (Cordeiro and Lemonte;, MO extended Weibull(Santos-Neto et al.;, the beta MO-G (Alizadeh et al. 2015), the MO generalized exponential (Ristić, & Kundu;, MO gamma-Weibull (Saboor and Pogány;, MO generalized-G , MO additive Weibull  and Weibull MO family (Korkmaz et al.;2019). This paper is sketched into the following sections. In Section 2, BIIIMO-G family is development via the T-X family technique. The basic structural properties and sub-models are also studied. In Section 3, two special models are studied. Section 4, deals with linear representations for the cdf and pdf of the BIIIMO-G family. In Section 5, moments, incomplete moments, inequality measures and some other properties are theoretically derived. In Section 6, stress-strength reliability and multicomponent stress-strength reliability of the model are studied. In Section 7, BIIIMO-G family is characterized via (i) conditional expectation; (ii) ratio of truncated moments and (iii) reverse hazard rate function. In Section 8, the maximum likelihood method is employed to estimate the parameters of the Burr III Marshall Olkin Weibull (BIIIMO-W) and Burr III Marshall Olkin Lindley (BIIIMO-L) distributions. In section 9, a simulation study is performed to illustrate the performance of the maximum likelihood estimates (MLEs). In Section 10, the potentiality of BIIIMO-G family is demonstrated by its application to real data sets: survival times of leukemia patients and bladder cancer patients' data. Goodness of fit of the probability distribution through different methods is studied. Section 11 contains concluding remarks. Alzaatreh et al. (2013) proposed a T-X family technique for the development of the wider families based on any probability density function (pdf). The cdf of the T-X family of distributions is given by

Development of BIIIMO-G family
where r(t) is the pdf of a random variable (rv) T, where T ∈ [a 1 , a 2 ] for − ∞ ≤ a 1 < a 2 < ∞ and W[G(x; ψ)] is a function of the baseline cumulative distribution function (cdf) of a rv X, depending on the vector parameter ψ and satisfies three conditions i) W[G(x; ψ)] ∈ [a 1 , a 2 ], ii) W[G(x; ψ)] is differentiable and monotonically increasing and iii) lim x→−∞ W ½Gðx; ψÞ →a 1 and lim x→∞ ½Gðx; ψÞ→a 2 . The pdf corresponding to (2) is In this article, the BIIIMO-G family is developed via the T-X family technique by setting.
Then, the cdf of BIIIMO-G family is where α > 0, β > 0, λ > 0 and ψ > 0 are parameters. The pdf corresponding to (4) is given by where g(x; ψ) is the baseline pdf. In future, a rv with pdf (5) is denoted by X~BIIIMO − G(α, β, λ, ψ). The dependence on the parameter vector ψ can be omitted and simply write as g(x) = g(x; ψ), Let T be a BIII random variable with shape parameters α, β. The BIIIMO-G rv with cdf (4) can be obtained from has the BIIIMO-G distribution. The quantile function (qf) of X is the solution of the non-linear equation is the qf of the baseline distribution. Hence, if U is a uniform rv on (0, 1), then X = Q(U) follows the BIIIMO-G family.

Transformations and compounding
The BIIIMO-G family is derived through (i) ratio of the exponential and gamma random variables and (ii) compounding generalized inverse Weibull-MO (GIW-MO) and gamma distributions.

Structural properties of BIIIMO-G family
The survival, hazard, cumulative hazard, reverse hazard functions and the Mills ratio of a random variable X with BIIIMO-G family are, respectively given, by and m x ð Þ ¼ The elasticity eðxÞ ¼ dlnFðxÞ The elasticity of BIIIMO-G family shows the behavior of the accumulation of probability in the domain of the random variable.

Sub-models
The BIIIMO-G family has the following sub models (Table 1).

Special BIIIMO-G models
The BIIIMO-G family density (5) produces greater flexibility than any baseline distribution for data modeling. It can be most tractable, when the functions g(x; ψ) and G(x; ψ) have simple analytic expressions. Then, two special sub-models of BIIIMO-G family are introduced.

