Discussion
Which of the following distributions does not belong to the Cramer family ?
Answer: A
The distribution outside the Cramer family is the exponential distribution with location parameter \(\theta\) and scale 1.
The Cramer (regular) family requires, among other things, that the support of the density does not depend on the parameter, so that differentiation and integration can be interchanged and the Cramer-Rao bound applies. The density \(e^{-(x - \theta)}\) for \(x \ge \theta\) has support \([\theta, \infty)\), which moves with \(\theta\). This is the classic non-regular case, where the MLE \(X_{(1)}\) converges at rate \(n\), faster than the regular \(\sqrt{n}\).
The Cauchy location family, \(N(\theta, \theta)\) and the Laplace scale family all have fixed support (the whole line) and densities smooth in \(\theta\), so they satisfy the regularity conditions.
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