The Unit Generalized Log Burr XII Distribution: Properties and The Unit Generalized Log Burr XII Distribution: Properties and Application Application

: In this paper, a three-parameter bounded unit distribution with a flexible hazard rate called the unit generalized log Burr XII (UGLBXII) distribution is derived. To show the importance of the proposed distribution, we establish some of its mathematical properties such as random number generator, ordinary moments, generalized TL moments, conditional moments, reliability and uncertainty measures. We characterize the UGLBXII distribution via innovative techniques. We also present the bivariate ‐ and multivariate ‐ type distributions via Morgenstern (Mor) family and via Clayton family. Six estimation methods such as the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling methods are adopted to estimate its unknown parameters. We perform simulation studies on the basis of the graphical results to see the performance of the above estimators. Two real data sets are considered to prove the empirical superiority of the proposed model.


Introduction
Statistical distributions bounded on (0,1) are useful tools to describe uncertainty phenomena.For modeling proportions, percentages, indices, rates and ratios measured on the unit interval, many new continuous unit distributions have been developed.In statistical literature, the most famous unit Integrating (3), we obtain ( ) ( ) Using condition ( ) , the normalizing constant is and the pdf of UGLBXII distribution is ( ) The cdf corresponding to (4) is ( ) (5) Volume 6, Issue 9, 10222-10252.

Relationship between the exponential and gamma variable
Here, we derive the UGLBXII distribution from a relationship between the exponential and gamma random variables, i.e., ( ) After simplification, we have ( ) , which is the UGLBXII density.

Structural properties
If X~ UGLBXII ( ) ,,    , the survival function, failure rate function, cumulative hazard function, reverse hazard function and elasticity are given, respectively, by The quantile function of the UGLBXII distribution for 01 q  is .
The random number generator of the UGLBXII distribution for U ~Uniform (0,1) is given by , the UGLBXII distribution becomes the unit log-exponential distribution.v).For 1 X − , the UGLBXII distribution reduces to the generalized log Burr XII (GLBXII) distribution [19].vi).For 1 X − and 1  = , the UGLBXII distribution reduces to the log Burr XII (LBXII) distribution.

Plots of the UGLBXII density and failure rate functions
We plot the pdf and failure rate functions of the UGLBXII distribution to the selected parameters values.Figure 1 displays that the UGLBXII density can take various shapes such as decreasing, unimodal, U-shaped, decreasing-increasing-decreasing (inverse N-shaped) and the possible pdf's regions of the UGLBXII density with 1  = .On the other hand, Figure 2 shows that failure rate function can be increasing, bathtub, N-shaped and bimodal.Therefore, the UGLBXII distribution is quite flexible and can be applied for numerous data sets.

Mathematical properties
We present some properties such as ordinary moments, generalized TL moments, conditional moments, reliability and uncertainty measures.We also present the bivariate and multivariate-type distributions.

Moments
The moments are significant tools for statistical analysis in pragmatic sciences.The r th ordinary moment of X is , we can write   ( ) The Mellin transformation =  is applied to get the moments of X as of the UGLBXII distribution for selected values of ,,    are listed in Table 1. Figure 3 shows the skewness and kurtosis of the distribution for the selected parameters values with 1.

Figure 3．
The skewness and kurtosis of the distribution for the selected parameters values with 1.

The generalized TL-moments
Elamir and Seheult [20] developed generalized TL-moments.The expression for r th generalized TL-moment is The pdf of ( rm +−) th order statistic : r m r m n X + − + + for the UGLBXII distribution is ( ) The k th moment about the origin of : r m r m n X + − + + for the UGLBXII model is , we arrive at The r th generalized TL-moment is obtained as The r th generalized TL-moment can be used to obtain the r th L-moment (0, 0), LL-moments (0, n), TL-moment (1, 1) and LH-moment (m, 0) for the UGLBXII distribution.The first two moments are employed to compute the location and dispersion of the data, respectively.
measures for skewness and kurtosis, respectively.

