CANCER CASE REPORTS - Volume 1; Issue 1, (2020)

Pages: 1-4

## The Burr XI Generated Family of Distribution with Illustration to Cancer Patients Data

Category: Cancer Research

Abstract:

The Burr system of distributions appeared in (Burr, 1942). On the other hand in (Eugene et al., 2002), they pioneered the beta-generated method of creating probability distributions. Inspired by these developments, we introduce the Burr XI generated method of creating probability distributions. We show a distribution arising from this method is a good fit to real-life data, indicating the new method of creating probability distributions should be applicable in various disciplines. Finally, we ask the reader to investigate some properties and applications of a so-called Burr VI generated family of distributions of two types.

Keywords: Burr system of distributions, Breast cancer, T-X(W) family of distributions, Beta-G

Full Text:

1. Introduction and the New Family

The Burr system of distributions (Burr, 1942) arise from the differential equation

where and g(x, y) is a nonnegative function for 0 ? y ? 1 and x is in the range over which the solution is to be used. The Burr XI distribution has CDF, and for example, see Table 1.1 (Momanyi, 2017), given by where r > 0, and 0 < x < 1. By differentiating the CDF above, we see the PDF of the Burr XI distribution is given by Now we introduce the Burr XI generated family of distributions via the following integral where r > 0 and G(x; ξ) is the CDF of some baseline distribution. The parameter space of ξ depends on the choosen baseline distribution. Evaluating this integral leads to the following

Proposition 1.1. The CDF of the Burr XI-G family of distributions is given by where x?, r > 0, and G(x; ξ) is the CDF of some baseline distribution. The parameter space of ξ depends on the choosen baseline distribution.

By differentiating the CDF above, we have the following

Proposition 1.2. The PDF of the Burr XI-G family of distributions is given by where x ?, r > 0, G(x; ξ) and g(x; ξ) are the CDF and PDF, respectively, of some base- line distribution. The parameter space of ξ depends on the choosen baseline distribution

The rest of this paper is organized as follows. In the next section we show a so-called Burr XI-Weibull distribution is a good fit to the breast cancer data (Girish and Jayakumar, 2017). The last section is devoted to the conclusions. As a further recommendation we suggest obtaining some properties and applications of a so-called Burr VI generated family of distributions of two types.

2 Practical Illustration

We assume the baseline distribution is Weibull distributed with the following CDF where x, a, b > 0. By Proposition 1.1, we have the following

Corollary 2.1. The CDF of the Burr XI-Weibull distribution is given by where x, a, b, r > 0

Notation 2.2. We write V ? BXIW (a, b, r), if V is a Burr XI-Weibull random variable. Figure 1. The CDF of BXIW (1.20333, 97.7151, 0.329319) fttted to the empirical distribution of the breast cancer data (Girish and Jayakumar, 2017).

Remark 2.3. The PDF of the Burr XI-Weibull distribution can be obtained by differen- tiating the CDF Figure 2. The PDF of BXIW (1.20333, 97.7151, 0.329319) fttted to the histogram of the breast cancer data (Girish and Jayakumar, 2017).

3 Concluding Remarks and Further Recommendations

The present paper has introduced a family of distributions arising from the Burr system of distributions. In particular, a so-called Burr XI-G family of distributions have been introduced, and a member of this class of distributions is shown to be a good fit to real life data in the health sciences. The Burr VI distribution has CDF where k, c, r > 0 and x ?. Inspired by the T - X(W) framework (Alzaatreh and Lee, 2013b), we ask the reader to investigate some properties and applications of a so-called Burr VI generated family of distributions of two types. We leave the reader with the following integral representations for their CDF's.

Definition 3.1. The CDF of the Burr VI generated family of distributions is given by where G(x; ξ) is the CDF of some baseline distribution. The parameter space of ξ and x

depends on the chosen baseline distribution, and k, c, r > 0.

(a) The Burr VI generated family of distributions is of type I, if we take for some α > 0

(b) The Burr VI generated family of distributions is of type II, if we take for some α > 0

References:

Burr IW. Cumulative frequency functions. Annals of Mathematical Statistics 1942; 13: 215-232

Eugene N, Lee C, Famoye F. Beta-normal distribution and its applica- tions. Communications in Statistics: theory and Methods 2002; 31: 497-512.

Momanyi ReinPeter Ondeyo. Generating Probability Distributions Based on Burr Differential Equation, Master Project in Mathematics, University of Nairobi, Kenya 2017.

Girish Babu Moolath and Jayakumar K. T-Transmuted X Family of Distributions, STA-TISTICA, anno LXXVII. 2017; 3.

Alzaatreh A, Lee C, Famoye F. A new method for generating families of continuous distributions. Metron 2013b; 71: 63-79.