The BIIIMO-Weibull (BIIIMO-W) distribution
The cdf and pdf of the Weibull random variable are gðx; bÞ Then, the pdf of the BIIIMO-W model is given by The following graphs show that shapes of BIIIMO-W density are arc, bimodal, exponential, left-skewed, right-skewed and symmetrical (Fig. 1). The BIIIMO-W distribution has uni-model, bimodal, increasing, increasing and decreasing, inverted bathtub and modified bathtub hazard rate function (hrf) Fig. 2.

The Burr III Marshal Olkin-Lindley (BIIIMO-L) distribution
The cdf and pdf of the Lindley random variable are gðx; Then, the pdf of the BIIIMO-L model are given by  The following graphs show that shapes of BIIIMO-L density are J, revers J, arc, exponential, left-skewed, right-skewed and symmetrical (Fig. 3). The BIIIMO-L distribution has unimodal, increasing, increasing and decreasing, decreasing-increasing-decreasing inverted bathtub, bathtub and modified bathtub hazard rate function (Fig. 4).

Useful expansions
In this sub-section, the linear representations for the cdf and pdf of the BIIIMO-G family are obtained. The cdf in (4) can be expressed as Following (Tahir et al. 2016 etc., and Then, we arrive at   Gradshteyn and Ryzhik 2014), By applying the power series to the quantity A, where |z| < 1 and b is a real non-integer, we obtain For 0 < λ ≤ 1, applying ð1−xÞ −n ¼ P ∞ ℓ¼0 ð nþℓ−1 n−1 Þx ℓ to the quantity B, we arrive is the cdf of the Lehmann type II Exp (1-G) model with power θ > 0. By differentiating (14) we get. and is the pdf of the Lehmann type II Exp (1-G) model with power θ > 0. Equations (14) and (15)

Moments
Moments, incomplete moments, inequality measures and some other properties are theoretically derived in this section.

Moments about origin
The r th moment of X, say μ ′ r , follows from (15) as Henceforth, Y m denotes the Lehmann type II exp-(1-G) distribution with power parameter m.
The n th central moment of X, say M n , is given by The cumulants (κ n ) of X follow recursively from where 1 , etc. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.

Generating function
The moment generating function (mgf) M X (t) = E(e t X ) of X is given by where M m (t) is the mgf of Y m . Hence, M X (t) can be determined from Lehmann type II exp-(1-G) generating function.

Incomplete moments
The s th incomplete moment, say I s (t), of X can be expressed from (15) as A general formula for the first incomplete moment, I 1 (t), can be derived from the last equation (with s = 1).
The first incomplete moment can be applied to construct Bonferroni and Lorenz curves defined for a given probability π by BðπÞ ¼ I 1 ðqÞ=ðπμ 0 1 Þ and LðπÞ ¼ I 1 ðqÞ=μ 0 1 , respectively, where μ 0 1 ¼ EðXÞ and q = Q(π) is the qf of X at π. The mean deviations about the mean ½δ 1 ¼ EðjX−μ 0 1 jÞ and about the median [δ 2 = E(|X − M|)] of X are given by 2 Þ is the median and Fðμ 0 1 Þ is easily calculated from (4). Table 2 shows the numerical measures of the median, mean, standard deviation, skewness and Kurtosis of the BIIIMO-W distribution for selected parameter values to describe their effect on these measures. Table 3 shows the numerical measures of the median, mean, standard deviation, skewness and Kurtosis of the BIIIMO-L distribution for selected parameter values to describe their effect on these measures.

Reliability measures
In this section, different reliability measures for the BIIIMO-G family are studied.
Multicomponent stress-strength reliability estimator R s, κ based on BIIIMO-G family Suppose a machine has at least "s" components working out of " κ " components. The strengths of all components of system are X 1 , X 2 , . …X κ and stress Y is applied to the system. Both the strengths X 1 , X 2 , . …X κ are i.i.d. and are independent of stress Y. F  and G are the cdf of X and Y respectively. The reliability of a machine is the probability that the machine functions properly. Let X ∼ BIIIMO − G(α 1 , β, λ, ψ), Y ∼ BIIIMO − G(α 2 , β, λ, ψ) with common parameters β, λ, ψ and unknown shape parameters α 1 and α 2 . The multicomponent stress-strength reliability for BIIIMO-G family is given by (Bhattacharyya and Johnson 1974). , then we obtain The probability R s, κ in (24) is called multicomponent stress-strength model reliability.