The TL-moments (1,1)
When only the extreme observations are trimmed from the array sample, then the r th generalized TL-moment befits r th TL-moment (m=n=1).The r th TL-moment is obtained as follows The L-moments [21] When no value is trimmed from the array sample, then the r th generalized TL-moment becomes r th L-moment (m=n=0).L-moments are used for estimation of the parameters.The r th L-moment for the UGLBXII distribution is The LH-moments [22] When lowermost m values are trimmed from the array sample, then the r th generalized TL-moment becomes r th LH-moment (m, 0).The r th LH-moments (m, 0) gives more weight to the upper part of data.The r th LH-moment for the UGLBXII distribution is

The LL-moments [23]
When the uppermost n values are trimmed from the array sample, then the r th generalized TL-moment becomes r th LH-moment (0, n).The r th LL-moments (0, n) gives more weight to the lowermost part of data.The r th LL-moment for the UGLBXII distribution is

Conditional moments
The r th conditional moment of X is ( ) ( ) ( ) where ( ) Sz is survival function and ( ) ;.,.Bz is the incomplete beta function.The k th conditional moment of X is where ( ) Fzis cdf and ( ) The mean deviation about the mean ( ) and about the median ( ) , respectively, where , where ( ) q Q p = .

Reliability in multicomponent stress-strength model
Consider a system with  identical elements, out of which s elements are operative.Let X , 1, 2... i i =  represent strengths of  elements with the cdf F while, the stress Y enforced on the elements has the cdf G.The strengths   and stress Y are independently and identically distributed (i.i.d.).The probability that system operates properly, is the reliability of the system, i.e., , [at the minimum" " (X , 1, 2.. Then, we can write this probability (from [24]) as follows:    with unknown 12 and  , common ,  where X and Y are independently distributed.The reliability in multicomponent stress-strength model for the UGLBXII distribution is ( ) The probability in ( 7) is called the reliability in a multicomponent stress-strength model.For s= =1, the multicomponent stress-strength model reduces to the stress-strength model [25] as ) , where  1 +  2 > 0.

Uncertainty measures
The measure of uncertainty of a random variable (rv) is called entropy.Rényi entropy generalizes Hartley, Min, Shannon and collision entropies.Entropies are useful to study daily temperature instabilities (climatic), abnormal diffusion, DNA structures, information content gestures, heart rate variability (HRV) and cardiac autonomic neuropathy (CAN).Here, we study Shannon entropy, Awad entropy, Rényi, Q, Havrda, Chavrat and Tsallis-entropies.

Bivariate and multivariate extensions of the UGLBXII distribution
Here, we derive the bivariate UGLBXII model via Morgenstern (Mor) family and via Clayton family [31].

Bivariate UGLBXII distribution via Mor family
The cdf of the bivariate UGLBXII model via Mor family for random vector ( )

Bivariate UGLBXII distribution via Clayton family
The cdf of the bivariate UGLBXII model via Clayton family for random vector ( ) ,,    .Then, setting ( ) The cdf of the bivariate UGLBXII distribution via Clayton family for random vector (W1, W2) is

The multivariate extension
Here, we derive the multivariate UGLBXII distribution.A straightforward d-dimensional extension from the above will be ( ) ( ) In future works, we could study various characteristics of the bivariate and the multivariate extensions of the UGLBXII model.

Characterizations
In this section, we characterize the UGLBXII distribution via; (i) ratio of truncated moments; (ii) reverse hazard function and (iii) Elasticity function.

Ratio of truncated moments
We employ ratio of truncated moments of X using a Theorem due to Glänzel [32] to characterize the UGLBXII distribution.0,1 → be a continuous rv and let ( ) and ( ) ( ) According to the Glänzel Theorem, the rv X has pdf (4), iff the function ( ) Proof.For the rv X with the pdf (4), we have ( ) Therefore, in light of Theorem of Glänzel [32], X has pdf (4).X: 0,1 → be a continuous rv and let ( ) Then, X has pdf (4) iff the functions ( ) x  and ( ) where D is a constant.Then, the reverse hazard function F r satisfies the differential equation

( )
Let X: 0,1 → be continuous rv.The pdf of X is (4) iff its reverse hazard function, F r satisfies the first order differential equation Proof.If X has pdf (4), then (10) holds.Now if (10) which is the reverse hazard function of the UGLBXII distribution.Then, the elasticity function ( )

Different estimation methods
In this section, we propose various estimators for estimating the unknown parameters of the UGLBXII distribution.We discuss maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimation methods and compare their performances on the basis of a simulated sample from the UGLBXII distribution.The details are as follows.

