Reparameterisation of a family of distributions may be necessary before we can identify location and scale parameters.

Rectangular distribution

Rectangular distributions are usually defined to have

\[ f(x) = \begin{cases} \frac {\large 1} {\large \beta-\alpha} & \quad\text{for } \alpha \lt x \lt \beta \\[0.2em] 0 & \quad\text{otherwise} \end{cases} \]

Neither \(\alpha\) nor \(\beta\) are location or scale parameters since

\[ Y = a + bX \sim \RectDistn(a + b\alpha, a + b\beta) \]

However if the distribution is reparameterised with

\[ f(x) = \begin{cases} \frac {\large 1} {\large \phi} & \quad\text{for } \alpha \lt x \lt \alpha + \phi\\[0.2em] 0 & \quad\text{otherwise} \end{cases} \]

then \(Y = a + bX\) has a rectangular distribution with pdf

\[ f_Y(y) = \begin{cases} \frac {\large 1} {\large b\phi} & \quad\text{for } (a + b\alpha) \lt y \lt (a + b\alpha) + b\phi\\[0.2em] 0 & \quad\text{otherwise} \end{cases} \]

This is a rectangular distribution with parameters \(\alpha^* = a + b\alpha\) and \(\phi^* = b\phi\) so \(\alpha\) and \(\phi\) are location and scale parameters.

In the following example, reparameterisation is again necessary before a scale parameter can be found.

Question

In a \(\GammaDistn(\alpha, \beta)\) distribution, show that \(\phi = \frac {\large 1}{\large \beta}\) is a scale parameter.

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