The New Size-Biased Kumaraswamy-G Distribution

The concept of weighted distributions is largely credited to Fisher [1,2] and Rao [3]. In the size biased schemes, the probability of sampling an individual is proportional to X^r provided that for all θ. In such situations, the weighted probability density function is defined as

where When the probability of observing a positive-valued random variable is proportional to the value of the variable the resulting distribution is termed size-biased with probability density function

The Kumaraswamy distribution [4] has probability density function

and the rth order raw moment is given by [1]

where is a beta function defined by the integral

Notation 1.1. We write X SBKD(a,b) , if X is a size biased Kumaraswamy random variable

Proposition 1.2. [1] The PDF of the SBKD(a, b) distribution is given by

where is a beta function defined by the integral

with a, b > 0

Theorem 1.3. [1] The CDF of the SBKD (a, b) distribution is given by

is the regularized incomplete beta function and is defined as the ratio of an incomplete beta function,

and the complete beta function,

Definition 2.1. [5] Let where α is the rate scale parameter and 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 distributions if the Cumulative Distribution Function (CDF) is given by

where the baseline distribution has CDF G(x;ξ ).

Proposition 2.2. [5] The PDF of the Proposition 2.2. [5] The PDF of the

where is the baseline cumulative distribution with

probability density function g (x;ξ )

By modifying the parameter space for α in the PT-Standard Uniform family and using it in Theorem 1.3, we have the following

Proposition 3.1. The CDF of the new size biased Kumaraswamy Distribution is given by

is the regularized incomplete beta function and is defined as the ratio of an incomplete beta function,

Using the above, we introduce the following

Definition 3.2. We say Z is a new size-biased Kumaraswamy-G random variable, and write SBKGD(a, b,α ,ξ ), if the CDF is given by

where F is given by the previous Proposition, a, b > 0, α ∈ R, α 6= 0, x ∈ R, and ξ is a vector of parameters in the baseline distribution with CDF G(x;ξ )

Remark 3.3. The PDF of the new size-biased Kumaraswamy-G family of distributions can be obtained by differentiating the CDF

We assume the baseline distribution is Normal with the following CDF

Thus from Definition 3.2 and Proposition 3.1, we have the following

Proposition 4.1. The CDF of the new size-biased Kumaraswamy- Normal distribution is given by

Remark 4.2. We write Q ∼ N SBKN D(a, b, α, c, d), if Q is a new size-biased Kumaraswamy-Normal random variable. When we fix a = 1, we denote the reduced distribution as N BIN D(b, α, c, d), and say Q is a new Beta-I-Normal random variable (Figure 1).

Remark 4.3. The PDF of the new size-biased Kumaraswamy- Normal distribution can be obtained by differentiating the CDF (Figure 2).

Size-biased distributions arise naturally in a range of sampling and modeling problems in forestry [7], as such we suggest to the reader to obtain some properties and applications of this new class of distributions as it pertains to forestry.

None.

No conflict of interest.

1. Dreamlee Sharma, Tapan Kumar Chakrabarty (2016) On Size Biased Kumaraswamy Distribution.
2. Fisher RA (1934) The effects of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics 6(1): 13-25.
3. Rao CR (1965) On discrete distributions arising out of methods of ascertainment. In: Patil GP (Ed.), Classical and Contagious Discrete Distributions. Pergamon Press and Statistical Publishing Society: Calcutta, pp. 320-332
4. Kumaraswamy PA (1980) Generalized probability density function for double-bounded ran-dom processes. Journal of Hydrology 46(1): 79-88.
5. 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
6. 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.
7. Gove JH (2003) Estimation and applications of size-biased distributions in forestry. In: Amaro A, Reed D, Soares P (Eds.), Modeling Forest Systems; CABI Publishing, pp. 201-212.