Maximum likelihood estimation
In this subsection we estimate the parameters of the UGLBXII distribution via the method of maximum likelihood estimation (MLE).Let 12 ,, The elements of the score vector, ( ) , are given as 2 1 ln ln 1 We can obtain the estimates of the unknown parameters by setting the score vector to zero,

Maximum product spacing estimates
The maximum product spacing (MPS) method is an alternative method to MLE for parameter estimation.This method was proposed by Cheng and Amin [33,34] as well as independently developed by Ranneby [35] as an approximation to the Kullback-Leibler measure of information.This method is based on the idea that differences (spacings) between the values of the cdf at consecutive data points should be identically distributed.Let ( )  be ordered sample of size n from UGLBXII distribution.The geometric mean of the differences is given by 1 1 1 The maximum product spacing estimates (MPSEs), say respectively, where ( )

Least squares estimates
be ordered sample of size n from UGLBXII distribution.Then, the expectation of the empirical cumulative distribution function is defined as The least square estimates (LSEs) say, LSE , ˆLSE and ˆLSE  , of ,  and  are obtained by minimizing Therefore, LSE , ˆLSE and ˆLSE  can be obtained as the simultaneous solution of the following non-linear equations: where the ( ) and ( ) are defined before.

Weighted least squares estimates
be ordered sample of size n from the UGLBXII distribution.The variance of the empirical cumulative distribution is defined as Ξ 0 Ξ 0

The Cramer-von Mises estimations
The Cramer-von Mises (CVM) minimum distance estimates, CVM CVM , and CVM  , of ,  and  are obtained by minimizing ( ) Therefore, the CVM CVM , and CVM  can be obtained as the simultaneous solution of the following non-linear equations: We refer the interested readers to Chen and Balakrishnan [37] for AD and CVM goodness-of-fits statistics.To solve the above equations, Eqs ( 13), ( 15), ( 16)-( 19) can be optimized either directly by using the R (optim and maxLik functions), SAS (PROC NLMIXED) and Ox package (sub-routine Max BFGS) or the non-linear optimization methods such as the quasi-Newton procedure to numerically optimize the ( )

Simulation experiments
In this Section, we perform a simulation study by using the UGLBXII to see the performance of the above estimators corresponding to this distribution and obtain the graphical results.We generate N=1000 samples of size n=20, 25, …, 1000 from the UGLBXII distribution with true parameter values 1, 3   =  = and 0.5

=
. The random numbers generation is obtained by its quantile function.In this simulation study, we calculate the empirical bias and mean square errors (MSEs) of all estimators to compare in terms of their biases and MSEs with varying sample size.The empirical bias and MSE are calculated by (for ,, respectively.We expect that the MSEs and biases are near zero.All related to estimations were obtained using optim-CG routine in the R programme.
The results of this simulation study are shown in Figures 4-6.These figures show that all estimators are to be consistent, since the MSE and biases decrease with increasing sample size.It is clear that the estimates of parameters are asymptotically unbiased.For all parameters estimations, the performances of all estimators are close.

