General Results for the Transmuted Family of Distributions and New Models

The transmuted family of distributions has been receiving increased attention over the last few years. For a baseline G distribution, we derive a simple representation for the transmuted-G family density function as a linear mixture of the G and exponentiatedG densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set.


Introduction
Adding parameters to a well-established distribution is a time honored device for obtaining more flexible new families of distributions.Shaw and Buckley [1] pioneered an interesting method of adding a new parameter to an existing distribution that would offer more distributional flexibility.They used the quadratic rank transmutation map (QRTM) in order to generate a flexible family of distributions.The generated family, also called the transmuted extended distribution, includes the parent distribution as a special case and gives more flexibility to model various types of data.
In the last three years, there has been a growing interest in transmuted distributions and several of them have been investigated.A significant amount of work has been attributed towards developing a new transmuted model and subsequently discussing its utilities as enhanced flexibility in modeling various types of real life data, where the parent model does not provide a good fit.Aryal and Tsokos [2] defined the transmuted generalized extreme value distribution and studied some basic mathematical characteristics of the transmuted Gumbel distribution and its applications to climate data.Aryal and Tsokos [3] presented a new generalized Weibull distribution called the transmuted Weibull distribution.Recently, Aryal [4] proposed and studied various structural properties of the transmuted log-logistic distribution.Khan and King [5] introduced the transmuted modified Weibull distribution, which extends the transmuted Weibull distribution [3], and studied its mathematical properties and maximum likelihood estimation of the model parameters.Elbatal [6] proposed the transmuted modified inverse Weibull distribution.Elbatal and Aryal [7] explored the transmuted additive Weibull model, which extends the additive Weibull distribution and some other distributions using the QRTM method [1].However, several published works did not investigate many properties such as finite mixture of the density function, Rényi and Shannon entropies, extreme values, probability weighted moments (PWMs), and bivariate and multivariate generalization.This paper aims to fill out this gap in the existing literature and contribute with general properties of the transmuted family.
This vast amount of literature merits for a detailed study for the most general transmuted family of distributions, which is our major motivation to carry out this work.In this paper, we derive general mathematical properties for the transmuted family, which hold for any baseline distribution, such as the ordinary, central, and incomplete moments, quantile and generating functions, mean deviations, Rényi and
A random variable  has the transmuted-G () family if the pdf and cdf are defined through the QRTM method by (for  ∈ where (; ) is the parent cdf and (; ) is the parent pdf.
Both functions depend on the parameter vector .For  = 0, it reduces to the parent model.Hereafter, the random variable  following (1) with parameter  and baseline vector of parameters  is denoted by  ∼ (, ).The computations for fitting family (1) to real data in practical problems can be easily performed using the AdequacyModel script in the R software.
Theorem 1.The density function of  can be expressed as the linear mixture where  = 1 + .
Theorem 1 is important to obtain some measures of  from those of exponentiated distributions.This result plays an important role in the paper, since we can obtain, for example, the moments, generating function, and mean deviations of .Established explicit expressions for these measures can be simpler than using numerical integration.

Asymptotes and Shapes
Proposition 3. The asymptotics of ( 1) and ( 2) as () → 0 are Proposition 4. The asymptotics of ( 1) and ( 2) as  → ∞ are The shapes of the density and hazard functions of  can be described analytically.The critical points of the  pdf are the roots of the equation There may be more than one root to (8).Let () = ( 2 / 2 ) log[()].We have If  =  0 is a root of ( 9), then it corresponds to a local maximum if () > 0 for all  <  0 and () < 0 for all  >  0 .It corresponds to a local minimum if () < 0 for all  <  0 and () > 0 for all  >  0 .It refers to a point of inflection if either () > 0 for all  ̸ =  0 or () < 0 for all  ̸ =  0 .The critical points of the hrf of  are obtained from Again, there may be two roots to (10).Let () = ( 2 / 2 ) log ℎ().We have If  =  0 is a root of ( 11), then it corresponds to a local maximum if () > 0 for all  <  0 and () < 0 for all  >  0 .It corresponds to a local minimum if () < 0 for all  <  0 and () > 0 for all  >  0 .It refers to a point of inflection if either () > 0 for all  ̸ =  0 or () < 0 for all  ̸ =  0 .

Quantile Function and Simulation
The qf of the  family is given by where   (; ) =  −1 (; ) is the inverse of the baseline cdf.
The  family is easily simulated by Algorithm 1. Table 1 gives   (; ) and the corresponding parameters for some special distributions.

Moments and Generating Function
Many of the important characteristics and features of a distribution are determined through the ordinary moments.The th ordinary moment of  is obtained from Theorem 1 as where   ∼ Exp-(; ) for  = 1, 2. Some moments obtained from ( 13) are reported in Table 2.
The central moments (  ) and cumulants (  ) of  follow from (13) as respectively, where  1 =   1 .Further, the skewness and kurtosis are obtained from the third and fourth standardized cumulants  1 =  3 / The moment generating function (mgf) of , say () = (  ), can be expressed from Theorem 1 as where The integrals  0 () and  1 () can be evaluated numerically for most parent distributions.

