For a group of policies, you are given:

1. Losses follow the distribution function
   [[math]] F(x) = 1-\theta/x, \, x \gt \theta [[/math]]
2. A sample of 20 losses resulted in the following:

   Interval	Number of losses
   (0,10]	9
   (10,25]	6
   (25,[math]\infty[/math])	5

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

• 5.00
• 5.50
• 5.75
• 6.00
• 6.25

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

Key: B

[[math]] \begin{aligned} L(\theta) &= \left( \frac{\theta}{10}\right)^9 \left( \frac{\theta}{10} - \frac{\theta}{25} \right)^6 \left( \frac{\theta}{25}\right)^5 \propto (10-\theta)^9 \theta^{11} \\ l(\theta) &= 9 \ln(10 − \theta ) + 11 \ln(\theta ) \\ l^{'}(\theta) &= - \frac{9}{10-\theta} + \frac{11}{\theta} = 0 \\ 11(10-\theta) &= 9 \theta \\ 110 &= 20\theta \\ \theta &= 110/20 = 5.5 \end{aligned} [[/math]]

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