Data Applications
We verify the potentiality of the UGLBXII model via two real data sets.The first data set represents monthly water capacity [38,39] for February (1991 to 2010) at Shasta reservoir (California, USA).The second data set is about the proportion of total milk production in the first birth of 107 cows (Carnaúba farm, Taperoá, Brazil) from SINDI race [40].Both data sets are converted to the interval (0, 1) using a transformation ( ) ( ) . We compare the UGLBXII distribution with models such as unit Log Burr XII (ULBXII), unit Log Lomax (ULLOM), unit modified Burr XII (UMBXII), Kumaraswamy (Kum), unit Weibull (UW), unit inverse Weibull (UIW), unit Gompertz (UGP) and beta.For the selection of the best fit distribution, we compute the estimate of likelihood ratio statistic ( 2 − ), Akaike information criterion (AIC), corrected Akaike information criterion (CAIC) and Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC) for all competing and sub-models.We also compute the MLEs along with their standard errors (SEs) in parentheses.Table 2 reports some descriptive measures for two data sets.
Table 2 shows that the monthly water capacity data set is significantly left-skewed, with high positive kurtosis.About the proportions of total milk production data set, it is left-skewed, with somewhat positive kurtosis.Figure 7 shows that both data sets are left-skewed.The nature of the two data sets differs in numerous features.Here, we study the statistical analysis by total time on test (TTT) for the two data sets in Figure 8.
In Figure 8, the TTT plots for both data sets are concave which suggests increasing failure intensity.So, the UGLBXII distribution is suitable to model these data sets.From the results presented in Tables 3 and 4, we see that the UGLBXII distribution is considered as the best model for monthly water capacity data because the values of all statistics are smaller for the proposed model.Figure 9 confirms this claim via the graphical display of fitted pdf, estimated cdf and PP plots of the UGLBXII distribution.From this figure, we can infer that the proposed model is closely fitted to monthly water capacity data.From the results presented in Tables 5 and 6, we see that the UGLBXII distribution is considered as the best model for proportions of total milk production data because the values of all statistics are smaller for the proposed model.

Conclusions
We derive and study the UGLBXII distribution.Some mathematical properties such as random number generator, sub-models, ordinary moments, generalized TL moments, conditional moments, reliability, uncertainty measures and characterizations are presented.We employ six different estimation methods to estimate the model parameters.We perform simulation studies on the basis of the graphical results to see the performances of the estimators of the UGLBXII distribution.We verify the potentiality of the UGLBXII distribution via two applications.In conclusion, it is expected that the UGLBXII model is the best fit for the monthly water capacity and the proportions of total milk production data analysis.The potentiality of the UGBLXII model illustrates that it is flexible, competitive and parsimonious to other existing distributions.Therefore, it should be included in the

Figure 1 .
Figure 1.Plots of pdf (left) and possible pdf's regions (right) of the UGLBXII density.

Figure 2 .
Figure 2. Plots of the hrf of the UGLBXII distribution.
and m and n are the probable trimming lowermost and uppermost values, respectively.
from(6).For specific probability p, Lorenz and Bonferroni curves are computed as ( ) rv with a twice continuously differential cdf ( ) Fx.
rv with a twice continuously differential cdf ( ) Fx.
be a random sample from the UGLBXII distribution with observed values 12 ,, be the vector of the model parameters.The log likelihood function for Ξ may be expressed as

, 2 z
 and  parameters.Since, these equations include non-linear equation systems, they must be solved with numerically methods.The equation (13) can be directly maximized by the different packet programs such as R, S-Plus, Mathematica and SAS.Moreover, for the UGLBXII distribution, all the second order derivatives exist.The interval estimation of the model parameters requires the 33  observed information matrix ( )   regularity conditions, can be used to provide approximate confidence intervals for the unknown parameters, where ( ) Ξ J is the total observed information matrix evaluated at Ξ .Then, approximate  percentile of the standard normal model and The ˆii J s are the i th diagonal elements of the ( ) from the authors.

Figure 7．
Figure 7． Boxplots of the (left) monthly water capacity (right) proportions of total milk production.

Figure 8 .
Figure 8. TTT plots of the (left) monthly water capacity (right) proportions of total milk production.

Figure 9 .
Figure 9. Fitted pdf (left), cdf (center) and PP (right) plots of the UGLBXII model for monthly water capacity.
Figure 10 confirms this claim via the graphical display of fitted pdf, estimated cdf and PP plots of the UGLBXII distribution.From this figure, we can infer that the proposed model is closely fitted to proportions of total milk production data.

Figure 10 .
Figure 10.Fitted pdf (left), cdf (center) and PP (right) plots of the UGLBXII model for milk production

Table 3
reports the MLEs, SEs (in parentheses) and measures W*, A*, K-S (p-values) for monthly water capacity data.Table 4 displays measures 2 − , AIC, CAIC, BIC and HQIC for water capacity data.

Table 3 .
MLEs, SEs and W*, A*, K-S (p-values) for monthly water capacity data.

Table 5 .
MLEs, SEs and W*, A*, K-S (p-values) for proportions of total milk production.