Gull Alpha Power of the Ampadu-Type: Properties and Applications

This paper introduces a new statistical distribution called Gull Alpha Power of the Ampadu Type (GAPA-G for short). The new distribution is inspired by the Gull Alpha Power of [1] and the Ampadu-G of [2]. Some properties with application are investigated.


Introduction
A random variable X is said to follow the Gull Alpha Power family of distributions if its CDF is given by G(x; α, ξ) = αF (x; ξ) α F (x;ξ) [1] where α > 1, ξ is a vector of parameters in the baseline distribution, and x ∈ Supp(F ). Obviously, the PDF can be obtained by differentiating the CDF. On the other hand a random variable J is said to follow the Ampadu-G family of distributions if its CDF is given by K(x; λ, ω) = 1 − e −λG(x;ω) 2 1 − e −λ [2] where λ ∈ R\{0}, x ∈ Supp(G), and ω is a vector of parameters in the baseline distribution. By a modification of Ampadu-G, this paper introduces a new kind of 3 The New Family Defined

The CDF of the New Family
The CDF of GAPA-G is given by where α ∈ R, G is a baseline CDF, x ∈ Supp(G) and the support of ξ depends on the chosen baseline distribution.

A Sub-Model of the New Family
The PDF of the sub-model is given by

Quantile Function
Theorem 5.1. The quantile function of GAPA-G is given by where W (z) gives the principal solution for w in z = we w ,0 < x < 1, and G −1 is the quantile of the baseline distribution with CDF G.
Proof. Let 0 < x < 1. We must solve the following equation for Q(x) Let y = G(Q(x)), using computer software like MATHEMATICA one can show that solving for y in the following equation below α + e α x e α + 1 . Thus where W (z) gives the principal solution for w in z = we w .

Random Number Generation
Random numbers from GAPA-G can be obtained via where W (z) gives the principal solution for w in z = we w , U ∼ Uniform(0, 1), and G −1 is the quantile of the baseline distribution with CDF G.

rth Non-Central Moments
Theorem 5.2. The rth non-central moments of GAPA-G can be expressed as where Ω i,k,n,j,m is defined as in the proof of the theorem, U ∼ Uniform(0, 1), and E[·] denotes an expectation.
Proof. According to [17], we can write where the coefficients are suitably chosen real numbers that depend on the parameters of the G(x) distribution. For a power series raised to a positive integer r ≥ 1, we have By the power series representation for the exponential function, and the binomial theorem, we can write Put

Renyi Entropy
Lemma 5.3. Let f (x) denote the PDF of GAPA-G, then for δ > 0, and δ = 1, where Ω q,m,v,l,r,j is defined as in the proof of the Lemma, g(x) and G(x) are the PDF and CDF associated with the baseline distribution.
Proof. By the power series representation for the exponential function and the binomial theorem, we can write We can also write On the other hand by the negative binomial series and the power series representation for the exponential function, we can write Put Thus, Let X be a random variable with PDF f (x). By definition, the Renyi entropy [19] is defined as where δ > 0, and δ = 1. From the above Lemma we have the following Theorem 5.4. The Renyi entropy of GAPA-G can be expressed as where δ > 0, and δ = 1, and g(x) and G(x) are the PDF and CDF associated with the baseline distribution, Ω q,m,v,l,r,j is defined as in the proof of the previous Lemma.
For computational purposes if we let u = F (x), then du = f (x)dx and x = F −1 (u), thus the Renyi entropy can be expressed as

Parameter Estimation
The method of maximum likelihood is used in this paper to estimate model parameters. Here we discuss this method for the GAPA-G family of distributions. Suppose x 1 , x 2 , · · · , x n is a random sample of size n from the GAPA-G family of distributions. It can be shown that the total log-likelihood function is given by ln(g(x; ξ)) + ln((e α + 1)) + ln(e α(G(x;ξ)−1) ) + ln(αG(x; ξ) + e αG(x;ξ) + 1) − ln((e αG(x;ξ) + 1) 2 ) where ξ is a vector of parameters associated with the baseline distribution, and α ∈ R. Partial differentiation of the total log-likelihood function with respect to model parameters gives the following as the score functions Equating the score functions to zero and numerically solving the system of equations using techniques such as the quasi Newton-Raphson method, gives the maximum likelihood estimates for the model parameters. Let = (α; ξ), for the purposes of constructing confidence intervals for the parameters in the GAPA-G family of distributions, the observed information matrix, call it J( ), can be used due to the complex nature of the expected information matrix. The observed information matrix is given by When the usual regularity conditions are satisfied and that the parameters are within the interior of the parameter space, but not on the boundary, the distribution of √ n( − ) converges to the multivariate normal distribution N p+1 (0, I −1 ( )), where I( ) is the expected information matrix, and it is assumed that ξ = (ξ 1 , · · · , ξ p ). The asymptotic behavior remains valid when I( ) is replaced by the observed information matrix evaluated at J( ). The asymptotic multivariate normal distribution N p+1 (0, J −1 ( )) is a very useful tool for constructing an approximate 100(1 − ψ)% two-sided confidence intervals for the model parameters, where ψ is the significance level.

