Estimating Parameters for Extension of Burr Type X Distribution by Using Conjugate Gradient in Unconstrained Optimization

1 Computer Department, Collage of Computer Science and Mathematics, University of Tikrit, Tikrit, Iraq. 2 Mathmatic Department, Collage of Computer Science and Mathematics, University of Tikrit, Tikrit, Iraq. 3 Department of General Administration, Collage of Administration and Economics, University of Tikrit, Tikrit, Iraq. 4 Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria. 1 zeyaemoh1978@tu.edu.iq, 2 mun880088@tu.edu.iq, 3 moudher@yahoo.com, 4 pelumi.oguntunde@covenantuniversity.edu.ng


Introduction:
Twelve distributions were introduced by [1] using differential equation approach; of this, Burr type XII and Burr Type X distributions have received adequate attention in the literature [2]. For instance, various extensions of the Burr type X (BX) distribution has been introduced in recent times, the two-parameter BX distribution [3], Beta BX distribution [4], Exponentiated Generalized BX distribution [2], Gamma BX distribution [5], Transmuted BX distribution [6] and several others are notable examples. The usefulness of these distributions have however been demonstrated using real life applications.
The interest of this research, a new method was presented of the parameter estimation by using the conjugate gradient method in unconstrained optimization. In additional to extend the Exponentiated BX distribution using the Marshall-Olkin's method [7] of generating new distributions because of its flexibility. Several other families of distributions are available in the literature, readers are referred to [8,9,10,11, and 12] for further details.
The remaining part of this paper is structured in the following manner; in section 2, the new model, Marshall-Olkin Exponentiated Burr X (MOEBX) distribution is derived including its statistical properties while real life applications are provided in section 3.

Marshall-Olkin Exponentiated Burr Type X (MOEBX) Distribution:
Suppose Y denote a random variable (R.V), the cumulative distribution (cdf) and the probability density functions (pdf) of the BX distribution are; Now, following the work of [13], the cdf of the Exponentiated Burr X (EBX) distribution is obtained as; The pdf of the EBX refers to; The corresponding pdf of the MOEBX is obtained as; The expression in Equation (8) can be re-written as; The plot for the pdf of MOEBX distribution is as illustrated in Fig. 1.

Expansion for the pdf:
The pdf of the MOEBX distribution was obtained in Equation (9) Substituting Equation (10) in Equation (9), we get: The expression in Equation (11) can be re-written as: is the pdf of Exponentiated Burr X (EBX) distribution with parameters  and *  .
For 1   , we can use the same argument as in Equation (12), after some algebraic calculations we obtain: where; Therefore, the pdf of the MOEBX distribution can be expressed as an infinite linear combination of Exponentiated Burr Type X pdf. Moreover, Equations (12) and (13) are used to find the mathematical properties such as the r-th moments of the MOEBX distribution.

Hazard and Survival Function:
Hazard function is obtained using: Therefore, the hazard function of the MOEBX distribution is: The plot for the hazard function of MOEBX distribution is as illustrated in Fig. 2. The survival function (S.F) on the other hand is obtained from: Therefore, we get the S.F function of the MOEBX distribution as: , , , 1

Quantile Function and Median:
The quantile function (Qf) which is otherwise known as the inverse cdf is obtained from: Therefore, Qf for the MOEBX distribution is: This indicates that random samples can be generated for the MOEBX distribution using eq (17) as follows: where 'U' is uniformly distributed with parameters (0,1). The first and third quartiles can also be obtained when U=0.25 and U=0.75 in Equation (16) respectively.
The median can be written by using u=0.5 as follows:

Moments:
By using the r-th moment of EBX distribution which is: We obtain the r-th moment for the MOEBX distribution as: The moment generating function (mgf) of the MOEBX is therefore given by;

Order Statistics:
, ..., n y y y are random samples from a cdf and pdf generated from the MOEBX distribution, the pdf of the ith order statistics of the MOEBX distribution is thus obtained as follows: Where ( )is represented Beta distribution.

Maximum Likelihood Estimation by Conjugate Gradient (CG) Method:
The parameters were estimated by using maximum likelihood function by CG method in unconstrained optimization (Fletcher-Reeves update) in R programme package "optim". The log likelihood function of the new distribution can be written as follows: The aforementioned equations cannot be solved analytically. The iterative method as the CG method must be used. So, the latter (conjugate gradient method) used the default function of R program, in which called ''optim'' function with '' Fletcher-Reeves update [14].

( )
" to obtain the MLEs of and by means of conjugate gradient method. In addition, the initial value of CG method was suggested as follows: Where is MLE of the Marshall -Olkin Exponentiated Burr type X distribution (MOEBX).

Applications:
The MOEBX distribution is applied to a real life dataset and its performance compared with other compound distributions like Gamma Burr X, Beta Burr X, Weibull Burr X, Exponentiated Burr X and Burr X distributions. The data set used relates to the strengths of 1.5cm glass fibres obtained from [15, 16,4 and 12]. The R software was used to compute the Negative log-likelihood (NLL) value, Maximum likelihood estimates, by means of conjugate gradient method in unconstrained optimization. Akaike Information Criteria (AIC), CAIC, Bayesian Information Criteria (BIC) and HQIC. The distribution that has the lowest value of these criteria is adjudged the best distribution. The result of the analysis is provided in Table   2.

Distributions Estimates (with Standard Error in Parentheses)
Gamma Burr X The plots in Fig. 3 also confirm the results in Table 2.

Conclusion:
The numerical experiment confirmed that the CG method was effective in the computational solving of unconstrained optimization problems, in which used to estimate in simulation section. The MOEBX distribution has been successfully studied and its various statistical properties have been established. The distribution is flexible and versatile; it performs better than the Gamma Burr X, Beta Burr X, Weibull Burr X, Exponentiated Burr X and Burr X distributions. The estimates of the parameters are quite stable and close to the true values as we increase the sample size. We hope that this newly introduced distribution would gain wider attention in modelling real life events in engineering, finance, medicine and so on.