Algo\(\left(0<\alpha<1\right)\)
- Set \(d\ =\ 1.0334-0.0766e^{2.2942\alpha},\ a=2^{\alpha}\left(1-e^{-\frac{d}{2}}\right)^{^{^{\alpha}}},\ b=\alpha d^{\alpha-1}e^{-d}\ and\ c\ =\ a+b\)
- Generate \(U\) from uniform(0,1).
- If \(U\le\frac{a}{a+b}\), then \(x=-2\ln\left[1-\left(cU\right)^{\frac{1}{\alpha}}\right]\), otherwise \(x=-\ln\left[\frac{c\left(1-U\right)}{\alpha d^{\alpha-1}}\right]\).
- Generate \(V\)from uniform(0,1). If \(x\le d\ and\ V\le\frac{x^{\alpha-1}e^{-\frac{x}{2}}}{2^{\alpha-1}\left(1-e^{-\frac{x}{2}}\right)^{^{^{\alpha-1}}}}\), then return x, otherwise go back to 2. If \(x>d\ and\ v\le\left(\frac{d}{x}\right)^{^{1-\alpha}}\), then retun x or go back to step 2.
Result
Refer to Table 1(\ref{my-label}) for proportion of rejection and time elapsed for different algorithms.