Mean Deviations
The th incomplete moment of , say   () = ∫  −∞   (; , ), is expressed as where The integral V , () can be determined analytically for some special models with closed form expressions for   (; ) or evaluated at least numerically for most baseline distributions.
It can also be obtained for several baseline  distributions using power series methods.These methods are at the heart of many aspects of applied mathematics and statistics.If this function does not have a closed form expression, it can be expressed as a power series: where coefficients   are suitably chosen real numbers.For some important distributions, such as the normal, Student t, gamma, and beta distributions,   (; ) does not have closed form but it can be expanded as in (20).For example, for the standard normal distribution, coefficients    s are given by where   = 0 (for  = 0, 2, 4, . ..) and   =  (−1)/2 (for  = 1, 3, 5, . ..), and the    s are determined recursively from Then,  0 = 1,  1 = 1/6,  2 = 7/120, and  3 = 127/7560, . . .We consider a result by Gradshteyn and Ryzhik [17] for a power series raised to a positive integer : where coefficients  , (for  = 1, 2, . ..) are determined from the recurrence equation and  ,0 =   0 .Coefficient  , can be obtained from quantities  0 , . . .,   in any analytical or numerical software.Hence, quantity V , () (for  = 1, 2) in ( 19) is given by An important application of the first incomplete moment of  in ( 18) is related to the Bonferroni and Lorenz curves.These curves are very useful in economics, reliability, demography, insurance, and medicine.For a given probability , they are given by () =  1 ()/(  1 ) and () =  1 ()/  1 , where  = (; , ) comes from (12).
The magnitude of dispersion associated with the population can be measured by the totality of deviations from the mean and median.Another application refers to the the deviations about the mean ( 1 = (| −   1 |)) and about the median ( 2 = (| − |)) of  given by respectively, where  is the median of ,   1 = () is determined from (13), (  1 ) is easily evaluated from (2), and Next, we provide two applications of ( 19) by taking for the baseline model the exponential (with parameter ) and standard logistic distributions listed in Table 1.By using the generalized binomial expansion, we obtain V 1, () (for  = 1 and 2) for the transmuted-exponential (TE) (with parameter ) and transmuted-standard logistic (TSL) as respectively.

Special Transmuted Models
The pdf and cdf of  in ( 1) and ( 2) will be most tractable when () and () have simple analytic expressions.In this section, we present two special  models.
The TBXII distribution includes an important special case when  = 1: the transmuted-log-logistic [4] distribution.Further, we obtain the transmuted Lomax [18] distribution when  = 1.Some plots of the TBXII density function are displayed in Figure 1.
The th ordinary moment of the TBXII model can be obtained from (13) as where (⋅, ⋅) is the beta function.() = 1 − (1 −   )  , respectively.Then, the TKw cdf is given by where  ∈ [−1, 1].The corresponding pdf is given by  (; , , ) Ahmad et al. [19] proposed the transmuted Kumaraswamy (TKw) distribution as an extension of the Kw distribution and obtained the density and cumulative functions.However, they did not investigate an application to real data and explore the qf.In Figure 2, we plot the TKw density function for some parameter values.
The th moment of TKw can be obtained from (13) as and the qf is given by (34)

Extreme Values
If  =  −1 ( 1 +  2 + ⋅ ⋅ ⋅ +   ) denotes the sample mean from iid random variables following (2), then by standard central limit theorem √( − ())/√Var() converges in distribution to the standard normal as  → ∞ under suitable conditions.However, one might be interested in the asymptotics of the extreme values  max = max( 1 , . . .,   ) and  min = min( 1 , . . .,   ).We consider the following: (i) Suppose that  belongs to the max.domain of attraction of the Gumbel extreme value distribution.Then, by Leadbetter et al. [20], there must be a strictly positive function, say ℎ(), such that lim for every  ∈ (−∞, ∞).In our case, we have lim for every  ∈ (−∞, ∞).Hence, it follows by Leadbetter et al. [20] that  also belongs to the max.
for some suitable norming constants   > 0 and   .
(ii) Again, suppose that  belongs to the max.domain of attraction of the Fréchet extreme value distribution.By Leadbetter et al. [20], there must exist a  > 0, such that lim for every  > 0. In our case, lim for every  > 0. Hence, it follows by Leadbetter et al. [20] that  also belongs to the max.domain attraction of the Fréchet extreme value distribution with lim for some suitable norming constants   > 0 and   .
(iii) Also, suppose that  belongs to the max.domain of attraction of the Weibull extreme value distribution.
By Leadbetter et al. [20], there must be an  > 0 such that lim for every  < 0. In our case, we have lim for every  < 0. Hence, it follows by Leadbetter et al. [20,Chapter 1] that  also belongs to the max.domain of attraction of the Weibull extreme value distribution with lim for some suitable norming constants   > 0 and   .
Similar arguments apply to min.domains of attraction.That is,  belongs to the same min.domain of attraction as that of .

