Review Article
The Ampadu APT − qT − X Family of Distributions Induced by V with an Illustration to Data in the Health Sciences
Clement Boateng Ampadu, Department of Mathematics, USA.
Received Date: March 05, 2019; Published Date: March 26, 2019
Abstract
The alpha power transform family of distributions ( APT −G for short) appeared in [1], and since then variants of it has been
proposed, and for example, see [2]. In this paper we propose another variant of the APT −G family of distributions called the Ampadu
APT −G family of distributions. Assuming G is given by the CDF of qT − X family of distributions induced by V , and for example, see [3] and [4], we show sub-models of the Ampadu APT − qT − X (V) family of distributions is significant in modeling data in the health sciences, in particular, the breast cancer patients data contained in [5]. As a further development of the APT −G family of distributions, we propose a so-called power transform family of distributions (
PT-G for short).
Keywords: Alpha power transform; Ampadu-G; Quantile generated family of distributions
Background on qT −X Family of Distributions Induced by V
This section is inspired by [3] and [4]
Definition 3.1. Let V be any function such that the following holds:
b) F (x) is differentiable and strictly increasing
then the CDF of the qT − X family induced by V is given by
![Click here to view Large equation 1 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E001.png)
where is the quantile density function of random variable
T ∈ [a,b] , for −∞ ≤ a < b ≤ ∞, and F (x) is the CDF of any random
variable X.
Theorem 3.2. The CDF of the qT − X family induced by V is given by
![Click here to view Large equation 2 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E002.png)
Proof. Follows from the previous definition and noting that
Theorem 3.3. The PDF of the qT − X family induced by V is given by
![Click here to view Large equation 3 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E003.png)
Proof. and K is given by Theorem 3.5.2
Remark 3.4. When the support of T is[a,∞), where a ≥ 0 , we can take V as follows
where α > 0
where α < 0
Remark 3.5. When the support of T is (−∞,∞) , we can take V as follows
where α > 0
where α > 0
The New Family of Distributions
The Ampadu APT-G aamily of distributions
Definition 4.1. A random variable X will be called Ampadu APT-G distributed if the
CDF is given by
![Click here to view Large equation 4 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E004.png)
and the PDF is given by
![Click here to view Large equation 5 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E005.png)
Where ,and G is some baseline
distribution.
Application to the qT − X family of distributions induced by V
Definition 4.2. A random variable J1 will be called Ampadu APT − qT − X distributed of type I if the CDF is given by
![Click here to view Large equation 6 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E006.png)
Where G is some baseline distribution associated with the
random variable X, β > 0 , and the random variable T with support [0, 1] has quantile QT
Remark 4.3. The PDF can be obtained by differentiating the CDF above
Definition 4.4. A random variable J2 will be called Ampadu APT − qT − X distributed of type II if the CDF is given either by
![Click here to view Large equation 7 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E007.png)
Or
![Click here to view Large equation 8 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E008.png)
where G is some baseline distribution associated with the random variable X, β > 0 , and the
random variable T with support [0,1] has quantile QT
Remark 4.5. The PDF can be obtained by differentiating the CDF immediately above
Definition 4.6. A random variable J3 will be called Ampadu APT − qT − X distributed
of type III if the CDF is given either by
![Click here to view Large equation 9 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E009.png)
Or
![Click here to view Large equation 10 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E010.png)
where G is some baseline distribution
associated with the
random varaible X, β > 0 , and the random variable T with support (−∞,∞) has quantileQT
Remark 4.7. The PDF can be obtained by differentiating the CDF immediately above
An Illustration to Breast Cancer Patients Data
Application of Definition 4.2
We assume β =1, T is uniform on [0, 1], and the random variable X follows the Weibull distribution, so that the CDF of X is given by
![Click here to view Large equation 11 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E011.png)
and the PDF of X is given by
![Click here to view Large equation 12 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E012.png)
Now from Definition 4.2, we have the following
Theorem 5.1. The CDF of the Ampadu APT-{Quantile Uniform- Weibull} distribution of type I is given by
![Click here to view Large equation 13 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E013.png)
Where
Remark 5.2. If a random variable B (say) has CDF given by the previous theorem we Write B ~ AAPTQCW(a,b,α ) (Figure 1)
![Click here to view Large Figure 1 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G001.png)
By differentiating the CDF given by Theorem 5.1, we get the following
Theorem 5.3. The PDF of the Ampadu APT-{Quantile Uniform- Weibull} distribution
of type I is given by
![Click here to view Large equation 14 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E014.png)
Where (Figure 2)
![Click here to view Large Figure 2 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G002.png)
Application of definition 4.4
We assume T is a fully specified exponential distribution. In particular, we assume T has CDF
1− e−0.714286t
so that the quantile of T in this case is given by
![Click here to view Large equation 15 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E015.png)
We assume β =1 and consider X to be Weibull distributed with CDF and PDF as defined
in the previous section. The second CDF in Definition 4.4 implies the following
Theorem 5.4. The CDF of the Ampadu APT-{Quantile Exponential-Weibull} distribution
of type II is given by
![Click here to view Large equation 16 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E016.png)
Remark 5.5. If a random variable L has CDF given by the theorem immediately above, we write
L ~ AAPTQEW(a,b,α) (Figure 3)
![Click here to view Large Figure 3 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G003.png)
Remark 5.6. By differentiating the CDF of the AAPTQEW distribution, the PDF can be obtained (Figure 4).
