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.
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.
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