Mini Review
A New T-X Family of Distributions
Clement Boateng Ampadu, Department of Biostatistics, USA.
Received Date: June 26, 2019; Published Date: August 26, 2019
Abstract
This paper introduces a new weight in the T – X (W ) framework, and shows a new family induced by this new weight is good in fitting real-life data.
Keywords: Exponential distribution; Normal distribution; T-X family of distributions
The T − X(W) Family of Distributions
This family of distributions is a generalization of the the betagenerated family of dis-tributions first proposed in [1] In particular, let r(t) be the PDF of the random variable T ∈ [a, b], −∞ ≤ a < b ≤ ∞, and let W (F (x)) be a monotonic and absolutely continu-ous function of the CDF F (x) of any random variable X. The CDF of a new family of distributions defined in [2] is given by
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E001.png)
where R(·) is the CDF of the random variable T and a ≥ 0
Remark 3.1: The PDF of the T − X(W ) family of distributions is obtained by differentiating the CDF above
Remark 3.2. When we set W (x) := − log(1 − x), then we use the term “T-X ” to de-scribe all sub-classes of the T-X(W) family of distributions induced by the weight function W (x)= − log(1 − x). A description of different weight functions that are appropriate given the support of the random variable T is discussed in [2]
A plethora of results studying properties and application of the T-X(W) family of distri-butions have appeared in the literature, and the research papers, assuming open access, can be easily obtained on the web via common search engines, like Google, etc
PT-G Family of Distributions" class="head-link">The PT-G Family of Distributions
Definition 4.1: [3] ξ > where α is the rate
scale parameter and ξ is a vector of parameters in the baseline
distribution all of whose entries are positive. A random variable Z is
said to follow the
power transform family of power transform
family of distributions if the Cumulative Distribution Function
(CDF) is given by
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E002.png)
Where is the baseline
cumulative distribution with probability density function g(x; ξ).
The New Weight and the Distribution
By modifying the parameter space for α in the PT-Standard
Uniform family and using it in Remark 1.2, we define the new
weight as
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E003.png)
where [0,1] . Using this we introduce
the following
Definition 5.1: Let X be a random variable with CDF F (x; ξ) and PDF f(x; ξ); and let T be a random variable with support [0, ∞) whose PDF is r(t; γ), and whose CDF is R(t; γ); let ξ and γ be vector of parameters in the distribution of the random variables X and T, Citation: Clement Boateng Ampadu. A New T-X Family of Distributions. Annal Biostat & Biomed Appli. 3(1): 2019. ABBA.MS.ID.000553. DOI: 10.33552/ABBA.2019.03.000553. Page 2 of 3 respectively. The new T −X family distributions has CDF defined by the following
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E004.png)
where
Remark 5.2: The PDF of the new T-X family of distributions can be obtained by differentiating the CDF above
Practical Illustration
Assume X is a normal random variable with CDF
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E005.png)
Where
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E006.png)
where t, b > 0 . Now the following is immediate from Definition 3.1
Proposition 6.1: The CDF of the new Exponential-Normal family of distributions is given by
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E007.png)
where and erfc
Remark 6.2: If a random variable M has CDF given by the previous Proposition we write M ∼ N EN (c, d, b, α) (Figure 1 & 2) [4]
The fit in Figure 1 and the approximately straight line in Figure 2 indicates the NEN distribution is a good fit.
![Click here to view Large Figure 1 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.G001.png)
![Click here to view Large Figure 2 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.G002.png)
By differentiating the CDF, we have the following
Proposition 6.3: The PDF of the new Exponential-Normal family of distributions is given by
where
![](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.E008.png)
![Click here to view Large Figure 3 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.G003.png)
![Click here to view Large Figure 4 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000553.G004.png)
As the box-whisker charts in Figure 3 and the quantile plots in Figure 4 appear almost identical, we have more evidence that the NEN distribution is a good fit.
Concluding Remarks
This paper has introduced a new T-X family of distributions via Definition 3.1 and has shown a member of this new class of distribtutions is a good fit to real life data. Our hope is that the researchers of distribution theory and its applications will further develop the properties and applications of the new T-X class of distributions.
Acknowledgement
None.
Conflict of Interest
No conflict of interest.
References
- Eugene N, Lee C, Famoye F (2002) The beta-normal distribution and its applications. Communications in Statistics-Theory and Methods 31(4): 497-512.
- Alzaatreh A, Lee C, Famoye F (2013b) A new method for generating families of continuous distributions. Metron 71(1): 63-79.
- Abdulzeid Yen Anafo (2019) The New Alpha Power Transform: Properties and Ap-plications, Master of Science in Mathematical Sciences Essay, African Institute for Mathematical Sciences (Ghana). Unpublished Manuscript.
- Ayman Alzaatreh, Carl Lee Felix Famoye (2014) T-normal family of distributions: a new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications 1:16.
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Clement Boateng Ampadu. A New T-X Family of Distributions. Annal Biostat & Biomed Appli. 3(1): 2019. ABBA.MS.ID.000553.
Exponential distribution; Normal distribution; T-X family of distributions
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