Which one of the following are valid link functions? Recall that a link function g is required to be monotone increasing and differentiable. (Choose all that apply.)
Note: The link function, in general, can be monotone increasing or monotone decreasing. In this class, we have chosen as convention to require it be monotone increasing.
g(\mu ) = \mu , \mu \in \mathbb {R}
g(\mu ) = -\frac{1}{\mu }, \mu > 0
g(\mu ) = \mu ^2, \mu \in \mathbb {R}
\ln \left(\frac{\mu ^3}{1-\mu ^3}\right),\, 0<\mu < 1
-\ln \left[-\ln \left(\frac{\mu }{n}\right)\right], 0 < \mu < n and n > 0 known
The valid link functions are:
g(\mu ) = \mu , \mu \in \mathbb {R}
g(\mu ) = \mu ^2, \mu \in \mathbb {R}
\ln \left(\frac{\mu ^3}{1-\mu ^3}\right),\, 0<\mu < 1
To determine which of the following link functions are valid, we need to check if they satisfy the requirements of being monotone increasing and differentiable.
1. g(μ) = μ, μ ∈ ℝ:
This function is a valid link function as it is both monotone increasing (since the derivative is 1) and differentiable.
2. g(μ) = -1/μ, μ > 0:
This function is not a valid link function because it is not monotone increasing. As μ increases, -1/μ decreases.
3. g(μ) = μ^2, μ ∈ ℝ:
This function is not a valid link function because it is not monotone increasing. As μ increases, μ^2 increases at an increasing rate.
4. ln((μ^3) / (1-μ^3)), 0 < μ < 1:
This function is a valid link function as it is both monotone increasing and differentiable within the given range.
5. -ln[-ln(μ/n)], 0 < μ < n and n > 0 known:
This function is a valid link function as it is both monotone increasing and differentiable within the given range.
Therefore, the valid link functions are:
- g(μ) = μ, μ ∈ ℝ
- ln((μ^3) / (1-μ^3)), 0 < μ < 1
- -ln[-ln(μ/n)], 0 < μ < n and n > 0 known