The Ampadu APT − qT − X Family of Distributions Induced by V with an Illustration to Data in the Health Sciences

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

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

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

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 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

and the PDF is given by

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

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 Q_T

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

Or

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

Or

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

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

and the PDF of X is given by

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

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)

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

Where (Figure 2)

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

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

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)

Remark 5.6. By differentiating the CDF of the AAPTQEW distribution, the PDF can be obtained (Figure 4).

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

so that the quantile of T is given by

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

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)

Remark 5.9. By differentiating the CDF of the AAPTQCW distribution, the PDF can be obtained (Figure 6)

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

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

For

Remark 6.4. If a random variable M has CDF given by the theorem immediately above,

we write (Figure 7)

Remark 6.5. By differentiating the CDF of the PTW distribution, the PDF can be obtained (Figure 8).

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.

None.

No conflict of interest.

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2. Zubair Ahmad, Muhammad Ilyas, GG Hamedani (2016) The Extended Alpha Power Transformed Family of Distributions: Properties and Applications. Journal of Data Science.
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4. 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.
5. Girish Babu Moolath, Jayakumar K(2017) T-Transmuted X Family of Distributions. STATISTICA 3.