We have discussed canonical exponential famlies parametrized by \theta, with the log-partition function b(\theta ) having the property that b'(\theta ) = \mu. Recall that in GLMs, the point of the link function is to assume g(\mu (\mathbf x)) = \mathbf x^ T {\boldsymbol \beta }, where \mu is the regression function : the mean of Y given \mathbf X=\mathbf x, \mathbb E[Y \; |\; \mathbf X=\mathbf x].
Let f_\theta = \exp \left( \frac{y \theta - b(\theta )}{\phi } + c(y,\phi ) \right) for \phi \neq 0 describe an exponential family. Which one of the following statements about the function g(\mu ) = \theta is false ?
The canonical link function always exists.
g is identical to (b')^{-1}.
If g strictly increases, then g^{-1} strictly increases.
Regardless of the value of \phi, g is strictly increasing.
The false statement is: Regardless of the value of φ, g is strictly increasing.
In an exponential family, the log-partition function b(θ) is related to the mean parameter μ through the equation b'(θ) = μ. Taking the derivative of both sides with respect to θ, we have b''(θ) = μ'.
The link function g(μ) is defined as g(μ) = θ. Taking the derivative of both sides with respect to μ, we have g'(μ) = θ'.
From the equations b''(θ) = μ' and g'(μ) = θ', we can see that g'(μ) = (b''(θ))^{-1}. Therefore, statement 2 is true: g is identical to (b')^{-1}.
If g is strictly increasing, it means that g'(μ) > 0 for all μ. Since g'(μ) = (b''(θ))^{-1}, this implies that b''(θ) > 0 for all θ. This means that the second derivative of the log-partition function is positive, indicating a concave log-partition function. Therefore, statement 3 is true: if g strictly increases, then g^{-1} strictly increases.
However, the statement "Regardless of the value of φ, g is strictly increasing" is false. The value of φ affects the shape of the exponential family distribution and consequently the shape of the log-partition function b(θ). It is possible for the log-partition function to have regions where its second derivative is negative, resulting in a non-strictly increasing link function g(μ). Therefore, this statement is false.
The false statement among the given options is "Regardless of the value of ϕ, g is strictly increasing."
Explanation:
In the given information, the function g(μ) = θ is the link function used in generalized linear models (GLMs). The link function connects the mean of the response variable (μ) to the linear predictor in the model.
Let's evaluate each statement one by one:
1. The canonical link function always exists: This statement is true. In exponential families, there exists a canonical link function that connects the natural parameter (θ) to the mean of the response variable (μ) given the predictors.
2. g is identical to (b')^(-1): This statement is true. In exponential families, the derivative of the log-partition function b(θ) with respect to θ represents the mean of the response variable (μ). Therefore, the link function g(μ) is the inverse of the derivative of b(θ) with respect to θ, which is (b')^(-1).
3. If g strictly increases, then g^(-1) strictly increases: This statement is true. If the link function g(μ) is strictly increasing (i.e., as μ increases, g(μ) also increases), then the inverse link function g^(-1)(θ) will also be strictly increasing.
4. Regardless of the value of ϕ, g is strictly increasing: This statement is false. The function g(μ) = θ is not always strictly increasing regardless of the value of ϕ. The strict monotonicity of g(μ) depends on the specific distribution and the choice of the function g(μ). In some cases, g(μ) may be strictly increasing, while in others, it may not be.
Hence, the false statement is "Regardless of the value of ϕ, g is strictly increasing."