Let (\{ 0, \ldots , K\} , \{ ext {Bin}(K, heta ) \} _{ heta \in (0,1)}) denote a binomial statistical model. Let X_1, \ldots , X_ n \stackrel{iid}{\sim } ext {Bin}(K, heta ^*) for some unknown parameter heta ^* \in (0,1).

Question

Let (\{ 0, \ldots , K\} , \{ 	ext {Bin}(K, 	heta ) \} _{	heta \in (0,1)}) denote a binomial statistical model. Let X_1, \ldots , X_ n \stackrel{iid}{\sim } 	ext {Bin}(K, 	heta ^*) for some unknown parameter 	heta ^* \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 	heta, A depends on \sum _{i = 1}^ n X_ i, and B depends on 	heta.

What is A?

Use Sigma to stand for \sum _{i = 1}^ n X_ i.

unanswered 
 
What is B?

GPT 3.5 · Answer

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.

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).

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).