Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2015 |
| Nomor Soal |
: |
21 |
SOAL
Anda mencocokkan sebuah model Autoregressive \(AR\left( 1 \right)\) terhadap data berikut ini
\({y_1} = 2,0\)
\({y_2} = – 1,8\)
\({y_3} = 1,4\)
\({y_4} = – 2,0\)
\({y_5} = 1,2\)
Anda memilih \({\varepsilon _1} = 0\), \(\mu = 0\), \({\rho _1} = 0,5\) sebagai nilai awal.
Hitunglah nilai dari jumlah kuadrat dari fungsi \(S = \sum {{{\left[ {\left. {{\varepsilon _t}} \right|{\varepsilon _1} = 0,\mu = 0,{\rho _1} = 0,5} \right]}^2}} \), yaitu nilai dari \(\sum\nolimits_{i = 1}^5 {\varepsilon _i^2} \) (dibulatkan 2 desimal)
- 25,26
- 18,87
- 20,42
- 28,21
- 26,63
| Diketahui |
\({y_1} = 2,0;{y_2} = – 1,8;{y_3} = 1,4;{y_4} = – 2,0;{y_5} = 1,2\)
\({\varepsilon _1} = 0\); \(\mu = 0\); \({\rho _1} = 0,5\) |
| Rumus yang digunakan |
Untuk \(AR\left( p \right)\)
\({y_t} = {\phi _1}{y_{t – 1}} + {\phi _2}{y_{t – 2}} + \cdots + {\phi _p}{y_{t – p}} + \delta + {\varepsilon _t}\)
\(\mu = \frac{\delta }{{1 – {\phi _1} – {\phi _2} – \cdots – {\phi _p}}}\)
\({\rho _k} = \phi _1^k\)
\({\varepsilon _t} = {y_t} – {\hat y_t} = {y_t} – {\phi _1}{y_{t – 1}} – {\phi _2}{y_{t – 2}} – \cdots – {\phi _p}{y_{t – p}} – \delta \) |
| Proses pengerjaan |
\({\rho _1} = {\phi _1}\)
\({\phi _1} = 0,5\) |
| \(\mu = \frac{\delta }{{1 – {\phi _1}}}\)
\(0 = \frac{\delta }{{1 – 0,5}}\)
\(\delta = 0\) |
Diperoleh model \(AR\left( 1 \right)\)
\({y_t} = 0,5{y_{t – 1}} + {\varepsilon _t}\)
| \(t\) |
\({y_t}\) |
\({\hat y_t}\) |
\({\varepsilon _t}\) |
\(\varepsilon _t^2\) |
| 1 |
2.00 |
|
0.00 |
0.00 |
| 2 |
-1.80 |
1.00 |
-2.80 |
7.84 |
| 3 |
1.40 |
-0.90 |
2.30 |
5.29 |
| 4 |
-2.00 |
0.70 |
-2.70 |
7.29 |
| 5 |
1.20 |
-1.00 |
2.20 |
4.84 |
| Total |
25.26 |
Jadi diperoleh \(\sum\nolimits_{i = 1}^5 {\varepsilon _i^2} = 25,26\) |
| Jawaban |
a. 25,26 |
