Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / O / 1 / Solution

For the continuous distribution in the question, $F$ maps the support monotonically from zero to one. For $0<t<1$, strict monotonicity givesThus $F(X)$ is uniform on $(0,1)$, the probability integral transform.

The order statistics are the sample values sorted increasingly, $X_{(1)}\le \cdots \le X_{(n)}$. Monotonicity of $F$ makes $F(X_{(i)})$ the $i$th order statistic of independent uniforms. Its density is $n!t^{i-1}(1-t{)}^{n-i}/[(i-1)!(n-i)!]$, givingThis follows either by beta integrals or by the equal expected lengths of the $n+1$ uniform spacings.

For the stated log-logistic cumulative distribution,Therefore plot $\log [p_i/(1-p_i)]$ vertically against $\log X_{(i)}$ horizontally, with plotting positions $p_i=i/(n+1)$. A plausible model gives an approximately straight line, with slope $\beta -1$ and intercept $-(\beta -1)\log \alpha$. This specifies the axes explicitly, since reversing them would invert the slope.