Even though logistic regression is formulated with continuous input data in mind, one can also try to apply it to categorical inputs. For example, consider the following set-up: We observe \, n \, samples \, Y_ i \in \{ 0, 1\} \,, \, i = 1, \dots , n \,, and covariates \, X_ i \in \{ 0, 1\} \,, \, i = 1, \dots , n \,. Moreover, assume that given \, X_ i \,, the \, Y_ i \, are independent.
First, let us apply regular maximum likelihood estimation. To this end, write
\displaystyle f_{00} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 0 \text { and } Y_ i = 0 \}
\displaystyle f_{01} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 0 \text { and } Y_ i = 1 \}
\displaystyle f_{10} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 1 \text { and } Y_ i = 0 \}
\displaystyle f_{11} = {} \displaystyle \frac{1}{n} \# \{ i : X_ i = 1 \text { and } Y_ i = 1 \}
and assume that \, f_{00}, f_{01}, f_{10}, f_{11} > 0 \,. We can parametrize this model in terms of
\displaystyle p_{01} = {} \displaystyle P(Y_ i = 1 | X_ i = 0)
\displaystyle p_{11} = {} \displaystyle P(Y_ i = 1 | X_ i = 1)
Compute the maximum likelihood estimators \, \widehat{p}_{01} \, and \, \widehat{p}_{11} \, for \, p_{01} \, and \, p_{11} \,, respectively. Express your answer in terms of f_{00} (enter “A"), f_{01} (enter “B"), f_{10} (enter “C"), f_{11} (enter “D") and n.
\widehat{p}_{01}
unanswered
\widehat{p}_{11}
To compute the maximum likelihood estimators for p_{01} and p_{11}, we can use the following formulas:
\widehat{p}_{01} = \frac{f_{01}}{f_{00} + f_{01}}
\widehat{p}_{11} = \frac{f_{11}}{f_{10} + f_{11}}
Therefore, the maximum likelihood estimator for p_{01} is \boxed{\frac{f_{01}}{f_{00} + f_{01}}}, and the maximum likelihood estimator for p_{11} is \boxed{\frac{f_{11}}{f_{10} + f_{11}}}.
To compute the maximum likelihood estimators for \(\widehat{p}_{01}\) and \(\widehat{p}_{11}\), we can use the following formulas:
\[
\widehat{p}_{01} = \frac{f_{01}}{f_{00} + f_{01}}
\]
\[
\widehat{p}_{11} = \frac{f_{11}}{f_{10} + f_{11}}
\]
where \(f_{00}\), \(f_{01}\), \(f_{10}\), and \(f_{11}\) are defined as follows:
\[
f_{00} = \frac{1}{n} \# \{ i : X_i = 0 \text{ and } Y_i = 0 \}
\]
\[
f_{01} = \frac{1}{n} \# \{ i : X_i = 0 \text{ and } Y_i = 1 \}
\]
\[
f_{10} = \frac{1}{n} \# \{ i : X_i = 1 \text{ and } Y_i = 0 \}
\]
\[
f_{11} = \frac{1}{n} \# \{ i : X_i = 1 \text{ and } Y_i = 1 \}
\]
Please note that the symbol "#" represents the count of the specified conditions.