Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Banyaknya klaim mengikuti distribusi negative binomial dengan parameter \(\beta \) dan \(r\) dimana \(\beta \) tidak diketahui dan \(r\) diketahui. Misalkan kamu diberi tugas untuk mengestimasi \(\beta \) dengan jumlah observasi sebanyak \(n\), yang mana diketahui \(\bar x\) adalah rataan dari observasi ini
Tentukan \(\beta \) dengan metode maximum likelihood
- \(\frac{{\bar x}}{{{r^2}}}\)
- \(\frac{{\bar x}}{r}\)
- \(\bar x\)
- \(r\bar x\)
- \({r^2}\bar x\)
| Diketahui |
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| Rumus yang digunakan | \({p\left( x \right) = \frac{{r\left( {r + 1} \right) \cdots \left( {r + x – 1} \right){\beta ^x}}}{{x!{{\left( {1 + \beta } \right)}^{r + x}}}},}\) \({L\left( \beta \right) = \prod\limits_{i = 1}^n {p\left( x \right)} ,}\) \({\frac{{d\ln \left[ {L\left( \beta \right)} \right]}}{{d\beta }} = 0}\) |
| Proses pengerjaan | \(L\left( \beta \right) = \prod\limits_{i = 1}^n {\frac{{r\left( {r + 1} \right) \cdots \left( {r + {x_i} – 1} \right){\beta ^{{x_i}}}}}{{{x_i}!{{\left( {1 + \beta } \right)}^{r + {x_i}}}}}} \propto \prod\limits_{i = 1}^n {\frac{{{\beta ^{{x_i}}}}}{{{{\left( {1 + \beta } \right)}^{r + {x_i}}}}}} \) \(\ln \left[ {L\left( \beta \right)} \right] = \sum\limits_{i = 1}^n {\left[ {{x_i}\ln \beta – \left( {r + {x_i}} \right)\ln \left( {1 + \beta } \right)} \right]} \) \(\frac{{d\ln \left[ {L\left( \beta \right)} \right]}}{{d\beta }} = \sum\limits_{i = 1}^n {\left[ {\frac{{{x_i}}}{\beta } – \frac{{r + {x_i}}}{{1 + \beta }}} \right]} = 0\) \(0 = \sum\limits_{i = 1}^n {\left[ {{x_i}\left( {1 + \beta } \right) – \beta \left( {r + {x_i}} \right)} \right]} \) \(0 = \sum\limits_{i = 1}^n {\left[ {{x_i} + {x_i}\beta – \beta r – {x_i}\beta } \right]} \) \(0 = \sum\limits_{i = 1}^n {{x_i}} – \sum\limits_{i = 1}^n {\beta r} \) \(0 = n\bar x – n\beta r\) \(\hat \beta = \frac{{\bar x}}{r}\) |
| Jawaban | b. \(\frac{{\bar x}}{r}\) |