![Click here to view Large Figure 4 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G004.png)
Application of Definition 1.6
We assume β =1, α =1.3 , X is Weibull distributed with CDF and PDF as defined in Section 5.1, and T is a fully specified Cauchy distribution. In particular, we assume T has CDF
![Click here to view Large equation 17 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E017.png)
so that the quantile of T is given by
![Click here to view Large equation 18 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E018.png)
The second CDF in Definition 3.6 implies the following
Theorem 5.7. The CDF of the Ampadu APT-{Quantile Cauchy- Weibull} distribution of type III is given by
![Click here to view Large equation 19 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E019.png)
where x, a, b > 0
Remark 5.8. If a random variable V has CDF given by the theorem immediately above,
we write V ~ AAPTQCW(a,b) (Figure 5)
![Click here to view Large Figure 5 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G005.png)
Remark 5.9. By differentiating the CDF of the AAPTQCW distribution, the PDF can be obtained (Figure 6)
![Click here to view Large Figure 6 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G006.png)
Further Developments and Concluding Remarks
Inspired by the APT-G distribution in [1], we ask the reader to
investigate properties and applications of a so-called power
transform distribution
PT-G for short)
Definition 6.1. A random variable Z will be called PT-G
distributed if the CDF is given by
![Click here to view Large equation 20 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E020.png)
For
Remark 6.2. The PDF can be obtained by differentiating the CDF
The new distribution appears practically significant in modeling real-life data as shown below. We assume G is given by the CDF of the Weibull distribution, as in Section 5.1. Thus, from the above definition the following is immediate
Theorem 6.3. The CDF of the PT-Weibull distribution is
given by
![Click here to view Large equation 21 irispublishers-openaccess-biostatistics-biometric-applications](../tables/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.E021.png)
For
Remark 6.4. If a random variable M has CDF given by the theorem immediately above,
we write (Figure 7)
![Click here to view Large Figure 7 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G007.png)
Remark 6.5. By differentiating the CDF of the PTW distribution, the PDF can be obtained (Figure 8).
![Click here to view Large Figure 8 irispublishers-openaccess-biostatistics-biometric-applications](../images/irispublishers-openaccess-biostatistics-biometric-applications.ID.000526.G008.png)
Whilst we have shown the distributions are practically significant in the health sciences, we hope they find applications in other disciplines. Also we hope the researchers will further develop the mathematical and statistical properties of these distributions.
Acknowledgement
None.
Conflict of interest
No conflict of interest.
References
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- Clement Ampadu, Results in Distribution Theory and Its Applications Inspired by Quantile Generated Probability Distributions, Lulu.com. ISBN: 0359249957,9780359249954.
- Clement Boateng Ampadu (2018) Quantile-Generated Family of Distributions: A New Method for Generating Continuous Distributions. Fundamental Journal of Mathematics and Mathematical Sciences, Volume 9(1): 13-34.
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Clement Boateng Ampadu. The Ampadu APT − qT − X Family of Distributions Induced by V with an Illustration to Data in the Health Sciences. Annal Biostat & Biomed Appli. 2(1): 2019. ABBA.MS.ID.000526.
Public Health, Health Science, Mathematics, Data Science, Statistics-Theory, Family of Distributions, Mathematical Sciences, Statistical properties, Breast cancer, Patients Data
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