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.

# Negative Log Likelihood Estimation or Maximum Likelihood Estimation Technique

All of the Binomial Mixture and Alternate Binomial distributions in this package use this technique. This is very simple, by using the probability mass function we can develop the Likelihood function and then maximize it or develop the Negative Log Likelihood function and minimize it. Maximizing or minimizing means finding appropriate values for the parameters in question which will provide a most maximum Likelihood value or least minimum Negative Log Likelihood value.

If we have one parameter to estimate it is possible to use brute force technique. When the number of parameters need to be estimated increases this is very time consuming. Similarly, if the parameter in estimation has a very broad domain also it would be difficult.

In situations with difficulty of such we can use functions such as optim,mle, mle2, nlm, nlminb. fitODBODis using the mle2 function of the bbmle package. There are situations where user defined functions without the help of mle2.

## Functions which does not need bbmle package

These situations occurring are very less in comparing to the number of distributions we have in fitODBOD. They genuinely occur when the parameter in need of estimation has a closed domain and that domain is very small. Here by small it is indicating a scenario of such as in-between zero and one or greater than zero.

# estimating parameter for p and alpha using Additive Binomial Distribution of Chromosome data
Chromosome_data
##   No.of.Asso fre
## 1          0  32
## 2          1 103
## 3          2 122
## 4          3  80

The distributions in relation are

1. from Alternate Binomial Distribution - Additive Binomial Distribution

There are two parameters to estimate, which are p and alpha. These parameters are unique values, therefore estimating them once is enough. Time related to estimation is dependent on the data. The outputs are of class mlAB and ml, where the outputs can be extracted by the methods print, summary, coef and AIC.

## Estimated parameter p for Chromosome data= 0.580941
## Estimated parameter alpha for Chromosome data= 0.088305
1. from Binomial Mixture Distribution - Triangular Binomial Distribution

There is only one parameter that needs to be estimated, which is mode or c. This is in-between zero and one. The outputs are of class mlTB and ml, where the outputs can be extracted by the methods print, summary, coef and AIC.

## Estimated parameter mode or c for Chromosome data= 0.707276

## Functions which does need bbmle package

By Using the vignette file or manual pdf file we can get to know how the bbmle package works. Specifically looking into mle2 function is necessary and also class of mle2. Notable points related are in point form as

• coef
• min
• nobs
• minuslogl
• method

It is preferable to try them and understand the functions capability and process. You can achieve proper results by looking at how the inputs should mentioned, because there is a specific format to follow.

Below are instances of using the mle2 function.

The distributions in relation are

### from Alternate Binomial distributions

1. Beta-Correlated Binomial distribution

Here there are three parameters to be estimated which are cov, a and b. It should be mentioned that cov is a unique value but alpha and beta can take specific combinations.

• cov is in the domain region of -infinity and +infinity.
• a, b are domain in the region of greater than zero.
## Estimated parameter covariance for Chromosome data= 0.01106996
## Estimated parameter a for Chromosome data= 12.9595
## Estimated parameter b for Chromosome data= 9.350303
1. Correlated Binomial distribution

Two parameters that needs to be estimated which are cov and p. They are unique values for the distribution.

• cov is in the domain region of -infinity and +infinity.
• p is in the domain region of zero and one.
## Estimated parameter p for Chromosome data= 0.5809409
## Estimated parameter covariance for Chromosome data= 0.02149907
1. COM Poisson Binomial distribution

Two parameters that needs to be estimated which are p and cov. They are unique values for the distribution.

• cov is in the domain region of -infinity and +infinity.
• p is in the domain region of zero and one.
## Estimated parameter p for Chromosome data= 0.5687794
## Estimated parameter covariance for Chromosome data= 0.7025567
1. Multiplicative Binomial distribution

Two parameters that needs to be estimated which are p and theta. Only p value is unique.

• p is in the domain region of zero and one.
• theta is in the domain region of greater than zero.
## Estimated parameter p for Chromosome data= 0.5687846
## Estimated parameter theta for Chromosome data= 0.8493023
1. Lovinson Multiplicative Binomial distribution

Two parameters that needs to be estimated which are p and phi. Only p value is unique.

• p is in the domain region of zero and one.
• phi is in the domain region of greater than zero.
## Estimated parameter p for Chromosome data= 0.5687846
## Estimated parameter phi for Chromosome data= 0.8493023

### from Binomial Mixture distributions

1. Beta-Binomial distribution

Two parameters to be estimated which are a and b. Together they have to represent a unique combination.

• a,b in the domain region of greater than zero.
## Estimated parameter a for Chromosome data= 6.131277
## Estimated parameter b for Chromosome data= 4.426927
1. Kumaraswamy Binomial distribution

Two parameters to be estimated which are a and b. Together they have to represent a unique combination.

• a,b in the domain region of greater than zero.
• it value or iteration value as a variable replacing the infinity in the summation.

Here the number of iterations has to be estimated as well, which is time consuming. therefore I haven’t estimated it.

## Estimated parameter a for Chromosome data= 4.220499
## Estimated parameter b for Chromosome data= 6.008058
## Estimated parameter iteration for Chromosome data= 7500
1. GHGBB distribution

Three parameters to be estimated which are a,b and c. They have to represent a unique combination as a whole.

• a,b,c in the domain region of greater than zero.
## Estimated parameter a for Chromosome data= 2.774708
## Estimated parameter b for Chromosome data= 23.16726
## Estimated parameter c for Chromosome data= 8.626376
1. McGBB distribution

Three parameters to be estimated which are a,b and c. They have to represent a unique combination as a whole.

• a,b,c in the domain region of greater than zero.
## Estimated parameter a for Chromosome data= 0.3865119
## Estimated parameter b for Chromosome data= 16.20521
## Estimated parameter c for Chromosome data= 9.362786
1. Gamma Binomial distribution

Two parameters to be estimated which are a and b. Together they have to represent a unique combination.

• c,l in the domain region of greater than zero.
## Estimated parameter c for Chromosome data= 7.470102
## Estimated parameter l for Chromosome data= 4.326182
1. Grassia II Binomial distribution

Two parameters to be estimated which are a and b. Together they have to represent a unique combination.

• a,b in the domain region of greater than zero.
## Estimated parameter a for Chromosome data= 13.88833
## Estimated parameter b for Chromosome data= 0.06473275