The modified beta transmuted family of distributions with applications using the exponential distribution

In this work, a new family of distributions, which extends the Beta transmuted family, was obtained, called the Modified Beta Transmuted Family of distribution. This derived family has the Beta Family of Distribution and the Transmuted family of distribution as subfamilies. The Modified beta transmuted frechet, modified beta transmuted exponential, modified beta transmuted gompertz and modified beta transmuted lindley were obtained as special cases. The analytical expressions were studied for some statistical properties of the derived family of distribution which includes the moments, moments generating function and order statistics. The estimates of the parameters of the family were obtained using the maximum likelihood estimation method. Using the exponential distribution as a baseline for the family distribution, the resulting distribution (modified beta transmuted exponential distribution) was studied and its properties. The modified beta transmuted exponential distribution was applied to a real life time data to assess its flexibility in which the results shows a better fit when compared to some competitive models.


Introduction
Due to complexity in distributions of real life data, there is need for developing distributions that are more flexible in fitting these data. The flexible distributions can be derived by addition of new parameters to the baseline distributions. Over years, many family of distributions has been developed. Examples like Beta-G [1], Weibull-G [2], Beta-Weibull-G [3], Modified Beta-G [4], Cubic Transmuted -G [5], Gompertz-G [6], Odd Lindley-G [7] e.t.c. Through these families of distributions, several models have been developed and applied to real life situations. [8] derived the transmuted-G family of distribution. In their work, they considered a baseline cumulative distribution function (cdf) G(x;γ) with corresponding probability density function (pdf) g(x;γ) and obtained the c.d.f of transmuted-G family of distribution P(x;γ) as with the probability distribution function p.d.f as where ϕ is the transmuted parameter. When ϕ = 0 in Eqs 1 and 2, gives the p.d.f and the c.d.f of the baseline distribution.
In this work, a new family of distribution was derived that will be more flexible than the transmuted-G family of distribution by the addition of three more parameters to the transmuted-G family of distribution [8]. This concept is inspired by the work of Nadarajah et al. (2014), who obtained the modified beta-G families of distributions. This study will derive another family of distributions called the modified beta transmuted family of distributions which is more flexible and model fitting than that of Nadarajah et.al.(2014). Another important and crucial motivation is the study of modeling and analyses of lifetime data. The fitness of the assumed lifetime distribution, on the other hand, has a significant impact on the quality of statistical analyses. In a bid to achieve this, the modified beta-G family of distribution [4] was used to obtain the modified beta transmuted family of distribution. Given the c.d.f of baseline distribution G(x;γ), the c.d.f of the modified beta-G family A(x;γ) of distribution is given as where r ¼ tðGðx;gÞÞ 1þððtÀ 1ÞGðx;gÞÞ and B(r; μ, σ) is an incomplete beta function. where μ and σ are shape parameters, I tGðx;gÞ 1þððtÀ 1ÞGðx;gÞÞ ða; bÞ is the incomplete beta function ratio. If μ = σ = τ = 1, it gives the g (x;γ) and G(x;γ) of baseline distribution. Therefore, in the section 2, the new family of distribution was derived. In Section 3, the mixture representation of the p.d.f and the c.d.f of the family of distribution was obtained, section 4 studied the statistical properties and the estimation of parameters of the family of distribution. Then, in Section 5, the family of distribution was studied using the exponential distribution as the baseline distribution. The properties were studied and applied to a real data to assess its performance when compared to some sub-models. Section 6 gives the conclusion of the work.

Sub-models of the MBTG family of distributions
In this section, three special models of the MBTG family of distribution is presented. These models generalize some models that are already existing in literatures. The models have baselines of Gompertz (G), Exponential(E) and Lindley(L) distributions.

Modified Beta Transmuted Lindley (MBTL) distribution
The pdf and cdf of lindley distribution are given as Now, the pdf f MBTL and hazard function h MBTL MBTL distribution is given as ðBðm; sÞ À Bðf ; m; sÞÞ 1 À ð1 À tÞ 1 À e À bx ð1þbþbxÞ The MBTL distribution includes the Transmuted Lindley(TL) [16] when θ = z = ϕ = 1. For θ = α = ρ = 1, the MBTL becomes Beta Lindley(BL) distribution [17]. For θ = z = 1, MBTL reduces to Exponentiated Transmuted Lindey(ETL) distribution [18]. Plots of the density function and the hazard function of the MBTL with various assigned parameter values are shown in Figs 5 and 6. From the plots of the submodels of the MBTG distribution, it shows that the proposed family of distribution can be rightly skewed, symmetric, reverse J shape and other forms of shape inferring that this family of distribution will be suitable in modeling different form of real life situations due to its flexibility.

