The Logarithmic Burr-Hatke Exponential Distribution for Modeling Reliability and Medical Data

In this work, we introduced a new one-parameter exponential distribution. Some of its structural properties are derived. The maximum likelihood method is used to estimate the model parameters by means of numerical Monte Carlo simulation study. The justification for the practicality of the new lifetime model is based on the wider use of the exponential model. The new model can be viewed as a mixture of the exponentiated exponential distribution. It can also be considered as a suitable model for fitting right skewed data. We prove empirically the importance and flexibility of the new model in modeling cancer patients data, the new model provides adequate fits as compared to other related models with small values for W∗ and A∗. The new model is much better than the Modified beta-Weibull, Weibull, exponentiated transmuted generalized Rayleig, the transmuted modified-Weibull, and transmuted additive Weibull models in modeling cancer patients data. We are also motivated to introduce this new model because it has only one parameter and we can generate some new families based on it such as the the odd Burr-Hatke exponential-G family of distributions, the logarithmic BurrHatke exponential-G family of distributions and the generalized Burr-Hatke exponential-G family of distributions, among others.


Introduction
A random variable (RV) X is said to have the exponential (E) distribution if its probability density function (PDF) and cumulative distribution function (CDF) are given by respectively, where λ > 0 and x > 0. Maniu and Voda (2008) introduced and studied the Burr-Hatke (BH) distribution with CDF and PDF given by respectivily.In this work, we propose a new one parameter E distribution called the Logarithmic Burr-Hatke exponential (LBH-E) distribution using the BH model.
The justification for the practicality of the one-parameter LBHE lifetime model is based on the wider use of the E model.
The new model can be viewed as a mixture of the exponentiated E (Exp-E) distribution.It can also be considered as a suitable model for fitting right skewed data (see Figure 1).The hazard rate function (HRF) of the new model exhibits the constant and the decreasing shapes (see Figure 1).We prove empirically the flexibility and the importance of the oneparameter LBHE model in modeling a real data set, the new model provides adequate fits as compared to other related models with small values for W * and A * and it is much better than the Weibull, Modified beta-Weibull, Transmuted modified-Weibull, exponentiated transmuted generalized Rayleig and transmuted additive Weibull models in modeling cancer patients data.
The paper is outlined as follows.In Section 2, we present the formulation, expansions, graphical presentation and motivation for the logarithmic Burr-Hatke exponential model .In Sec. 3, we derive some its mathematical properties.In Sec. 4, the model parameter is estimated by using maximum likelihood (ML) method.In Sec. 5, we assess the performance of the maximum likelihood estimators by means of a simulation study.An application to real data is given in Sec.6 to illustrate the flexibility of the logarithmic Burr-Hatke exponential model.Finally, some concluding remarks are presented in Sec. 7.

Formulation and Motivation
Starting from (2) and replacing t by the argument , then the CDF of the new one parameter LBH-E distributions is defined by The PDF corresponding to (2) is The reliability function (rf) and HRF of new LBHE model are given by We are also motivated to introduce this new model because it has only one parameter and we can generate some new families based on it such as the the odd Burr-Hatke exponential-G (OBrHE-G) family, the logarithmic Burr-Hatke exponential-G (LogBrHE-G) family and the generalized Burr-Hatke exponential-G (GBrHE-G) with the following CDFs respectively.Where G (x; Θ) is the baseline CDF and Θ is the vector of parameters.As a future work we will study these new families in a separate articles.

Expansions
Consider the following expansions, The CDF (2) can be rewritten as where applying (5) for 1 + λ x, still in (2), we get where c 0 = a o /b 0 and, for k ≥ 1, we have Finally the CDF (2) can be expressed as where is the CDF of the Exp-E model with power parameter η.By differentiating (6), we obtain the same mixture representation where is the Exp-E PDF with power parameter η.Equation ( 7) reveals that the PDF of the LBHE is a linear combination of Exp-E density.Thus, some of its structural properties can be immediately

Graphical Presentation
In this subsection, we show the flexibility of the LBHE model using the graphical presentation for the PDF and HRF as follows

Asymptotics
Let a = inf{x|F(x) > 0} the asymptotics of CDF, PDF and HRF as x → a are given by and The asymptotics of CDF, PDF and HRF as x → ∞ are given by The effect of the parameters on tails of distribution can be evaluated by means of above equations.

Moments
The r (th) moment (about the zero) of X is given by Then, we have where which can be used to get the moment about the mean.The skewness ( √ β 1 ) and kurtosis (β 2 ) measures also can be calculated from the ordinary moments using the well-known relationships.For the skewness and kurtosis coefficients, we have respectively.Numerically, we prove that the LBHE distribution provides better fits than five models with (two, three, four and five parameters) (see Section 6) so the new model is a good alternative for modeling cancer patients data.Further, the LBHE density can be only right-skewed (see Figure1 and column 4 in Table 1).Whereas the LBHE HRF can be monotonically decreasing and constant (see Figure 2).The skewness of the LBHE distribution can range in the interval (0.45, 2.53), whereas the kurtosis of the LBHE distribution varies only in the interval (1.5, 13) also that the mean of X decreases as λ increases, the skewness is always positive (see Table 1).

