Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
April 2019 |
| Nomor Soal |
: |
23 |
SOAL
Dari tabel nomor 22. Tentukan interval kepercayaan 95% berdistribusi log transformed untuk H(3) berdasarkan estimasi Nelson-Aalen
- \(\left( {0,3:0,9} \right)\)
- \(\left( {0,31:1,54} \right)\)
- \(\left( {0,39:0,99} \right)\)
- \(\left( {0,56:0,79} \right)\)
- \(\left( {0,44:1,07} \right)\)
| Diketahui |
| Waktu (t) |
Jumlah yang beresiko saat t |
Jumlah kegagalan saat t |
| 1 |
30 |
5 |
| 2 |
27 |
9 |
| 3 |
32 |
6 |
| 4 |
25 |
5 |
| 5 |
20 |
4 |
|
| Rumus yang digunakan |
\(\hat H\left( {{t_k}} \right) = \sum\limits_{j = 1}^k {\frac{{{d_j}}}{{{r_j}}}} ,{\rm{ }}\) \({\rm{ }}{t_k} \le t < {t_{k + 1}}\)
\(\widehat {Var\left( {\hat H\left( {{t_k}} \right)} \right)} = \sum\limits_{j = 1}^k {\frac{{{d_j}}}{{r_j^2}}} ,{\rm{ }}{t_k} \le t < {t_{k + 1}}\)
Log-Transformed confidence untuk estimasi Nelson-Aaen
\(\left( {\frac{{\hat H\left( {{t_k}} \right)}}{U}:U \cdot \hat H\left( {{t_k}} \right)} \right)\) dengan \(U = \exp \left( {\frac{{{z_{\frac{{p + 1}}{2}}}\sqrt {\widehat {Var}\left[ {\hat H\left( {{t_k}} \right)} \right]} }}{{\hat H\left( {{t_k}} \right)}}} \right)\) |
| Proses pengerjaan |
\(\hat H\left( 3 \right) = \sum\limits_{j = 1}^3 {\frac{{{d_j}}}{{{r_j}}}} = \frac{5}{{30}} + \frac{9}{{27}} + \frac{6}{{32}} = 0,6875\) |
| \(\widehat {Var}\left[ {\hat H\left( 3 \right)} \right] = \sum\limits_{j = 1}^3 {\frac{{{d_j}}}{{r_j^2}}} = \frac{5}{{{{30}^2}}} + \frac{9}{{{{27}^2}}} + \frac{6}{{{{32}^2}}} = 0,0238\) |
| \(U = \exp \left( {\frac{{{z_{\frac{{0,95 + 1}}{2}}}\sqrt {\widehat {Var}\left[ {\hat H\left( 3 \right)} \right]} }}{{\hat H\left( 3 \right)}}} \right) = \exp \left( {\frac{{1,96\sqrt {0,0238} }}{{0,6875}}} \right) = 1,5519\) |
| \(\left( {\frac{{\hat H\left( 3 \right)}}{U}:U \cdot \hat H\left( 3 \right)} \right) = \left( {\frac{{0,6875}}{{1,5519}}:\left( {0,6875} \right)\left( {1,5519} \right)} \right) = \left( {0,443:1,0669} \right)\) |
| Jawaban |
e. \(\left( {0,44:1,07} \right)\) |