Characterizations
In this section, BIIIMO-G family is characterized via: (i) conditional expectation; (ii) ratio of truncated moments and (iii) reverse hazard rate function.

Characterization based on conditional expectation
Here BIIIMO-G family is characterized via conditional expectation.

Proposition
Let X : Ω → (0, ∞) be a continuous random variable with cdf F(x) ( 0 < F(x) < 1 for x ≥ 0), then for α > 1, X has cdf (4) if and only if ; for t > 0 Proof. If pdf of X is (5), then Upon integration by parts and simplification, we obtain ; t > 0: Conversely, if (25) holds, then Differentiating (26) with respect to t, we obtain. After simplification and integration we arrive at

Characterization of BIIIMO-G family through ratio of truncated moments
Here BIIIMO-G family is characterized using Theorem G (Glänzel;1987) on the basis of a simple relationship between two truncated moments of X.

Proposition
Let X : Ω → ℝ be a continuous random variable.
The random variable X has pdf (5)  Proof. For random variable X with pdf (5), ; x∈ℝ: ; x∈R: The differential equation ; x∈ℝ: Therefore, in the light of theorem G, X has pdf (5).

Corollary
Let X : Ω → ℝ be a continuous random variable and let. p(x) and h 1 (x) defined in Theorem G, satisfying the differential equation : Proof If X has pdf (5), then (29) holds. Now if (29) holds, then which is the reverse hazard function of the BIIIMO-G Family.

Maximum likelihood estimation
In this section, parameter estimates are derived using maximum likelihood method. The log-likelihood function for the vector of parameters Φ = (α, β, λ, ψ τ ) of BIIIMO-G family is In order to compute the estimates of the parameters α, β, λ, ψ τ , the following nonlinear equations must be solved simultaneously: where g 0 ðx i ; ψÞ ¼ δ δψ gðx i ; ψÞ:

Simulation studies
In this Section, we perform the simulation study to see the performance of MLE's of BIIIMO-L distribution. The random number generation is obtained with inverse of its cdf. The MLEs, say ðα i ;β i ;λ i ;b i Þ for i = 1,2,…,N, have been obtained by CG routine in R programme.
The simulation study is based on graphical results. We generate N = 1000 samples of sizes n = 20,30,…,1000 from BIIIMOL distribution and get true values of α = 1.5, β = 2.55, λ = 0.095 and b = 5 for this simulation study. We also calculate the mean, standard deviations (sd), bias and mean square error (MSE) of the MLEs. The bias and MSE are calculated by (for h = α, β, λ, b) The results are given by Fig. 5. Figure 5 reveals that the empirical means tend to the true parameter values and that the sds, biases and MSEs decrease when the sample size increases as expected. These results are in agreement with first-order asymptotic theory.

Applications
The
We can also perceive that the BIIIMO-L distribution is best fitted model than other sub-models and competing models because BIIIMO-L distribution offers the closer fit to empirical data (Fig. 7).

Concluding remarks
We have developed the BIIIMO-G family via the T-X family technique. We have studied properties such as sub-models; descriptive measures based on the quantiles, moments, inequality measures, stress-strength reliability and multicomponent stress-strength reliability model. The BIIIMO-G family is characterized via different techniques. The MLEs for the BIIIMO-G family have been computed. A simulation studies is performed to illustrate the performance of the maximum likelihood estimates (MLEs). Applications of the BIIIMO-G model to real data sets (survival times of leukemia patients and bladder cancer patients data) are presented to show the significance and flexibility of the BIIIMO-G family. Goodness of fit shows that the BIIIMO-G family is a better fit. We have demonstrated that the BIIIMO-G family is empirically better for lifetime applications.

PP Plot for BIIIMO-Lindley distribution for Survival Times of Cancer patients Data
Observed Probabil ites Expected Probabilites Fig. 7 Fitted pdf, cdf, survival and pp. plots of the BIIIMO-Lindley distribution for survival times data