Revision as of 14:26, 13 May 2023 by Admin (Created page with "'''Key: B''' <math display="block">\begin{aligned} & L=F(1000)^{7}[F(2000)-F(1000)]^{6}[1-F(2000)]^{7} \\ & =\left(1-e^{-1000 / \theta}\right)^{7}\left(e^{-1000 / \theta}-e^{...")
Exercise
May 13'23
Answer
Key: B
[[math]]\begin{aligned}
& L=F(1000)^{7}[F(2000)-F(1000)]^{6}[1-F(2000)]^{7} \\
& =\left(1-e^{-1000 / \theta}\right)^{7}\left(e^{-1000 / \theta}-e^{-2000 / \theta}\right)^{6}\left(e^{-2000 / \theta}\right)^{7} \\
& =(1-p)^{7}\left(p-p^{2}\right)^{6}\left(p^{2}\right)^{7}=p^{20}(1-p)^{13}
\end{aligned}[[/math]]
where [math]p=e^{-1000 / \theta}[/math]. The maximum occurs at [math]p=20 / 33[/math] and so [math]\hat{\theta}=-1000 / \ln (20 / 33)=1996.90[/math].
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.