Monte Carlo Simulation Study
In this section we show that the method of maximum likelihood is adequate in estimating the parameters in the GAPA-G distribution. For this, a Monte Carlo simulation study is carried out to assess the performance of the estimation method in the Gull Alpha Power Ampadu-Logisitic(GAPAL) sub-model. Samples of sizes 200, 400, 500, and 700, are drawn from the GAPAL distribution, and the samples have been drawn for the following set of parameters The maximum likelihood estimators for the parameters a and b are obtained. The procedure has been repeated 400 times, and the mean and standard deviation for the estimates are computed, and the results are summarized in Table 4-6 below for each of sets I, and II, respectively, considered above From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as can been seen by plotting the standard deviation against the sample size. From the table above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated standard deviation consistently decrease with increasing sample size as can been seen by plotting the standard deviation against the sample size.
Overall the simulation study conducted, indicated that using the method of maximum likelihood in estimating model parameters is adequate.
Here we demonstrate usefulness of the new family to the breaking stress of carbon fibers data, Table 2 [20]. The PDF's and CDF's of the first submodel that we consider is already given in Section 4. The second submodel has PDF given by and CDF given by where x, a, c ∈ R, b > 0, and erfc gives the complementary error function. We call the distribution with the above PDF and CDF the Gull Alpha Power Ampadu-Normal family. We write AN (a, b, c) if JK is a Gull Alpha Power Ampadu-Normal random variable. Using the R software, we report below in Table 3, the estimates for the parameters in each of the two distributions alongside their standard errors.   Table 4 below. Whilst it appears from the fits above, that all the distributions are competitive in fitting the breaking stress of carbon fibers data, Table 4 reveals that the GAPAL distribution is most compatible with this data set, and hence can be considered the best in this instance.

A Characterization Theorem
It is well known that the hazard function, h F , of a twice differentiable function, F , satisfies the first order differential equation In this section we present a Kumaraswamy-GAPA-G type distribution. The result here is inspired by [21]. First let us introduce the following Definition 9.1. We say a random variable X follows a Kumaraswamy-G type distribution if its CDF is given by where G is some baseline distribution, x ∈ Supp(G), and ξ is a vector of parameters in the baseline distribution whose support depends on G.
Remark 9.2. Note that if we take λ = 1 and ϕ = 2 in equation (1) of [22], then we get the CDF in the above definition.
The PDF of the Kumaraswamy-G type distribution is given by where g is the PDF of the baseline distribution. Clearly the hazard rate function of the Kumaraswamy-G type distribution is given by h(x; ξ) = 2g(x; ξ) (1 − G(x; ξ)) .
Theorem 9.3. Let X : Ω → R be a continuous random variable. The PDF of X is 2g(x; ξ)(1 − G(x; ξ)) for some baseline distribution with PDF g and CDF G if and only if its hazard rate function h(x) satisfies the following differential equation h(x) = 2g(x) 2 (1 − G(x)) 2 with boundary condition h(0) = 2g(0).
Proof. If X has PDF as stated in the theorem, then the differential equation as stated holds. Now if the stated differential equation holds, then which is the hazard rate function of the Kumaraswamy-G type distribution.
Clearly, a characterization of the Kumaraswamy-GAPA-G type distribution. is obtained from the above theorem by letting the baseline PDF be given as in Section 3.2, and letting the baseline CDF be given as in Section 3.1.
As a further recommendation we suggest obtaining some properties and applications of a so-called Type II Gull Alpha Power Transform family of distributions. We leave the reader with the following. Obiviously, the PDF can be obtained by differentiating the CDF below Definition 10.1. A random variable X will be called Type II Gull Alpha Power if its CDF can be expressed as (e α + 1)F (x; ξ) e αF (x;ξ) 2 + 1 where α ∈ R, ξ is a vector of parameters in the baseline distribution F , and x ∈ Supp(F ).