CAS MAS-IModern Actuarial Statistics I

Losses follow an exponential distribution. A sample yields 3, 5, 7, 9, 11. Calculate the estimated variance of the maximum likelihood estimator of the mean.

Claims arrive according to a Poisson process at 3 per hour. Calculate the probability that no claim arrives in a given 30-minute period.

A stationary AR(1) process satisfies $X_t=0.6X_{t-1}+\varepsilon_t$ with $\operatorname{Var}(\varepsilon_t)=4$. Calculate $\operatorname{Var}(X_t)$.

Model A (5 parameters) has log-likelihood −1,200; nested model B (8 parameters) has log-likelihood −1,195. At the 5% level ($\chi^2_{3,0.95}=7.81$), which statement is correct?

In a Poisson GLM with log link, the fitted coefficient for territory B relative to base territory A is 0.18. Calculate the indicated frequency relativity for territory B.

Claim counts for two regions are Poisson with rates $\lambda_1,\lambda_2$ per exposure. Region 1 has 40 claims on 500 exposures; region 2 has 30 claims on 250 exposures. Derive the likelihood-ratio test of $H_0:\lambda_1=\lambda_2$ and state your conclusion at the 5% level.

A driver's status transitions between Good (G) and Bad (B) rating classes each year according to a Markov chain with transition matrix $P=\begin{pmatrix}0.9&0.1\\0.4&0.6\end{pmatrix}$ (rows = from G, B). Calculate the long-run (stationary) probability that a driver is in class G.

A chi-square goodness-of-fit test compares observed claim counts by class (50, 30, 20) to expected counts under a proposed model (40, 40, 20). Calculate the test statistic and state the conclusion at the 5% level ($\chi^2_{2,0.95}=5.99$).

A Poisson frequency GLM with a log link fits a coefficient of $-0.25$ for Territory C relative to the base territory. Calculate the indicated frequency relativity for Territory C.

A time series follows $X_t=\varepsilon_t+0.4\varepsilon_{t-1}$ with $\operatorname{Var}(\varepsilon_t)=9$. Calculate $\operatorname{Var}(X_t)$ and the lag-1 autocorrelation $\rho_1$.

Claims of type A arrive as a Poisson process at rate 4 per day and, independently, claims of type B arrive as a Poisson process at rate 6 per day. (a) Identify the distribution of the combined arrival process and its rate. (b) Given that a claim has just arrived, find the probability it is type A. (c) Find the probability that at least 2 claims (of either type) arrive in a given half-day.