The Generalized Ampadu-G Family of Distributions: Properties, Applications and Characterizations

This paper introduces a new class of distributions called the generalized Ampadu-G (GAG for short) family of distributions, and with a certain restriction on the parameter space, the family is shown to be a life-time distribution. The shape of the density function and hazard rate function of the GA-G family is described analytically. When G follows the Weibull distribution, the generalized Ampadu-Weibull (GA-W for short) is presented along with its hazard and survival function. Several sub-models of the GA-W family are presented. The transformation technique is applied to this new family of distributions, and we obtain the quantile function of the new family. Power series representations for the cumulative distribution function (CDF) and probability density function (PDF) are also obtained. The rth non-central moments, moment generating function, and Renyi entropy associated with the new family of distributions are derived. Characterization theorems based on two truncated moments and conditional expectation are also presented. A simulation study is also conducted, and we find that using the method of maximum likelihood to estimate model parameters is adequate. The GA-W family of distributions is shown to be practically significant in modeling real life data, and is shown to be superior to some non-trivial generalizations of the Weibull distribution. A further development concludes the paper. Clement Boateng Ampadu and Abdulzeid Yen Anafo http://www.earthlinepublishers.com 140


Introduction
Statistical distributions have been developed in recent years by researchers in order to model and predict real world data. There has been a great number of flexible distributions developed by researchers for modelling data. Modelling techniques used are based statistical distributions and this has prompted many researchers in developing new family of distributions through various different methods for defining these families of distributions. Many attempts have been made to propose new families of distributions in the statistical papers by the use of some baseline distributions. Gupta et al. [1] introduced the exponentiated family where the introduction of an extra parameter was introduced. The cumulative distribution function (CDF) of the Gupta et al. [1] proposed exponentiated family is given by, where ψ is a parameter vector of the baseline distribution ( ).
; ψ x G In relation to [1], many lifetime distributions have been proposed. Some these distributions include, Exponentiated Exponential in Gupta and Kundu [2], Exponentiated beta in Nadarajah [3], exponentiated lognormal in Shirke and Kakade [4], exponentiated Kumaraswamy in Lemonte et al. [5], Exponentiated Power Lindley in Ashour and Eltehiwy [6] and exponentiated Weibull-Pareto in Afify et al. [7]. Another prominent method in this area, is Marshall-Olkin family of distributions proposed by Marshall and Olkin [8] defined by the CDF, where ψ is a parameter vector of the baseline distribution ( ).
; ψ x G in relation to [8], many distributions have been generalized. These include, Marshall Olkin extended Weibull of Ghitany et al. [9], Marshall Olkin gamma Marshall Olkin extended log-Logistic Weibull of Lepetu et al. [10] and Marshall Olkin gamma Weibull of Saboor and Pogány [11]. Recently, [12] developed the Ampadu-G family of distributions and defined its CDF as

Description of the New Family of Distributions
This section presents the GA-G family of distributions and applies the proposed family of distributions to the Weibull distribution and this new distribution is named the generalized Ampadu-Weibull distribution (GA-W for short). The sub-models of the GA-W distribution are also presented.

The GA-G family of distributions
ξ is a vector of parameters in the baseline distribution with CDF G, and . 0 > β Proposition 2.2. The PDF of the generalized Ampadu-G family of distributions is given by family of distributions can be recovered [13]. In both instances the parameter space is a subspace of the generalized Ampadu-G family of distributions

Physical interpretation of the GA-G family
Suppose in Definition 2.1, the parameter space of λ is restricted to ( ).
and ξ is a vector of parameters in the baseline distribution with CDF G. If j Z is the failure time of the jth subsystem and X represents the time to failure of the first out of the M operating subsystems such that Then the conditional CDF of X given M can be shown to be given by Thus, the marginal CDF of X is given by and ξ is a vector of parameters in the baseline distribution with CDF G, and . 0 > β With this physical interpretation, and restriction on the parameter , λ the generalized Ampadu-G becomes the Poisson Exponentiated G family of distributions. The PDF of the Poisson Exponentiated G family of distributions can be obtained by differentiating the CDF immediately above.

