Personal auto property damage claims in a certain region are known to follow the Weibull distribution:

[[math]] F(x) = 1 - \exp \left[-(\frac{x}{\theta})^{0.2}\right], \, x \gt 0 [[/math]]

A sample of four claims is:

130  240  300  540

The values of two additional claims are known to exceed 1000.

Calculate the maximum likelihood estimate of [math]\theta[/math].

• Less than 300
• At least 300, but less than 1200
• At least 1200, but less than 2100
• At least 2100, but less than 3000
• At least 3000

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: E

The density function is

[[math]] f(x) = \frac{0.2x^{-0.8}}{\theta^{0.2}}e^{-(x/\theta)^{0.2}}. [[/math]]

The likelihood function is

[[math]] \begin{aligned} L(\theta) &= f (130) f (240) f (300) f (540)[1 − F (1000)]^2 \\ &= \frac{0.2(130)^{-0.8}}{\theta^{0.2}} e^{-(130/\theta)^{0.2}} \frac{0.2(240)^{-0.8}}{\theta^{0.2}} e^{-(240/\theta)^{0.2}} \frac{0.2(300)^{-0.8}}{\theta^{0.2}} e^{-(300/\theta)^{0.2}} \frac{0.2(540)^{-0.8}}{\theta^{0.2}} e^{-(540/\theta)^{0.2}}e^{-(1000/\theta)^{0.2}}e^{-(1000/\theta)^{0.2}} \\ & \propto \theta^{-0.8} e^{ −\theta^{0.2} (130^{0.2} + 240^{0.2} + 300^{0.2} + 540^{0.2} +1000^{0.2} +1000^{0.2} )} \\ l(\theta) &= -0.8\ln(\theta) - \theta^{-0.2}(130^{0.2} + 240^{0.2} + 300^{0.2} + 540^{0.2} + 1000^{0.2} + 1000^{0.2}) \\ &= −0.8\ln(\theta ) − 20.2505 \theta^{−0.2} , \\ l^{'}(\theta) &= -0.8\theta^{-1} + 0.2(20.2505)\theta^{-1.2} = 0, \\ \theta^{-0.2} &= 0.197526, \, \hat{\theta} = 3,325.67. \end{aligned} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.