Mixture representation
In this section, the mixture representation of the p.d.f of the MBTG family of distribution is derived. Having this expression simplifies the derivation of some statistical properties of MBTG family.
Using the binomial expression, as written in Wolfram Statistics such that |z| < 1 and k > 0 real non-integer. From Eq 8, Considering By the application of the binomial expression, Eq 12 is Likewise considering and using the binomial expression, Eq 14 is Applying Eqs 13 and 15 to Eq 8, the mixture representation of the p.d.f of the MBTG family is Furthermore, Eq 16 can written in form of the exponentiated transmuted G as where From Eq 17, the corresponding c.d.f of the MBTG family of distribution is β μ+k+l is the c.d.f of the exponentiated transmuted-G family of distribution with index parameters μ+k+l.

Statistical properties
In this section, some statistical properties of the MBTG family of distribution are studied. The properties include order statistics, moments, moment generating function, shanon entropy and the quantile function.

Order statistics
Order statistics make their appearance in many areas of statistical theory and practice. Let X, X 2 , X 3 , X 4 , . . ., X n be random sample generated from the MBTG family of distributions. The p. d.f of i th order statistic, X i:n , can be written as In terms of the mixture representation, order statistics of the MBTG family of distribution is a ði:nÞ ðx; gÞ ¼ n and the first order marginal p.d.f and last order marginal p.d.f given as

Moments
The r th moment of X, say c 0 r follows from Eq 17 as Therefore E½P r mþkþl � is the r-th moment of the exp-Transmuted G family. The n th central moment of X, say M n is given by

Moment generating function
Using the expression as in Eq 17, the moment generating function of the MBTG family of distribution is where M μ+k+l (t) is the moment generating function of the exp-Transmuted G family of distribution.

Quantile function
The quantile function of the distribution is discussed here. If X MBTG(μ, σ, τ, ϕ, γ), then the quantile function of X can be simulated as ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where
Under conditions that are fulfilled for parameters, the asymptotic distribution of ffi ffi ffi n p ðd À dÞ is N 6 ð0; JðdÞ À 1 Þ distribution of δ can be used to construct approximate confidence intervals and confidence regions for the parameters and for the hazard and survival functions. The asymptotic normality is also useful for testing goodness of fit of the beta type I generalized half logistic distribution and for comparing this distribution with some of its special sub-models using one of these two well known asymptotically equivalent test statistics-namely, the likelihood ratio statistic and Wald statistic. An asymptotic confidence interval with significance level τ for each parameter δ i is given by where Jd ;d is the i t h diagonal element of K n ðdÞ À 1 for i = 1, 2, 3, 4, 5, 6 and z τ/2 is the quantile of the standard normal distribution.

The modified beta transmuted exponential distribution
In this section the exponential distribution is considered as a baseline distribution of the MBTG family of distribution. The exponential has been studied and many generalizations have been made by different authors. Some of these works employed the use of transmutation approach to derived the generalization of the exponential distribution. Such works includes the transmuted exponential, exponentiated transmuted exponential, exponentiated cubic exponential e.t.c. The p.d.f of the exponential distribution is with c.d.f as where λ is a scale parameter. Therefore inserting the Eq 38 into Eq 8, the p.d.f of the Modified Beta Transmuted Exponential Distribution q E (x; γ) is derived as where Mðx; lÞ ¼ tð1À e À lx þ�e À lx À �e À 2lx Þð1À �þ2�e À 2lx Þ 1þððtÀ 1Þð1À e À lx Þð1þ�e À lx ÞÞ and B(M(x; γ);μ, σ) is an incomplete beta function.

Mixture representation of the MBTED
In this subsection, the mixture representation of the MBTED is derived. This will help derive the analytical expression of the distribution and will be useful in obtaining some properties of the MBTED. Inserting Eqs 38 and 39 in Eq 16, the mixture representation of the p.d.f of MBTED is obtained as Re-writing Eq 42 in terms of the p.d.f of exp-transmuted exponential distribution, it gives where χ μ+k+l is the p.d.f of the exponentiated transmuted exponential distribution with index parameters μ+k+l as derived by [15]. From Eq 43, the corresponding c.d.f of the MBTG family of distribution is and the hazard function as hðx; lÞ ¼ t m e À lx ð1 À � þ 2�e À 2lx Þð1 À e À lx þ �e À lx À �e À 2lx Þ mÀ 1 ðe À lx À �e À lx þ �e À 2lx Þ ðBðMðx; lÞ; m; sÞÞ½1 À ð1 À tÞð1 À e À lx þ �e À lx À �e À 2lx Þ� ð46Þ

