The New X r T Family of Distributions: Some Properties with Applications

The X r T family of distributions induced by V which have been introduced in [1] is further explored in this paper. In particular, we have obtained some basic mathematical properties of this new family. The simulation study shows the method of maximum likelihood is adequate in estimating the unknown parameters in sub-models of this new class of statistical distributions. Further, the application shows that sub-models of this new family of distributions are useful in material science engineering and related disciplines that call for modeling and forecasting of related data sets. Finally, inspired by the Ampadu-G family of distributions [2], we propose a new class of distributions that have never appeared in the literature, and ask the reader to investigate some properties and applications of this new class of distributions.


Introduction and Preliminaries
We begin by recalling the X r T − family of distributions induced by V which have been introduced in [1]. In particular we start with the following x F Remark 1.4. By differentiating the CDF above, the PDF of the new X r T − family of distributions of type I can be obtained.

The Approximation to the New Family
It is well known [4] that the principal solution for w in , gives an approximation to the X r T − family of distributions, in particular we define the generic approximated distribution as follows: has PDF , T r and the random variable X has CDF ( ) x F and PDF ( ).
x f

Some Sub-models of the New Family
Assuming the random variable T is Exponentially distributed so that its CDF is given by ( ) then we have the following from Theorem 1.3.
Theorem 3.4. The CDF of the new Exponential-Weibull distribution of type I is given by  Assuming the random variable X is normally distributed, so that the CDF is given by ( )

Basic Mathematical Properties induced by the Approximated Family
an expectation, and the random variable X has PDF f and quantile . X Q Theorem 4.6. The CDF of the exponentiated approximated new standard exponential-X family of distributions of type I which is given by is the CDF of the random variable X, admit the following Theorem 4.7. The PDF of the exponentiated approximated new standard exponential-X family of distributions of type I which is given by are the CDF and PDF, respectively, of the random variable X, admit the following expansion

Parameter Estimation in a Sub-model of the New Family
In this section we discuss estimating the unknown parameters in the new Exponential-X family of distributions by using the methods of least squares estimation, weighted least squares estimation and the maximum likelihood estimation.

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Observe we have the following from Theorem 1.3

Corollary 5.1. The CDF of the new Exponential-X family of distributions of type I is given by
; ; ) ( ( ( ) )) ( ( ) ) . ; ; The system of nonlinear equations can be solved using Newton's method or fixed point iteration techniques.
Note that the weighted least squares estimates WLS â and WLS ξ of a and , ξ respectively, are obtained by minimizing the equation

Simulation Study Induced by the Approximated Family
In this section, a Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimation method in the new X r T − family of distributions of type I by using its approximated family. At first we demonstrate the equivalence of the approximated new Recall the CDF of the new Exponential-Logistic distribution of type I is given by Theorem 3.1, it follows from Corollary 2.1, that its approximation is given by the following:  The maximum likelihood estimators for the parameters m, s, and a are obtained. The procedure has been repeated 1000 times, and the mean and mean square error for the estimates are computed, and the results are summarized in Tables 1-3 below for each of  sets I, II, and III, respectively, considered above.      Table 2, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated mean square errors (MSEs) consistently decrease with increasing sample size as seen in Figures 11-13.    From Table 3, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated mean square errors (MSEs) consistently decrease with increasing sample size as seen in Figures 14-16.

Application based on the New Family
In this section, we illustrate the usefulness of the new X r T − family of distributions of type I. We compare the fits of three sub-models of this family of distributions discussed in Section 3 to the carbon fibers data, Section 6 [5].