Let (\{ 0, \ldots , K\} , \{ \text {Bin}(K, \theta ) \} _{\theta \in (0,1)}) denote a binomial statistical model. Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \text {Bin}(K, \theta ^*) for some unknown parameter \theta ^* \in (0,1).
The log-likelihood of this statistical model can be written
C + A \log B + (n K - A ) \log (1 - B)
where C is independent of \theta, A depends on \sum _{i = 1}^ n X_ i, and B depends on \theta.
What is A?
Use Sigma to stand for \sum _{i = 1}^ n X_ i.
unanswered
What is B?
To find the value of A, we need to determine what A depends on in the expression given for the log-likelihood.
From the expression, we can see that A depends on the sum of the random variables X_1, X_2, ..., X_n, denoted as Sigma. Therefore, A is dependent on the value of Sigma, which is the sum of the observed data in the sample.
To find B, we need to analyze the expression for the log-likelihood:
C + A log B + (nK - A) log(1 - B)
From the given information, we know that B depends on θ. So, we need to figure out how θ is related to B.
To do that, let's consider the binomial probability mass function (pmf) for a single observation:
P(X = k) = (K choose k) θ^k (1 - θ)^(K-k)
In this expression, θ^k is equivalent to the term B^k in the log-likelihood expression. Therefore, we can assume that B represents the value of θ.
In summary:
B = θ
Hence, B represents the parameter θ in the log-likelihood expression for the binomial statistical model.