Quantile function
Inverting q E (x; λ) = U, the quantile function of the MBTED is determined as

Order statistics of MBTED
Let X 1 , X 2 , X 3 , X 4 , . . ., X n be random sample generated from the MBTED distributions. The p. d.f of i th order statistic, X i:n , can be written as Inserting Eqs 40 and 41 in 48, the order statistics of the MBTED has the expression as Bðm; sÞ½1 À ð1 À tÞð1 À e À lx þ �e À lx À �e À 2lx Þ� In terms of the mixture representation, order statistics of the MBTG family of distribution can be written as and the first order marginal p.d.f and last order marginal p.d.f given as

Moments of MBTED
The moments of the Exponential Transmuted exponential distribution, as established by [15] is the moments of the MBTED is derived as From the expression in Eq 54, the mean E[X], second moment E[X 2 ], Variance, Kurtosis and Skewness can be derived.

Moment generating function of MBTED
Using the moment generating function as established by [15], to have the moment generating function of MBTED as

Shanon entropy
Entropy measures the uncertainty of a random variable X. The entropy of the MBTED is This can be estimated iteratively.

Simulation study
In this section, a simulation study was performed using the MBTED in orfer to assess the performance of the maximum likelihood estimates of the distribution. To conduct this, 1000 samples of sizes 30,100,200 were generated from the quantile function of the MBTED for parameter values (2,3,2.5,-0.7,2),(3.2,1.3,1.5,0.5,0.5) and (3,3,3.5,0.2,2). The results of the simulation study are presented in Tables 1-3. These results show that the estimates for the mean is close to the parameter values as the sample sizes increase. Also, the mean square error decreases as the sample size increases.

Application to real data
In this section, applications to two real data(Medicine and Behavioral datasets) are presented to illustrate the importance and the fit of the MBTED. The maximum likelihood estimates (M. L.E) of the distribution and that of the competitive distributions will be obtained. The goodness of fit of the distributins was assessed using the log-likelihood, Akaike's information  [20], Beta Burr XII [21], Modified Beta Gompertz(MBG) [22], Exponential, Exponentiated Transmuted Exponential(ETED) [15]. The p.d.fs of these distributions are as follows: EGW ¼ abðtg t x tÀ 1 e À ðgxÞ t Þð1 À e À ðgxÞ t Þ aÀ 1 1 À ð1 À e À ðgxÞ t Þ a � � bÀ 1

Survival times of breast cancer patients.
The real data set represent the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938 [23]. The data are: 0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6. 8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21. Table 4 shows the summary statistics for the real data. Fig 7 is the TTT plots of the dataset which shows a non decreasing curve. Fig 8 shows the fitted plot of the data using the MBTED and the competitive distributions. This indicated that the model fits the data. Table 5 reveals that the modified beta transmuted exponential distribution gives the best fit when compared to its submodels, due to lowest values of AIC, BIC, CAIC and HQIC therefore making it the preferred model to consider for this data. 5.9.2 Recidivism failure time data. The second data consists of 61 observed recidivism failure times (in days) revealed by correctional institutions in Columbia USA by [24].  Table 6 shows the summary statistics for the real data. Fig 9 is the TTT plots of the dataset which shows a non decreasing curve. Fig 10 shows the fitted plot of the data using the MBTED and the competitive distributions. This indicated that the model fits the data. Table 7 reveals that the modified beta transmuted exponential distribution gives the best fit when compared to its submodels, due to lowest values of AIC, BIC, CAIC and HQIC therefore making it the preferred model to consider for this data. Clearly, based on the values of the criteria used, all of the two applications provided indicate that the MBTED distribution is superior to the other models. It has lower values for the LL, AIC, CAIC, BIC, and HQIC than it does for the others.

Conclusion
In this article, a new family distribution called the Modified Beta Transmuted-G family is introduced. The properties of the family such as moments, generating functions, quantile function, random number generation, reliability function and order statistics were extensively studied. Furthermore, expressions for the the maximum likelihood estimation of parameters for the Modified Beta Transmuted-G family of distribution were derived. An exponential distribution was applied as a baseline distribution for the modified beta transmuted-G to derive the modified beta transmuted exponential distribution. The properties of the modified beta transmuted exponential distribution were also been discussed and estimation of parameters done using the maximum likelihood estimation method. The modified beta transmuted exponential distribution was applied on a real data set in which it was observed that the modified beta transmuted exponential distribution provides better fit than its submodels. We anticipate that the proposed model will be used to investigate a wider range of applications in diverse areas of applied research in the future, and that it will be considered a superior alternative to the baseline model. The model could also be applied in other fields such as machine learning and artificial intelligence.