BETA CHI-SQUARE DISTRIBUTION PROPERTIES AND APPLICATIONS

: In this paper we introduce the distribution of three parameters, called Beta Chi-square distribution (BCHI), is presented and contains the Chi-square distribution as a sub-model. Its density function can be expressed as a linear combination of Chi-square density function. Some structural properties of this distribution, how moments, and hazard function are presented. Estimates of the model parameters are performed using the Maximum Likelihood method. We obtain the observed information matrix and discuss inference methods. In order to demonstrate the utility of the distribution, a real data set is analyzed.

1 Introduction Lancaster (1966) observed that Bienaymé (1838) obtained the Chi-square distribution as the convergence in distribution of the random variable k i=1 (N i − np i ) 2 /np i , where N 1 , · · · , N k has a multinomial joint distribution with the parameters n, p 1 , · · · , p k . It is also known that U 1 , , U 2 , · · · , U k are independent standard normal variables, so k i=1 U 2 i has a Chi-square distribution with k degrees of freedom. Pearson's Chi-square distribution also appeared in (1900) as the approximate distribution for Chi-square statistics used for various tests on contingency tables (of course the distribution exact of the statistic is discrete). The use of the Chi-square distribution to approximate the distribution in a quadratic form (particularly the positive defined one) in multinormal variables is

The Model Definition
If G(x) denotes the cumulative distribution function (cdf) of a random variable X, then the generalized class of distributions, as defined by Eugene et al. (2002), can be define as for a > 0 e b > 0, where denotes the incomplete beta ration, and the incomplete beta function is given by and The probability density function corresponding to (1) is given by where g(x) is the pdf corresponding to G(x).
In this article, we consider the case when G(x) is the cumulative distribution function of the Chi-square distribution with parameter α. Thus, the random variable X follows the beta Chi-square distribution BCHI(α, a, b), with probability density function (pdf) where γ(α/2, x/2) = x/2 0 t α/2−1 exp(−t)dt is the incomplete Gamma function.
By using Equation (1), the cdf of the BCHI(α, a, b) is given by The BCHI distribution for a = b = 1 reduces to the Chi-square distribution with parameter α. The Fig. 1 illustrates the distribution function (6) and the density function (5), respectively, of BCHI, for different values of α.

General formula for moments
In statistical analysis, it is essential to study the moments, some of the most important characteristics of a distribution can be studied using moments, such as trend, dispersion, asymmetry and kurtosis. According to Cordeiro and Nadarajah (2011), we must assume that X has pdf of any primitive G distribution, in our case the Chi-square distribution, and Y follows the Beta Chi-square distribution function. Here, we use power series to rewrite Chi-square beta distribution, as follows Repeating this process, we can rewrite the equation (4) as follows on what More details can be found in Cordeiro and Nadarajah (2011). In order to find ordinary moments, we use the probability-weighted moments (PWM) method, proposed by Greenword, et al. (1979) and later applied to the Betas distributions by Cordeiro and Nadarajah (2011).
According to Greenword, et al. (1979), a distribution function F = F (x) = P (X ≤ x) can be characterized by PWM, when it is defined as Consider that X has pdf of any G distribution function and Y follows the pdf of the beta G distribution. So, we have to a ∈ Z is for a real non integer, 3.1 Moments of the beta Chi-square The sth moment of the Y can be expressed in terms of the (s, r)th PWM of X; i.e., M s,r . We should still consider ∞ r=0 w r = 1 and ∞ l,j=0 j r=0 w l,j,r = 1. With the formulas above obtained by Cordeiro and Nadarajah (2011), we can find the moments of Beta Chi-square.

Hazard functions
Let X be a continuous random variable with distribution function F , and probability density function (pdf) f ,then the hazard function is given by h for x ≥ 0, α > 0, a > 0 and b > 0. The plots at Fig. 2 show various shapes including monotonically decreasing, monotonically increasing with four combinations of values of the parameters. This flexibility makes the BCHI hazard rate function useful and suitable for behaviors which are more likely to be encountered or observed in the reality.