Shape of density and hazard function of the GA-G family
The shapes of the density function and hazard rate function of the GA-G family can be described analytically. The critical points of the density function can be shown to be roots of the equation The above equation may have more than one root. If It can be shown that the hazard rate function of the GA-G family is given by where the parameter space is the same as in the GA-G family. In a similar way the critical points of the hazard function can be shown to be roots of the equation The above equation may have more than one root. If Proposition 2.5. The PDF of the generalized Ampadu-Weibull distribution is given by, The survival function and hazard rate function of the generalized Ampadu-Weibull distribution is given by, respectively.
Using the estimated parameters in the GA-W distribution recorded in Table 1, we display the survival function and hazard rate function in Appendix H

Sub-models of the GA-W family of distributions
Several families can be derived from the generalized Ampadu-Weibull distribution for different choice of the parameters, and we list some of them below the generalized Ampadu-Weibull distribution reduces to the generalized Ampadu-Exponential Distribution the generalized Ampadu-Weibull distribution reduces to the generalized Ampadu-Standard Exponential Distribution

Quantile function
For some choices of ( ),

Expansion of CDF
Let r be arbitrary positive integer, by the Binomial Theorem, we can write By the power series representation for the exponential function, we can write the equation immediately above can be written as Thus we have the following x and ξ is a vector of parameters in the baseline distribution with CDF ( ).

Expansion of PDF
and ξ is a vector of parameters in the baseline distribution with CDF ( ) ξ ; x G and PDF ( ). can be expressed as and ξ is a vector of parameters in the baseline distribution with CDF ( ) ξ ; x G and PDF ( ).
; ξ x g Given a random variable X with PDF ( ), Using this representation we display the first ten moments of the generalized Ampadu-Standard Uniform distribution for some choice parameters in Appendix G.

Moment generating function
and ξ is a vector of parameters in the baseline distribution with CDF ( ) ξ ; x G and PDF ( ).  ; ξ x g

Simulation Study
In this simulation study samples of sizes 400, 550, 700, and 900, are drawn from the generalized Ampadu-Weibull family of distributions. Overall, the simulation study conducted indicates that using the method of maximum likelihood to estimates the parameters in the generalized Ampadu-G family of distributions is adequate.

Maximum likelihood estimation in the GA-G distribution
In this section, we obtain the maximum likelihood estimators (MLEs) for the parameters in the generalized Ampadu-G family of distributions. For this, let From the above the log-likelihood function is given by Obtaining the partial derivatives of the equation immediately above, we get the following By setting the three equations immediately above to zero and solving simultaneously we obtain the maximum likelihood estimates.

Performance of GA-W distribution
In this section we investigate the performance of the generalized Ampadu-Weibull distribution (GA-W) in fitting real-life data. We consider the data-set on the survival times of 72 guinea pigs infected with virulent tubercle bacilli [14]. The performance of the generalized Ampadu-Weibull distribution is compared with two other generalizations of the Weibull distribution, that is, the Marshall-Olkin Weibull (MOW) and the exponentiated Weibull (EW) distributions. The measures of goodness of fit we consider include Akaike information criterion (AIC), Bayesian information criterion (BIC), -2Log-Likelihood, and second-order Akaike Information Criterion (AICc). The Weibull distribution can be written in different forms, however in defining the MOW distribution and the EW distribution, we use the form of the Weibull distribution as employed in Definition 2.4. In particular, using the form of the Weibull distribution in Definition 2.4 in equation (1.1) of [8], and then differentiating, we have the following as the PDF of the MOW distribution ( )  The idea of powering a positive real number (say) to the CDF of any distribution is popular technique in distribution theory, and is due to [1]  The parameter estimates in the GA-W, MOW, and EW distributions along with their standard errors are summarized in Table 1 above. An inspection of Table 2 below shows that the EW distribution is preferred in comparison with the other distributions in fitting the same data. Among the non-trivial generalizations of the Weibull distribution, it can be seen that the GA-W distribution is preferred to the MOW distribution in fitting the same data. The GA-W distribution as a non-trivial generalization of the Weibull distribution should play significant role in fitting real-life data from various disciplines.

Table 2. Goodness-of-fit measures
The estimated PDFs to the histogram and the estimated CDFs to the empirical distribution are displayed in Appendix E.

Characterization Theorems
The characterization of statistical distributions plays a major role in stochastic modeling. In this section we present some characterizations of the generalized Ampadu-G family of distributions. Our first characterization theorem is based on a simple relationship between two truncated moments, and for related works in this direction, the reader is referred to [15]- [20].
At first, we recall the following which will be useful in the sequel η Assume that where the function s is a solution of the differential equation Remark 6.2. The characterization based on the ratio of two truncated moments is stable in the sense of weak convergence, and for more details see [21] Our second characterization result employs a single function ψ of X and states a characterization result in terms of ( ).

X ψ
The following known results is useful for our purposes here.

Characterization based on two truncated moments
The main result here is the following is a vector of parameters in the baseline distribution with CDF G and PDF g, and we deduce the following is a vector of parameters in the baseline distribution with CDF G and PDF g, and . 0 > β Remark 6.7. The general solution of the differential equation in the above Corollary is given by

Further Development
In [23] they introduced a new method of generating continuous distributions based on the alpha power transformation family of [24] is a vector of parameters in the baseline distribution with CDF F, and . 0 > β In particular, we ask the reader to investigate some properties and applications of this new class of within quantile distributions.