Reparameterisation of a family of distributions

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

Rectangular distribution

If the family of rectangular distributions is defined with parameters \(\alpha\) and \(\beta\) so that the pdf of \(X\) is

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

then neither \(\alpha\) nor \(\beta\) can be considered to be location or scale parameters.

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

However if the distribution is reparameterised as having pdf

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

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.

Example

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

If \(X \sim \GammaDistn(\alpha, \beta)\), it has pdf

\[ f_X(x) \;\;=\;\; \begin{cases} \dfrac {\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-x\beta}& \quad\text{if }x \gt 0 \\[0.2em] 0 & \quad\text{otherwise} \end{cases} \]

In terms of parameters \(\alpha\) and \(\phi\), this can be written as

\[ f_X(x) \;\;=\;\; \begin{cases} \dfrac {1}{\phi^\alpha\Gamma(\alpha)} x^{\alpha - 1} e^{-\frac x{\phi}}& \quad\text{if }x \gt 0 \\[0.2em] 0 & \quad\text{otherwise} \end{cases} \]

The pdf of \(Y = aX\) can be found from the general result about the pdf of a monotonic transformation of a variable.

\[ \begin{align} f_Y(y) \;\;=\;\;\frac 1 a f_X\left(\frac y a\right) \;\;&=\;\; \dfrac {1}{a \phi^\alpha \Gamma(\alpha)} \left(\frac y a\right)^{\alpha - 1} e^{-\frac {y}{a \phi}} \\[0.4em] &=\;\; \dfrac {1}{(a \phi)^\alpha \Gamma(\alpha)} y^{\alpha - 1} e^{-\frac {y}{a \phi}} \end{align} \]

for \(y \gt 0\). This is the pdf of a Gamma distribution with the parameter \(\phi\) replaced by \((a\phi)\), showing that \(\phi\) is a scale parameter.