Which of the following characteristics is/are true about the distribution of the response variable in a logistic regression model?
(Check all that apply.)
It is a Bernoulli distribution
It can be any distribution over the interval (0,1).
It can be any distribution from the exponential family.
It is a distribution from the canonical exponential family.
In the logistic regression model, which of the following is the canonical link function?
the logistic function
the probit link function
In the logistic regression model, the maximum likelihood estimator is always unique.
True
False
The correct answers are:
- It can be any distribution from the exponential family.
- It is a distribution from the canonical exponential family.
- The logistic function is the canonical link function.
- The statement "the maximum likelihood estimator is always unique" is false.
For the first question, the correct choices are:
- It can be any distribution from the exponential family.
- It is a distribution from the canonical exponential family.
The response variable in a logistic regression model follows a distribution from the exponential family, which includes various distributions like the Bernoulli, binomial, and normal among others. So, the first option is not necessarily true.
The second statement is true because the response variable follows a distribution from the exponential family, and logistic regression is a member of this family.
For the second question, the correct choice is:
- the logistic function
The logistic function is the canonical link function for logistic regression. It is the natural link function that connects the linear predictor to the probability of a binary outcome.
For the third question, the correct choice is:
- True
The maximum likelihood estimator in logistic regression is indeed unique. It maximizes the likelihood function based on the observed data, and under certain assumptions, it leads to a unique solution.