Question

ANSWER QUESTION 2

1. The size of claims made on an insurance policy are modelled through the following distribu- tion: λ+1 You are interested in estimating the parameter λ > 0, using the following observations 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of when > 1 (c) Prove that the method of moments estimator of λ is λί- . Calculate its observed value (d) Find the likelihood and loglikelihood functions (e) Find the score function. (f) Find the observed Fisher information. (g) Prove that the Maximum Likelihood Estimator (MLE) of λ is λ2-1/5, where ỹ = 1 log (xi) /n. Calculate its observed value (h) Find the expected Fisher information. (i) Approximate the probability that the MLE takes value in the interval (λ-0.1, λ +0.1) (i) Use a Gaussian density to approximate the likelihood function around the MLE, and use R to illustrate the approximation (include both the figure and the code used) 2. Consider the statistical model of exercise 1, Investigate empirically the properties of λ| and λ2, assuming that λ = 10. In particular, estimate the bias, the expectation, the standard error, and the mean squared error for both estimators. Based on these results, decide whether the estimators are equally good or not. Use R to perform the simulations, and include the code used and any plot that you create. Note: to generate random values from the population density you can use the following R function:

Answer 1

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