Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Permodelan dan Teori Risiko |
| Periode Ujian |
: |
November 2018 |
| Nomor Soal |
: |
7 |
SOAL
Sebuah sampel acak berukuran \(n\) diambil dari distribusi dengan fungsi kepadatan peluang
\(\begin{array}{*{20}{c}} {f\left( x \right) = \frac{\theta }{{{{\left( {\theta + x} \right)}^2}}},}&{0 < x < \infty ,}&{\theta > 0} \end{array}\)
Hitung variansi asimtotik dari maximum likelihood estimator \(\theta \)
- \(\frac{{3{\theta ^2}}}{n}\)
- \(\frac{1}{{3n{\theta ^2}}}\)
- \(\frac{3}{{n{\theta ^2}}}\)
- \(\frac{n}{{3{\theta ^2}}}\)
- \(\frac{1}{{3{\theta ^2}}}\)
| Diketahui |
Sebuah sampel acak berukuran \(n\) diambil dari distribusi dengan fungsi kepadatan peluang
\(\begin{array}{*{20}{c}} {f\left( x \right) = \frac{\theta }{{{{\left( {\theta + x} \right)}^2}}},}&{0 < x < \infty ,}&{\theta > 0} \end{array}\) |
| Rumus yang digunakan |
\(I\left( \theta \right)Var\left( \theta \right) = 1\)
\(I\left( \theta \right) = – nE\left[ {\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}}} \right]\) |
| Proses pengerjaan |
\(\ln f\left( x \right) = \ln \left( \theta \right) – 2\ln \left( {\theta + x} \right)\)
\(\frac{{\partial \ln f\left( x \right)}}{{\partial \theta }} = \frac{1}{\theta } – \frac{2}{{\theta + x}}\)
\(\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}} = – \frac{1}{{{\theta ^2}}} + \frac{2}{{{{\left( {\theta + x} \right)}^2}}}\) |
|
\(E\left[ {\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}}} \right] = – \frac{1}{{{\theta ^2}}} + \int_0^\infty {\frac{{2\theta }}{{{{\left( {\theta + x} \right)}^4}}}dx} \)
\(E\left[ {\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}}} \right] = – \frac{1}{{{\theta ^2}}} + \left[ {\left. { – \frac{{2\theta }}{{3{{\left( {\theta + x} \right)}^3}}}} \right|_0^\infty } \right] = – \frac{1}{{{\theta ^2}}} + \frac{2}{{3{\theta ^2}}} = – \frac{1}{{3{\theta ^2}}}\) |
|
\(I\left( \theta \right) = – nE\left[ {\frac{{{\partial ^2}\ln f\left( x \right)}}{{\partial {\theta ^2}}}} \right] = \frac{n}{{3{\theta ^2}}}\)
\(Var\left( \theta \right) = \frac{1}{{I\left( \theta \right)}} = \frac{{3{\theta ^2}}}{n}\) |
| Jawaban |
a. \(\frac{{3{\theta ^2}}}{n}\) |
