91Ó°ÊÓ

Consider the Uniform distribution: \[ f(y | \alpha, \beta)=\frac{1}{\beta-\alpha} I(y \in[\alpha, \beta]) \] The exponential family form is given by: \[ f(y | \theta) = h(y) \exp{( \eta(\theta)^T u(y) - A(\theta) )} \] For the Uniform distribution, the pdf is not easily expressed in this form. There is no natural parameter \(\theta\) and sufficient statistics \(u(y)\) that would fit the exponential family form directly, indicating it cannot be written in this format.

Since the Uniform distribution doesn't fit in the exponential family form generally, consider if a special case can. For example, if \(\alpha = 0\) and \(\beta = 1\), then the distribution becomes the standard uniform distribution \(U(0,1)\): \[ f(y) = 1, \quad y \in[0, 1] \] This still can't be expressed in the exponential family form as there is no natural parameter \(\theta\) or sufficient statistics \(u(y)\) simplifying to this form.

Consider the Weibull distribution: \[ f(y | \alpha, \beta)=\alpha \beta^{-\alpha} y^{\alpha-1} e^{-(y / \beta)^{\alpha}} \] To put this into the exponential family form, attempt to rewrite the pdf: \[ f(y | \alpha, \beta)= \exp{(\log(\alpha) - \alpha \log(\beta) + (\alpha-1) \log(y) - (y/\beta)^{\alpha})} \] Now compare this with \(f(y | \theta) = h(y) \exp{( \eta(\theta)^T u(y) - A(\theta) )} \): \[ \eta(\theta) = \begin{pmatrix} \alpha \log(\beta) \ \log(y) \ -y^\alpha / \beta^\alpha \end{pmatrix} \quad u(y) = \begin{pmatrix} 1 \ y^{\alpha-1} \ \exp(-(y/\beta)^{\alpha}) \end{pmatrix} \quad A(\theta) = \log(\alpha) \] This shows that the Weibull distribution can indeed be transformed into the exponential family form.

Consider the Gumbel distribution: \[ f(y | \alpha, \beta)=\frac{1}{\beta} \exp{((y-\alpha) / \beta -\exp{((y-\alpha) / \beta)})} \] Attempt to rewrite the pdf in the exponential family form: \[ f(y | \alpha, \beta)=\frac{1}{\beta} \exp{((y-\alpha) / \beta)} \exp{(-\exp{((y-\alpha) / \beta)})} \] This fits the form: \[ f(y | \theta) = \exp\{ -\frac{y-\alpha}{\beta} - \exp\left(-\frac{y-\alpha}{\beta}\right) - \log\beta \} \] Letting \(\eta = \frac{1}{\beta} \) and \( \theta = \alpha \), we have: \[ f(y | \alpha, \beta) = h(y) \exp{( \eta(\theta)^T u(y) - A(\theta) )} \] Hence, the Gumbel distribution fits into the exponential family form.