The Generalized Sigmoid Almost Power Law Family of Distributions

Contents

The New Family

At first, we recall the following

Proposition

[1] Let and X is a random variable with CDF FX, then the CDF of U is

Given the cumulative distribution function (CDF) of some baseline distribution as F(x), we defined the generalized sigmoid generated family of distributions with the following CDF

where It follows from

Proposition 1.1:

Proposition

Suppose the random variable Z has CDF F(z), then, the CDF of the generalized sigmoid almost power law family of distributions is given by

Practical illustration of the new family

We assume Z is a Weibull random variable, so that

where a, b, z > 0. It now follows from Proposition 1.2, that we have the following

Theorem

The CDF of the generalized sigmoid Weibull almost power law family of distributions is given by

Remark

We write if M is a random variable with CDF given by the theorem immediately above. The generalized sigmoid Weibull almost power law family of distributions is practically significant in modelling real-life data as shown below (Figure 1).

Figure 1: The CDF of GSWAPL (1.40849, 165.612, 2.4847, 0.3) fitted to the empirical distribution of the breast cancer data [3].

Remark

Figure 2: The PDF of GSWAPL (1.40849, 165.612, 2.4847, 0.3) fitted to the histogram of the breast cancer data [3].

The PDF of the generalized sigmoid almost power law family of distributions can be obtained by differentiating the CDF (Figure 2).

Concluding Remarks

In this short communication, we introduced a new family of almost power law distributions, and showed a sub-model is practically significant in fitting data in the health sciences. A future interesting problem is to investigate some properties and additional applications of this class of statistical distributions [3].

References

• Clement Boateng Ampadu (2020) A Class of Almost Power Law Distributions. Unpublished.
• Clement Boateng Ampadu (2019) The Generalized Sigmoid Generated Family of Distributions with Illustration to Breast Cancer Data. Unpublished.
• K Jayakumar, M Girish Babu (2017) T-Transmuted X Family of Distributions. Statistica anno LXXVII 77(3): 251-276.