Inference
Let x = (x 1 , · · · , x n ) be a random sample of the BCHI distribution with unknown parameter vector θ = (α, a, b). The log likelihood ℓ = ℓ(θ; x) for θ is The maximum likelihood estimate (MLE)θ of θ is calculated numerically from the nonlinear equations U θ = 0 using the EM algorithm. The components of the score vector U θ = (U α , U a , U b ) T , consider α 2 = λ, are given by the MeijerG and PolyGamma functions were defined by Fields (1972) and Wolfram (1988), respectively.
The EML'sθ = (λ,â,b) ⊤ in θ = (λ, a, b) ⊤ are simultaneously the solutions of the equations U λ = U a = U b = 0 and can be obtained numerically using the Newton-Raphson method. The observed information matrix can be obtained by J n (θ) = −∂ 2 ℓ(θ) ∂(θ)∂θ ⊤ = −U ij , for i, j = λ, a and b, We can calculate the likelihood ratio (LR) test to test some sub-models of the BCHI distribution. For example, we can use LR to check whatever the fit using the BCHI distribution is statistically "best" than an fit using the χ 2 distribution, for a given data set. Consider the partition θ = θ ⊤ 1 , θ ⊤ 2 the vector of parameters of the BCHI distribution, where θ 1 is a subset of parameters of interest and θ 2 is a subset of perturbation parameter vectors. The LR statistic for testing null hypotheses H 0 : θ 1 = θ (0) 1 versus the alternative hypothesis H 1 : θ 1 ̸ = θ (0) 1 it is given by w = 2 ℓ(θ) − ℓ(θ) , beingθ andθ are the MLE's under the null hypothesis and the alternative, respectively, and θ (0) 1 is a specified parameter vector. The w statistic is asymptotically distributed (n → ∞) for χ 2 k , where k is the size of the subset of interest θ 1 . Then, we can compare the BCHI model against the model χ 2 to test H 0 : a = b = 1 versus H 1 : a ̸ = b ̸ = 1 and the LR statistic becomes w = 2 ℓ λ ,â,b − ℓ λ , 1, 1 , whereλ,â andb they are the MLE's in H 1 eλ,ã eb are MLE's under H 0 .
Non-nested distributions can be compared based on the Akaike information criterion given by the formula AIC = −2ℓ(θ) + 2p and the Bayesian information criterion defined by BIC = 2ℓ(θ) + p log(n), being p the number of parameters in the model. The lowest value distribution of any of these criteria (among all the distributions considered) is generally considered the best choice to describe the data set.

Application
In this section, the Beta Chi-square distribution was adjusted using a real database and then compared with the Chi-square distribution in order to compare them and verify their potentiality. The data set represents the survival times, in weeks, of 33 patients suffering from acute Myelogeneous Leukemia. These data have been analyzed by Feigl and Zelen (1965). The data set was recently studied by Woll et al. (2014) and Altun et al. (2021). Obtaining the maximum likelihood estimates (MLEs) for the distribution parameters, the maxLik function in maxLik-package of the statistical software R was used, and the iteration method was Newton Raphson. The estimated values of the parameters, the Hannan-Quinn information criterion (HQC), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) are presented in the Table 1.
The Figure 3 shows the fits of the BCHI and Chi-square. According to the illustration, the good fit of the BCHI distribution is observed. Table 1 lists the MLEs of the models parameters BCHI and Chi-square, and the statistics AIC and BIC. These results show that the BCHI distribution has the lowest statistics and so it could be chosen as the best model.

Conclusions
In this work, we define a new model called Beta Chi-square distribution. It is observed the new distribution of three parameters is quite similar in nature to the distribution Chi-square. Although the new distribution is more flexible due the fact it has more parameters than the primitive distribution.