IT WOULD BE CLEARLY BENEFICIAL FOR YOU BY USING THE RMD FILES IN THE GITHUB DIRECTORY FOR FURTHER EXPLANATION OR UNDERSTANDING OF THE R CODE FOR THE RESULTS OBTAINED IN THE VIGNETTES.

# Estimating the parameters using Moment Generating Function

In the eleven Binomial Mixture and Alternate Binomial Distributions only Beta-Binomial Distribution is related to this technique. Moment Generating function only exists to Beta-Binomial Distribution.

Let $$Y=[Y_1,Y_2,...,Y_N]^T$$ be a random sample of size $$N$$ from Beta-Binomial distribution with the probability mass function. $$n$$ is fixed for all clusters. Therefore, shape parameters $$\alpha$$(a) and $$\beta$$(b) are estimated using the below equations as $$\hat{\alpha}$$ and $$\hat{\beta}$$.

$\hat{\alpha}= \frac{(n*m_1 -m_2)m_1}{n(m_2-m_1-{m_1}^2)+{m_1}^2}$

$\hat{\beta}= \frac{(n*m_1-m_2)(n-m_1)}{n(m_2-m_1-{m_1}^2)+{m_1}^2}$
where $$m_1=\sum_{i=1}^{N} \frac{y_i}{N}$$ and $$m_2= \sum_{i=1}^{N} \frac{{Y_i}^2}{N}$$ are the first two sample moments.

These equations produce unique values for $$\alpha$$ (a) and $$\beta$$ (b).

Below is the code for estimating shape parameters using this technique and function used for this is EstMGFBetaBin.

## Using the Chromosome data provided in the package.

This EstMGFBetaBin function is of output of class mgf, where outputs include estimated a ,b parameters, minimized Negative Log Likelihood value min, Akaike Information Criterion (AIC) and function call with input arguments.

##   No.of.Asso fre
## 1          0  32
## 2          1 103
## 3          2 122
## 4          3  80
## Estimated alpha parameter for Chromosome data = 6.167982
## Estimated beta parameter for Chromosome data = 4.455237

## Using the Male Children data provided in the package

##    No_of_Males freq
## 1            0    3
## 2            1   24
## 3            2  104
## 4            3  286
## 5            4  670
## 6            5 1033
## 7            6 1343
## 8            7 1112
## 9            8  829
## 10           9  478
## 11          10  181
## 12          11   45
## 13          12    7
## Estimated alpha parameter Male_children data= 34.13502
## Estimated beta parameter Male_children data= 31.60849

Further, we can use the above estimated parameters in the fitBetaBin function and check how good the Beta-Binomial Distribution is fitted for a given Binomial outcome data.