Probability Weighted Moments
The PWMs of a baseline model can be very useful to determine the moments of more complex distributions.Distributions that can be expressed in inverse form may present problems in estimating their parameters as functions of ordinary moments.For these distributions, the relations between the PWMs and the parameters have simpler analytical structure than those between the ordinary moments and the parameters.The PWMs are also widely used for estimating parameters of distributions from complete or censored samples.We demonstrate that the (, )th PWM of , say  , = {  (; , )  } (for ,  = 0, 1, . ..), can be expressed as linear combinations of the baseline PWMs defined by  , = ∫ ∞ −∞   (; )  .First, we can write  , from ( 20) and ( 23) by interchanging the sum and the integral Further, we have and then using the binomial expansion, we can express the PWMs of  as where  ,− and  ,+1− are obtained from (44).

Order Statistics
Order statistics are required in many fields, such as climatology, engineering, and industry.Further, they play an important role in Statistical Inference and Nonparametric Statistics.In this section, we present some results with respect to the order statistics.We obtain an expression for the density of the th order statistic and the large sample distribution of the minimum and maximum when a random sample of size  is drawn from the  family.The density function of the th order statistic, say  : , from a random sample of size  drawn from ( 5) is given by where  = ∫ +−1  can be evaluated numerically for most parent distributions using statistical software.

Information Theory
11.1.Entropies.An entropy is a measure of variation or uncertainty of a random variable .Two well-known entropy measures are the Rényi and Shannon entropies.The Rényi entropy of a random variable  with pdf () is defined by Next, we consider the Rényi entropy.From (1), we have which can be computed numerically. where Thus,  3 = (1/2 1 ) 3 (; ).Also, in this case,  2 =  4 .Hence, the KL divergence is given by

Bivariate and Multivariate Generalization
where (, ; ) is a bivariate continuous distribution with marginal cdfs  1 (; ) and  2 (; ).The corresponding joint pdf is given by The marginal cdfs are given by The marginal pdfs are given by  0.21, 0.13, 0.01, 0.01, 0.17, 0.01, 0.01, 0.21, 0.13, 0.69, 0.25, 0.01, 0.01, 0.09, 0.13, 0.01, 0.05, 0.01, 0.01, 0.29, 0.25, 0.49, 0.01, and 0.01.Table 3 lists some descriptive statistics of these data.Table 4 gives the MLEs (with corresponding standard errors in parentheses) and some goodness of fit measures for the fitted TKw and Kw distributions.Since the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC), and consistent Akaike information criterion (CAIC) are smaller for the TKw distribution compared to those values of the Kw model, the new distribution is a very competitive model to these data.
In order to test the null hypothesis  0 :  = 0 against the alternative hypothesis  1 :  ̸ = 0, we obtain a confidence interval (with 5% significance level).The 95% confidence interval for  is (0.2159, 0.9202) and since it does not contain zero, we can reject the null hypothesis in favor of the alternative hypothesis that the data set is generated from a TKw model with  ̸ = 0. Further, the LR statistic (see Section 13) to test  0 : Kw against  1 : TKw is 7.2 ( value < 0.01).Thus, using any usual significance level, we reject the null hypothesis in favor of the TKw distribution, that is, this distribution is significantly better than the Kw distribution to explain the current data.
Plots of the pdf and cdf of the fitted TKw and Kw models to these data are displayed in Figure 3.They indicate that the TKw model is superior to the Kw model in terms of model fitting.

Conclusions
In this paper, we explore general properties of the transmuted- family of distributions.This family is obtained by adding shape parameters to an existing well-known distribution by using the transmutation map approach [1].We investigate some of its general mathematical properties including ordinary and incomplete moments, generating function, probability weighted moments, Shannon and Rényi entropies, Kullback-Leibler divergence, and limiting behavior of the extremum values.The existing literature does not include these general properties.We also provide a real life application to prove empirically the usefulness of the transmuted- family.The application indicates that the transmuted model performs better compared to the parent model.Consequently, it merits further a thorough study in terms of its extension to a bivariate and subsequently multivariate set-up (we have discussed briefly this topic in Section 12).Needless to say, inferential procedures, especially ii) () ∼ (1 + )(), (iii) ℎ() ∼ (1 + )()/(), where () = 1 − ().

2 Figure 2 :
Figure 2: Plots of the TKw density function for some parameter values.

Figure 3 :
Figure 3: Estimated pdf and cdf from the fitted TKw and Kw models for the data.

Table 1 :
(3)Repeat steps (1) to (2) until the required amount of random numbers to be completed.Algorithm 1: Random number generator for the TG distribution.Distributions and corresponding qfs.

Table 2 :
Distributions and their moments.

Table 4 :
Estimates of the parameters (standard errors in parentheses), goodness-of-fit statistics, and confidence intervals with significance level at 5% for the data set.