Generating Function
The moment generating function Clearly, the first one can be derived using Equation (7) as

Order Statistics
Suppose X 1:n , X 2:n , . . ., X n:n , is a random sample (RS) from the LBHE model.Let X i:n denote the i (th) order statistic.The PDF of X i:n can be expressed as Following Gradshteyn and Ryzhik (2000) the result 0.314 for a power series raised to a positive integer n (for n ≥ 1) where the coefficients c n,i ∀ i = 1, 2, . . .are determined from the recurrence equation (with c n,0 = a n 0 ) The i (th) order statistic of the LBHE model can be expressed as where π τ+k+1 (x) denotes the Exp-E density function with parameter (τ + k + 1) and and d τ+1 is given in subsection 3.2 and the quantities ς j+i−1,k can be determined with f j+i−1,0 = d j+i−1 0 and recursively for k ≥ 1 the moments of X i:n can be expressed as where

Quantile Spread Ordering
The quantile spread (QS) of the RV W ∼LBHE(λ) having CDF (2) is given by where is the survival function.The QS of a any distribution describes how the probability mass is placed symmetrically about its median and hence it can be used to formalize concepts such as peakedness and tail weight traditionally associated with the kurtosis.So, it allows us to separate concepts of the kurtosis and peakedness for asymmetric models.
Let W 1 and W 2 be two RVs following the LBHE model with QSs QS W 1 and QS W 2 , respectively.Then W 1 is called smaller than W 2 in quantile spread order, denoted as Following are properties of the QS order can be obtained: • The order ≤ (QS ) is dilative • Let F W 1 and F W 2 be symmetric, then • The order ≤ (QS ) implies ordering of the mean absolute deviation around the median, say ψ(W i ), i = 1, 2. where Finally

Moments of Residual Life
The n (th) moment of the residual life, say ], the n (th) moment of the residual life of X is given by for the LBHE model

Moment of the Reversed Residual Life
The n (th) moment of the reversed residual life, say ] uniquely determines the CDF, the we have The n (th) moment of the reversed residual life of X becomes

Estimation
Let x 1 , . . ., x n be a random sample from the LBHE distribution the log-likelihood function is given by The maximum likelihood estimation (MLE) estimate λ of λ is the solution of the non-linear equation where x is the sample mean.To obtain the MLE of λ, we can maximize (12) directly with respect to λ or we can solve the non-linear equation ( 13).Note that maximum likelihood estimation (MLE) of the λ cannot be solved analytically; numerical iteration techniques, such as the Newton-Raphson algorithm, are adopted to solve the log-likelihood equation for which (12) is maximized.

Simulation Study
Here, in this Section, we will perform the simulation study using the LBHE distribution.To see the performance of MLEs of this distribution, we generate 1, 000 samples of sizes 20, 50, 100 and 300 from LBHE using inverse of the its CDF.We also compute the biases and mean squared errors (MSE) of the MLEs with respectively.To obtain the inverse of the CDF, We used the uniroot routine in the R programme for the random generation and used the optim routine for MLEs.Results of the simulation are reported in Table 2. From Table 2, we observe that when the sample size increases, biases  We provide an application to show empirically the potentiality of the new model.In order to compare the fits of the LBH-E distribution with other competing distributions, we consider the Cramér-von Mises (W * ) and the Anderson-Darling (A * ) statistics.These two statistics are widely used to determine how closely a specific CDF fits the empirical distribution of a given data set.These statistics are given by and ) respectively, where ) and the y j 's values are the ordered observations.The smaller these statistics are, the better the fit.The required computations are carried out using the R software.The MLEs and the corresponding standard errors (in parentheses) of the model parameters are given in Table 2.The numerical values of the statistics W* and A* are listed in Table3.The Estimated PDF, CDF, HRF and P-P plot of the new model are displayed in Figure 3.The cancer patients data set represents the remission times (in months) of a random sample of 128 bladder cancer patients as reported in Lee and Wang (2003).This data is given in Appendix A. We will compare the fits of the LBHE distribution with other competitive models, namely: the Weibull W (Weibull, 1951), the etransmuted additive Weibull distribution (TA-W) (Elbatal and Aryal, 2013), the Modified beta-Weibull (MB-W) (Khan, 2015), the transmuted modified-Weibull (TM-W) (Khan and King, 2013) and the exponentiated transmuted generalized Rayleigh (ETGR) (Afify et al., 2015) distributions.0.12511 0.76028 MB-W (α,β,a,b,c) 0.10679 0.72074 TA-W (α,β,η,θ,λ) 0.11288 0.70326 ETG-R (α,β,δ,λ) 0.39794 2.36077 Based on Table 4 we note that the new model is much better than the TM-W, TA-W, MB-W, ETG-R and W, models with small values for W * and A * in modeling cancer patients data.

Figure 1 .
Figure 1.The PDF of the LBHE model

Figure 2 .
Figure 2. The HRF of the LBHE model

Figure 3 .
Figure 3.Estimated PDF, CDF, HRF and P-P plot for the cancer patients data

Table 1 .
Mean, variance, skewness and kurtosis of the LBHE distribution with different values of λ

Table 2 .
Emprical mean, Bias and MSE of the estimator λ

Table 3 .
MLEs (standard errors in parentheses) for cancer patients data

Table 4 .
W * and A * for cancer patients data